LMPO

Execution Time (secs)	0.408
Answer	MAYBE
Input	Transformed CSR 04 PEANO complete GM

MAYBE

We consider the following Problem:

  Strict Trs:
    {  a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
     , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
     , a__U13(tt()) -> tt()
     , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
     , a__U22(tt()) -> tt()
     , a__U31(tt(), N) -> mark(N)
     , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
     , a__and(tt(), X) -> mark(X)
     , a__isNat(0()) -> tt()
     , a__isNat(plus(V1, V2)) ->
       a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
     , a__isNatKind(0()) -> tt()
     , a__isNatKind(plus(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2))
     , a__isNatKind(s(V1)) -> a__isNatKind(V1)
     , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
     , a__plus(N, s(M)) ->
       a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
     , mark(isNat(X)) -> a__isNat(X)
     , mark(U13(X)) -> a__U13(mark(X))
     , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
     , mark(U22(X)) -> a__U22(mark(X))
     , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
     , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
     , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
     , mark(and(X1, X2)) -> a__and(mark(X1), X2)
     , mark(isNatKind(X)) -> a__isNatKind(X)
     , mark(tt()) -> tt()
     , mark(s(X)) -> s(mark(X))
     , mark(0()) -> 0()
     , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U12(X1, X2) -> U12(X1, X2)
     , a__isNat(X) -> isNat(X)
     , a__U13(X) -> U13(X)
     , a__U21(X1, X2) -> U21(X1, X2)
     , a__U22(X) -> U22(X)
     , a__U31(X1, X2) -> U31(X1, X2)
     , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
     , a__plus(X1, X2) -> plus(X1, X2)
     , a__and(X1, X2) -> and(X1, X2)
     , a__isNatKind(X) -> isNatKind(X)}
  StartTerms: basic terms
  Strategy: innermost

Certificate: MAYBE

Proof:
  The input cannot be shown compatible

Arrrr..

MPO

Execution Time (secs)	0.586
Answer	MAYBE
Input	Transformed CSR 04 PEANO complete GM

MAYBE

We consider the following Problem:

  Strict Trs:
    {  a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
     , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
     , a__U13(tt()) -> tt()
     , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
     , a__U22(tt()) -> tt()
     , a__U31(tt(), N) -> mark(N)
     , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
     , a__and(tt(), X) -> mark(X)
     , a__isNat(0()) -> tt()
     , a__isNat(plus(V1, V2)) ->
       a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
     , a__isNatKind(0()) -> tt()
     , a__isNatKind(plus(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2))
     , a__isNatKind(s(V1)) -> a__isNatKind(V1)
     , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
     , a__plus(N, s(M)) ->
       a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
     , mark(isNat(X)) -> a__isNat(X)
     , mark(U13(X)) -> a__U13(mark(X))
     , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
     , mark(U22(X)) -> a__U22(mark(X))
     , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
     , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
     , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
     , mark(and(X1, X2)) -> a__and(mark(X1), X2)
     , mark(isNatKind(X)) -> a__isNatKind(X)
     , mark(tt()) -> tt()
     , mark(s(X)) -> s(mark(X))
     , mark(0()) -> 0()
     , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U12(X1, X2) -> U12(X1, X2)
     , a__isNat(X) -> isNat(X)
     , a__U13(X) -> U13(X)
     , a__U21(X1, X2) -> U21(X1, X2)
     , a__U22(X) -> U22(X)
     , a__U31(X1, X2) -> U31(X1, X2)
     , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
     , a__plus(X1, X2) -> plus(X1, X2)
     , a__and(X1, X2) -> and(X1, X2)
     , a__isNatKind(X) -> isNatKind(X)}
  StartTerms: basic terms
  Strategy: innermost

Certificate: MAYBE

Proof:
  The input cannot be shown compatible

Arrrr..

