TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60006 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (3077ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (1146ms), PolynomialLinearRange4iUR (timeout), DependencyGraph (937ms), PolynomialLinearRange8NegiUR (30004ms), DependencyGraph (946ms), ReductionPairSAT (timeout)].
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| | – Problem 4 was processed with processor DependencyGraph (4ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U72(X1, X2, X3)) | → | a__U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) | → | a__x#(mark(X1), mark(X2)) | | mark#(U32(X)) | → | mark#(X) |
| a__plus#(N, s(M)) | → | a__U51#(a__isNat(M), M, N) | | mark#(U21(X)) | → | mark#(X) |
| a__U72#(tt, M, N) | → | mark#(M) | | a__U41#(tt, N) | → | mark#(N) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__U51#(tt, M, N) | → | a__U52#(a__isNat(N), M, N) | | mark#(U41(X1, X2)) | → | a__U41#(mark(X1), X2) |
| a__U52#(tt, M, N) | → | a__plus#(mark(N), mark(M)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | a__U52#(mark(X1), X2, X3) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U61(X)) | → | mark#(X) |
| a__x#(N, s(M)) | → | a__U71#(a__isNat(M), M, N) | | mark#(U12(X)) | → | mark#(X) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | a__U72#(tt, M, N) | → | a__plus#(a__x(mark(N), mark(M)), mark(N)) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | a__U71#(tt, M, N) | → | a__U72#(a__isNat(N), M, N) |
| a__U72#(tt, M, N) | → | a__x#(mark(N), mark(M)) | | a__plus#(N, 0) | → | a__U41#(a__isNat(N), N) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | a__U52#(tt, M, N) | → | mark#(N) |
| mark#(U51(X1, X2, X3)) | → | a__U51#(mark(X1), X2, X3) | | mark#(x(X1, X2)) | → | mark#(X2) |
| mark#(U71(X1, X2, X3)) | → | a__U71#(mark(X1), X2, X3) | | a__U52#(tt, M, N) | → | mark#(M) |
| a__U72#(tt, M, N) | → | mark#(N) | | mark#(x(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, V2) | → | a__U32(a__isNat(V2)) |
| a__U32(tt) | → | tt | | a__U41(tt, N) | → | mark(N) |
| a__U51(tt, M, N) | → | a__U52(a__isNat(N), M, N) | | a__U52(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__U61(tt) | → | 0 | | a__U71(tt, M, N) | → | a__U72(a__isNat(N), M, N) |
| a__U72(tt, M, N) | → | a__plus(a__x(mark(N), mark(M)), mark(N)) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) | | a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) |
| a__isNat(x(V1, V2)) | → | a__U31(a__isNat(V1), V2) | | a__plus(N, 0) | → | a__U41(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U51(a__isNat(M), M, N) | | a__x(N, 0) | → | a__U61(a__isNat(N)) |
| a__x(N, s(M)) | → | a__U71(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U32(X)) | → | a__U32(mark(X)) | | mark(U41(X1, X2)) | → | a__U41(mark(X1), X2) |
| mark(U51(X1, X2, X3)) | → | a__U51(mark(X1), X2, X3) | | mark(U52(X1, X2, X3)) | → | a__U52(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(U61(X)) | → | a__U61(mark(X)) |
