MAYBE Time: 9.432865 TRS: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} DP: DP: { U12# mark X -> U12# X, U12# ok X -> U12# X, isNat# ok X -> isNat# X, active# U12 X -> U12# active X, active# U12 X -> active# X, active# isNat s V1 -> isNat# V1, active# isNat s V1 -> U21# isNat V1, active# isNat plus(V1, V2) -> isNat# V1, active# isNat plus(V1, V2) -> U11#(isNat V1, V2), active# isNat x(V1, V2) -> isNat# V1, active# isNat x(V1, V2) -> U31#(isNat V1, V2), active# U11(X1, X2) -> active# X1, active# U11(X1, X2) -> U11#(active X1, X2), active# U11(tt(), V2) -> U12# isNat V2, active# U11(tt(), V2) -> isNat# V2, active# U21 X -> active# X, active# U21 X -> U21# active X, active# U32 X -> active# X, active# U32 X -> U32# active X, active# U31(X1, X2) -> active# X1, active# U31(X1, X2) -> U31#(active X1, X2), active# U31(tt(), V2) -> isNat# V2, active# U31(tt(), V2) -> U32# isNat V2, active# U41(X1, X2) -> active# X1, active# U41(X1, X2) -> U41#(active X1, X2), active# U52(X1, X2, X3) -> active# X1, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3), active# U52(tt(), M, N) -> s# plus(N, M), active# U52(tt(), M, N) -> plus#(N, M), active# U51(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3), active# U51(tt(), M, N) -> isNat# N, active# U51(tt(), M, N) -> U52#(isNat N, M, N), active# s X -> active# X, active# s X -> s# active X, active# plus(N, s M) -> isNat# M, active# plus(N, s M) -> U51#(isNat M, M, N), active# plus(N, 0()) -> isNat# N, active# plus(N, 0()) -> U41#(isNat N, N), active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2, active# plus(X1, X2) -> plus#(X1, active X2), active# plus(X1, X2) -> plus#(active X1, X2), active# U61 X -> active# X, active# U61 X -> U61# active X, active# U72(X1, X2, X3) -> active# X1, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3), active# U72(tt(), M, N) -> plus#(x(N, M), N), active# U72(tt(), M, N) -> x#(N, M), active# U71(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3), active# U71(tt(), M, N) -> isNat# N, active# U71(tt(), M, N) -> U72#(isNat N, M, N), active# x(N, s M) -> isNat# M, active# x(N, s M) -> U71#(isNat M, M, N), active# x(N, 0()) -> isNat# N, active# x(N, 0()) -> U61# isNat N, active# x(X1, X2) -> active# X1, active# x(X1, X2) -> active# X2, active# x(X1, X2) -> x#(X1, active X2), active# x(X1, X2) -> x#(active X1, X2), U11#(mark X1, X2) -> U11#(X1, X2), U11#(ok X1, ok X2) -> U11#(X1, X2), U21# mark X -> U21# X, U21# ok X -> U21# X, U32# mark X -> U32# X, U32# ok X -> U32# X, U31#(mark X1, X2) -> U31#(X1, X2), U31#(ok X1, ok X2) -> U31#(X1, X2), U41#(mark X1, X2) -> U41#(X1, X2), U41#(ok X1, ok X2) -> U41#(X1, X2), U52#(mark X1, X2, X3) -> U52#(X1, X2, X3), U52#(ok X1, ok X2, ok X3) -> U52#(X1, X2, X3), U51#(mark X1, X2, X3) -> U51#(X1, X2, X3), U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3), s# mark X -> s# X, s# ok X -> s# X, plus#(X1, mark X2) -> plus#(X1, X2), plus#(mark X1, X2) -> plus#(X1, X2), plus#(ok X1, ok X2) -> plus#(X1, X2), U61# mark X -> U61# X, U61# ok X -> U61# X, U72#(mark X1, X2, X3) -> U72#(X1, X2, X3), U72#(ok X1, ok X2, ok X3) -> U72#(X1, X2, X3), U71#(mark X1, X2, X3) -> U71#(X1, X2, X3), U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3), x#(X1, mark X2) -> x#(X1, X2), x#(mark X1, X2) -> x#(X1, X2), x#(ok X1, ok X2) -> x#(X1, X2), proper# U12 X -> U12# proper X, proper# U12 X -> proper# X, proper# isNat X -> isNat# proper X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> U11#(proper X1, proper X2), proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> U21# proper X, proper# U21 X -> proper# X, proper# U32 X -> U32# proper X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> U31#(proper X1, proper X2), proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> U41#(proper X1, proper X2), proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3), proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3), proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> s# proper X, proper# s X -> proper# X, proper# plus(X1, X2) -> plus#(proper X1, proper X2), proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2, proper# U61 X -> U61# proper X, proper# U61 X -> proper# X, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3), proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3), proper# U71(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> x#(proper X1, proper X2), proper# x(X1, X2) -> proper# X1, proper# x(X1, X2) -> proper# X2, top# mark X -> proper# X, top# mark X -> top# proper X, top# ok X -> active# X, top# ok X -> top# active X} TRS: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} UR: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2)} EDG: { (active# U21 X -> U21# active X, U21# ok X -> U21# X) (active# U21 X -> U21# active X, U21# mark X -> U21# X) (active# s X -> s# active X, s# ok X -> s# X) (active# s X -> s# active X, s# mark X -> s# X) (proper# U12 X -> U12# proper X, U12# ok X -> U12# X) (proper# U12 X -> U12# proper X, U12# mark X -> U12# X) (proper# U21 X -> U21# proper X, U21# ok X -> U21# X) (proper# U21 X -> U21# proper X, U21# mark X -> U21# X) (proper# s X -> s# proper X, s# ok X -> s# X) (proper# s X -> s# proper X, s# mark X -> s# X) (top# mark X -> top# proper X, top# ok X -> top# active X) (top# mark X -> top# proper X, top# ok X -> active# X) (top# mark X -> top# proper X, top# mark X -> top# proper X) (top# mark X -> top# proper X, top# mark X -> proper# X) (active# U51(tt(), M, N) -> U52#(isNat N, M, N), U52#(ok X1, ok X2, ok X3) -> U52#(X1, X2, X3)) (active# U51(tt(), M, N) -> U52#(isNat N, M, N), U52#(mark X1, X2, X3) -> U52#(X1, X2, X3)) (active# U71(tt(), M, N) -> U72#(isNat N, M, N), U72#(ok X1, ok X2, ok X3) -> U72#(X1, X2, X3)) (active# U71(tt(), M, N) -> U72#(isNat N, M, N), U72#(mark X1, X2, X3) -> U72#(X1, X2, X3)) (active# isNat s V1 -> U21# isNat V1, U21# ok X -> U21# X) (active# isNat s V1 -> U21# isNat V1, U21# mark X -> U21# X) (active# U11(tt(), V2) -> U12# isNat V2, U12# ok X -> U12# X) (active# U11(tt(), V2) -> U12# isNat V2, U12# mark X -> U12# X) (active# U11(tt(), V2) -> isNat# V2, isNat# ok X -> isNat# X) (active# U11(X1, X2) -> U11#(active X1, X2), U11#(ok X1, ok X2) -> U11#(X1, X2)) (active# U11(X1, X2) -> U11#(active X1, X2), U11#(mark X1, X2) -> U11#(X1, X2)) (active# U41(X1, X2) -> U41#(active X1, X2), U41#(ok X1, ok X2) -> U41#(X1, X2)) (active# U41(X1, X2) -> U41#(active X1, X2), U41#(mark X1, X2) -> U41#(X1, X2)) (active# x(X1, X2) -> x#(active X1, X2), x#(ok X1, ok X2) -> x#(X1, X2)) (active# x(X1, X2) -> x#(active X1, X2), x#(mark X1, X2) -> x#(X1, X2)) (active# x(X1, X2) -> x#(active X1, X2), x#(X1, mark X2) -> x#(X1, X2)) (active# U72(tt(), M, N) -> x#(N, M), x#(ok X1, ok X2) -> x#(X1, X2)) (active# U72(tt(), M, N) -> x#(N, M), x#(mark X1, X2) -> x#(X1, X2)) (active# U72(tt(), M, N) -> x#(N, M), x#(X1, mark X2) -> x#(X1, X2)) (active# x(X1, X2) -> active# X2, active# x(X1, X2) -> x#(active X1, X2)) (active# x(X1, X2) -> active# X2, active# x(X1, X2) -> x#(X1, active X2)) (active# x(X1, X2) -> active# X2, active# x(X1, X2) -> active# X2) (active# x(X1, X2) -> active# X2, active# x(X1, X2) -> active# X1) (active# x(X1, X2) -> active# X2, active# x(N, 0()) -> U61# isNat N) (active# x(X1, X2) -> active# X2, active# x(N, 0()) -> isNat# N) (active# x(X1, X2) -> active# X2, active# x(N, s M) -> U71#(isNat M, M, N)) (active# x(X1, X2) -> active# X2, active# x(N, s M) -> isNat# M) (active# x(X1, X2) -> active# X2, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# x(X1, X2) -> active# X2, active# U71(tt(), M, N) -> isNat# N) (active# x(X1, X2) -> active# X2, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# x(X1, X2) -> active# X2, active# U71(X1, X2, X3) -> active# X1) (active# x(X1, X2) -> active# X2, active# 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(proper# U71(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U71(X1, X2, X3) -> proper# X2, proper# U61 X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X2, proper# U61 X -> U61# proper X) (proper# U71(X1, X2, X3) -> proper# X2, proper# plus(X1, X2) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X2, proper# plus(X1, X2) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X2, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U71(X1, X2, X3) -> proper# X2, proper# s X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X2, proper# s X -> s# proper X) (proper# U71(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3) (proper# U71(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U71(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3) (proper# U71(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U71(X1, X2, X3) -> proper# X2, proper# U41(X1, X2) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X2, proper# U41(X1, X2) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X2, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U71(X1, X2, X3) -> proper# X2, proper# U31(X1, X2) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X2, proper# U31(X1, X2) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X2, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U71(X1, X2, X3) -> proper# X2, proper# U32 X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X2, proper# U32 X -> U32# proper X) (proper# U71(X1, X2, X3) -> proper# X2, proper# U21 X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X2, proper# U21 X -> U21# proper X) (proper# U71(X1, X2, X3) -> proper# X2, proper# U11(X1, X2) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X2, proper# U11(X1, X2) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X2, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U71(X1, X2, X3) -> proper# X2, proper# isNat X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X2, proper# isNat X -> isNat# proper X) (proper# U71(X1, X2, X3) -> proper# X2, proper# U12 X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X2, proper# U12 X -> U12# proper X) (U11#(mark X1, X2) -> U11#(X1, X2), U11#(ok X1, ok X2) -> U11#(X1, X2)) (U11#(mark X1, X2) -> U11#(X1, X2), U11#(mark X1, X2) -> U11#(X1, X2)) (U31#(mark X1, X2) -> U31#(X1, X2), U31#(ok X1, ok X2) -> U31#(X1, X2)) (U31#(mark X1, X2) -> U31#(X1, X2), U31#(mark X1, X2) -> U31#(X1, X2)) (U41#(mark X1, X2) -> U41#(X1, X2), U41#(ok X1, ok X2) -> U41#(X1, X2)) (U41#(mark X1, X2) -> U41#(X1, X2), U41#(mark X1, X2) -> U41#(X1, X2)) (plus#(X1, mark X2) -> plus#(X1, X2), plus#(ok X1, ok X2) -> plus#(X1, X2)) (plus#(X1, mark X2) -> plus#(X1, X2), plus#(mark X1, X2) -> plus#(X1, X2)) (plus#(X1, mark X2) -> plus#(X1, X2), plus#(X1, mark X2) -> plus#(X1, X2)) (plus#(ok X1, ok X2) -> plus#(X1, X2), plus#(ok X1, ok X2) -> plus#(X1, X2)) (plus#(ok X1, ok X2) -> plus#(X1, X2), plus#(mark X1, X2) -> plus#(X1, X2)) (plus#(ok X1, ok X2) -> plus#(X1, X2), plus#(X1, mark X2) -> plus#(X1, X2)) (x#(mark X1, X2) -> x#(X1, X2), x#(ok X1, ok X2) -> x#(X1, X2)) (x#(mark X1, X2) -> x#(X1, X2), x#(mark X1, X2) -> x#(X1, X2)) (x#(mark X1, X2) -> x#(X1, X2), x#(X1, mark X2) -> x#(X1, X2)) (active# plus(N, s M) -> isNat# M, isNat# ok X -> isNat# X) (proper# U52(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X2) (proper# U52(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X1) (proper# U52(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U52(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X3) (proper# U52(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X2) (proper# U52(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X1) (proper# U52(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U52(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X3) (proper# U52(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X2) (proper# U52(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X1) (proper# U52(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U52(X1, X2, X3) -> proper# X3, proper# U61 X -> proper# X) (proper# U52(X1, X2, X3) -> proper# X3, proper# U61 X -> U61# proper X) (proper# U52(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> proper# X2) (proper# U52(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> proper# X1) (proper# U52(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U52(X1, X2, X3) -> proper# X3, proper# s X -> proper# X) (proper# U52(X1, X2, X3) -> proper# X3, proper# s X -> s# proper X) (proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X3) (proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X2) (proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1) (proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U52(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> proper# X3) (proper# U52(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> proper# X2) (proper# U52(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> proper# X1) (proper# U52(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U52(X1, X2, X3) -> proper# X3, proper# U41(X1, X2) -> proper# X2) (proper# U52(X1, X2, X3) -> proper# X3, proper# U41(X1, X2) -> proper# X1) (proper# U52(X1, X2, X3) -> proper# X3, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U52(X1, X2, X3) -> proper# X3, proper# U31(X1, X2) -> proper# X2) (proper# U52(X1, X2, X3) -> proper# X3, proper# U31(X1, X2) -> proper# X1) (proper# U52(X1, X2, X3) -> proper# X3, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U52(X1, X2, X3) -> proper# X3, proper# U32 X -> proper# X) (proper# U52(X1, X2, X3) -> proper# X3, proper# U32 X -> U32# proper X) (proper# U52(X1, X2, X3) -> proper# X3, proper# U21 X -> proper# X) (proper# U52(X1, X2, X3) -> proper# X3, proper# U21 X -> U21# proper X) (proper# U52(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> proper# X2) (proper# U52(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> proper# X1) (proper# U52(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U52(X1, X2, X3) -> proper# X3, proper# isNat X -> proper# X) (proper# U52(X1, X2, X3) -> proper# X3, proper# isNat X -> isNat# proper X) (proper# U52(X1, X2, X3) -> proper# X3, proper# U12 X -> proper# X) (proper# U52(X1, X2, X3) -> proper# X3, proper# U12 X -> U12# proper X) (proper# U72(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U72(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U72(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U72(X1, X2, X3) -> proper# X3, proper# U61 X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X3, proper# U61 X -> U61# proper X) (proper# U72(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U72(X1, X2, X3) -> proper# X3, proper# s X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X3, proper# s X -> s# proper X) (proper# U72(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, 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X3) -> proper# X3, proper# U21 X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X3, proper# U21 X -> U21# proper X) (proper# U72(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U72(X1, X2, X3) -> proper# X3, proper# isNat X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X3, proper# isNat X -> isNat# proper X) (proper# U72(X1, X2, X3) -> proper# X3, proper# U12 X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X3, proper# U12 X -> U12# proper X) (active# plus(N, 0()) -> U41#(isNat N, N), U41#(ok X1, ok X2) -> U41#(X1, X2)) (active# plus(N, 0()) -> U41#(isNat N, N), U41#(mark X1, X2) -> U41#(X1, X2)) (active# U11(X1, X2) -> active# X1, active# x(X1, X2) -> x#(active X1, X2)) (active# U11(X1, X2) -> active# X1, active# x(X1, X2) -> x#(X1, active X2)) (active# U11(X1, X2) -> active# X1, active# x(X1, X2) -> active# X2) (active# U11(X1, X2) -> active# X1, active# x(X1, X2) -> active# X1) (active# U11(X1, X2) -> active# X1, active# x(N, 0()) -> U61# isNat N) (active# U11(X1, X2) -> active# X1, active# x(N, 0()) -> isNat# N) (active# U11(X1, X2) -> active# X1, active# x(N, s M) -> U71#(isNat M, M, N)) (active# U11(X1, X2) -> active# X1, active# x(N, s M) -> isNat# M) (active# U11(X1, X2) -> active# X1, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# U11(X1, X2) -> active# X1, active# U71(tt(), M, N) -> isNat# N) (active# U11(X1, X2) -> active# X1, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# U11(X1, X2) -> active# X1, active# U71(X1, X2, X3) -> active# X1) (active# U11(X1, X2) -> active# X1, active# U72(tt(), M, N) -> x#(N, M)) (active# U11(X1, X2) -> active# X1, active# U72(tt(), M, N) -> plus#(x(N, M), N)) (active# U11(X1, X2) -> active# X1, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3)) (active# U11(X1, X2) -> active# X1, active# U72(X1, X2, X3) -> active# X1) (active# U11(X1, X2) -> active# X1, active# U61 X -> U61# active X) (active# U11(X1, X2) -> active# X1, active# U61 X -> active# X) (active# U11(X1, X2) -> active# X1, active# plus(X1, X2) -> plus#(active X1, X2)) (active# U11(X1, X2) -> active# X1, active# plus(X1, X2) -> plus#(X1, active X2)) (active# U11(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2) (active# U11(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X1) (active# U11(X1, X2) -> active# X1, active# plus(N, 0()) -> U41#(isNat N, N)) (active# U11(X1, X2) -> active# X1, active# plus(N, 0()) -> isNat# N) (active# U11(X1, X2) -> active# X1, active# plus(N, s M) -> U51#(isNat M, M, N)) (active# U11(X1, X2) -> active# X1, active# plus(N, s M) -> isNat# M) (active# U11(X1, X2) -> active# X1, active# s X -> s# active X) (active# U11(X1, X2) -> active# X1, active# s X -> active# X) (active# U11(X1, X2) -> active# X1, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (active# U11(X1, X2) -> active# X1, 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X1, active# U32 X -> U32# active X) (active# U11(X1, X2) -> active# X1, active# U32 X -> active# X) (active# U11(X1, X2) -> active# X1, active# U21 X -> U21# active X) (active# U11(X1, X2) -> active# X1, active# U21 X -> active# X) (active# U11(X1, X2) -> active# X1, active# U11(tt(), V2) -> isNat# V2) (active# U11(X1, X2) -> active# X1, active# U11(tt(), V2) -> U12# isNat V2) (active# U11(X1, X2) -> active# X1, active# U11(X1, X2) -> U11#(active X1, X2)) (active# U11(X1, X2) -> active# X1, active# U11(X1, X2) -> active# X1) (active# U11(X1, X2) -> active# X1, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# U11(X1, X2) -> active# X1, active# isNat x(V1, V2) -> isNat# V1) (active# U11(X1, X2) -> active# X1, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# U11(X1, X2) -> active# X1, active# isNat plus(V1, V2) -> isNat# V1) (active# U11(X1, X2) -> active# X1, active# isNat s V1 -> U21# isNat V1) (active# U11(X1, X2) -> active# X1, active# isNat s V1 -> isNat# V1) (active# U11(X1, X2) -> active# X1, active# U12 X -> active# X) (active# U11(X1, X2) -> active# X1, active# U12 X -> U12# active X) (active# U41(X1, X2) -> active# X1, active# x(X1, X2) -> x#(active X1, X2)) (active# U41(X1, X2) -> active# X1, active# x(X1, X2) -> x#(X1, active X2)) (active# U41(X1, X2) -> active# X1, active# x(X1, X2) -> active# X2) (active# U41(X1, X2) -> active# X1, active# x(X1, X2) -> active# X1) (active# U41(X1, X2) -> active# X1, active# x(N, 0()) -> U61# isNat N) (active# U41(X1, X2) -> active# X1, active# x(N, 0()) -> isNat# N) (active# U41(X1, X2) -> active# X1, active# x(N, s M) -> U71#(isNat M, M, N)) (active# U41(X1, X2) -> active# X1, active# x(N, s M) -> isNat# M) (active# U41(X1, X2) -> active# X1, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# U41(X1, X2) -> active# X1, active# U71(tt(), M, N) -> isNat# N) (active# U41(X1, X2) -> active# X1, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# U41(X1, X2) -> active# X1, active# U71(X1, X2, X3) -> active# X1) (active# U41(X1, X2) -> active# X1, active# U72(tt(), M, N) -> x#(N, M)) (active# U41(X1, X2) -> active# X1, active# U72(tt(), M, N) -> plus#(x(N, M), N)) (active# U41(X1, X2) -> active# X1, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3)) (active# U41(X1, X2) -> active# X1, active# U72(X1, X2, X3) -> active# X1) (active# U41(X1, X2) -> active# X1, active# U61 X -> U61# active X) (active# U41(X1, X2) -> active# X1, active# U61 X -> active# X) (active# U41(X1, X2) -> active# X1, active# plus(X1, X2) -> plus#(active X1, X2)) (active# U41(X1, X2) -> active# X1, active# plus(X1, X2) -> plus#(X1, active X2)) (active# U41(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2) (active# U41(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X1) (active# U41(X1, X2) -> active# X1, active# plus(N, 0()) -> U41#(isNat N, N)) (active# U41(X1, X2) -> active# X1, active# plus(N, 0()) -> isNat# N) (active# U41(X1, X2) -> active# X1, active# plus(N, s M) -> U51#(isNat M, M, N)) (active# U41(X1, X2) -> active# X1, active# plus(N, s M) -> isNat# M) (active# U41(X1, X2) -> active# X1, active# s X -> s# active X) (active# U41(X1, X2) -> active# X1, active# s X -> active# X) (active# U41(X1, X2) -> active# X1, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (active# U41(X1, X2) -> active# X1, active# U51(tt(), M, N) -> isNat# N) (active# U41(X1, X2) -> active# X1, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3)) (active# U41(X1, X2) -> active# X1, active# U51(X1, X2, X3) -> active# X1) (active# U41(X1, X2) -> active# X1, active# U52(tt(), M, N) -> plus#(N, M)) (active# U41(X1, X2) -> active# X1, active# U52(tt(), M, N) -> s# plus(N, M)) (active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1) (active# U41(X1, X2) -> active# X1, active# U41(X1, X2) -> U41#(active