(STRATEGY
    INNERMOST)

(VAR
    M N V1 V2 X X1 X2 X3)
(DATATYPES
    A = µX.< tt, mark(X), 0, ok(X) >)
(SIGNATURES
    active :: [A] -> A
    U11 :: [A x A x A] -> A
    U12 :: [A x A] -> A
    U13 :: [A] -> A
    U21 :: [A x A] -> A
    U22 :: [A] -> A
    U31 :: [A x A] -> A
    U41 :: [A x A x A] -> A
    s :: [A] -> A
    plus :: [A x A] -> A
    and :: [A x A] -> A
    proper :: [A] -> A
    isNat :: [A] -> A
    isNatKind :: [A] -> A
    top :: [A] -> A)
(RULES
    active(U11(tt(),V1,V2)) ->
      mark(U12(isNat(V1),V2))
    active(U12(tt(),V2)) ->
      mark(U13(isNat(V2)))
    active(U13(tt())) -> mark(tt())
    active(U21(tt(),V1)) ->
      mark(U22(isNat(V1)))
    active(U22(tt())) -> mark(tt())
    active(U31(tt(),N)) -> mark(N)
    active(U41(tt(),M,N)) ->
      mark(s(plus(N,M)))
    active(and(tt(),X)) -> mark(X)
    active(isNat(0())) -> mark(tt())
    active(isNat(plus(V1,V2))) ->
      mark(U11(and(isNatKind(V1)
                  ,isNatKind(V2))
              ,V1
              ,V2))
    active(isNat(s(V1))) ->
      mark(U21(isNatKind(V1),V1))
    active(isNatKind(0())) ->
      mark(tt())
    active(isNatKind(plus(V1
                         ,V2))) -> mark(and(isNatKind(V1)
                                           ,isNatKind(V2)))
    active(isNatKind(s(V1))) ->
      mark(isNatKind(V1))
    active(plus(N,0())) ->
      mark(U31(and(isNat(N)
                  ,isNatKind(N))
              ,N))
    active(plus(N,s(M))) ->
      mark(U41(and(and(isNat(M)
                      ,isNatKind(M))
                  ,and(isNat(N),isNatKind(N)))
              ,M
              ,N))
    active(U11(X1,X2,X3)) ->
      U11(active(X1),X2,X3)
    active(U12(X1,X2)) ->
      U12(active(X1),X2)
    active(U13(X)) -> U13(active(X))
    active(U21(X1,X2)) ->
      U21(active(X1),X2)
    active(U22(X)) -> U22(active(X))
    active(U31(X1,X2)) ->
      U31(active(X1),X2)
    active(U41(X1,X2,X3)) ->
      U41(active(X1),X2,X3)
    active(s(X)) -> s(active(X))
    active(plus(X1,X2)) ->
      plus(active(X1),X2)
    active(plus(X1,X2)) -> plus(X1
                               ,active(X2))
    active(and(X1,X2)) ->
      and(active(X1),X2)
    U11(mark(X1),X2,X3) ->
      mark(U11(X1,X2,X3))
    U12(mark(X1),X2) -> mark(U12(X1
                                ,X2))
    U13(mark(X)) -> mark(U13(X))
    U21(mark(X1),X2) -> mark(U21(X1
                                ,X2))
    U22(mark(X)) -> mark(U22(X))
    U31(mark(X1),X2) -> mark(U31(X1
                                ,X2))
    U41(mark(X1),X2,X3) ->
      mark(U41(X1,X2,X3))
    s(mark(X)) -> mark(s(X))
    plus(mark(X1),X2) ->
      mark(plus(X1,X2))
    plus(X1,mark(X2)) ->
      mark(plus(X1,X2))
    and(mark(X1),X2) -> mark(and(X1
                                ,X2))
    proper(U11(X1,X2,X3)) ->
      U11(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(tt()) -> ok(tt())
    proper(U12(X1,X2)) ->
      U12(proper(X1),proper(X2))
    proper(isNat(X)) ->
      isNat(proper(X))
    proper(U13(X)) -> U13(proper(X))
    proper(U21(X1,X2)) ->
      U21(proper(X1),proper(X2))
    proper(U22(X)) -> U22(proper(X))
    proper(U31(X1,X2)) ->
      U31(proper(X1),proper(X2))
    proper(U41(X1,X2,X3)) ->
      U41(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(s(X)) -> s(proper(X))
    proper(plus(X1,X2)) ->
      plus(proper(X1),proper(X2))
    proper(and(X1,X2)) ->
      and(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    proper(isNatKind(X)) ->
      isNatKind(proper(X))
    U11(ok(X1),ok(X2),ok(X3)) ->
      ok(U11(X1,X2,X3))
    U12(ok(X1),ok(X2)) -> ok(U12(X1
                                ,X2))
    isNat(ok(X)) -> ok(isNat(X))
    U13(ok(X)) -> ok(U13(X))
    U21(ok(X1),ok(X2)) -> ok(U21(X1
                                ,X2))
    U22(ok(X)) -> ok(U22(X))
    U31(ok(X1),ok(X2)) -> ok(U31(X1
                                ,X2))
    U41(ok(X1),ok(X2),ok(X3)) ->
      ok(U41(X1,X2,X3))
    s(ok(X)) -> ok(s(X))
    plus(ok(X1),ok(X2)) ->
      ok(plus(X1,X2))
    and(ok(X1),ok(X2)) -> ok(and(X1
                                ,X2))
    isNatKind(ok(X)) ->
      ok(isNatKind(X))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X)))