| mark(U71(X1, X2, X3)) | → | a__U71(mark(X1), X2, X3) | | mark(U72(X1, X2, X3)) | → | a__U72(mark(X1), X2, X3) |
| mark(x(X1, X2)) | → | a__x(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U32(X) | → | U32(X) |
| a__U41(X1, X2) | → | U41(X1, X2) | | a__U51(X1, X2, X3) | → | U51(X1, X2, X3) |
| a__U52(X1, X2, X3) | → | U52(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
| a__U61(X) | → | U61(X) | | a__U71(X1, X2, X3) | → | U71(X1, X2, X3) |
| a__U72(X1, X2, X3) | → | U72(X1, X2, X3) | | a__x(X1, X2) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: a__U51, a__U52, isNat, U61, U41, a__U41, a__U72, a__U71, a__U21, U21, plus, a__plus, mark, U71, U72, 0, U51, s, tt, a__isNat, U52, a__U32, a__U31, U11, a__U12, U12, a__U11, U31, U32, a__U61, a__x, x
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | a__U72#(tt, M, N) | → | mark#(M) |
| a__x#(N, 0) | → | a__U61#(a__isNat(N)) | | a__U51#(tt, M, N) | → | a__U52#(a__isNat(N), M, N) |
| mark#(U21(X)) | → | a__U21#(mark(X)) | | mark#(U41(X1, X2)) | → | a__U41#(mark(X1), X2) |
| a__U52#(tt, M, N) | → | a__plus#(mark(N), mark(M)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | a__x#(N, 0) | → | a__isNat#(N) |
| a__isNat#(x(V1, V2)) | → | a__U31#(a__isNat(V1), V2) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| a__U72#(tt, M, N) | → | a__plus#(a__x(mark(N), mark(M)), mark(N)) | | a__U31#(tt, V2) | → | a__U32#(a__isNat(V2)) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | a__U31#(tt, V2) | → | a__isNat#(V2) |
| a__U11#(tt, V2) | → | a__U12#(a__isNat(V2)) | | a__U72#(tt, M, N) | → | a__x#(mark(N), mark(M)) |
| a__plus#(N, 0) | → | a__U41#(a__isNat(N), N) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| a__U52#(tt, M, N) | → | mark#(N) | | a__U51#(tt, M, N) | → | a__isNat#(N) |
| mark#(x(X1, X2)) | → | mark#(X2) | | mark#(U71(X1, X2, X3)) | → | a__U71#(mark(X1), X2, X3) |
| a__plus#(N, 0) | → | a__isNat#(N) | | a__U72#(tt, M, N) | → | mark#(N) |
| a__isNat#(x(V1, V2)) | → | a__isNat#(V1) | | mark#(U72(X1, X2, X3)) | → | a__U72#(mark(X1), X2, X3) |
| mark#(isNat(X)) | → | a__isNat#(X) | | mark#(x(X1, X2)) | → | a__x#(mark(X1), mark(X2)) |
| mark#(U32(X)) | → | a__U32#(mark(X)) | | mark#(U32(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | a__U31#(mark(X1), X2) | | a__isNat#(s(V1)) | → | a__isNat#(V1) |
| a__plus#(N, s(M)) | → | a__U51#(a__isNat(M), M, N) | | a__U41#(tt, N) | → | mark#(N) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| a__isNat#(plus(V1, V2)) | → | a__U11#(a__isNat(V1), V2) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| a__isNat#(s(V1)) | → | a__U21#(a__isNat(V1)) | | a__U11#(tt, V2) | → | a__isNat#(V2) |
| a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) | | a__plus#(N, s(M)) | → | a__isNat#(M) |
| mark#(U52(X1, X2, X3)) | → | a__U52#(mark(X1), X2, X3) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | a__U61#(mark(X)) | | mark#(U61(X)) | → | mark#(X) |
| a__x#(N, s(M)) | → | a__U71#(a__isNat(M), M, N) | | mark#(U12(X)) | → | mark#(X) |
| mark#(U12(X)) | → | a__U12#(mark(X)) | | a__U71#(tt, M, N) | → | a__U72#(a__isNat(N), M, N) |