X1, X2)) (active# U41(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1) (active# U41(X1, X2) -> active# X1, active# U31(tt(), V2) -> U32# isNat V2) (active# U41(X1, X2) -> active# X1, active# U31(tt(), V2) -> isNat# V2) (active# U41(X1, X2) -> active# X1, active# U31(X1, X2) -> U31#(active X1, X2)) (active# U41(X1, X2) -> active# X1, active# U31(X1, X2) -> active# X1) (active# U41(X1, X2) -> active# X1, active# U32 X -> U32# active X) (active# U41(X1, X2) -> active# X1, active# U32 X -> active# X) (active# U41(X1, X2) -> active# X1, active# U21 X -> U21# active X) (active# U41(X1, X2) -> active# X1, active# U21 X -> active# X) (active# U41(X1, X2) -> active# X1, active# U11(tt(), V2) -> isNat# V2) (active# U41(X1, X2) -> active# X1, active# U11(tt(), V2) -> U12# isNat V2) (active# U41(X1, X2) -> active# X1, active# U11(X1, X2) -> U11#(active X1, X2)) (active# U41(X1, X2) -> active# X1, active# U11(X1, X2) -> active# X1) (active# U41(X1, X2) -> active# X1, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# U41(X1, X2) -> active# X1, active# isNat x(V1, V2) -> isNat# V1) (active# U41(X1, X2) -> active# X1, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# U41(X1, X2) -> active# X1, active# isNat plus(V1, V2) -> isNat# V1) (active# U41(X1, X2) -> active# X1, active# isNat s V1 -> U21# isNat V1) (active# U41(X1, X2) -> active# X1, active# isNat s V1 -> isNat# V1) (active# U41(X1, X2) -> active# X1, active# U12 X -> active# X) (active# U41(X1, X2) -> active# X1, active# U12 X -> U12# active X) (active# U51(X1, X2, X3) -> active# X1, active# x(X1, X2) -> x#(active X1, X2)) (active# U51(X1, X2, X3) -> active# X1, active# x(X1, X2) -> x#(X1, active X2)) (active# U51(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X2) (active# U51(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X1) (active# U51(X1, X2, X3) -> active# X1, active# x(N, 0()) -> U61# isNat N) (active# U51(X1, X2, X3) -> active# X1, active# x(N, 0()) -> isNat# N) (active# U51(X1, X2, X3) -> active# X1, active# x(N, s M) -> U71#(isNat M, M, N)) (active# U51(X1, X2, X3) -> active# X1, active# x(N, s M) -> isNat# M) (active# U51(X1, X2, X3) -> active# X1, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# U51(X1, X2, X3) -> active# X1, active# U71(tt(), M, N) -> isNat# N) (active# U51(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# U51(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> active# X1) (active# U51(X1, X2, X3) -> active# X1, active# U72(tt(), M, N) -> x#(N, M)) (active# U51(X1, X2, X3) -> active# X1, active# U72(tt(), M, N) -> plus#(x(N, M), N)) (active# U51(X1, X2, X3) -> active# X1, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3)) (active# U51(X1, X2, X3) -> active# X1, active# U72(X1, X2, X3) -> active# X1) (active# U51(X1, X2, X3) -> active# X1, active# U61 X -> U61# active X) (active# U51(X1, X2, X3) -> active# X1, active# U61 X -> active# X) (active# U51(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> plus#(active X1, X2)) (active# U51(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> plus#(X1, active X2)) (active# U51(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> active# X2) (active# U51(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> active# X1) (active# U51(X1, X2, X3) -> active# X1, active# plus(N, 0()) -> U41#(isNat N, N)) (active# U51(X1, X2, X3) -> active# X1, active# plus(N, 0()) -> isNat# N) (active# U51(X1, X2, X3) -> active# X1, active# plus(N, s M) -> U51#(isNat M, M, N)) (active# U51(X1, X2, X3) -> active# X1, active# plus(N, s M) -> isNat# M) (active# U51(X1, X2, X3) -> active# X1, active# s X -> s# active X) (active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X) (active# U51(X1, X2, X3) -> active# X1, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (active# U51(X1, X2, X3) -> active# X1, active# U51(tt(), M, N) -> isNat# N) (active# U51(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3)) (active# U51(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1) (active# U51(X1, X2, X3) -> active# X1, active# U52(tt(), M, N) -> plus#(N, M)) (active# U51(X1, X2, X3) -> active# X1, active# U52(tt(), M, N) -> s# plus(N, M)) (active# U51(X1, X2, X3) -> active# X1, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (active# U51(X1, X2, X3) -> active# X1, active# U52(X1, X2, X3) -> active# X1) (active# U51(X1, X2, X3) -> active# X1, active# U41(X1, X2) -> U41#(active X1, X2)) (active# U51(X1, X2, X3) -> active# X1, active# U41(X1, X2) -> active# X1) (active# U51(X1, X2, X3) -> active# X1, active# U31(tt(), V2) -> U32# isNat V2) (active# U51(X1, X2, X3) -> active# X1, active# U31(tt(), V2) -> isNat# V2) (active# U51(X1, X2, X3) -> active# X1, active# U31(X1, X2) -> U31#(active X1, X2)) (active# U51(X1, X2, X3) -> active# X1, active# U31(X1, X2) -> active# X1) (active# U51(X1, X2, X3) -> active# X1, active# U32 X -> U32# active X) (active# U51(X1, X2, X3) -> active# X1, active# U32 X -> active# X) (active# U51(X1, X2, X3) -> active# X1, active# U21 X -> U21# active X) (active# U51(X1, X2, X3) -> active# X1, active# U21 X -> active# X) (active# U51(X1, X2, X3) -> active# X1, active# U11(tt(), V2) -> isNat# V2) (active# U51(X1, X2, X3) -> active# X1, active# U11(tt(), V2) -> U12# isNat V2) (active# U51(X1, X2, X3) -> active# X1, active# U11(X1, X2) -> U11#(active X1, X2)) (active# U51(X1, X2, X3) -> active# X1, active# U11(X1, X2) -> active# X1) (active# U51(X1, X2, X3) -> active# X1, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# U51(X1, X2, X3) -> active# X1, active# isNat x(V1, V2) -> isNat# V1) (active# U51(X1, X2, X3) -> active# X1, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# U51(X1, X2, X3) -> active# X1, active# isNat plus(V1, V2) -> isNat# V1) (active# U51(X1, X2, X3) -> active# X1, active# isNat s V1 -> U21# isNat V1) (active# U51(X1, X2, X3) -> active# X1, active# isNat s V1 -> isNat# V1) (active# U51(X1, X2, X3) -> active# X1, active# U12 X -> active# X) (active# U51(X1, X2, X3) -> active# X1, active# U12 X -> U12# active X) (active# U72(X1, X2, X3) -> active# X1, active# x(X1, X2) -> x#(active X1, X2)) (active# U72(X1, X2, X3) -> active# X1, active# x(X1, X2) -> x#(X1, active X2)) (active# U72(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X2) (active# U72(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X1) (active# U72(X1, X2, X3) -> active# X1, active# x(N, 0()) -> U61# isNat N) (active# U72(X1, X2, X3) -> active# X1, active# x(N, 0()) -> isNat# N) (active# U72(X1, X2, X3) -> active# X1, active# x(N, s M) -> U71#(isNat M, M, N)) (active# U72(X1, X2, X3) -> active# X1, active# x(N, s M) -> isNat# M) (active# U72(X1, X2, X3) -> active# X1, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# U72(X1, X2, X3) -> active# X1, active# U71(tt(), M, N) -> isNat# N) (active# U72(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# U72(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> active# X1) (active# U72(X1, X2, X3) -> 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-> active# X, active# plus(N, s M) -> U51#(isNat M, M, N)) (active# U61 X -> active# X, active# plus(N, s M) -> isNat# M) (active# U61 X -> active# X, active# s X -> s# active X) (active# U61 X -> active# X, active# s X -> active# X) (active# U61 X -> active# X, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (active# U61 X -> active# X, active# U51(tt(), M, N) -> isNat# N) (active# U61 X -> active# X, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3)) (active# U61 X -> active# X, active# U51(X1, X2, X3) -> active# X1) (active# U61 X -> active# X, active# U52(tt(), M, N) -> plus#(N, M)) (active# U61 X -> active# X, active# U52(tt(), M, N) -> s# plus(N, M)) (active# U61 X -> active# X, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (active# U61 X -> active# X, active# U52(X1, X2, X3) -> active# X1) (active# U61 X -> active# X, active# U41(X1, X2) -> U41#(active X1, X2)) (active# U61 X -> active# X, active# U41(X1, X2) -> active# X1) (active# U61 X -> active# X, active# U31(tt(), 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U51(X1, X2, X3) -> proper# X2) (proper# isNat X -> proper# X, proper# U51(X1, X2, X3) -> proper# X1) (proper# isNat X -> proper# X, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# isNat X -> proper# X, proper# U52(X1, X2, X3) -> proper# X3) (proper# isNat X -> proper# X, proper# U52(X1, X2, X3) -> proper# X2) (proper# isNat X -> proper# X, proper# U52(X1, X2, X3) -> proper# X1) (proper# isNat X -> proper# X, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# isNat X -> proper# X, proper# U41(X1, X2) -> proper# X2) (proper# isNat X -> proper# X, proper# U41(X1, X2) -> proper# X1) (proper# isNat X -> proper# X, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# isNat X -> proper# X, proper# U31(X1, X2) -> proper# X2) (proper# isNat X -> proper# X, proper# U31(X1, X2) -> proper# X1) (proper# isNat X -> proper# X, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# isNat X -> proper# X, proper# U32 X -> proper# X) (proper# isNat X -> proper# X, proper# U32 X -> U32# proper X) (proper# isNat X -> proper# X, proper# U21 X -> proper# X) (proper# isNat X -> proper# X, proper# U21 X -> U21# proper X) (proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X2) (proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1) (proper# isNat X -> proper# X, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# isNat X -> proper# X, proper# isNat X -> proper# X) (proper# isNat X -> proper# X, proper# isNat X -> isNat# proper X) (proper# isNat X -> proper# X, proper# U12 X -> proper# X) (proper# isNat X -> proper# X, proper# U12 X -> U12# proper X) (proper# U32 X -> proper# X, proper# x(X1, X2) -> proper# X2) (proper# U32 X -> proper# X, proper# x(X1, X2) -> proper# X1) (proper# U32 X -> proper# X, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U32 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X3) (proper# U32 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X2) (proper# U32 X -> 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-> U32# proper X) (proper# U32 X -> proper# X, proper# U21 X -> proper# X) (proper# U32 X -> proper# X, proper# U21 X -> U21# proper X) (proper# U32 X -> proper# X, proper# U11(X1, X2) -> proper# X2) (proper# U32 X -> proper# X, proper# U11(X1, X2) -> proper# X1) (proper# U32 X -> proper# X, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U32 X -> proper# X, proper# isNat X -> proper# X) (proper# U32 X -> proper# X, proper# isNat X -> isNat# proper X) (proper# U32 X -> proper# X, proper# U12 X -> proper# X) (proper# U32 X -> proper# X, proper# U12 X -> U12# proper X) (proper# U61 X -> proper# X, proper# x(X1, X2) -> proper# X2) (proper# U61 X -> proper# X, proper# x(X1, X2) -> proper# X1) (proper# U61 X -> proper# X, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U61 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X3) (proper# U61 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X2) (proper# U61 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X1) (proper# 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-> isNat# M) (top# ok X -> active# X, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (top# ok X -> active# X, active# U71(tt(), M, N) -> isNat# N) (top# ok X -> active# X, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (top# ok X -> active# X, active# U71(X1, X2, X3) -> active# X1) (top# ok X -> active# X, active# U72(tt(), M, N) -> x#(N, M)) (top# ok X -> active# X, active# U72(tt(), M, N) -> plus#(x(N, M), N)) (top# ok X -> active# X, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3)) (top# ok X -> active# X, active# U72(X1, X2, X3) -> active# X1) (top# ok X -> active# X, active# U61 X -> U61# active X) (top# ok X -> active# X, active# U61 X -> active# X) (top# ok X -> active# X, active# plus(X1, X2) -> plus#(active X1, X2)) (top# ok X -> active# X, active# plus(X1, X2) -> plus#(X1, active X2)) (top# ok X -> active# X, active# plus(X1, X2) -> active# X2) (top# ok X -> active# X, active# plus(X1, X2) -> active# X1) (top# ok X -> active# X, active# plus(N, 0()) -> U41#(isNat N, N)) (top# ok X -> active# X, active# plus(N, 0()) -> isNat# N) (top# ok X -> active# X, active# plus(N, s M) -> U51#(isNat M, M, N)) (top# ok X -> active# X, active# plus(N, s M) -> isNat# M) (top# ok X -> active# X, active# s X -> s# active X) (top# ok X -> active# X, active# s X -> active# X) (top# ok X -> active# X, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (top# ok X -> active# X, active# U51(tt(), M, N) -> isNat# N) (top# ok X -> active# X, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3)) (top# ok X -> active# X, active# U51(X1, X2, X3) -> active# X1) (top# ok X -> active# X, active# U52(tt(), M, N) -> plus#(N, M)) (top# ok X -> active# X, active# U52(tt(), M, N) -> s# plus(N, M)) (top# ok X -> active# X, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (top# ok X -> active# X, active# U52(X1, X2, X3) -> active# X1) (top# ok X -> active# X, active# U41(X1, X2) -> U41#(active X1, X2)) (top# ok X -> active# X, active# U41(X1, X2) -> active# X1) (top# ok X -> active# X, active# U31(tt(), V2) -> U32# isNat V2) (top# ok X -> active# X, active# U31(tt(), V2) -> isNat# V2) (top# ok X -> active# X, active# U31(X1, X2) -> U31#(active X1, X2)) (top# ok X -> active# X, active# U31(X1, X2) -> active# X1) (top# ok X -> active# X, active# U32 X -> U32# active X) (top# ok X -> active# X, active# U32 X -> active# X) (top# ok X -> active# X, active# U21 X -> U21# active X) (top# ok X -> active# X, active# U21 X -> active# X) (top# ok X -> active# X, active# U11(tt(), V2) -> isNat# V2) (top# ok X -> active# X, active# U11(tt(), V2) -> U12# isNat V2) (top# ok X -> active# X, active# U11(X1, X2) -> U11#(active X1, X2)) (top# ok X -> active# X, active# U11(X1, X2) -> active# X1) (top# ok X -> active# X, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (top# ok X -> active# X, active# isNat x(V1, V2) -> isNat# V1) (top# ok X -> active# X, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (top# ok X -> active# X, active# isNat plus(V1, V2) -> isNat# V1) (top# ok X -> active# X, active# isNat s V1 -> U21# isNat V1) (top# ok X -> active# X, active# isNat s V1 -> isNat# V1) (top# ok X -> active# X, active# U12 X -> active# X) (top# ok X -> active# X, active# U12 X -> U12# active X) (active# isNat plus(V1, V2) -> isNat# V1, isNat# ok X -> isNat# X) (proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3), U52#(ok X1, ok X2, ok X3) -> U52#(X1, X2, X3)) (proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3), U52#(mark X1, X2, X3) -> U52#(X1, X2, X3)) (proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3), U72#(ok X1, ok X2, ok X3) -> U72#(X1, X2, X3)) (proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3), U72#(mark X1, X2, X3) -> U72#(X1, X2, X3)) (active# U52(tt(), M, N) -> s# plus(N, M), s# ok X -> s# X) (active# U52(tt(), M, N) -> s# plus(N, M), s# mark X -> s# X) (proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3), U71#(mark X1, X2, X3) -> U71#(X1, X2, X3)) (proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3), U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3)) (proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3), U51#(mark X1, X2, X3) -> U51#(X1, X2, X3)) (proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3), U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3)) (active# isNat x(V1, V2) -> isNat# V1, isNat# ok X -> isNat# X) (active# isNat s V1 -> isNat# V1, isNat# ok X -> isNat# X) (top# mark X -> proper# X, proper# U12 X -> U12# proper X) (top# mark X -> proper# X, proper# U12 X -> proper# X) (top# mark X -> proper# X, proper# isNat X -> isNat# proper X) (top# mark X -> proper# X, proper# isNat X -> proper# X) (top# mark X -> proper# X, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (top# mark X -> proper# X, proper# U11(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# U11(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# U21 X -> U21# proper X) (top# mark X -> proper# X, proper# U21 X -> proper# X) (top# mark X -> proper# X, proper# U32 X -> U32# proper X) (top# mark X -> proper# X, proper# U32 X -> proper# X) (top# mark X -> proper# X, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (top# mark X -> proper# X, proper# U31(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# U31(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (top# mark X -> proper# X, proper# U41(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# U41(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (top# mark X -> proper# X, proper# U52(X1, X2, X3) -> proper# X1) (top# mark X -> proper# X, proper# U52(X1, X2, X3) -> proper# X2) (top# mark X -> proper# X, proper# U52(X1, X2, X3) -> proper# X3) (top# mark X -> proper# X, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (top# mark X -> proper# X, proper# U51(X1, X2, X3) -> proper# X1) (top# mark X -> proper# X, proper# U51(X1, X2, X3) -> proper# X2) (top# mark X -> proper# X, proper# U51(X1, X2, X3) -> proper# X3) (top# mark X -> proper# X, proper# s X -> s# proper X) (top# mark X -> proper# X, proper# s X -> proper# X) (top# mark X -> proper# X, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (top# mark X -> proper# X, proper# plus(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# plus(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# U61 X -> U61# proper X) (top# mark X -> proper# X, proper# U61 X -> proper# X) (top# mark X -> proper# X, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (top# mark X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1) (top# mark X -> proper# X, proper# U72(X1, X2, X3) -> proper# X2) (top# mark X -> proper# X, proper# U72(X1, X2, X3) -> proper# X3) (top# mark X -> proper# X, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (top# mark X -> proper# X, proper# U71(X1, X2, X3) -> proper# X1) (top# mark X -> proper# X, proper# U71(X1, X2, X3) -> proper# X2) (top# mark X -> proper# X, proper# U71(X1, X2, X3) -> proper# X3) (top# mark X -> proper# X, proper# x(X1, X2) -> x#(proper X1, proper X2)) (top# mark X -> proper# X, proper# x(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# x(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# U12 X -> U12# proper X) (proper# s X -> proper# X, proper# U12 X -> proper# X) (proper# s X -> proper# X, proper# isNat X -> isNat# proper X) (proper# s X -> proper# X, proper# isNat X -> proper# X) (proper# s X -> proper# X, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# s X -> proper# X, proper# U11(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# U11(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# U21 X -> U21# proper X) (proper# s X -> proper# X, proper# U21 X -> proper# X) (proper# s X -> proper# X, proper# U32 X -> U32# proper X) (proper# s X -> proper# X, proper# U32 X -> proper# X) (proper# s X -> proper# X, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# s X -> proper# X, proper# U31(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# U31(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# s X -> proper# X, proper# U41(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# U41(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# s X -> proper# X, proper# U52(X1, X2, X3) -> proper# X1) (proper# s X -> proper# X, proper# U52(X1, X2, X3) -> proper# X2) (proper# s X -> proper# X, proper# U52(X1, X2, X3) -> proper# X3) (proper# s X -> proper# X, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# s X -> proper# X, proper# U51(X1, X2, X3) -> proper# X1) (proper# s X -> proper# X, proper# U51(X1, X2, X3) -> proper# X2) (proper# s X -> proper# X, proper# U51(X1, X2, X3) -> proper# X3) (proper# s X -> proper# X, proper# s X -> s# proper X) (proper# s X -> proper# X, proper# s X -> proper# X) (proper# s X -> proper# X, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# U61 X -> U61# proper X) (proper# s X -> proper# X, proper# U61 X -> proper# X) (proper# s X -> proper# X, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# s X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1) (proper# s X -> proper# X, proper# U72(X1, X2, X3) -> proper# X2) (proper# s X -> proper# X, proper# U72(X1, X2, X3) -> proper# X3) (proper# s X -> proper# X, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# s X -> proper# X, proper# U71(X1, X2, X3) -> proper# X1) (proper# s X -> proper# X, proper# U71(X1, X2, X3) -> proper# X2) (proper# s X -> proper# X, proper# U71(X1, X2, X3) -> proper# X3) (proper# s X -> proper# X, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# s X -> proper# X, proper# x(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# x(X1, X2) -> proper# X2) (proper# U21 X -> proper# X, proper# U12 X -> U12# proper X) (proper# U21 X -> proper# X, proper# U12 X -> proper# X) (proper# U21 X -> proper# X, proper# isNat X -> isNat# proper X) (proper# U21 X -> proper# X, proper# isNat X -> proper# X) (proper# U21 X -> proper# X, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U21 X -> proper# X, proper# U11(X1, X2) -> proper# X1) (proper# U21 X -> proper# X, proper# U11(X1, X2) -> proper# X2) (proper# U21 X -> proper# X, proper# U21 X -> U21# proper X) (proper# U21 X -> proper# X, proper# U21 X -> proper# X) (proper# U21 X -> proper# X, proper# U32 X -> U32# proper X) (proper# U21 X -> proper# X, proper# U32 X -> proper# X) (proper# U21 X -> proper# X, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U21 X -> proper# X, proper# U31(X1, X2) -> proper# X1) (proper# U21 X -> proper# X, proper# U31(X1, X2) -> proper# X2) (proper# U21 X -> proper# X, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U21 X -> proper# X, proper# U41(X1, X2) -> proper# X1) (proper# U21 X -> proper# X, proper# U41(X1, X2) -> proper# X2) (proper# U21 X -> proper# X, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U21 X -> proper# X, proper# U52(X1, X2, X3) -> proper# X1) (proper# U21 X -> proper# X, proper# U52(X1, X2, X3) -> proper# X2) (proper# U21 X -> proper# X, proper# U52(X1, X2, X3) -> proper# X3) (proper# U21 X -> proper# X, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U21 X -> proper# X, proper# U51(X1, X2, X3) -> proper# X1) (proper# U21 X -> proper# X, proper# U51(X1, X2, X3) -> proper# X2) (proper# U21 X -> proper# X, proper# U51(X1, X2, X3) -> proper# X3) (proper# U21 X -> proper# X, proper# s X -> s# proper X) (proper# U21 X -> proper# X, proper# s X -> proper# X) (proper# U21 X -> proper# X, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U21 X -> proper# X, proper# plus(X1, X2) -> proper# X1) (proper# U21 X -> proper# X, proper# plus(X1, X2) -> proper# X2) (proper# U21 X -> proper# X, proper# U61 X -> U61# proper X) (proper# U21 X -> proper# X, proper# U61 X -> proper# X) (proper# U21 X -> proper# X, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U21 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1) (proper# U21 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X2) (proper# U21 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X3) (proper# U21 X -> proper# X, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U21 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X1) (proper# U21 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X2) (proper# U21 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X3) (proper# U21 X -> proper# X, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U21 X -> proper# X, proper# x(X1, X2) -> proper# X1) (proper# U21 X -> proper# X, proper# x(X1, X2) -> proper# X2) (proper# U12 X -> proper# X, proper# U12 X -> U12# proper X) (proper# U12 X -> proper# X, proper# U12 X -> proper# X) (proper# U12 X -> proper# X, proper# isNat X -> isNat# proper X) (proper# U12 X -> proper# X, proper# isNat X -> proper# X) (proper# U12 X -> proper# X, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U12 X -> proper# X, proper# U11(X1, X2) -> proper# X1) (proper# U12 X -> proper# X, proper# U11(X1, X2) -> proper# X2) (proper# U12 X -> proper# X, proper# U21 X -> U21# proper X) (proper# U12 X -> proper# X, proper# U21 X -> proper# X) (proper# U12 X -> proper# X, proper# U32 X -> U32# proper X) (proper# U12 X -> proper# X, proper# U32 X -> proper# X) (proper# U12 X -> proper# X, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U12 X -> proper# X, proper# U31(X1, X2) -> proper# X1) (proper# U12 X -> proper# X, proper# U31(X1, X2) -> proper# X2) (proper# U12 X -> proper# X, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U12 X -> proper# X, proper# U41(X1, X2) -> proper# X1) (proper# U12 X -> proper# X, proper# U41(X1, X2) -> proper# X2) (proper# U12 X -> proper# X, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U12 X -> proper# X, proper# U52(X1, X2, X3) -> proper# X1) (proper# U12 X -> proper# X, proper# U52(X1, X2, X3) -> proper# X2) (proper# U12 X -> proper# X, proper# U52(X1, X2, X3) -> proper# X3) (proper# U12 X -> proper# X, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U12 X -> proper# X, proper# U51(X1, X2, X3) -> proper# X1) (proper# U12 X -> proper# X, proper# U51(X1, X2, X3) -> proper# X2) (proper# U12 X -> proper# X, proper# U51(X1, X2, X3) -> proper# X3) (proper# U12 X -> proper# X, proper# s X -> s# proper X) (proper# U12 X -> proper# X, proper# s X -> proper# X) (proper# U12 X -> proper# X, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U12 X -> proper# X, proper# plus(X1, X2) -> proper# X1) (proper# U12 X -> proper# X, proper# plus(X1, X2) -> proper# X2) (proper# U12 X -> proper# X, proper# U61 X -> U61# proper X) (proper# U12 X -> proper# X, proper# U61 X -> proper# X) (proper# U12 X -> proper# X, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U12 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1) (proper# U12 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X2) (proper# U12 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X3) (proper# U12 X -> proper# X, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U12 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X1) (proper# U12 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X2) (proper# U12 X -> proper# X, proper# U71(X1, X2, X3) -> proper# X3) (proper# U12 X -> proper# X, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U12 X -> proper# X, proper# x(X1, X2) -> proper# X1) (proper# U12 X -> proper# X, proper# x(X1, X2) -> proper# X2) (U61# mark X -> U61# X, U61# mark X -> U61# X) (U61# mark X -> U61# X, U61# ok X -> U61# X) (s# mark X -> s# X, s# mark X -> s# X) (s# mark X -> s# X, s# ok X -> s# X) (U32# mark X -> U32# X, U32# mark X -> U32# X) (U32# mark X -> U32# X, U32# ok X -> U32# X) (U21# mark X -> U21# X, U21# mark X -> U21# X) (U21# mark X -> U21# X, U21# ok X -> U21# X) (active# s X -> active# X, active# U12 X -> U12# active X) (active# s X -> active# X, active# U12 X -> active# X) (active# s X -> active# X, active# isNat s V1 -> isNat# V1) (active# s X -> active# X, active# isNat s V1 -> U21# isNat V1) (active# s X -> active# X, active# isNat plus(V1, V2) -> isNat# V1) (active# s X -> active# X, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# s X -> active# X, active# isNat x(V1, V2) -> isNat# V1) (active# s X -> active# X, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# s X -> active# X, active# U11(X1, X2) -> active# X1) (active# s X -> active# X, active# U11(X1, X2) -> U11#(active X1, X2)) (active# s X -> active# X, active# U11(tt(), V2) -> U12# isNat V2) (active# s X -> active# X, active# U11(tt(), V2) -> isNat# V2) (active# s X -> active# X, active# U21 X -> active# X) (active# s X -> active# X, active# U21 X -> U21# active X) (active# s X -> active# X, active# U32 X -> active# X) (active# s X -> active# X, active# U32 X -> U32# active X) (active# s X -> active# X, active# U31(X1, X2) -> active# X1) (active# s X -> active# X, active# U31(X1, X2) -> U31#(active X1, X2)) (active# s X -> active# X, active# U31(tt(), V2) -> isNat# V2) (active# s X -> active# X, active# U31(tt(), V2) -> U32# isNat V2) (active# s X -> active# X, active# U41(X1, X2) -> active# X1) (active# s X -> active# X, active# U41(X1, X2) -> U41#(active X1, X2)) (active# s X -> active# X, active# U52(X1, X2, X3) -> active# X1) (active# s X -> active# X, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (active# s X -> active# X, active# U52(tt(), M, N) -> s# 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s X -> active# X, active# x(X1, X2) -> active# X2) (active# s X -> active# X, active# x(X1, X2) -> x#(X1, active X2)) (active# s X -> active# X, active# x(X1, X2) -> x#(active X1, X2)) (active# U21 X -> active# X, active# U12 X -> U12# active X) (active# U21 X -> active# X, active# U12 X -> active# X) (active# U21 X -> active# X, active# isNat s V1 -> isNat# V1) (active# U21 X -> active# X, active# isNat s V1 -> U21# isNat V1) (active# U21 X -> active# X, active# isNat plus(V1, V2) -> isNat# V1) (active# U21 X -> active# X, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# U21 X -> active# X, active# isNat x(V1, V2) -> isNat# V1) (active# U21 X -> active# X, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# U21 X -> active# X, active# U11(X1, X2) -> active# X1) (active# U21 X -> active# X, active# U11(X1, X2) -> U11#(active X1, X2)) (active# U21 X -> active# X, active# U11(tt(), V2) -> U12# isNat V2) (active# U21 X -> active# X, active# U11(tt(), V2) -> isNat# V2) 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(active# isNat x(V1, V2) -> U31#(isNat V1, V2), U31#(ok X1, ok X2) -> U31#(X1, X2)) (active# x(N, 0()) -> isNat# N, isNat# ok X -> isNat# X) (active# plus(N, 0()) -> isNat# N, isNat# ok X -> isNat# X) (U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3), U71#(mark X1, X2, X3) -> U71#(X1, X2, X3)) (U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3), U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3)) (U72#(ok X1, ok X2, ok X3) -> U72#(X1, X2, X3), U72#(mark X1, X2, X3) -> U72#(X1, X2, X3)) (U72#(ok X1, ok X2, ok X3) -> U72#(X1, X2, X3), U72#(ok X1, ok X2, ok X3) -> U72#(X1, X2, X3)) (U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3), U51#(mark X1, X2, X3) -> U51#(X1, X2, X3)) (U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3), U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3)) (U52#(ok X1, ok X2, ok X3) -> U52#(X1, X2, X3), U52#(mark X1, X2, X3) -> U52#(X1, X2, X3)) (U52#(ok X1, ok X2, ok X3) -> U52#(X1, X2, X3), U52#(ok X1, ok X2, ok X3) -> U52#(X1, X2, X3)) (active# U71(X1, X2, X3) -> U71#(active X1, X2, X3), U71#(mark X1, X2, X3) -> U71#(X1, X2, X3)) (active# U71(X1, X2, X3) -> U71#(active X1, X2, X3), U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3)) (active# U51(X1, X2, X3) -> U51#(active X1, X2, X3), U51#(mark X1, X2, X3) -> U51#(X1, X2, X3)) (active# U51(X1, X2, X3) -> U51#(active X1, X2, X3), U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3)) (proper# x(X1, X2) -> proper# X1, proper# U12 X -> U12# proper X) (proper# x(X1, X2) -> proper# X1, proper# U12 X -> proper# X) (proper# x(X1, X2) -> proper# X1, proper# isNat X -> isNat# proper X) (proper# x(X1, X2) -> proper# X1, proper# isNat X -> proper# X) (proper# x(X1, X2) -> proper# X1, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# x(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X1) (proper# x(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2) (proper# x(X1, X2) -> proper# X1, proper# U21 X -> U21# proper X) (proper# x(X1, X2) -> proper# X1, proper# U21 X -> proper# X) (proper# x(X1, X2) -> proper# X1, proper# U32 X 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-> proper# X1, proper# U52(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U72(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X1, proper# s X -> s# proper X) (proper# U72(X1, X2, X3) -> proper# X1, proper# s X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X1, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U72(X1, X2, X3) -> proper# X1, proper# plus(X1, X2) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X1, proper# plus(X1, X2) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X1, proper# U61 X -> U61# proper X) (proper# U72(X1, X2, X3) -> proper# X1, proper# U61 X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U72(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X1, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U72(X1, X2, X3) -> proper# X1, proper# x(X1, X2) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X1, proper# x(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X1, proper# U12 X -> U12# proper X) (proper# U51(X1, X2, X3) -> proper# X1, proper# U12 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X1, proper# isNat X -> isNat# proper X) (proper# U51(X1, X2, X3) -> proper# X1, proper# isNat X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X1, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X1, proper# U11(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X1, proper# U11(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X1, proper# U21 X -> U21# proper X) (proper# U51(X1, X2, X3) -> proper# X1, proper# U21 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X1, proper# U32 X -> U32# proper X) (proper# U51(X1, X2, X3) -> proper# X1, proper# U32 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X1, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X1, proper# U31(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X1, proper# U31(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X1, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X1, proper# U41(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X1, proper# U41(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X1, proper# s X -> s# proper X) (proper# U51(X1, X2, X3) -> proper# X1, proper# s X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X1, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X1, proper# plus(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X1, proper# plus(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X1, proper# U61 X -> U61# proper X) (proper# U51(X1, X2, X3) -> proper# X1, proper# U61 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X1, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X1, proper# x(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X1, proper# x(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X1, proper# U12 X -> U12# proper X) (proper# U41(X1, X2) -> proper# X1, proper# U12 X -> proper# X) (proper# U41(X1, X2) -> proper# X1, proper# isNat X -> isNat# proper X) (proper# U41(X1, X2) -> proper# X1, proper# isNat X -> proper# X) (proper# U41(X1, X2) -> proper# X1, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X1, proper# U21 X -> U21# proper X) (proper# U41(X1, X2) -> proper# X1, proper# U21 X -> proper# X) (proper# U41(X1, X2) -> proper# X1, proper# U32 X -> U32# proper X) (proper# U41(X1, X2) -> proper# X1, proper# U32 X -> proper# X) (proper# U41(X1, X2) -> proper# X1, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X1, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U41(X1, X2) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X1) (proper# U41(X1, X2) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2) (proper# U41(X1, X2) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X3) (proper# U41(X1, X2) -> proper# X1, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U41(X1, X2) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X1) (proper# U41(X1, X2) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2) (proper# U41(X1, X2) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X3) (proper# U41(X1, X2) -> proper# X1, proper# s X -> s# proper X) (proper# U41(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# U41(X1, X2) -> proper# X1, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X1, proper# U61 X -> U61# proper X) (proper# U41(X1, X2) -> proper# X1, proper# U61 X -> proper# X) (proper# U41(X1, X2) -> proper# X1, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U41(X1, X2) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X1) (proper# U41(X1, X2) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2) (proper# U41(X1, X2) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X3) (proper# U41(X1, X2) -> proper# X1, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U41(X1, X2) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X1) (proper# U41(X1, X2) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X2) (proper# U41(X1, X2) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X3) (proper# U41(X1, X2) -> proper# X1, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X1, proper# x(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X1, proper# x(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X1, proper# U12 X -> U12# proper X) (proper# U11(X1, X2) -> proper# X1, proper# U12 X -> proper# X) (proper# U11(X1, X2) -> proper# X1, proper# isNat X -> isNat# proper X) (proper# U11(X1, X2) -> proper# X1, proper# isNat X -> proper# X) (proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X1, proper# U21 X -> U21# proper X) (proper# U11(X1, X2) -> proper# X1, proper# U21 X -> proper# X) (proper# U11(X1, X2) -> proper# X1, proper# U32 X -> U32# proper X) (proper# U11(X1, X2) -> proper# X1, proper# U32 X -> proper# X) (proper# U11(X1, X2) -> proper# X1, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X1, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X1, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U11(X1, X2) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X1) (proper# U11(X1, X2) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2) (proper# U11(X1, X2) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X3) (proper# U11(X1, X2) -> proper# X1, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U11(X1, X2) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X1) (proper# U11(X1, X2) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2) (proper# U11(X1, X2) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X3) (proper# U11(X1, X2) -> proper# X1, proper# s X -> s# proper X) (proper# U11(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# U11(X1, X2) -> proper# X1, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X1, proper# U61 X -> U61# proper X) (proper# U11(X1, X2) -> proper# X1, proper# U61 X -> proper# X) (proper# U11(X1, X2) -> proper# X1, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U11(X1, X2) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X1) (proper# U11(X1, X2) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2) (proper# U11(X1, X2) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X3) (proper# U11(X1, X2) -> proper# X1, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U11(X1, X2) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X1) (proper# U11(X1, X2) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X2) (proper# U11(X1, X2) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X3) (proper# U11(X1, X2) -> proper# X1, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X1, proper# x(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X1, proper# x(X1, X2) -> proper# X2) (active# U71(X1, X2, X3) -> active# X1, active# U12 X -> U12# active X) (active# U71(X1, X2, X3) -> active# X1, active# U12 X -> active# X) (active# U71(X1, X2, X3) -> active# X1, active# isNat s V1 -> isNat# V1) (active# U71(X1, X2, X3) -> active# X1, active# isNat s V1 -> U21# isNat V1) (active# U71(X1, X2, X3) -> active# X1, active# isNat plus(V1, V2) -> isNat# V1) (active# U71(X1, X2, X3) -> active# X1, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# U71(X1, X2, X3) -> active# X1, active# isNat x(V1, V2) -> isNat# V1) (active# U71(X1, X2, X3) -> active# X1, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# U71(X1, X2, X3) -> active# X1, active# U11(X1, X2) -> active# X1) (active# U71(X1, X2, X3) -> active# X1, active# U11(X1, X2) -> U11#(active X1, X2)) (active# U71(X1, X2, X3) -> active# X1, active# U11(tt(), V2) -> U12# isNat V2) (active# U71(X1, X2, X3) -> active# X1, active# U11(tt(), V2) -> isNat# V2) (active# U71(X1, X2, X3) -> active# X1, active# U21 X -> active# X) (active# U71(X1, X2, X3) -> active# X1, active# U21 X -> U21# active X) (active# U71(X1, X2, X3) -> active# X1, active# U32 X -> active# X) (active# U71(X1, X2, X3) -> active# X1, active# U32 X -> U32# active X) (active# U71(X1, X2, X3) -> active# X1, active# U31(X1, X2) -> active# X1) (active# U71(X1, X2, X3) -> active# X1, active# U31(X1, X2) -> U31#(active X1, X2)) (active# U71(X1, X2, X3) -> active# X1, active# U31(tt(), V2) -> isNat# V2) (active# U71(X1, X2, X3) -> active# X1, active# U31(tt(), V2) -> U32# isNat V2) (active# U71(X1, X2, X3) -> active# X1, active# U41(X1, X2) -> active# X1) (active# U71(X1, X2, X3) -> active# X1, active# U41(X1, X2) -> U41#(active X1, X2)) (active# U71(X1, X2, X3) -> active# X1, active# U52(X1, X2, X3) -> active# X1) (active# U71(X1, X2, X3) -> active# X1, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (active# U71(X1, X2, X3) -> active# X1, active# U52(tt(), M, N) -> s# plus(N, M)) (active# U71(X1, X2, X3) -> active# X1, active# U52(tt(), M, N) -> plus#(N, M)) (active# U71(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1) (active# U71(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3)) (active# U71(X1, X2, X3) -> active# X1, active# U51(tt(), M, N) -> isNat# N) (active# U71(X1, X2, X3) -> active# X1, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (active# U71(X1, X2, X3) -> active# X1, active# s X -> active# X) (active# U71(X1, X2, X3) -> active# X1, active# s X -> s# active X) (active# U71(X1, X2, X3) -> active# X1, active# plus(N, s M) -> isNat# M) (active# U71(X1, X2, X3) -> active# X1, active# plus(N, s M) -> U51#(isNat M, M, N)) (active# U71(X1, X2, X3) -> active# X1, active# plus(N, 0()) -> isNat# N) (active# U71(X1, X2, X3) -> active# X1, active# plus(N, 0()) -> U41#(isNat N, N)) (active# U71(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> active# X1) (active# U71(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> active# X2) (active# U71(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> plus#(X1, active X2)) (active# U71(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> plus#(active