POP*

Execution Time (secs)	0.341
Answer	MAYBE
Input	Transformed CSR 04 PEANO complete GM

MAYBE

We consider the following Problem:

  Strict Trs:
    {  a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
     , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
     , a__U13(tt()) -> tt()
     , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
     , a__U22(tt()) -> tt()
     , a__U31(tt(), N) -> mark(N)
     , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
     , a__and(tt(), X) -> mark(X)
     , a__isNat(0()) -> tt()
     , a__isNat(plus(V1, V2)) ->
       a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
     , a__isNatKind(0()) -> tt()
     , a__isNatKind(plus(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2))
     , a__isNatKind(s(V1)) -> a__isNatKind(V1)
     , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
     , a__plus(N, s(M)) ->
       a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
     , mark(isNat(X)) -> a__isNat(X)
     , mark(U13(X)) -> a__U13(mark(X))
     , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
     , mark(U22(X)) -> a__U22(mark(X))
     , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
     , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
     , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
     , mark(and(X1, X2)) -> a__and(mark(X1), X2)
     , mark(isNatKind(X)) -> a__isNatKind(X)
     , mark(tt()) -> tt()
     , mark(s(X)) -> s(mark(X))
     , mark(0()) -> 0()
     , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U12(X1, X2) -> U12(X1, X2)
     , a__isNat(X) -> isNat(X)
     , a__U13(X) -> U13(X)
     , a__U21(X1, X2) -> U21(X1, X2)
     , a__U22(X) -> U22(X)
     , a__U31(X1, X2) -> U31(X1, X2)
     , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
     , a__plus(X1, X2) -> plus(X1, X2)
     , a__and(X1, X2) -> and(X1, X2)
     , a__isNatKind(X) -> isNatKind(X)}
  StartTerms: basic terms
  Strategy: innermost

Certificate: MAYBE

Proof:
  The input cannot be shown compatible

Arrrr..

POP* (PS)

Execution Time (secs)	0.237
Answer	MAYBE
Input	Transformed CSR 04 PEANO complete GM

MAYBE

We consider the following Problem:

  Strict Trs:
    {  a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
     , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
     , a__U13(tt()) -> tt()
     , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
     , a__U22(tt()) -> tt()
     , a__U31(tt(), N) -> mark(N)
     , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
     , a__and(tt(), X) -> mark(X)
     , a__isNat(0()) -> tt()
     , a__isNat(plus(V1, V2)) ->
       a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
     , a__isNatKind(0()) -> tt()
     , a__isNatKind(plus(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2))
     , a__isNatKind(s(V1)) -> a__isNatKind(V1)
     , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
     , a__plus(N, s(M)) ->
       a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
     , mark(isNat(X)) -> a__isNat(X)
     , mark(U13(X)) -> a__U13(mark(X))
     , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
     , mark(U22(X)) -> a__U22(mark(X))
     , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
     , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
     , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
     , mark(and(X1, X2)) -> a__and(mark(X1), X2)
     , mark(isNatKind(X)) -> a__isNatKind(X)
     , mark(tt()) -> tt()
     , mark(s(X)) -> s(mark(X))
     , mark(0()) -> 0()
     , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U12(X1, X2) -> U12(X1, X2)
     , a__isNat(X) -> isNat(X)
     , a__U13(X) -> U13(X)
     , a__U21(X1, X2) -> U21(X1, X2)
     , a__U22(X) -> U22(X)
     , a__U31(X1, X2) -> U31(X1, X2)
     , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
     , a__plus(X1, X2) -> plus(X1, X2)
     , a__and(X1, X2) -> and(X1, X2)
     , a__isNatKind(X) -> isNatKind(X)}
  StartTerms: basic terms
  Strategy: innermost

Certificate: MAYBE

Proof:
  The input cannot be shown compatible

Arrrr..