| mark#(U51(X1, X2, X3)) | → | a__U51#(mark(X1), X2, X3) | | a__x#(N, s(M)) | → | a__isNat#(M) |
| a__U52#(tt, M, N) | → | mark#(M) | | mark#(U11(X1, X2)) | → | a__U11#(mark(X1), X2) |
| a__U71#(tt, M, N) | → | a__isNat#(N) | | mark#(x(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, V2) | → | a__U32(a__isNat(V2)) |
| a__U32(tt) | → | tt | | a__U41(tt, N) | → | mark(N) |
| a__U51(tt, M, N) | → | a__U52(a__isNat(N), M, N) | | a__U52(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__U61(tt) | → | 0 | | a__U71(tt, M, N) | → | a__U72(a__isNat(N), M, N) |
| a__U72(tt, M, N) | → | a__plus(a__x(mark(N), mark(M)), mark(N)) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) | | a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) |
| a__isNat(x(V1, V2)) | → | a__U31(a__isNat(V1), V2) | | a__plus(N, 0) | → | a__U41(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U51(a__isNat(M), M, N) | | a__x(N, 0) | → | a__U61(a__isNat(N)) |
| a__x(N, s(M)) | → | a__U71(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U32(X)) | → | a__U32(mark(X)) | | mark(U41(X1, X2)) | → | a__U41(mark(X1), X2) |
| mark(U51(X1, X2, X3)) | → | a__U51(mark(X1), X2, X3) | | mark(U52(X1, X2, X3)) | → | a__U52(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(U61(X)) | → | a__U61(mark(X)) |
| mark(U71(X1, X2, X3)) | → | a__U71(mark(X1), X2, X3) | | mark(U72(X1, X2, X3)) | → | a__U72(mark(X1), X2, X3) |
| mark(x(X1, X2)) | → | a__x(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U32(X) | → | U32(X) |
| a__U41(X1, X2) | → | U41(X1, X2) | | a__U51(X1, X2, X3) | → | U51(X1, X2, X3) |
| a__U52(X1, X2, X3) | → | U52(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
| a__U61(X) | → | U61(X) | | a__U71(X1, X2, X3) | → | U71(X1, X2, X3) |
| a__U72(X1, X2, X3) | → | U72(X1, X2, X3) | | a__x(X1, X2) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: a__U51, a__U52, isNat, U61, U41, a__U41, a__U72, a__U71, U21, a__U21, plus, a__plus, mark, U71, U72, 0, U51, s, tt, a__isNat, U52, a__U32, a__U31, U11, U12, a__U12, U31, a__U11, a__U61, U32, a__x, x
Strategy
The following SCCs where found
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U72(X1, X2, X3)) → a__U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) → a__x#(mark(X1), mark(X2)) | mark#(U32(X)) → mark#(X) |
| a__U41#(tt, N) → mark#(N) | a__U72#(tt, M, N) → mark#(M) |
| mark#(U21(X)) → mark#(X) | a__plus#(N, s(M)) → a__U51#(a__isNat(M), M, N) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → mark#(X2) |
| mark#(U41(X1, X2)) → a__U41#(mark(X1), X2) | a__U51#(tt, M, N) → a__U52#(a__isNat(N), M, N) |
| a__U52#(tt, M, N) → a__plus#(mark(N), mark(M)) | mark#(U52(X1, X2, X3)) → a__U52#(mark(X1), X2, X3) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| mark#(U41(X1, X2)) → mark#(X1) | mark#(U61(X)) → mark#(X) |