X1, X2)) (active# U71(X1, X2, X3) -> active# X1, active# U61 X -> active# X) (active# U71(X1, X2, X3) -> active# X1, active# U61 X -> U61# active X) (active# U71(X1, X2, X3) -> active# X1, active# U72(X1, X2, X3) -> active# X1) (active# U71(X1, X2, X3) -> active# X1, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3)) (active# U71(X1, X2, X3) -> active# X1, active# U72(tt(), M, N) -> plus#(x(N, M), N)) (active# U71(X1, X2, X3) -> active# X1, active# U72(tt(), M, N) -> x#(N, M)) (active# U71(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> active# X1) (active# U71(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# U71(X1, X2, X3) -> active# X1, active# U71(tt(), M, N) -> isNat# N) (active# U71(X1, X2, X3) -> active# X1, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# U71(X1, X2, X3) -> active# X1, active# x(N, s M) -> isNat# M) (active# U71(X1, X2, X3) -> active# X1, active# x(N, s M) -> U71#(isNat M, M, N)) (active# U71(X1, X2, X3) -> active# X1, active# x(N, 0()) -> isNat# N) (active# U71(X1, X2, X3) -> active# X1, active# x(N, 0()) -> U61# isNat N) (active# U71(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X1) (active# U71(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X2) (active# U71(X1, X2, X3) -> active# X1, active# x(X1, X2) -> x#(X1, active X2)) (active# U71(X1, X2, X3) -> active# X1, active# x(X1, X2) -> x#(active X1, X2)) (active# plus(X1, X2) -> active# X1, active# U12 X -> U12# active X) (active# plus(X1, X2) -> active# X1, active# U12 X -> active# X) (active# plus(X1, X2) -> active# X1, active# isNat s V1 -> isNat# V1) (active# plus(X1, X2) -> active# X1, active# isNat s V1 -> U21# isNat V1) (active# plus(X1, X2) -> active# X1, active# isNat plus(V1, V2) -> isNat# V1) (active# plus(X1, X2) -> active# X1, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# plus(X1, X2) -> active# X1, active# isNat x(V1, V2) -> isNat# V1) (active# plus(X1, X2) -> active# X1, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# plus(X1, X2) -> active# X1, active# U11(X1, X2) -> active# X1) (active# plus(X1, X2) -> active# X1, active# U11(X1, X2) -> U11#(active X1, X2)) (active# plus(X1, X2) -> active# X1, active# U11(tt(), V2) -> U12# isNat V2) (active# plus(X1, X2) -> active# X1, active# U11(tt(), V2) -> isNat# V2) (active# plus(X1, X2) -> active# X1, active# U21 X -> active# X) (active# plus(X1, X2) -> active# X1, active# U21 X -> U21# active X) (active# plus(X1, X2) -> active# X1, active# U32 X -> active# X) (active# plus(X1, X2) -> active# X1, active# U32 X -> U32# active X) (active# plus(X1, X2) -> active# X1, active# U31(X1, X2) -> active# X1) (active# plus(X1, X2) -> active# X1, active# U31(X1, X2) -> U31#(active X1, X2)) (active# plus(X1, X2) -> active# X1, active# U31(tt(), V2) -> isNat# V2) (active# plus(X1, X2) -> active# X1, active# U31(tt(), V2) -> U32# isNat V2) (active# plus(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1) (active# plus(X1, X2) -> active# X1, active# U41(X1, X2) -> U41#(active X1, X2)) (active# plus(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1) (active# plus(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (active# plus(X1, X2) -> active# X1, active# U52(tt(), M, N) -> s# plus(N, M)) (active# plus(X1, X2) -> active# X1, active# U52(tt(), M, N) -> plus#(N, M)) (active# plus(X1, X2) -> active# X1, active# U51(X1, X2, X3) -> active# X1) (active# plus(X1, X2) -> active# X1, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3)) (active# plus(X1, X2) -> active# X1, active# U51(tt(), M, N) -> isNat# N) (active# plus(X1, X2) -> active# X1, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (active# plus(X1, X2) -> active# X1, active# s X -> active# X) (active# plus(X1, X2) -> active# X1, active# s X -> s# active X) (active# plus(X1, X2) -> active# X1, active# plus(N, s M) -> isNat# M) (active# plus(X1, X2) -> active# X1, active# plus(N, s M) -> U51#(isNat M, M, N)) (active# plus(X1, X2) -> active# X1, active# plus(N, 0()) -> isNat# N) (active# plus(X1, X2) -> active# X1, active# plus(N, 0()) -> U41#(isNat N, N)) (active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X1) (active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2) (active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> plus#(X1, active X2)) (active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> plus#(active X1, X2)) (active# plus(X1, X2) -> active# X1, active# U61 X -> active# X) (active# plus(X1, X2) -> active# X1, active# U61 X -> U61# active X) (active# plus(X1, X2) -> active# X1, active# U72(X1, X2, X3) -> active# X1) (active# plus(X1, X2) -> active# X1, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3)) (active# plus(X1, X2) -> active# X1, active# U72(tt(), M, N) -> plus#(x(N, M), N)) (active# plus(X1, X2) -> active# X1, active# U72(tt(), M, N) -> x#(N, M)) (active# plus(X1, X2) -> active# X1, active# U71(X1, X2, X3) -> active# X1) (active# plus(X1, X2) -> active# X1, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# plus(X1, X2) -> active# X1, active# U71(tt(), M, N) -> isNat# N) (active# plus(X1, X2) -> active# X1, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# plus(X1, X2) -> active# X1, active# x(N, s M) -> isNat# M) (active# plus(X1, X2) -> active# X1, active# x(N, s M) -> U71#(isNat M, M, N)) (active# plus(X1, X2) -> active# X1, active# x(N, 0()) -> isNat# N) (active# plus(X1, X2) -> active# X1, active# x(N, 0()) -> U61# isNat N) (active# plus(X1, X2) -> active# X1, active# x(X1, X2) -> active# X1) (active# plus(X1, X2) -> active# X1, active# x(X1, X2) -> active# X2) (active# plus(X1, X2) -> active# X1, active# x(X1, X2) -> x#(X1, active X2)) (active# plus(X1, X2) -> active# X1, active# x(X1, X2) -> x#(active X1, X2)) (active# U52(X1, X2, X3) -> active# X1, active# U12 X -> U12# active X) (active# U52(X1, X2, X3) -> active# X1, active# U12 X -> active# X) (active# U52(X1, X2, X3) -> active# X1, active# isNat s V1 -> isNat# V1) (active# U52(X1, X2, X3) -> active# X1, active# isNat s V1 -> U21# isNat V1) (active# U52(X1, X2, X3) -> active# X1, active# isNat plus(V1, V2) -> isNat# V1) (active# U52(X1, X2, X3) -> active# X1, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# U52(X1, X2, X3) -> active# X1, active# isNat x(V1, V2) -> isNat# V1) (active# U52(X1, X2, X3) -> active# X1, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# U52(X1, X2, X3) -> active# X1, active# U11(X1, X2) -> active# X1) (active# U52(X1, X2, X3) -> active# X1, active# U11(X1, X2) -> U11#(active X1, X2)) (active# U52(X1, X2, X3) -> active# X1, active# U11(tt(), V2) -> U12# isNat V2) (active# U52(X1, X2, X3) -> active# X1, active# U11(tt(), V2) -> isNat# V2) (active# U52(X1, X2, X3) -> active# X1, active# U21 X -> active# X) (active# U52(X1, X2, X3) -> active# X1, active# U21 X -> U21# active X) (active# U52(X1, X2, X3) -> active# X1, active# U32 X -> active# X) (active# U52(X1, X2, X3) -> active# X1, active# U32 X -> U32# active X) (active# U52(X1, X2, X3) -> active# X1, active# U31(X1, X2) -> active# X1) (active# U52(X1, X2, X3) -> active# X1, active# U31(X1, X2) -> U31#(active X1, X2)) (active# U52(X1, X2, X3) -> active# X1, active# U31(tt(), V2) -> isNat# V2) (active# U52(X1, X2, X3) -> active# X1, active# U31(tt(), V2) -> U32# isNat V2) (active# U52(X1, X2, X3) -> active# X1, active# U41(X1, X2) -> active# X1) (active# U52(X1, X2, X3) -> active# X1, active# U41(X1, X2) -> U41#(active X1, X2)) (active# U52(X1, X2, X3) -> active# X1, active# U52(X1, X2, X3) -> active# X1) (active# U52(X1, X2, X3) -> active# X1, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (active# U52(X1, X2, X3) -> active# X1, active# U52(tt(), M, N) -> s# plus(N, M)) (active# U52(X1, X2, X3) -> active# X1, active# U52(tt(), M, N) -> plus#(N, M)) (active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1) (active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3)) (active# U52(X1, X2, X3) -> active# X1, active# U51(tt(), M, N) -> isNat# N) (active# U52(X1, X2, X3) -> active# X1, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (active# U52(X1, X2, X3) -> active# X1, active# s X -> active# X) (active# U52(X1, X2, X3) -> active# X1, active# s X -> s# active X) (active# U52(X1, X2, X3) -> active# X1, active# plus(N, s M) -> isNat# M) (active# U52(X1, X2, X3) -> active# X1, active# plus(N, s M) -> U51#(isNat M, M, N)) (active# U52(X1, X2, X3) -> active# X1, active# plus(N, 0()) -> isNat# N) (active# U52(X1, X2, X3) -> active# X1, active# plus(N, 0()) -> U41#(isNat N, N)) (active# U52(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> active# X1) (active# U52(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> active# X2) (active# U52(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> plus#(X1, active X2)) (active# U52(X1, X2, X3) -> active# X1, active# plus(X1, X2) -> plus#(active X1, X2)) (active# U52(X1, X2, X3) -> active# X1, active# U61 X -> active# X) (active# U52(X1, X2, X3) -> active# X1, active# U61 X -> U61# active X) (active# U52(X1, X2, X3) -> active# X1, active# U72(X1, X2, X3) -> active# X1) (active# U52(X1, X2, X3) -> active# X1, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3)) (active# U52(X1, X2, X3) -> active# X1, active# U72(tt(), M, N) -> plus#(x(N, M), N)) (active# U52(X1, X2, X3) -> active# X1, active# U72(tt(), M, N) -> x#(N, M)) (active# U52(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> active# X1) (active# U52(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# U52(X1, X2, X3) -> active# X1, active# U71(tt(), M, N) -> isNat# N) (active# U52(X1, X2, X3) -> active# X1, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# U52(X1, X2, X3) -> active# X1, active# x(N, s M) -> isNat# M) (active# U52(X1, X2, X3) -> active# X1, active# x(N, s M) -> U71#(isNat M, M, N)) (active# U52(X1, X2, X3) -> active# X1, active# x(N, 0()) -> isNat# N) (active# U52(X1, X2, X3) -> active# X1, active# x(N, 0()) -> U61# isNat N) (active# U52(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X1) (active# U52(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X2) (active# U52(X1, X2, X3) -> active# X1, active# x(X1, X2) -> x#(X1, active X2)) (active# U52(X1, X2, X3) -> active# X1, active# x(X1, X2) -> x#(active X1, X2)) (active# U31(X1, X2) -> active# X1, active# U12 X -> U12# active X) (active# U31(X1, X2) -> active# X1, active# U12 X -> active# X) (active# U31(X1, X2) -> active# X1, active# isNat s V1 -> isNat# V1) (active# U31(X1, X2) -> active# X1, active# isNat s V1 -> U21# isNat V1) (active# U31(X1, X2) -> active# X1, active# isNat plus(V1, V2) -> isNat# V1) (active# U31(X1, X2) -> active# X1, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# U31(X1, X2) -> active# X1, active# isNat x(V1, V2) -> isNat# V1) (active# U31(X1, X2) -> active# X1, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# U31(X1, X2) -> active# X1, active# U11(X1, X2) -> active# X1) (active# U31(X1, X2) -> active# X1, active# U11(X1, X2) -> U11#(active X1, X2)) (active# U31(X1, X2) -> active# X1, active# U11(tt(), V2) -> U12# isNat V2) (active# U31(X1, X2) -> active# X1, active# U11(tt(), V2) -> isNat# V2) (active# U31(X1, X2) -> active# X1, active# U21 X -> active# X) (active# U31(X1, X2) -> active# X1, active# U21 X -> U21# active X) (active# U31(X1, X2) -> active# X1, active# U32 X -> active# X) (active# U31(X1, X2) -> active# X1, active# U32 X -> U32# active X) (active# U31(X1, X2) -> active# X1, active# U31(X1, X2) -> active# X1) (active# U31(X1, X2) -> active# X1, active# U31(X1, X2) -> U31#(active X1, X2)) (active# U31(X1, X2) -> active# X1, active# U31(tt(), V2) -> isNat# V2) (active# U31(X1, X2) -> active# X1, active# U31(tt(), V2) -> U32# isNat V2) (active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1) (active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> U41#(active X1, X2)) (active# U31(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1) (active# U31(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (active# U31(X1, X2) -> active# X1, active# U52(tt(), M, N) -> s# plus(N, M)) (active# U31(X1, X2) -> active# X1, active# U52(tt(), M, N) -> plus#(N, M)) (active# U31(X1, X2) -> active# X1, active# U51(X1, X2, X3) -> active# X1) (active# U31(X1, X2) -> active# X1, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3)) (active# U31(X1, X2) -> active# X1, active# U51(tt(), M, N) -> isNat# N) (active# U31(X1, X2) -> active# X1, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (active# U31(X1, X2) -> active# X1, active# s X -> active# X) (active# U31(X1, X2) -> active# X1, active# s X -> s# active X) (active# U31(X1, X2) -> active# X1, active# plus(N, s M) -> isNat# M) (active# U31(X1, X2) -> active# X1, active# plus(N, s M) -> U51#(isNat M, M, N)) (active# U31(X1, X2) -> active# X1, active# plus(N, 0()) -> isNat# N) (active# U31(X1, X2) -> active# X1, active# plus(N, 0()) -> U41#(isNat N, N)) (active# U31(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X1) (active# U31(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2) (active# U31(X1, X2) -> active# X1, active# plus(X1, X2) -> plus#(X1, active X2)) (active# U31(X1, X2) -> active# X1, active# plus(X1, X2) -> plus#(active X1, X2)) (active# U31(X1, X2) -> active# X1, active# U61 X -> active# X) (active# U31(X1, X2) -> active# X1, active# U61 X -> U61# active X) (active# U31(X1, X2) -> active# X1, active# U72(X1, X2, X3) -> active# X1) (active# U31(X1, X2) -> active# X1, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3)) (active# U31(X1, X2) -> active# X1, active# U72(tt(), M, N) -> plus#(x(N, M), N)) (active# U31(X1, X2) -> active# X1, active# U72(tt(), M, N) -> x#(N, M)) (active# U31(X1, X2) -> active# X1, active# U71(X1, X2, X3) -> active# X1) (active# U31(X1, X2) -> active# X1, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# U31(X1, X2) -> active# X1, active# U71(tt(), M, N) -> isNat# N) (active# U31(X1, X2) -> active# X1, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# U31(X1, X2) -> active# X1, active# x(N, s M) -> isNat# M) (active# U31(X1, X2) -> active# X1, active# x(N, s M) -> U71#(isNat M, M, N)) (active# U31(X1, X2) -> active# X1, active# x(N, 0()) -> isNat# N) (active# U31(X1, X2) -> active# X1, active# x(N, 0()) -> U61# isNat N) (active# U31(X1, X2) -> active# X1, active# x(X1, X2) -> active# X1) (active# U31(X1, X2) -> active# X1, active# x(X1, X2) -> active# X2) (active# U31(X1, X2) -> active# X1, active# x(X1, X2) -> x#(X1, active X2)) (active# U31(X1, X2) -> active# X1, active# x(X1, X2) -> x#(active X1, X2)) (active# U72(tt(), M, N) -> plus#(x(N, M), N), plus#(X1, mark X2) -> plus#(X1, X2)) (active# U72(tt(), M, N) -> plus#(x(N, M), N), plus#(mark X1, X2) -> plus#(X1, X2)) (active# U72(tt(), M, N) -> plus#(x(N, M), N), plus#(ok X1, ok X2) -> plus#(X1, X2)) (proper# U71(X1, X2, X3) -> proper# X3, proper# U12 X -> U12# proper X) (proper# U71(X1, X2, X3) -> proper# X3, proper# U12 X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X3, proper# isNat X -> isNat# proper X) (proper# U71(X1, X2, X3) -> proper# X3, proper# isNat X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U71(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X3, proper# U21 X -> U21# proper X) (proper# U71(X1, X2, X3) -> proper# X3, proper# U21 X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X3, proper# U32 X -> U32# proper X) (proper# U71(X1, X2, X3) -> proper# X3, proper# U32 X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X3, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U71(X1, X2, X3) -> proper# X3, proper# U31(X1, X2) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X3, proper# U31(X1, X2) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X3, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U71(X1, X2, X3) -> proper# X3, proper# U41(X1, X2) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X3, proper# U41(X1, X2) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U71(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> proper# X3) (proper# U71(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U71(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X3) (proper# U71(X1, X2, X3) -> proper# X3, proper# s X -> s# proper X) (proper# U71(X1, X2, X3) -> proper# X3, proper# s X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U71(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X3, proper# U61 X -> U61# proper X) (proper# U71(X1, X2, X3) -> proper# X3, proper# U61 X -> proper# X) (proper# U71(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U71(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X3) (proper# U71(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U71(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X2) (proper# U71(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X3) (proper# U71(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U71(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X1) (proper# U71(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X3, proper# U12 X -> U12# proper X) (proper# U51(X1, X2, X3) -> proper# X3, proper# U12 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X3, proper# isNat X -> isNat# proper X) (proper# U51(X1, X2, X3) -> proper# X3, proper# isNat X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X3, proper# U11(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X3, proper# U21 X -> U21# proper X) (proper# U51(X1, X2, X3) -> proper# X3, proper# U21 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X3, proper# U32 X -> U32# proper X) (proper# U51(X1, X2, X3) -> proper# X3, proper# U32 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X3, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X3, proper# U31(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X3, proper# U31(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X3, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X3, proper# U41(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X3, proper# U41(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X3, proper# U52(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> s# proper X) (proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X3, proper# plus(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X3, proper# U61 X -> U61# proper X) (proper# U51(X1, X2, X3) -> proper# X3, proper# U61 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X3, proper# U72(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X2) (active# x(N, s M) -> isNat# M, isNat# ok X -> isNat# X) (x#(ok X1, ok X2) -> x#(X1, X2), x#(X1, mark X2) -> x#(X1, X2)) (x#(ok X1, ok X2) -> x#(X1, X2), x#(mark X1, X2) -> x#(X1, X2)) (x#(ok X1, ok X2) -> x#(X1, X2), x#(ok X1, ok X2) -> x#(X1, X2)) (x#(X1, mark X2) -> x#(X1, X2), x#(X1, mark X2) -> x#(X1, X2)) (x#(X1, mark X2) -> x#(X1, X2), x#(mark X1, X2) -> x#(X1, X2)) (x#(X1, mark X2) -> x#(X1, X2), x#(ok X1, ok X2) -> x#(X1, X2)) (plus#(mark X1, X2) -> plus#(X1, X2), plus#(X1, mark X2) -> plus#(X1, X2)) (plus#(mark X1, X2) -> plus#(X1, X2), plus#(mark X1, X2) -> plus#(X1, X2)) (plus#(mark X1, X2) -> plus#(X1, X2), plus#(ok X1, ok X2) -> plus#(X1, X2)) (U41#(ok X1, ok X2) -> U41#(X1, X2), U41#(mark X1, X2) -> U41#(X1, X2)) (U41#(ok X1, ok X2) -> U41#(X1, X2), U41#(ok X1, ok X2) -> U41#(X1, X2)) (U31#(ok X1, ok X2) -> U31#(X1, X2), U31#(mark X1, X2) -> U31#(X1, X2)) (U31#(ok X1, ok X2) -> U31#(X1, X2), U31#(ok X1, ok X2) -> U31#(X1, X2)) (U11#(ok X1, ok X2) -> U11#(X1, X2), U11#(mark X1, X2) -> U11#(X1, X2)) (U11#(ok X1, ok X2) -> U11#(X1, X2), U11#(ok X1, ok X2) -> U11#(X1, X2)) (proper# x(X1, X2) -> proper# X2, proper# U12 X -> U12# proper X) (proper# x(X1, X2) -> proper# X2, proper# U12 X -> proper# X) (proper# x(X1, X2) -> proper# X2, proper# isNat X -> isNat# proper X) (proper# x(X1, X2) -> proper# X2, proper# isNat X -> proper# X) (proper# x(X1, X2) -> proper# X2, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# x(X1, X2) -> proper# X2, proper# U11(X1, X2) -> proper# X1) (proper# x(X1, X2) -> proper# X2, proper# U11(X1, X2) -> proper# X2) (proper# x(X1, X2) -> proper# X2, proper# U21 X -> U21# proper X) (proper# x(X1, X2) -> proper# X2, proper# U21 X -> proper# X) (proper# x(X1, X2) -> proper# X2, proper# U32 X -> U32# proper X) (proper# x(X1, X2) -> proper# X2, proper# U32 X -> proper# X) (proper# x(X1, X2) -> proper# X2, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# x(X1, X2) -> proper# X2, proper# U31(X1, X2) -> proper# X1) (proper# x(X1, X2) -> proper# X2, proper# U31(X1, X2) -> proper# X2) (proper# x(X1, X2) -> proper# X2, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# x(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1) (proper# x(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X2) (proper# x(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# x(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1) (proper# x(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X2) (proper# x(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3) (proper# x(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# x(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X1) (proper# x(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X2) (proper# x(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3) (proper# x(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# x(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# x(X1, X2) -> proper# X2, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# x(X1, X2) -> proper# X2, proper# plus(X1, X2) -> proper# X1) (proper# x(X1, X2) -> proper# X2, proper# plus(X1, X2) -> proper# X2) (proper# x(X1, X2) -> proper# X2, proper# U61 