Small POP*

Execution Time (secs)	0.305
Answer	MAYBE
Input	Transformed CSR 04 PEANO complete GM

MAYBE

We consider the following Problem:

  Strict Trs:
    {  a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
     , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
     , a__U13(tt()) -> tt()
     , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
     , a__U22(tt()) -> tt()
     , a__U31(tt(), N) -> mark(N)
     , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
     , a__and(tt(), X) -> mark(X)
     , a__isNat(0()) -> tt()
     , a__isNat(plus(V1, V2)) ->
       a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
     , a__isNatKind(0()) -> tt()
     , a__isNatKind(plus(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2))
     , a__isNatKind(s(V1)) -> a__isNatKind(V1)
     , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
     , a__plus(N, s(M)) ->
       a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
     , mark(isNat(X)) -> a__isNat(X)
     , mark(U13(X)) -> a__U13(mark(X))
     , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
     , mark(U22(X)) -> a__U22(mark(X))
     , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
     , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
     , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
     , mark(and(X1, X2)) -> a__and(mark(X1), X2)
     , mark(isNatKind(X)) -> a__isNatKind(X)
     , mark(tt()) -> tt()
     , mark(s(X)) -> s(mark(X))
     , mark(0()) -> 0()
     , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U12(X1, X2) -> U12(X1, X2)
     , a__isNat(X) -> isNat(X)
     , a__U13(X) -> U13(X)
     , a__U21(X1, X2) -> U21(X1, X2)
     , a__U22(X) -> U22(X)
     , a__U31(X1, X2) -> U31(X1, X2)
     , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
     , a__plus(X1, X2) -> plus(X1, X2)
     , a__and(X1, X2) -> and(X1, X2)
     , a__isNatKind(X) -> isNatKind(X)}
  StartTerms: basic terms
  Strategy: innermost

Certificate: MAYBE

Proof:
  The input cannot be shown compatible

Arrrr..

Small POP* (PS)

Execution Time (secs)	0.423
Answer	MAYBE
Input	Transformed CSR 04 PEANO complete GM

MAYBE

We consider the following Problem:

  Strict Trs:
    {  a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
     , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
     , a__U13(tt()) -> tt()
     , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
     , a__U22(tt()) -> tt()
     , a__U31(tt(), N) -> mark(N)
     , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
     , a__and(tt(), X) -> mark(X)
     , a__isNat(0()) -> tt()
     , a__isNat(plus(V1, V2)) ->
       a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
     , a__isNatKind(0()) -> tt()
     , a__isNatKind(plus(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2))
     , a__isNatKind(s(V1)) -> a__isNatKind(V1)
     , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N)
     , a__plus(N, s(M)) ->
       a__U41(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
     , mark(isNat(X)) -> a__isNat(X)
     , mark(U13(X)) -> a__U13(mark(X))
     , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
     , mark(U22(X)) -> a__U22(mark(X))
     , mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
     , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
     , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
     , mark(and(X1, X2)) -> a__and(mark(X1), X2)
     , mark(isNatKind(X)) -> a__isNatKind(X)
     , mark(tt()) -> tt()
     , mark(s(X)) -> s(mark(X))
     , mark(0()) -> 0()
     , a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U12(X1, X2) -> U12(X1, X2)
     , a__isNat(X) -> isNat(X)
     , a__U13(X) -> U13(X)
     , a__U21(X1, X2) -> U21(X1, X2)
     , a__U22(X) -> U22(X)
     , a__U31(X1, X2) -> U31(X1, X2)
     , a__U41(X1, X2, X3) -> U41(X1, X2, X3)
     , a__plus(X1, X2) -> plus(X1, X2)
     , a__and(X1, X2) -> and(X1, X2)
     , a__isNatKind(X) -> isNatKind(X)}
  StartTerms: basic terms
  Strategy: innermost

Certificate: MAYBE

Proof:
  The input cannot be shown compatible

Arrrr..

Problem Transformed CSR 04 PEANO complete GM

LMPO

MPO

POP*

POP* (PS)

Small POP*

Small POP* (PS)