| a__x#(N, s(M)) → a__U71#(a__isNat(M), M, N) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| mark#(U12(X)) → mark#(X) | mark#(plus(X1, X2)) → a__plus#(mark(X1), mark(X2)) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | a__U72#(tt, M, N) → a__plus#(a__x(mark(N), mark(M)), mark(N)) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | a__U71#(tt, M, N) → a__U72#(a__isNat(N), M, N) |
| a__plus#(N, 0) → a__U41#(a__isNat(N), N) | a__U72#(tt, M, N) → a__x#(mark(N), mark(M)) |
| a__U52#(tt, M, N) → mark#(N) | mark#(plus(X1, X2)) → mark#(X1) |
| mark#(U51(X1, X2, X3)) → a__U51#(mark(X1), X2, X3) | mark#(x(X1, X2)) → mark#(X2) |
| mark#(U71(X1, X2, X3)) → a__U71#(mark(X1), X2, X3) | a__U52#(tt, M, N) → mark#(M) |
| a__U72#(tt, M, N) → mark#(N) | mark#(x(X1, X2)) → mark#(X1) |
| a__isNat#(plus(V1, V2)) → a__U11#(a__isNat(V1), V2) | a__U11#(tt, V2) → a__isNat#(V2) |
| a__isNat#(plus(V1, V2)) → a__isNat#(V1) | a__U31#(tt, V2) → a__isNat#(V2) |
| a__isNat#(s(V1)) → a__isNat#(V1) | a__isNat#(x(V1, V2)) → a__isNat#(V1) |
| a__isNat#(x(V1, V2)) → a__U31#(a__isNat(V1), V2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| a__isNat#(plus(V1, V2)) | → | a__U11#(a__isNat(V1), V2) | | a__U11#(tt, V2) | → | a__isNat#(V2) |
| a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) | | a__U31#(tt, V2) | → | a__isNat#(V2) |
| a__isNat#(s(V1)) | → | a__isNat#(V1) | | a__isNat#(x(V1, V2)) | → | a__isNat#(V1) |
| a__isNat#(x(V1, V2)) | → | a__U31#(a__isNat(V1), V2) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, V2) | → | a__U32(a__isNat(V2)) |
| a__U32(tt) | → | tt | | a__U41(tt, N) | → | mark(N) |
| a__U51(tt, M, N) | → | a__U52(a__isNat(N), M, N) | | a__U52(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__U61(tt) | → | 0 | | a__U71(tt, M, N) | → | a__U72(a__isNat(N), M, N) |
| a__U72(tt, M, N) | → | a__plus(a__x(mark(N), mark(M)), mark(N)) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) | | a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) |
| a__isNat(x(V1, V2)) | → | a__U31(a__isNat(V1), V2) | | a__plus(N, 0) | → | a__U41(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U51(a__isNat(M), M, N) | | a__x(N, 0) | → | a__U61(a__isNat(N)) |
| a__x(N, s(M)) | → | a__U71(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U32(X)) | → | a__U32(mark(X)) | | mark(U41(X1, X2)) | → | a__U41(mark(X1), X2) |
| mark(U51(X1, X2, X3)) | → | a__U51(mark(X1), X2, X3) | | mark(U52(X1, X2, X3)) | → | a__U52(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(U61(X)) | → | a__U61(mark(X)) |
| mark(U71(X1, X2, X3)) | → | a__U71(mark(X1), X2, X3) | | mark(U72(X1, X2, X3)) | → | a__U72(mark(X1), X2, X3) |
| mark(x(X1, X2)) | → | a__x(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U32(X) | → | U32(X) |
| a__U41(X1, X2) | → | U41(X1, X2) | | a__U51(X1, X2, X3) | → | U51(X1, X2, X3) |
| a__U52(X1, X2, X3) | → | U52(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
| a__U61(X) | → | U61(X) | | a__U71(X1, X2, X3) | → | U71(X1, X2, X3) |
| a__U72(X1, X2, X3) | → | U72(X1, X2, X3) | | a__x(X1, X2) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: a__U51, a__U52, isNat, U61, U41, a__U41, a__U72, a__U71, U21, a__U21, plus, a__plus, mark, U71, U72, 0, U51, s, tt, a__isNat, U52, a__U32, a__U31, U11, U12, a__U12, U31, a__U11, a__U61, U32, a__x, x