X -> U61# proper X) (proper# x(X1, X2) -> proper# X2, proper# U61 X -> proper# X) (proper# x(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# x(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X1) (proper# x(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X2) (proper# x(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3) (proper# x(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# x(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X1) (proper# x(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X2) (proper# x(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X3) (proper# x(X1, X2) -> proper# X2, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# x(X1, X2) -> proper# X2, proper# x(X1, X2) -> proper# X1) (proper# x(X1, X2) -> proper# X2, proper# x(X1, X2) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X2, proper# U12 X -> U12# proper X) (proper# U72(X1, X2, X3) -> proper# X2, proper# U12 X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X2, proper# isNat X -> isNat# proper X) (proper# U72(X1, X2, X3) -> proper# X2, proper# isNat X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X2, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U72(X1, X2, X3) -> proper# X2, proper# U11(X1, X2) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X2, proper# U11(X1, X2) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X2, proper# U21 X -> U21# proper X) (proper# U72(X1, X2, X3) -> proper# X2, proper# U21 X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X2, proper# U32 X -> U32# proper X) (proper# U72(X1, X2, X3) -> proper# X2, proper# U32 X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X2, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U72(X1, X2, X3) -> proper# X2, proper# U31(X1, X2) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X2, proper# U31(X1, X2) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X2, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U72(X1, X2, X3) -> proper# X2, proper# U41(X1, X2) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X2, proper# U41(X1, X2) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U72(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U72(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X2, proper# s X -> s# proper X) (proper# U72(X1, X2, X3) -> proper# X2, proper# s X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X2, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U72(X1, X2, X3) -> proper# X2, proper# plus(X1, X2) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X2, proper# plus(X1, X2) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X2, proper# U61 X -> U61# proper X) (proper# U72(X1, X2, X3) -> proper# X2, proper# U61 X -> proper# X) (proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X1) (proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X2) (proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3) (proper# U72(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U72(X1, X2, X3) -> 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proper# X2, proper# U21 X -> U21# proper X) (proper# U51(X1, X2, X3) -> proper# X2, proper# U21 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X2, proper# U32 X -> U32# proper X) (proper# U51(X1, X2, X3) -> proper# X2, proper# U32 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X2, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X2, proper# U31(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X2, proper# U31(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X2, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X2, proper# U41(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X2, proper# U41(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X2, proper# s X -> s# proper X) (proper# U51(X1, X2, X3) -> proper# X2, proper# s X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X2, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X2, proper# plus(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X2, proper# plus(X1, X2) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X2, proper# U61 X -> U61# proper X) (proper# U51(X1, X2, X3) -> proper# X2, proper# U61 X -> proper# X) (proper# U51(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U51(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X2) (proper# U51(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X3) (proper# U51(X1, X2, X3) -> proper# X2, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U51(X1, X2, X3) -> proper# X2, proper# x(X1, X2) -> proper# X1) (proper# U51(X1, X2, X3) -> proper# X2, proper# x(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X2, proper# U12 X -> U12# proper X) (proper# U41(X1, X2) -> proper# X2, proper# U12 X -> proper# X) (proper# U41(X1, X2) -> proper# X2, proper# isNat X -> isNat# proper X) (proper# U41(X1, X2) -> proper# X2, proper# isNat X -> proper# X) (proper# U41(X1, X2) -> proper# X2, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X2, proper# U11(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X2, proper# U11(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X2, proper# U21 X -> U21# proper X) (proper# U41(X1, X2) -> proper# X2, proper# U21 X -> proper# X) (proper# U41(X1, X2) -> proper# X2, proper# U32 X -> U32# proper X) (proper# U41(X1, X2) -> proper# X2, proper# U32 X -> proper# X) (proper# U41(X1, X2) -> proper# X2, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X2, proper# U31(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X2, proper# U31(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X2, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1) (proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X2) (proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3) (proper# U41(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U41(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X1) (proper# U41(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X2) (proper# U41(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3) (proper# U41(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# U41(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# U41(X1, X2) -> proper# X2, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X2, proper# plus(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X2, proper# plus(X1, X2) -> proper# X2) (proper# U41(X1, X2) -> proper# X2, proper# U61 X -> U61# proper X) (proper# U41(X1, X2) -> proper# X2, proper# U61 X -> proper# X) (proper# U41(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U41(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X1) (proper# U41(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X2) (proper# U41(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3) (proper# U41(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U41(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X1) (proper# U41(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X2) (proper# U41(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X3) (proper# U41(X1, X2) -> proper# X2, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U41(X1, X2) -> proper# X2, proper# x(X1, X2) -> proper# X1) (proper# U41(X1, X2) -> proper# X2, proper# x(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X2, proper# U12 X -> U12# proper X) (proper# U11(X1, X2) -> proper# X2, proper# U12 X -> proper# X) (proper# U11(X1, X2) -> proper# X2, proper# isNat X -> isNat# proper X) (proper# U11(X1, X2) -> proper# X2, proper# isNat X -> proper# X) (proper# U11(X1, X2) -> proper# X2, proper# U11(X1, X2) -> U11#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X2, proper# U11(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X2, proper# U11(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X2, proper# U21 X -> U21# proper X) (proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X) (proper# U11(X1, X2) -> proper# X2, proper# U32 X -> U32# proper X) (proper# U11(X1, X2) -> proper# X2, proper# U32 X -> proper# X) (proper# U11(X1, X2) -> proper# X2, proper# U31(X1, X2) -> U31#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X2, proper# U31(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X2, proper# U31(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X2, proper# U41(X1, X2) -> U41#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> U52#(proper X1, proper X2, proper X3)) (proper# U11(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1) (proper# U11(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X2) (proper# U11(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3) (proper# U11(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> U51#(proper X1, proper X2, proper X3)) (proper# U11(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X1) (proper# U11(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X2) (proper# U11(X1, X2) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3) (proper# U11(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# U11(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# U11(X1, X2) -> proper# X2, proper# plus(X1, X2) -> plus#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X2, proper# plus(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X2, proper# plus(X1, X2) -> proper# X2) (proper# U11(X1, X2) -> proper# X2, proper# U61 X -> U61# proper X) (proper# U11(X1, X2) -> proper# X2, proper# U61 X -> proper# X) (proper# U11(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> U72#(proper X1, proper X2, proper X3)) (proper# U11(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X1) (proper# U11(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X2) (proper# U11(X1, X2) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3) (proper# U11(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> U71#(proper X1, proper X2, proper X3)) (proper# U11(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X1) (proper# U11(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X2) (proper# U11(X1, X2) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X3) (proper# U11(X1, X2) -> proper# X2, proper# x(X1, X2) -> x#(proper X1, proper X2)) (proper# U11(X1, X2) -> proper# X2, proper# x(X1, X2) -> proper# X1) (proper# U11(X1, X2) -> proper# X2, proper# x(X1, X2) -> proper# X2) (active# plus(X1, X2) -> active# X2, active# U12 X -> U12# active X) (active# plus(X1, X2) -> active# X2, active# U12 X -> active# X) (active# plus(X1, X2) -> active# X2, active# isNat s V1 -> isNat# V1) (active# plus(X1, X2) -> active# X2, active# isNat s V1 -> U21# isNat V1) (active# plus(X1, X2) -> active# X2, active# isNat plus(V1, V2) -> isNat# V1) (active# plus(X1, X2) -> active# X2, active# isNat plus(V1, V2) -> U11#(isNat V1, V2)) (active# plus(X1, X2) -> active# X2, active# isNat x(V1, V2) -> isNat# V1) (active# plus(X1, X2) -> active# X2, active# isNat x(V1, V2) -> U31#(isNat V1, V2)) (active# plus(X1, X2) -> active# X2, active# 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X2) -> active# X2, active# U52(X1, X2, X3) -> active# X1) (active# plus(X1, X2) -> active# X2, active# U52(X1, X2, X3) -> U52#(active X1, X2, X3)) (active# plus(X1, X2) -> active# X2, active# U52(tt(), M, N) -> s# plus(N, M)) (active# plus(X1, X2) -> active# X2, active# U52(tt(), M, N) -> plus#(N, M)) (active# plus(X1, X2) -> active# X2, active# U51(X1, X2, X3) -> active# X1) (active# plus(X1, X2) -> active# X2, active# U51(X1, X2, X3) -> U51#(active X1, X2, X3)) (active# plus(X1, X2) -> active# X2, active# U51(tt(), M, N) -> isNat# N) (active# plus(X1, X2) -> active# X2, active# U51(tt(), M, N) -> U52#(isNat N, M, N)) (active# plus(X1, X2) -> active# X2, active# s X -> active# X) (active# plus(X1, X2) -> active# X2, active# s X -> s# active X) (active# plus(X1, X2) -> active# X2, active# plus(N, s M) -> isNat# M) (active# plus(X1, X2) -> active# X2, active# plus(N, s M) -> U51#(isNat M, M, N)) (active# plus(X1, X2) -> active# X2, active# plus(N, 0()) -> isNat# N) (active# plus(X1, X2) -> active# X2, active# plus(N, 0()) -> U41#(isNat N, N)) (active# plus(X1, X2) -> active# X2, active# plus(X1, X2) -> active# X1) (active# plus(X1, X2) -> active# X2, active# plus(X1, X2) -> active# X2) (active# plus(X1, X2) -> active# X2, active# plus(X1, X2) -> plus#(X1, active X2)) (active# plus(X1, X2) -> active# X2, active# plus(X1, X2) -> plus#(active X1, X2)) (active# plus(X1, X2) -> active# X2, active# U61 X -> active# X) (active# plus(X1, X2) -> active# X2, active# U61 X -> U61# active X) (active# plus(X1, X2) -> active# X2, active# U72(X1, X2, X3) -> active# X1) (active# plus(X1, X2) -> active# X2, active# U72(X1, X2, X3) -> U72#(active X1, X2, X3)) (active# plus(X1, X2) -> active# X2, active# U72(tt(), M, N) -> plus#(x(N, M), N)) (active# plus(X1, X2) -> active# X2, active# U72(tt(), M, N) -> x#(N, M)) (active# plus(X1, X2) -> active# X2, active# U71(X1, X2, X3) -> active# X1) (active# plus(X1, X2) -> active# X2, active# U71(X1, X2, X3) -> U71#(active X1, X2, X3)) (active# plus(X1, X2) -> active# X2, active# U71(tt(), M, N) -> isNat# N) (active# plus(X1, X2) -> active# X2, active# U71(tt(), M, N) -> U72#(isNat N, M, N)) (active# plus(X1, X2) -> active# X2, active# x(N, s M) -> isNat# M) (active# plus(X1, X2) -> active# X2, active# x(N, s M) -> U71#(isNat M, M, N)) (active# plus(X1, X2) -> active# X2, active# x(N, 0()) -> isNat# N) (active# plus(X1, X2) -> active# X2, active# x(N, 0()) -> U61# isNat N) (active# plus(X1, X2) -> active# X2, active# x(X1, X2) -> active# X1) (active# plus(X1, X2) -> active# X2, active# x(X1, X2) -> active# X2) (active# plus(X1, X2) -> active# X2, active# x(X1, X2) -> x#(X1, active X2)) (active# plus(X1, X2) -> active# X2, active# x(X1, X2) -> x#(active X1, X2)) (active# U52(tt(), M, N) -> plus#(N, M), plus#(X1, mark X2) -> plus#(X1, X2)) (active# U52(tt(), M, N) -> plus#(N, M), plus#(mark X1, X2) -> plus#(X1, X2)) (active# U52(tt(), M, N) -> plus#(N, M), plus#(ok X1, ok X2) -> plus#(X1, X2)) (active# plus(X1, X2) -> plus#(active X1, X2), plus#(X1, mark X2) -> plus#(X1, X2)) (active# plus(X1, X2) -> plus#(active X1, X2), plus#(mark X1, X2) -> plus#(X1, X2)) (active# plus(X1, X2) -> plus#(active X1, X2), plus#(ok X1, ok X2) -> plus#(X1, X2)) (active# U31(X1, X2) -> U31#(active X1, X2), U31#(mark X1, X2) -> U31#(X1, X2)) (active# U31(X1, X2) -> U31#(active X1, X2), U31#(ok X1, ok X2) -> U31#(X1, X2)) (active# U31(tt(), V2) -> isNat# V2, isNat# ok X -> isNat# X) (active# U31(tt(), V2) -> U32# isNat V2, U32# mark X -> U32# X) (active# U31(tt(), V2) -> U32# isNat V2, U32# ok X -> U32# X) (active# x(N, 0()) -> U61# isNat N, U61# mark X -> U61# X) (active# x(N, 0()) -> U61# isNat N, U61# ok X -> U61# X) (active# x(N, s M) -> U71#(isNat M, M, N), U71#(mark X1, X2, X3) -> U71#(X1, X2, X3)) (active# x(N, s M) -> U71#(isNat M, M, N), U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3)) (active# plus(N, s M) -> U51#(isNat M, M, N), U51#(mark X1, X2, X3) -> U51#(X1, X2, X3)) (active# plus(N, s M) -> U51#(isNat M, M, N), U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3)) (top# ok X -> top# active X, top# mark X -> proper# X) (top# ok X -> top# active X, top# mark X -> top# proper X) (top# ok X -> top# active X, top# ok X -> active# X) (top# ok X -> top# active X, top# ok X -> top# active X) (proper# U61 X -> U61# proper X, U61# mark X -> U61# X) (proper# U61 X -> U61# proper X, U61# ok X -> U61# X) (proper# U32 X -> U32# proper X, U32# mark X -> U32# X) (proper# U32 X -> U32# proper X, U32# ok X -> U32# X) (proper# isNat X -> isNat# proper X, isNat# ok X -> isNat# X) (active# U61 X -> U61# active X, U61# mark X -> U61# X) (active# U61 X -> U61# active X, U61# ok X -> U61# X) (active# U32 X -> U32# active X, U32# mark X -> U32# X) (active# U32 X -> U32# active X, U32# ok X -> U32# X) (active# U12 X -> U12# active X, U12# mark X -> U12# X) (active# U12 X -> U12# active X, U12# ok X -> U12# X) } STATUS: arrows: 0.869377 SCCS (18): Scc: {top# mark X -> top# proper X, top# ok X -> top# active X} Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2, active# U61 X -> active# X, active# U72(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X1, active# x(X1, X2) -> active# X2} Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2, proper# U61 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X1, proper# x(X1, X2) -> proper# X2} Scc: { x#(X1, mark X2) -> x#(X1, X2), x#(mark X1, X2) -> x#(X1, X2), x#(ok X1, ok X2) -> x#(X1, X2)} Scc: { U71#(mark X1, X2, X3) -> U71#(X1, X2, X3), U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3)} Scc: { U72#(mark X1, X2, X3) -> U72#(X1, X2, X3), U72#(ok X1, ok X2, ok X3) -> U72#(X1, X2, X3)} Scc: { plus#(X1, mark X2) -> plus#(X1, X2), plus#(mark X1, X2) -> plus#(X1, X2), plus#(ok X1, ok X2) -> plus#(X1, X2)} Scc: { U51#(mark X1, X2, X3) -> U51#(X1, X2, X3), U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3)} Scc: { U52#(mark X1, X2, X3) -> U52#(X1, X2, X3), U52#(ok X1, ok X2, ok X3) -> U52#(X1, X2, X3)} Scc: { U41#(mark X1, X2) -> U41#(X1, X2), U41#(ok X1, ok X2) -> U41#(X1, X2)} Scc: { U31#(mark X1, X2) -> U31#(X1, X2), U31#(ok X1, ok X2) -> U31#(X1, X2)} Scc: { U11#(mark X1, X2) -> U11#(X1, X2), U11#(ok X1, ok X2) -> U11#(X1, X2)} Scc: {isNat# ok X -> isNat# X} Scc: {U61# mark X -> U61# X, U61# ok X -> U61# X} Scc: {s# mark X -> s# X, s# ok X -> s# X} Scc: {U32# mark X -> U32# X, U32# ok X -> U32# X} Scc: {U21# mark X -> U21# X, U21# ok X -> U21# X} Scc: {U12# mark X -> U12# X, U12# ok X -> U12# X} SCC (2): Strict: {top# mark X -> top# proper X, top# ok X -> top# active X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} Fail SCC (16): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2, active# U61 X -> active# X, active# U72(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> active# X1, active# x(X1, X2) -> active# X1, active# x(X1, X2) -> active# X2} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1, [U51](x0, x1, x2) = x0 + x1, [U72](x0, x1, x2) = x0 + x1 + x2, [U71](x0, x1, x2) = x0 + x1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = x0 + x1, [x](x0, x1) = x0 + x1 + 1, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0 + 1, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = x0, [proper](x0) = x0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [active#](x0) = x0 Strict: active# x(X1, X2) -> active# X2 1 + 1X1 + 1X2 >= 0 + 1X2 active# x(X1, X2) -> active# X1 1 + 1X1 + 1X2 >= 0 + 1X1 active# U71(X1, X2, X3) -> active# X1 0 + 1X1 + 1X2 + 0X3 >= 0 + 1X1 active# U72(X1, X2, X3) -> active# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 active# U61 X -> active# X 0 + 1X >= 0 + 1X active# plus(X1, X2) -> active# X2 0 + 1X1 + 1X2 >= 0 + 1X2 active# plus(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# s X -> active# X 0 + 1X >= 0 + 1X active# U51(X1, X2, X3) -> active# X1 0 + 1X1 + 1X2 + 0X3 >= 0 + 1X1 active# U52(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 1X3 >= 0 + 1X1 active# U41(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# U31(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 + 1X2 + 1X3 proper U61 X -> U61 proper X 0 + 1X >= 0 + 1X proper 0() -> ok 0() 1 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 proper s X -> s proper X 0 + 1X >= 0 + 1X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 1X1 + 0X2 + 1X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 proper U32 X -> U32 proper X 0 + 1X >= 0 + 1X proper U21 X -> U21 proper X 0 + 1X >= 0 + 1X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 proper isNat X -> isNat proper X 1 + 1X >= 1 + 1X proper U12 X -> U12 proper X 0 + 1X >= 0 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 1X2 + 0X3 >= 1 + 1X1 + 1X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 1 + 1X >= 1 + 1X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 1X2 + 0X3 >= 1 + 1X1 + 1X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 1X3 >= 1 + 1X1 + 0X2 + 1X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 1 + 1X >= 1 + 1X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 1 + 1X >= 1 + 1X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 1 + 1X >= 1 + 1X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2, active# U61 X -> active# X, active# U72(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> active# X1} SCC (14): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2, active# U61 X -> active# X, active# U72(X1, X2, X3) -> active# X1, active# U71(X1, X2, X3) -> active# X1} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0, [U51](x0, x1, x2) = x0, [U72](x0, x1, x2) = x0, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0, [U31](x0, x1) = x0, [U41](x0, x1) = x0, [plus](x0, x1) = x0 + x1, [x](x0, x1) = x0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = x0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [active#](x0) = x0 Strict: active# U71(X1, X2, X3) -> active# X1 1 