(STRATEGY
    INNERMOST)

(VAR
    M N V1 V2 X X1 X2 X3)
(DATATYPES
    A = µX.< tt, s(X), 0, plus(X, X), U11(X, X, X), U12(X, X, X), isNatKind(X), U13(X, X, X), U14(X, X, X), U15(X, X), isNat(X), U16(X), U21(X, X), U22(X, X), U23(X), U31(X, X), U32(X), U41(X), U51(X, X), U52(X, X), U61(X, X, X), U62(X, X, X), U63(X, X, X), U64(X, X, X) >)
(SIGNATURES
    a__U11 :: [A x A x A] -> A
    a__U12 :: [A x A x A] -> A
    a__U13 :: [A x A x A] -> A
    a__U14 :: [A x A x A] -> A
    a__U15 :: [A x A] -> A
    a__U16 :: [A] -> A
    a__U21 :: [A x A] -> A
    a__U22 :: [A x A] -> A
    a__U23 :: [A] -> A
    a__U31 :: [A x A] -> A
    a__U32 :: [A] -> A
    a__U41 :: [A] -> A
    a__U51 :: [A x A] -> A
    a__U52 :: [A x A] -> A
    a__U61 :: [A x A x A] -> A
    a__U62 :: [A x A x A] -> A
    a__U63 :: [A x A x A] -> A
    a__U64 :: [A x A x A] -> A
    a__isNat :: [A] -> A
    a__isNatKind :: [A] -> A
    a__plus :: [A x A] -> A
    mark :: [A] -> A)
(RULES
    a__U11(tt(),V1,V2) ->
      a__U12(a__isNatKind(V1),V1,V2)
    a__U12(tt(),V1,V2) ->
      a__U13(a__isNatKind(V2),V1,V2)
    a__U13(tt(),V1,V2) ->
      a__U14(a__isNatKind(V2),V1,V2)
    a__U14(tt(),V1,V2) ->
      a__U15(a__isNat(V1),V2)
    a__U15(tt(),V2) ->
      a__U16(a__isNat(V2))
    a__U16(tt()) -> tt()
    a__U21(tt(),V1) ->
      a__U22(a__isNatKind(V1),V1)
    a__U22(tt(),V1) ->
      a__U23(a__isNat(V1))
    a__U23(tt()) -> tt()
    a__U31(tt(),V2) ->
      a__U32(a__isNatKind(V2))
    a__U32(tt()) -> tt()
    a__U41(tt()) -> tt()
    a__U51(tt(),N) ->
      a__U52(a__isNatKind(N),N)
    a__U52(tt(),N) -> mark(N)
    a__U61(tt(),M,N) ->
      a__U62(a__isNatKind(M),M,N)
    a__U62(tt(),M,N) ->
      a__U63(a__isNat(N),M,N)
    a__U63(tt(),M,N) ->
      a__U64(a__isNatKind(N),M,N)
    a__U64(tt(),M,N) ->
      s(a__plus(mark(N),mark(M)))
    a__isNat(0()) -> tt()
    a__isNat(plus(V1,V2)) ->
      a__U11(a__isNatKind(V1),V1,V2)
    a__isNat(s(V1)) ->
      a__U21(a__isNatKind(V1),V1)
    a__isNatKind(0()) -> tt()
    a__isNatKind(plus(V1,V2)) ->
      a__U31(a__isNatKind(V1),V2)
    a__isNatKind(s(V1)) ->
      a__U41(a__isNatKind(V1))
    a__plus(N,0()) ->
      a__U51(a__isNat(N),N)
    a__plus(N,s(M)) ->
      a__U61(a__isNat(M),M,N)
    mark(U11(X1,X2,X3)) ->
      a__U11(mark(X1),X2,X3)
    mark(U12(X1,X2,X3)) ->
      a__U12(mark(X1),X2,X3)
    mark(isNatKind(X)) ->
      a__isNatKind(X)
    mark(U13(X1,X2,X3)) ->
      a__U13(mark(X1),X2,X3)
    mark(U14(X1,X2,X3)) ->
      a__U14(mark(X1),X2,X3)
    mark(U15(X1,X2)) ->
      a__U15(mark(X1),X2)
    mark(isNat(X)) -> a__isNat(X)
    mark(U16(X)) -> a__U16(mark(X))
    mark(U21(X1,X2)) ->
      a__U21(mark(X1),X2)
    mark(U22(X1,X2)) ->
      a__U22(mark(X1),X2)
    mark(U23(X)) -> a__U23(mark(X))
    mark(U31(X1,X2)) ->
      a__U31(mark(X1),X2)
    mark(U32(X)) -> a__U32(mark(X))
    mark(U41(X)) -> a__U41(mark(X))
    mark(U51(X1,X2)) ->
      a__U51(mark(X1),X2)
    mark(U52(X1,X2)) ->
      a__U52(mark(X1),X2)
    mark(U61(X1,X2,X3)) ->
      a__U61(mark(X1),X2,X3)
    mark(U62(X1,X2,X3)) ->
      a__U62(mark(X1),X2,X3)
    mark(U63(X1,X2,X3)) ->
      a__U63(mark(X1),X2,X3)
    mark(U64(X1,X2,X3)) ->
      a__U64(mark(X1),X2,X3)
    mark(plus(X1,X2)) ->
      a__plus(mark(X1),mark(X2))
    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,X3) -> U12(X1
                           ,X2
                           ,X3)
    a__isNatKind(X) -> isNatKind(X)
    a__U13(X1,X2,X3) -> U13(X1
                           ,X2
                           ,X3)
    a__U14(X1,X2,X3) -> U14(X1
                           ,X2
                           ,X3)
    a__U15(X1,X2) -> U15(X1,X2)
    a__isNat(X) -> isNat(X)
    a__U16(X) -> U16(X)
    a__U21(X1,X2) -> U21(X1,X2)
    a__U22(X1,X2) -> U22(X1,X2)
    a__U23(X) -> U23(X)
    a__U31(X1,X2) -> U31(X1,X2)
    a__U32(X) -> U32(X)
    a__U41(X) -> U41(X)
    a__U51(X1,X2) -> U51(X1,X2)
    a__U52(X1,X2) -> U52(X1,X2)
    a__U61(X1,X2,X3) -> U61(X1
                           ,X2
                           ,X3)
    a__U62(X1,X2,X3) -> U62(X1
                           ,X2
                           ,X3)
    a__U63(X1,X2,X3) -> U63(X1
                           ,X2
                           ,X3)
    a__U64(X1,X2,X3) -> U64(X1
                           ,X2
                           ,X3)
    a__plus(X1,X2) -> plus(X1,X2))