Strategy
Projection
The following projection was used:
- π (a__U11#): 2
- π (a__U31#): 2
- π (a__isNat#): 1
Thus, the following dependency pairs are removed:
| a__isNat#(plus(V1, V2)) | → | a__U11#(a__isNat(V1), V2) | | a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) |
| a__isNat#(s(V1)) | → | a__isNat#(V1) | | a__isNat#(x(V1, V2)) | → | a__isNat#(V1) |
| a__isNat#(x(V1, V2)) | → | a__U31#(a__isNat(V1), V2) |
Problem 4: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__U11#(tt, V2) | → | a__isNat#(V2) | | a__U31#(tt, V2) | → | a__isNat#(V2) |
Rewrite Rules
| a__U11(tt, V2) | → | a__U12(a__isNat(V2)) | | a__U12(tt) | → | tt |
| a__U21(tt) | → | tt | | a__U31(tt, V2) | → | a__U32(a__isNat(V2)) |
| a__U32(tt) | → | tt | | a__U41(tt, N) | → | mark(N) |
| a__U51(tt, M, N) | → | a__U52(a__isNat(N), M, N) | | a__U52(tt, M, N) | → | s(a__plus(mark(N), mark(M))) |
| a__U61(tt) | → | 0 | | a__U71(tt, M, N) | → | a__U72(a__isNat(N), M, N) |
| a__U72(tt, M, N) | → | a__plus(a__x(mark(N), mark(M)), mark(N)) | | a__isNat(0) | → | tt |
| a__isNat(plus(V1, V2)) | → | a__U11(a__isNat(V1), V2) | | a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) |
| a__isNat(x(V1, V2)) | → | a__U31(a__isNat(V1), V2) | | a__plus(N, 0) | → | a__U41(a__isNat(N), N) |
| a__plus(N, s(M)) | → | a__U51(a__isNat(M), M, N) | | a__x(N, 0) | → | a__U61(a__isNat(N)) |
| a__x(N, s(M)) | → | a__U71(a__isNat(M), M, N) | | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) |
| mark(U12(X)) | → | a__U12(mark(X)) | | mark(isNat(X)) | → | a__isNat(X) |
| mark(U21(X)) | → | a__U21(mark(X)) | | mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) |
| mark(U32(X)) | → | a__U32(mark(X)) | | mark(U41(X1, X2)) | → | a__U41(mark(X1), X2) |
| mark(U51(X1, X2, X3)) | → | a__U51(mark(X1), X2, X3) | | mark(U52(X1, X2, X3)) | → | a__U52(mark(X1), X2, X3) |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | | mark(U61(X)) | → | a__U61(mark(X)) |
| mark(U71(X1, X2, X3)) | → | a__U71(mark(X1), X2, X3) | | mark(U72(X1, X2, X3)) | → | a__U72(mark(X1), X2, X3) |
| mark(x(X1, X2)) | → | a__x(mark(X1), mark(X2)) | | mark(tt) | → | tt |
| mark(s(X)) | → | s(mark(X)) | | mark(0) | → | 0 |
| a__U11(X1, X2) | → | U11(X1, X2) | | a__U12(X) | → | U12(X) |
| a__isNat(X) | → | isNat(X) | | a__U21(X) | → | U21(X) |
| a__U31(X1, X2) | → | U31(X1, X2) | | a__U32(X) | → | U32(X) |
| a__U41(X1, X2) | → | U41(X1, X2) | | a__U51(X1, X2, X3) | → | U51(X1, X2, X3) |
| a__U52(X1, X2, X3) | → | U52(X1, X2, X3) | | a__plus(X1, X2) | → | plus(X1, X2) |
| a__U61(X) | → | U61(X) | | a__U71(X1, X2, X3) | → | U71(X1, X2, X3) |
| a__U72(X1, X2, X3) | → | U72(X1, X2, X3) | | a__x(X1, X2) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: a__U51, a__U52, isNat, U61, U41, a__U41, a__U72, a__U71, a__U21, U21, plus, a__plus, mark, U71, U72, 0, U51, s, tt, a__isNat, U52, a__U32, a__U31, U11, a__U12, U12, a__U11, U31, U32, a__U61, a__x, x
Strategy
There are no SCCs!