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U72(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U61 X -> active# X 0 + 1X >= 0 + 1X active# plus(X1, X2) -> active# X2 0 + 1X1 + 1X2 >= 0 + 1X2 active# plus(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# s X -> active# X 0 + 1X >= 0 + 1X active# U51(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U52(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U41(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U31(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2, active# U61 X -> active# X, active# U72(X1, X2, X3) -> active# X1} SCC (13): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2, active# U61 X -> active# X, active# U72(X1, X2, X3) -> active# X1} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0, [U51](x0, x1, x2) = x0, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0, [U31](x0, x1) = x0, [U41](x0, x1) = x0, [plus](x0, x1) = x0 + x1, [x](x0, x1) = x0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = x0, [proper](x0) = 0, [ok](x0) = x0, [top](x0) = 0, [tt] = 0, [0] = 0, [active#](x0) = x0 Strict: active# U72(X1, X2, X3) -> active# X1 1 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U61 X -> active# X 0 + 1X >= 0 + 1X active# plus(X1, X2) -> active# X2 0 + 1X1 + 1X2 >= 0 + 1X2 active# plus(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# s X -> active# X 0 + 1X >= 0 + 1X active# U51(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U52(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U41(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U31(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 1X >= 0 + 1X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 1X >= 0 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 1X >= 0 + 1X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 1X >= 0 + 1X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 1X >= 0 + 1X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2, active# U61 X -> active# X} SCC (12): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2, active# U61 X -> active# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0, [U51](x0, x1, x2) = x0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0, [U31](x0, x1) = x0, [U41](x0, x1) = x0, [plus](x0, x1) = x0 + x1, [x](x0, x1) = x0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [active#](x0) = x0 Strict: active# U61 X -> active# X 1 + 1X >= 0 + 1X active# plus(X1, X2) -> active# X2 0 + 1X1 + 1X2 >= 0 + 1X2 active# plus(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# s X -> active# X 0 + 1X >= 0 + 1X active# U51(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U52(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U41(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U31(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 1 + 0X >= 0 + 0X U61 mark X -> mark U61 X 1 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 1 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2} SCC (11): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X, active# plus(X1, X2) -> active# X1, active# plus(X1, X2) -> active# X2} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0, [U51](x0, x1, x2) = x0 + x1, [U72](x0, x1, x2) = x0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0, [U31](x0, x1) = x0, [U41](x0, x1) = x0, [plus](x0, x1) = x0 + x1 + 1, [x](x0, x1) = x0 + 1, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = x0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = 0, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 1, [active#](x0) = x0 Strict: active# plus(X1, X2) -> active# X2 1 + 1X1 + 1X2 >= 0 + 1X2 active# plus(X1, X2) -> active# X1 1 + 1X1 + 1X2 >= 0 + 1X1 active# s X -> active# X 0 + 1X >= 0 + 1X active# U51(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 1X3 >= 0 + 1X1 active# U52(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U41(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U31(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 1 + 0X >= 0 + 0X proper 0() -> ok 0() 2 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 2 + 1X1 + 1X2 >= 3 + 1X1 + 1X2 proper s X -> s proper X 1 + 1X >= 1 + 1X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 1 + 1X1 + 0X2 + 1X3 >= 2 + 1X1 + 0X2 + 1X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper U32 X -> U32 proper X 1 + 1X >= 1 + 1X proper U21 X -> U21 proper X 1 + 1X >= 1 + 1X proper tt() -> ok tt() 1 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper isNat X -> isNat proper X 1 + 0X >= 0 + 0X proper U12 X -> U12 proper X 1 + 1X >= 1 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 1N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 1N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active plus(N, 0()) -> mark U41(isNat N, N) 2 + 1N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 1N + 1M >= 0 + 0N + 0M active s X -> s active X 0 + 1X >= 0 + 1X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 1N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 1X1 + 0X2 + 1X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 1X >= 0 + 1X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 1X >= 0 + 1X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 1X >= 0 + 1X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X} SCC (9): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1, active# s X -> active# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1, [U51](x0, x1, x2) = x0, [U72](x0, x1, x2) = x0 + x1, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0, [plus](x0, x1) = 0, [x](x0, x1) = x0 + x1 + 1, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0 + 1, [U61](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 1, [active#](x0) = x0 Strict: active# s X -> active# X 1 + 1X >= 0 + 1X active# U51(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U52(X1, X2, X3) -> active# X1 0 + 1X1 + 1X2 + 0X3 >= 0 + 1X1 active# U41(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U31(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 2 + 1X1 + 1X2 >= 3 + 1X1 + 1X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 proper U61 X -> U61 proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 2 + 1X >= 2 + 1X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper U32 X -> U32 proper X 1 + 1X >= 1 + 1X proper U21 X -> U21 proper X 1 + 1X >= 1 + 1X proper tt() -> ok tt() 1 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper isNat X -> isNat proper X 1 + 0X >= 0 + 0X proper U12 X -> U12 proper X 1 + 1X >= 1 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 1 + 0X >= 0 + 0X U61 mark X -> mark U61 X 1 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 1 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1} SCC (8): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1, active# U51(X1, X2, X3) -> active# X1} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = 1, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0, [U31](x0, x1) = x0, [U41](x0, x1) = x0, [plus](x0, x1) = 1, [x](x0, x1) = x0 + 1, [mark](x0) = x0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = x0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0 + 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [active#](x0) = x0 Strict: active# U51(X1, X2, X3) -> active# X1 1 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U52(X1, X2, X3) -> active# X1 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U41(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U31(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 1X >= 1 + 1X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 1X >= 0 + 1X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 1X >= 0 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 2 + 0N + 1M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 1 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 1 + 0N + 0M active s X -> s active X 1 + 1X >= 1 + 1X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 1 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 2 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 1X >= 0 + 1X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 1X >= 0 + 1X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 1X >= 0 + 1X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 1X >= 0 + 1X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1} SCC (7): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1, active# U52(X1, X2, X3) -> active# X1} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0, [U31](x0, x1) = x0, [U41](x0, x1) = x0, [plus](x0, x1) = 1, [x](x0, x1) = x0 + 1, [mark](x0) = x0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0 + 1, [U61](x0) = 0, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [active#](x0) = x0 Strict: active# U52(X1, X2, X3) -> active# X1 1 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 active# U41(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U31(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 1 + 0X >= 0 + 0X proper 0() -> ok 0() 1 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 2 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 2 + 1X >= 2 + 1X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper U32 X -> U32 proper X 1 + 1X >= 1 + 1X proper U21 X -> U21 proper X 1 + 1X >= 1 + 1X proper tt() -> ok tt() 1 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper isNat X -> isNat proper X 1 + 0X >= 0 + 0X proper U12 X -> U12 proper X 1 + 1X >= 1 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 1X >= 1 + 1X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 1X >= 0 + 1X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 1X >= 0 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 2 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 1X >= 0 + 1X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1} SCC (6): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1, active# U41(X1, X2) -> active# X1} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0, [U31](x0, x1) = x0, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 1, [x](x0, x1) = x0 + 1, [mark](x0) = x0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0 + 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [active#](x0) = x0 Strict: active# U41(X1, X2) -> active# X1 1 + 1X1 + 0X2 >= 0 + 1X1 active# U31(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 1X >= 1 + 1X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 1X >= 0 + 1X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 1X >= 0 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 1 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 2 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 1X >= 0 + 1X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1} SCC (5): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X, active# U31(X1, X2) -> active# X1} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = x0, [U11](x0, x1) = x0, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = x0 + 1, [x](x0, x1) = x0 + x1 + 1, [mark](x0) = 1, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0 + 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 1, [active#](x0) = x0 Strict: active# U31(X1, X2) -> active# X1 1 + 1X1 + 0X2 >= 0 + 1X1 active# U32 X -> active# X 0 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 2 + 1X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 1 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 2 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 2 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 1 + 0X >= 1 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 1 + 0X >= 1 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 1 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 1 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 1 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 1 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 1 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 1 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 1 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 1 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 1 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 1 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 1 + 0V1 active U12 tt() -> mark tt() 0 >= 1 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 1 + 0X >= 1 + 0X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X} SCC (4): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X, active# U32 X -> active# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0, [U31](x0, x1) = 0, [U41](x0, x1) = 0, [plus](x0, x1) = x0 + 1, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0 + 1, [s](x0) = x0 + 1, [U61](x0) = 0, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 1, [active#](x0) = x0 Strict: active# U32 X -> active# X 1 + 1X >= 0 + 1X active# U21 X -> active# X 0 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 1 + 0X >= 0 + 0X proper 0() -> ok 0() 2 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper s X -> s proper X 2 + 1X >= 2 + 1X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 2 + 1X >= 2 + 1X proper U21 X -> U21 proper X 1 + 1X >= 1 + 1X proper tt() -> ok tt() 1 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper isNat X -> isNat proper X 1 + 0X >= 0 + 0X proper U12 X -> U12 proper X 1 + 1X >= 1 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 1 + 0X >= 0 + 0X U32 mark X -> mark U32 X 1 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 1 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X} SCC (3): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1, active# U21 X -> active# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 1, [U11](x0, x1) = x0, [U31](x0, x1) = 0, [U41](x0, x1) = 0, [plus](x0, x1) = 1, [x](x0, x1) = x0 + 1, [mark](x0) = x0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = x0 + 1, [U32](x0) = 0, [s](x0) = x0 + 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [active#](x0) = x0 Strict: active# U21 X -> active# X 1 + 1X >= 0 + 1X active# U11(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 1X >= 1 + 1X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 1 + 0X >= 0 + 0X U21 mark X -> mark U21 X 1 + 1X >= 1 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 2 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 1 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 1 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 1X >= 0 + 1X SCCS (1): Scc: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1} SCC (2): Strict: { active# U12 X -> active# X, active# U11(X1, X2) -> active# X1} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = x0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = 0, [U41](x0, x1) = 0, [plus](x0, x1) = x0 + 1, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = 0, [active](x0) = 0, [U21](x0) = 0, [U32](x0) = 0, [s](x0) = x0 + 1, [U61](x0) = 0, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 1, [active#](x0) = x0 Strict: active# U11(X1, X2) -> active# X1 1 + 1X1 + 0X2 >= 0 + 1X1 active# U12 X -> active# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 1 + 0X >= 0 + 0X proper 0() -> ok 0() 2 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper s X -> s proper X 2 + 1X >= 2 + 1X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 1 + 0X >= 0 + 0X proper U21 X -> U21 proper X 1 + 0X >= 0 + 0X proper tt() -> ok tt() 1 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 proper isNat X -> isNat proper X 1 + 0X >= 0 + 0X proper U12 X -> U12 proper X 1 + 1X >= 1 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: {active# U12 X -> active# X} SCC (1): Strict: {active# U12 X -> active# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [active#](x0) = x0 Strict: active# U12 X -> active# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (28): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2, proper# U61 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X3, proper# x(X1, X2) -> proper# X1, proper# x(X1, X2) -> proper# X2} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1 + x2, [U51](x0, x1, x2) = x0 + x1 + x2, [U72](x0, x1, x2) = x0 + x1 + x2, [U71](x0, x1, x2) = x0 + x1 + x2, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = x0 + x1, [x](x0, x1) = x0 + x1 + 1, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = x0, [proper](x0) = x0 + 1, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# x(X1, X2) -> proper# X2 1 + 1X1 + 1X2 >= 0 + 1X2 proper# x(X1, X2) -> proper# X1 1 + 1X1 + 1X2 >= 0 + 1X1 proper# U71(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U71(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U71(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U72(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U72(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U72(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U61 X -> proper# X 0 + 1X >= 0 + 1X proper# plus(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# plus(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# U51(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U51(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U51(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U52(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U52(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U52(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U41(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U41(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U31(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 2 + 1X1 + 1X2 >= 3 + 1X1 + 1X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U61 X -> U61 proper X 1 + 1X >= 1 + 1X proper 0() -> ok 0() 1 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper s X -> s proper X 1 + 1X >= 1 + 1X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper U32 X -> U32 proper X 1 + 1X >= 1 + 1X proper U21 X -> U21 proper X 1 + 1X >= 1 + 1X proper tt() -> ok tt() 1 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper isNat X -> isNat proper X 1 + 1X >= 1 + 1X proper U12 X -> U12 proper X 1 + 1X >= 1 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 1 + 1X >= 1 + 1X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 1 + 1X >= 1 + 1X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 1 + 1X >= 1 + 1X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 1 + 1X >= 1 + 1X U12 ok X -> ok U12 X 1 + 1X >= 1 + 1X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2, proper# U61 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X3} SCC (26): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2, proper# U61 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3, proper# U71(X1, X2, X3) -> proper# X1, proper# U71(X1, X2, X3) -> proper# X2, proper# U71(X1, X2, X3) -> proper# X3} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1 + x2, [U51](x0, x1, x2) = x0 + x1 + x2, [U72](x0, x1, x2) = x0 + x1 + x2, [U71](x0, x1, x2) = x0 + x1 + x2 + 1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = x0 + x1, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = x0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U71(X1, X2, X3) -> proper# X3 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U71(X1, X2, X3) -> proper# X2 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U71(X1, X2, X3) -> proper# X1 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U72(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U72(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U72(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U61 X -> proper# X 0 + 1X >= 0 + 1X proper# plus(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# plus(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# U51(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U51(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U51(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U52(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U52(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U52(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U41(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U41(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U31(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 4 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 1 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 1 + 1X >= 1 + 1X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 1 + 1X >= 1 + 1X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 1 + 1X >= 1 + 1X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 1X2 + 1X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 1 + 1X >= 1 + 1X U12 ok X -> ok U12 X 1 + 1X >= 1 + 1X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2, proper# U61 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3} SCC (23): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2, proper# U61 X -> proper# X, proper# U72(X1, X2, X3) -> proper# X1, proper# U72(X1, X2, X3) -> proper# X2, proper# U72(X1, X2, X3) -> proper# X3} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1 + x2, [U51](x0, x1, x2) = x0 + x1 + x2, [U72](x0, x1, x2) = x0 + x1 + x2 + 1, [U71](x0, x1, x2) = x0 + x1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = x0 + x1, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = x0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U72(X1, X2, X3) -> proper# X3 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U72(X1, X2, X3) -> proper# X2 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U72(X1, X2, X3) -> proper# X1 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U61 X -> proper# X 0 + 1X >= 0 + 1X proper# plus(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# plus(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# U51(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U51(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U51(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U52(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U52(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U52(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U41(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U41(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U31(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 1X2 + 0X3 >= 1 + 1X1 + 1X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 4 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 1 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 1 + 1X >= 1 + 1X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 1 + 1X >= 1 + 1X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 1 + 1X >= 1 + 1X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 1X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 1 + 1X >= 1 + 1X U12 ok X -> ok U12 X 1 + 1X >= 1 + 1X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2, proper# U61 X -> proper# X} SCC (20): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2, proper# U61 X -> proper# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1 + x2, [U51](x0, x1, x2) = x0 + x1 + x2, [U72](x0, x1, x2) = x0 + x1 + 1, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = x0 + x1, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U61 X -> proper# X 1 + 1X >= 0 + 1X proper# plus(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# plus(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# U51(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U51(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U51(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U52(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U52(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U52(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U41(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U41(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U31(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 1 + 1X1 + 0X2 + 1X3 >= 1 + 1X1 + 0X2 + 1X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 1 + 1X >= 1 + 1X U61 mark X -> mark U61 X 1 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 1X >= 0 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 + 1X2 + 1X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 + 1X2 + 1X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 1X >= 0 + 1X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 1X >= 0 + 1X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 1 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 1X >= 0 + 1X U12 ok X -> ok U12 X 0 + 1X >= 0 + 1X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2} SCC (19): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# plus(X1, X2) -> proper# X1, proper# plus(X1, X2) -> proper# X2} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1 + x2, [U51](x0, x1, x2) = x0 + x1 + x2, [U72](x0, x1, x2) = x0 + x1 + x2 + 1, [U71](x0, x1, x2) = x0 + x1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = x0 + x1 + 1, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0, [U61](x0) = 0, [proper](x0) = x0 + 1, [ok](x0) = x0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# plus(X1, X2) -> proper# X2 1 + 1X1 + 1X2 >= 0 + 1X2 proper# plus(X1, X2) -> proper# X1 1 + 1X1 + 1X2 >= 0 + 1X1 proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# U51(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U51(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U51(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U52(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U52(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U52(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U41(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U41(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U31(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 2 + 1X1 + 1X2 + 1X3 >= 4 + 1X1 + 1X2 + 1X3 proper U61 X -> U61 proper X 1 + 0X >= 0 + 0X proper 0() -> ok 0() 1 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 2 + 1X1 + 1X2 >= 3 + 1X1 + 1X2 proper s X -> s proper X 1 + 1X >= 1 + 1X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper U32 X -> U32 proper X 1 + 1X >= 1 + 1X proper U21 X -> U21 proper X 1 + 1X >= 1 + 1X proper tt() -> ok tt() 1 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper isNat X -> isNat proper X 1 + 1X >= 1 + 1X proper U12 X -> U12 proper X 1 + 1X >= 1 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 1 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 1 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 1X >= 0 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 + 1X2 + 1X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 + 1X2 + 1X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 1X >= 0 + 1X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 1X >= 0 + 1X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 1X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 1X >= 0 + 1X U12 ok X -> ok U12 X 0 + 1X >= 0 + 1X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X} SCC (17): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3, proper# s X -> proper# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1 + x2, [U51](x0, x1, x2) = x0 + x1 + x2, [U72](x0, x1, x2) = x0 + x1 + x2, [U71](x0, x1, x2) = x0 + x1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = x0 + 1, [U61](x0) = 0, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# s X -> proper# X 1 + 1X >= 0 + 1X proper# U51(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U51(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U51(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U52(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U52(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U52(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U41(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U41(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U31(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U61 X -> U61 proper X 1 + 0X >= 0 + 0X proper 0() -> ok 0() 1 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 2 + 1X >= 2 + 1X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper U32 X -> U32 proper X 1 + 1X >= 1 + 1X proper U21 X -> U21 proper X 1 + 1X >= 1 + 1X proper tt() -> ok tt() 1 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper isNat X -> isNat proper X 1 + 1X >= 1 + 1X proper U12 X -> U12 proper X 1 + 1X >= 1 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3} SCC (16): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3, proper# U51(X1, X2, X3) -> proper# X1, proper# U51(X1, X2, X3) -> proper# X2, proper# U51(X1, X2, X3) -> proper# X3} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1 + x2, [U51](x0, x1, x2) = x0 + x1 + x2 + 1, [U72](x0, x1, x2) = x0 + x1, [U71](x0, x1, x2) = x0 + x1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = 0, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U51(X1, X2, X3) -> proper# X3 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U51(X1, X2, X3) -> proper# X2 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U51(X1, X2, X3) -> proper# X1 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U52(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U52(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U52(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U41(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U41(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U31(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 1 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 1X2 + 1X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3} SCC (13): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X1, proper# U52(X1, X2, X3) -> proper# X2, proper# U52(X1, X2, X3) -> proper# X3} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1 + x2 + 1, [U51](x0, x1, x2) = x0 + x1, [U72](x0, x1, x2) = x0 + x1, [U71](x0, x1, x2) = x0 + x1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = 0, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U52(X1, X2, X3) -> proper# X3 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# U52(X1, X2, X3) -> proper# X2 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# U52(X1, X2, X3) -> proper# X1 1 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# U41(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U41(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U31(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 1 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 1X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2} SCC (10): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2, proper# U41(X1, X2) -> proper# X1, proper# U41(X1, X2) -> proper# X2} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1, [U51](x0, x1, x2) = x0 + x1, [U72](x0, x1, x2) = x0 + x1, [U71](x0, x1, x2) = x0 + x1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1, [U41](x0, x1) = x0 + x1 + 1, [plus](x0, x1) = 1, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = 0, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = x0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U41(X1, X2) -> proper# X2 1 + 1X1 + 1X2 >= 0 + 1X2 proper# U41(X1, X2) -> proper# X1 1 + 1X1 + 1X2 >= 0 + 1X1 proper# U31(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 1X1 + 0X2 + 1X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 1X1 + 0X2 + 1X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 U41(mark X1, X2) -> mark U41(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 1X >= 0 + 1X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 1X >= 0 + 1X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 1X >= 0 + 1X U12 ok X -> ok U12 X 0 + 1X >= 0 + 1X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2} SCC (8): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X, proper# U31(X1, X2) -> proper# X1, proper# U31(X1, X2) -> proper# X2} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1, [U51](x0, x1, x2) = x0 + x1, [U72](x0, x1, x2) = x0 + x1, [U71](x0, x1, x2) = x0 + x1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = x0 + x1 + 1, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0, [s](x0) = 0, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U31(X1, X2) -> proper# X2 1 + 1X1 + 1X2 >= 0 + 1X2 proper# U31(X1, X2) -> proper# X1 1 + 1X1 + 1X2 >= 0 + 1X1 proper# U32 X -> proper# X 0 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X} SCC (6): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X, proper# U32 X -> proper# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1, [U51](x0, x1, x2) = x0 + x1, [U72](x0, x1, x2) = x0 + x1, [U71](x0, x1, x2) = x0 + x1, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = 0, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = 1, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = x0, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U32 X -> proper# X 1 + 1X >= 0 + 1X proper# U21 X -> proper# X 0 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 0 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 1 + 0X1 + 1X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 1 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 1 + 0X1 + 1X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U32 ok X -> ok U32 X 1 + 0X >= 0 + 0X U32 mark X -> mark U32 X 2 + 0X >= 1 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 1 + 0X >= 1 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 1 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 1X3 active U61 tt() -> mark 0() 0 >= 1 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 1 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 1 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 1 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 1 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 1 active U32 X -> U32 active X 0 + 0X >= 1 + 0X active U21 tt() -> mark tt() 0 >= 1 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 1 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 1 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 1 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 1 + 0V1 active U12 tt() -> mark tt() 0 >= 1 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 1 + 0X >= 1 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X} SCC (5): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2, proper# U21 X -> proper# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 1, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + x1, [U31](x0, x1) = 0, [U41](x0, x1) = 0, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = x0, [U21](x0) = x0 + 1, [U32](x0) = 0, [s](x0) = 0, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U21 X -> proper# X 1 + 1X >= 0 + 1X proper# U11(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 0 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 1 + 0X >= 0 + 0X U21 mark X -> mark U21 X 1 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 1 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 1 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 1 >= 0 active U21 X -> U21 active X 1 + 1X >= 1 + 1X active U11(tt(), V2) -> mark U12 isNat V2 0 + 1V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 1X >= 0 + 1X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2} SCC (4): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X, proper# U11(X1, X2) -> proper# X1, proper# U11(X1, X2) -> proper# X2} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + x1 + x2, [U51](x0, x1, x2) = x0 + x1, [U72](x0, x1, x2) = x0 + x1 + x2, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + x1 + 1, [U31](x0, x1) = 0, [U41](x0, x1) = x0 + x1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = 0, [U12](x0) = x0, [isNat](x0) = x0, [active](x0) = 0, [U21](x0) = 0, [U32](x0) = 0, [s](x0) = 0, [U61](x0) = x0, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# U11(X1, X2) -> proper# X2 1 + 1X1 + 1X2 >= 0 + 1X2 proper# U11(X1, X2) -> proper# X1 1 + 1X1 + 1X2 >= 0 + 1X1 proper# isNat X -> proper# X 0 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 1 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U61 X -> U61 proper X 1 + 1X >= 1 + 1X proper 0() -> ok 0() 1 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 1 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 1 + 1X1 + 1X2 + 1X3 >= 3 + 1X1 + 1X2 + 1X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U32 X -> U32 proper X 1 + 0X >= 0 + 0X proper U21 X -> U21 proper X 1 + 0X >= 0 + 0X proper tt() -> ok tt() 1 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 2 + 1X1 + 1X2 >= 3 + 1X1 + 1X2 proper isNat X -> isNat proper X 1 + 1X >= 1 + 1X proper U12 X -> U12 proper X 1 + 1X >= 1 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 0 + 0X >= 0 + 0X U21 mark X -> mark U21 X 0 + 0X >= 0 + 0X U11(ok X1, ok X2) -> ok U11(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 0 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U61 tt() -> mark 0() 0 >= 0 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 0 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 0N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 0 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 0 + 0V2 + 0V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 0 + 0V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 0 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: { proper# U12 X -> proper# X, proper# isNat X -> proper# X} SCC (2): Strict: { proper# U12 X -> proper# X, proper# isNat X -> proper# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 1, [x](x0, x1) = 1, [mark](x0) = x0, [U12](x0) = x0, [isNat](x0) = x0 + 1, [active](x0) = 0, [U21](x0) = x0 + 1, [U32](x0) = 0, [s](x0) = 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [tt] = 0, [0] = 1, [proper#](x0) = x0 Strict: proper# isNat X -> proper# X 1 + 1X >= 0 + 1X proper# U12 X -> proper# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 0 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 0 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 0 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 0 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 1 + 0X >= 0 + 0X U21 mark X -> mark U21 X 1 + 1X >= 1 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 1 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 1 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 1 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 2 + 1V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 1 + 0X >= 0 + 0X U12 ok X -> ok U12 X 0 + 0X >= 0 + 0X U12 mark X -> mark U12 X 0 + 1X >= 0 + 1X SCCS (1): Scc: {proper# U12 X -> proper# X} SCC (1): Strict: {proper# U12 X -> proper# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [proper#](x0) = x0 Strict: proper# U12 X -> proper# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (3): Strict: { x#(X1, mark X2) -> x#(X1, X2), x#(mark X1, X2) -> x#(X1, X2), x#(ok X1, ok X2) -> x#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [x#](x0, x1) = x0 + 1 Strict: x#(ok X1, ok X2) -> x#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 x#(mark X1, X2) -> x#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 x#(X1, mark X2) -> x#(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 1 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 2 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X SCCS (1): Scc: {x#(X1, mark X2) -> x#(X1, X2)} SCC (1): Strict: {x#(X1, mark X2) -> x#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [x#](x0, x1) = x0 Strict: x#(X1, mark X2) -> x#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: { U71#(mark X1, X2, X3) -> U71#(X1, X2, X3), U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [U71#](x0, x1, x2) = x0 Strict: U71#(ok X1, ok X2, ok X3) -> U71#(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 1X3 U71#(mark X1, X2, X3) -> U71#(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 1X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 1 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 1 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 1 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 2 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X SCCS (1): Scc: {U71#(mark X1, X2, X3) -> U71#(X1, X2, X3)} SCC (1): Strict: {U71#(mark X1, X2, X3) -> U71#(X1, X2, X3)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U71#](x0, x1, x2) = x0 Strict: U71#(mark X1, X2, X3) -> U71#(X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: { U72#(mark X1, X2, X3) -> U72#(X1, X2, X3), U72#(ok X1, ok X2, ok X3) -> U72#(X1, X2, X3)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [U72#](x0, x1, x2) = x0 Strict: U72#(ok X1, ok X2, ok X3) -> U72#(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 1X3 U72#(mark X1, X2, X3) -> U72#(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 1X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 1 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 1 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 1 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 2 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X SCCS (1): Scc: {U72#(mark X1, X2, X3) -> U72#(X1, X2, X3)} SCC (1): Strict: {U72#(mark X1, X2, X3) -> U72#(X1, X2, X3)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U72#](x0, x1, x2) = x0 Strict: U72#(mark X1, X2, X3) -> U72#(X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (3): Strict: { plus#(X1, mark X2) -> plus#(X1, X2), plus#(mark X1, X2) -> plus#(X1, X2), plus#(ok X1, ok X2) -> plus#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [plus#](x0, x1) = x0 + 1 Strict: plus#(ok X1, ok X2) -> plus#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 plus#(mark X1, X2) -> plus#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 plus#(X1, mark X2) -> plus#(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 1 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 2 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X SCCS (1): Scc: {plus#(X1, mark X2) -> plus#(X1, X2)} SCC (1): Strict: {plus#(X1, mark X2) -> plus#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [plus#](x0, x1) = x0 Strict: plus#(X1, mark X2) -> plus#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: { U51#(mark X1, X2, X3) -> U51#(X1, X2, X3), U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [U51#](x0, x1, x2) = x0 Strict: U51#(ok X1, ok X2, ok X3) -> U51#(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 1X3 U51#(mark X1, X2, X3) -> U51#(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 1X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 1 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 1 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 1 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 2 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X SCCS (1): Scc: {U51#(mark X1, X2, X3) -> U51#(X1, X2, X3)} SCC (1): Strict: {U51#(mark X1, X2, X3) -> U51#(X1, X2, X3)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U51#](x0, x1, x2) = x0 Strict: U51#(mark X1, X2, X3) -> U51#(X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: { U52#(mark X1, X2, X3) -> U52#(X1, X2, X3), U52#(ok X1, ok X2, ok X3) -> U52#(X1, X2, X3)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [U52#](x0, x1, x2) = x0 Strict: U52#(ok X1, ok X2, ok X3) -> U52#(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 1X3 U52#(mark X1, X2, X3) -> U52#(X1, X2, X3) 0 + 0X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 1X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 1 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 1 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 1 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 2 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X SCCS (1): Scc: {U52#(mark X1, X2, X3) -> U52#(X1, X2, X3)} SCC (1): Strict: {U52#(mark X1, X2, X3) -> U52#(X1, X2, X3)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U52#](x0, x1, x2) = x0 Strict: U52#(mark X1, X2, X3) -> U52#(X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 0X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: { U41#(mark X1, X2) -> U41#(X1, X2), U41#(ok X1, ok X2) -> U41#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + x1 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [U41#](x0, x1) = x0 Strict: U41#(ok X1, ok X2) -> U41#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 U41#(mark X1, X2) -> U41#(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 3 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 2M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 1M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 1 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 2 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X SCCS (1): Scc: {U41#(mark X1, X2) -> U41#(X1, X2)} SCC (1): Strict: {U41#(mark X1, X2) -> U41#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U41#](x0, x1) = x0 Strict: U41#(mark X1, X2) -> U41#(X1, X2) 1 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: { U31#(mark X1, X2) -> U31#(X1, X2), U31#(ok X1, ok X2) -> U31#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + x1 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [U31#](x0, x1) = x0 Strict: U31#(ok X1, ok X2) -> U31#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 U31#(mark X1, X2) -> U31#(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 3 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 2M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 1M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 1 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 2 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X SCCS (1): Scc: {U31#(mark X1, X2) -> U31#(X1, X2)} SCC (1): Strict: {U31#(mark X1, X2) -> U31#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U31#](x0, x1) = x0 Strict: U31#(mark X1, X2) -> U31#(X1, X2) 1 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: { U11#(mark X1, X2) -> U11#(X1, X2), U11#(ok X1, ok X2) -> U11#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + x1 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [U11#](x0, x1) = x0 Strict: U11#(ok X1, ok X2) -> U11#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 U11#(mark X1, X2) -> U11#(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 3 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 2M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 1M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 1X2 + 0X3 >= 2 + 1X1 + 1X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 1 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 2 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X SCCS (1): Scc: {U11#(mark X1, X2) -> U11#(X1, X2)} SCC (1): Strict: {U11#(mark X1, X2) -> U11#(X1, X2)} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U11#](x0, x1) = x0 Strict: U11#(mark X1, X2) -> U11#(X1, X2) 1 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (1): Strict: {isNat# ok X -> isNat# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [isNat#](x0) = x0 Strict: isNat# ok X -> isNat# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 1 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 proper U32 X -> U32 proper X 2 + 1X >= 2 + 1X proper U21 X -> U21 proper X 2 + 1X >= 2 + 1X proper tt() -> ok tt() 1 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 proper isNat X -> isNat proper X 2 + 1X >= 2 + 1X proper U12 X -> U12 proper X 2 + 1X >= 2 + 1X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: {U61# mark X -> U61# X, U61# ok X -> U61# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 1, [x](x0, x1) = 1, [mark](x0) = x0, [U12](x0) = 0, [isNat](x0) = x0 + 1, [active](x0) = 0, [U21](x0) = x0 + 1, [U32](x0) = 0, [s](x0) = 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U61#](x0) = x0 Strict: U61# ok X -> U61# X 1 + 1X >= 0 + 1X U61# mark X -> U61# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 1 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 2 + 0X s mark X -> mark s X 1 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 1 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 1 + 1X >= 1 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 1 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 1 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 2 + 1V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 0 + 0X >= 1 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: {U61# mark X -> U61# X} SCC (1): Strict: {U61# mark X -> U61# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U61#](x0) = x0 Strict: U61# mark X -> U61# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: {s# mark X -> s# X, s# ok X -> s# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 1, [x](x0, x1) = 1, [mark](x0) = x0, [U12](x0) = 0, [isNat](x0) = x0 + 1, [active](x0) = 0, [U21](x0) = x0 + 1, [U32](x0) = 0, [s](x0) = 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [s#](x0) = x0 Strict: s# ok X -> s# X 1 + 1X >= 0 + 1X s# mark X -> s# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 1 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 2 + 0X s mark X -> mark s X 1 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 1 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 1 + 1X >= 1 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 1 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 1 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 2 + 1V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 0 + 0X >= 1 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: {s# mark X -> s# X} SCC (1): Strict: {s# mark X -> s# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [s#](x0) = x0 Strict: s# mark X -> s# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: {U32# mark X -> U32# X, U32# ok X -> U32# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 1, [x](x0, x1) = 1, [mark](x0) = x0, [U12](x0) = 0, [isNat](x0) = x0 + 1, [active](x0) = 0, [U21](x0) = x0 + 1, [U32](x0) = 0, [s](x0) = 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U32#](x0) = x0 Strict: U32# ok X -> U32# X 1 + 1X >= 0 + 1X U32# mark X -> U32# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 1 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 2 + 0X s mark X -> mark s X 1 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 1 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 1 + 1X >= 1 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 1 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 1 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 2 + 1V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 0 + 0X >= 1 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: {U32# mark X -> U32# X} SCC (1): Strict: {U32# mark X -> U32# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U32#](x0) = x0 Strict: U32# mark X -> U32# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: {U21# mark X -> U21# X, U21# ok X -> U21# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 1, [x](x0, x1) = 1, [mark](x0) = x0, [U12](x0) = 0, [isNat](x0) = x0 + 1, [active](x0) = 0, [U21](x0) = x0 + 1, [U32](x0) = 0, [s](x0) = 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U21#](x0) = x0 Strict: U21# ok X -> U21# X 1 + 1X >= 0 + 1X U21# mark X -> U21# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 1 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 2 + 0X s mark X -> mark s X 1 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 1 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 1 + 1X >= 1 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 1 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 1 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 2 + 1V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 0 + 0X >= 1 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: {U21# mark X -> U21# X} SCC (1): Strict: {U21# mark X -> U21# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U21#](x0) = x0 Strict: U21# mark X -> U21# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed SCC (2): Strict: {U12# mark X -> U12# X, U12# ok X -> U12# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = 0, [U51](x0, x1, x2) = 0, [U72](x0, x1, x2) = 0, [U71](x0, x1, x2) = 0, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = 0, [plus](x0, x1) = 1, [x](x0, x1) = 1, [mark](x0) = x0, [U12](x0) = 0, [isNat](x0) = x0 + 1, [active](x0) = 0, [U21](x0) = x0 + 1, [U32](x0) = 0, [s](x0) = 1, [U61](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U12#](x0) = x0 Strict: U12# ok X -> U12# X 1 + 1X >= 0 + 1X U12# mark X -> U12# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 1 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 0 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 0 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 0 + 0X >= 1 + 0X U61 mark X -> mark U61 X 0 + 0X >= 0 + 0X plus(ok X1, ok X2) -> ok plus(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 1 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 1 + 0X >= 2 + 0X s mark X -> mark s X 1 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 U32 ok X -> ok U32 X 0 + 0X >= 1 + 0X U32 mark X -> mark U32 X 0 + 0X >= 0 + 0X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 1 + 1X >= 1 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 0 + 0N >= 0 + 0N active x(N, s M) -> mark U71(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 0 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 0 >= 1 active U61 X -> U61 active X 0 + 0X >= 0 + 0X active plus(X1, X2) -> plus(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 0 + 0N >= 0 + 0N active plus(N, s M) -> mark U51(isNat M, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active s X -> s active X 0 + 0X >= 1 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 0 + 0N + 0M >= 0 + 0N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 0 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 0 + 0N >= 0 + 1N active U41(X1, X2) -> U41(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 0 + 0V2 >= 0 + 0V2 active U31(X1, X2) -> U31(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active U32 tt() -> mark tt() 0 >= 0 active U32 X -> U32 active X 0 + 0X >= 0 + 0X active U21 tt() -> mark tt() 0 >= 0 active U21 X -> U21 active X 0 + 0X >= 1 + 0X active U11(tt(), V2) -> mark U12 isNat V2 0 + 0V2 >= 0 + 0V2 active U11(X1, X2) -> U11(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat 0() -> mark tt() 0 >= 0 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 0 + 0V2 + 0V1 >= 2 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 0 + 0V1 >= 2 + 1V1 active U12 tt() -> mark tt() 0 >= 0 active U12 X -> U12 active X 0 + 0X >= 0 + 0X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 0 + 0X >= 1 + 0X U12 mark X -> mark U12 X 0 + 0X >= 0 + 0X SCCS (1): Scc: {U12# mark X -> U12# X} SCC (1): Strict: {U12# mark X -> U12# X} Weak: { U12 mark X -> mark U12 X, U12 ok X -> ok U12 X, isNat ok X -> ok isNat X, active U12 X -> U12 active X, active U12 tt() -> mark tt(), active isNat s V1 -> mark U21 isNat V1, active isNat plus(V1, V2) -> mark U11(isNat V1, V2), active isNat 0() -> mark tt(), active isNat x(V1, V2) -> mark U31(isNat V1, V2), active U11(X1, X2) -> U11(active X1, X2), active U11(tt(), V2) -> mark U12 isNat V2, active U21 X -> U21 active X, active U21 tt() -> mark tt(), active U32 X -> U32 active X, active U32 tt() -> mark tt(), active U31(X1, X2) -> U31(active X1, X2), active U31(tt(), V2) -> mark U32 isNat V2, active U41(X1, X2) -> U41(active X1, X2), active U41(tt(), N) -> mark N, active U52(X1, X2, X3) -> U52(active X1, X2, X3), active U52(tt(), M, N) -> mark s plus(N, M), active U51(X1, X2, X3) -> U51(active X1, X2, X3), active U51(tt(), M, N) -> mark U52(isNat N, M, N), active s X -> s active X, active plus(N, s M) -> mark U51(isNat M, M, N), active plus(N, 0()) -> mark U41(isNat N, N), active plus(X1, X2) -> plus(X1, active X2), active plus(X1, X2) -> plus(active X1, X2), active U61 X -> U61 active X, active U61 tt() -> mark 0(), active U72(X1, X2, X3) -> U72(active X1, X2, X3), active U72(tt(), M, N) -> mark plus(x(N, M), N), active U71(X1, X2, X3) -> U71(active X1, X2, X3), active U71(tt(), M, N) -> mark U72(isNat N, M, N), active x(N, s M) -> mark U71(isNat M, M, N), active x(N, 0()) -> mark U61 isNat N, active x(X1, X2) -> x(X1, active X2), active x(X1, X2) -> x(active X1, X2), U11(mark X1, X2) -> mark U11(X1, X2), U11(ok X1, ok X2) -> ok U11(X1, X2), U21 mark X -> mark U21 X, U21 ok X -> ok U21 X, U32 mark X -> mark U32 X, U32 ok X -> ok U32 X, U31(mark X1, X2) -> mark U31(X1, X2), U31(ok X1, ok X2) -> ok U31(X1, X2), U41(mark X1, X2) -> mark U41(X1, X2), U41(ok X1, ok X2) -> ok U41(X1, X2), U52(mark X1, X2, X3) -> mark U52(X1, X2, X3), U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3), U51(mark X1, X2, X3) -> mark U51(X1, X2, X3), U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3), s mark X -> mark s X, s ok X -> ok s X, plus(X1, mark X2) -> mark plus(X1, X2), plus(mark X1, X2) -> mark plus(X1, X2), plus(ok X1, ok X2) -> ok plus(X1, X2), U61 mark X -> mark U61 X, U61 ok X -> ok U61 X, U72(mark X1, X2, X3) -> mark U72(X1, X2, X3), U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3), U71(mark X1, X2, X3) -> mark U71(X1, X2, X3), U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3), x(X1, mark X2) -> mark x(X1, X2), x(mark X1, X2) -> mark x(X1, X2), x(ok X1, ok X2) -> ok x(X1, X2), proper U12 X -> U12 proper X, proper isNat X -> isNat proper X, proper U11(X1, X2) -> U11(proper X1, proper X2), proper tt() -> ok tt(), proper U21 X -> U21 proper X, proper U32 X -> U32 proper X, proper U31(X1, X2) -> U31(proper X1, proper X2), proper U41(X1, X2) -> U41(proper X1, proper X2), proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3), proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3), proper s X -> s proper X, proper plus(X1, X2) -> plus(proper X1, proper X2), proper 0() -> ok 0(), proper U61 X -> U61 proper X, proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3), proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3), proper x(X1, X2) -> x(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [U52](x0, x1, x2) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U72](x0, x1, x2) = x0 + 1, [U71](x0, x1, x2) = x0 + 1, [U11](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U41](x0, x1) = x0 + 1, [plus](x0, x1) = 0, [x](x0, x1) = 0, [mark](x0) = x0 + 1, [U12](x0) = x0 + 1, [isNat](x0) = x0 + 1, [active](x0) = x0 + 1, [U21](x0) = x0 + 1, [U32](x0) = x0 + 1, [s](x0) = 0, [U61](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [U12#](x0) = x0 Strict: U12# mark X -> U12# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper x(X1, X2) -> x(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper U71(X1, X2, X3) -> U71(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U72(X1, X2, X3) -> U72(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U61 X -> U61 proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper plus(X1, X2) -> plus(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper U51(X1, X2, X3) -> U51(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U52(X1, X2, X3) -> U52(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper U41(X1, X2) -> U41(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U31(X1, X2) -> U31(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper U32 X -> U32 proper X 0 + 0X >= 1 + 0X proper U21 X -> U21 proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper U11(X1, X2) -> U11(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper U12 X -> U12 proper X 0 + 0X >= 1 + 0X x(ok X1, ok X2) -> ok x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(mark X1, X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 x(X1, mark X2) -> mark x(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 U71(ok X1, ok X2, ok X3) -> ok U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U71(mark X1, X2, X3) -> mark U71(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(ok X1, ok X2, ok X3) -> ok U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U72(mark X1, X2, X3) -> mark U72(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U61 ok X -> ok U61 X 2 + 1X >= 2 + 1X U61 mark X -> mark U61 X 2 + 1X >= 2 + 1X plus(ok X1, ok X2) -> ok plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(mark X1, X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 plus(X1, mark X2) -> mark plus(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X U51(ok X1, ok X2, ok X3) -> ok U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U51(mark X1, X2, X3) -> mark U51(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(ok X1, ok X2, ok X3) -> ok U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U52(mark X1, X2, X3) -> mark U52(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 U41(ok X1, ok X2) -> ok U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U41(mark X1, X2) -> mark U41(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(ok X1, ok X2) -> ok U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U31(mark X1, X2) -> mark U31(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U32 ok X -> ok U32 X 2 + 1X >= 2 + 1X U32 mark X -> mark U32 X 2 + 1X >= 2 + 1X U21 ok X -> ok U21 X 2 + 1X >= 2 + 1X U21 mark X -> mark U21 X 2 + 1X >= 2 + 1X U11(ok X1, ok X2) -> ok U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 U11(mark X1, X2) -> mark U11(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active x(X1, X2) -> x(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(X1, X2) -> x(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active x(N, 0()) -> mark U61 isNat N 1 + 0N >= 3 + 1N active x(N, s M) -> mark U71(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active U71(tt(), M, N) -> mark U72(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U71(X1, X2, X3) -> U71(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U72(tt(), M, N) -> mark plus(x(N, M), N) 2 + 0N + 0M >= 1 + 0N + 0M active U72(X1, X2, X3) -> U72(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U61 tt() -> mark 0() 2 >= 2 active U61 X -> U61 active X 2 + 1X >= 2 + 1X active plus(X1, X2) -> plus(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(X1, X2) -> plus(X1, active X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active plus(N, 0()) -> mark U41(isNat N, N) 1 + 0N >= 3 + 1N active plus(N, s M) -> mark U51(isNat M, M, N) 1 + 0N + 0M >= 3 + 0N + 1M active s X -> s active X 1 + 0X >= 0 + 0X active U51(tt(), M, N) -> mark U52(isNat N, M, N) 2 + 0N + 0M >= 3 + 1N + 0M active U51(X1, X2, X3) -> U51(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U52(tt(), M, N) -> mark s plus(N, M) 2 + 0N + 0M >= 1 + 0N + 0M active U52(X1, X2, X3) -> U52(active X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 2 + 1X1 + 0X2 + 0X3 active U41(tt(), N) -> mark N 2 + 0N >= 1 + 1N active U41(X1, X2) -> U41(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U31(tt(), V2) -> mark U32 isNat V2 2 + 0V2 >= 3 + 1V2 active U31(X1, X2) -> U31(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active U32 tt() -> mark tt() 2 >= 1 active U32 X -> U32 active X 2 + 1X >= 2 + 1X active U21 tt() -> mark tt() 2 >= 1 active U21 X -> U21 active X 2 + 1X >= 2 + 1X active U11(tt(), V2) -> mark U12 isNat V2 2 + 0V2 >= 3 + 1V2 active U11(X1, X2) -> U11(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active isNat x(V1, V2) -> mark U31(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat 0() -> mark tt() 3 >= 1 active isNat plus(V1, V2) -> mark U11(isNat V1, V2) 2 + 0V2 + 0V1 >= 3 + 0V2 + 1V1 active isNat s V1 -> mark U21 isNat V1 2 + 0V1 >= 3 + 1V1 active U12 tt() -> mark tt() 2 >= 1 active U12 X -> U12 active X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X U12 ok X -> ok U12 X 2 + 1X >= 2 + 1X U12 mark X -> mark U12 X 2 + 1X >= 2 + 1X Qed