TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60002 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (31934ms).
| – Problem 2 was processed with processor SubtermCriterion (3ms).
| – Problem 3 was processed with processor DependencyGraph (6833ms).
| | – Problem 60 remains open; application of the following processors failed [].
| – Problem 4 was processed with processor DependencyGraph (7091ms).
| | – Problem 61 remains open; application of the following processors failed [].
| – Problem 5 was processed with processor SubtermCriterion (2ms).
| | – Problem 51 remains open; application of the following processors failed [].
| – Problem 6 was processed with processor DependencyGraph (7463ms).
| | – Problem 62 remains open; application of the following processors failed [].
| | – Problem 63 remains open; application of the following processors failed [].
| – Problem 7 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (timeout)].
| – Problem 8 was processed with processor SubtermCriterion (2ms).
| | – Problem 52 remains open; application of the following processors failed [].
| – Problem 9 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 10 remains open; application of the following processors failed [SubtermCriterion (2ms)].
| – Problem 11 was processed with processor SubtermCriterion (2ms).
| – Problem 12 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 13 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 14 was processed with processor SubtermCriterion (2ms).
| | – Problem 53 remains open; application of the following processors failed [].
| – Problem 15 was processed with processor SubtermCriterion (2ms).
| – Problem 16 remains open; application of the following processors failed [SubtermCriterion (2ms)].
| – Problem 17 was processed with processor SubtermCriterion (3ms).
| | – Problem 54 remains open; application of the following processors failed [].
| – Problem 18 was processed with processor SubtermCriterion (3ms).
| – Problem 19 remains open; application of the following processors failed [SubtermCriterion (2ms)].
| – Problem 20 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 21 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 22 was processed with processor SubtermCriterion (2ms).
| – Problem 23 remains open; application of the following processors failed [SubtermCriterion (2ms)].
| – Problem 24 was processed with processor SubtermCriterion (2ms).
| | – Problem 55 remains open; application of the following processors failed [].
| – Problem 25 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 26 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 27 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 28 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 29 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 30 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 31 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 32 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 33 remains open; application of the following processors failed [SubtermCriterion (5ms)].
| – Problem 34 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 35 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 36 remains open; application of the following processors failed [SubtermCriterion (63ms)].
| – Problem 37 remains open; application of the following processors failed [SubtermCriterion (2ms)].
| – Problem 38 was processed with processor SubtermCriterion (2ms).
| | – Problem 56 remains open; application of the following processors failed [].
| – Problem 39 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 40 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 41 remains open; application of the following processors failed [SubtermCriterion (2ms)].
| – Problem 42 remains open; application of the following processors failed [SubtermCriterion (2ms)].
| – Problem 43 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 44 was processed with processor SubtermCriterion (1ms).
| | – Problem 59 remains open; application of the following processors failed [].
| – Problem 45 was processed with processor SubtermCriterion (2ms).
| – Problem 46 was processed with processor SubtermCriterion (2ms).
| | – Problem 57 remains open; application of the following processors failed [].
| – Problem 47 was processed with processor SubtermCriterion (2ms).
| | – Problem 58 remains open; application of the following processors failed [].
| – Problem 48 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 49 remains open; application of the following processors failed [SubtermCriterion (1ms)].
| – Problem 50 remains open; application of the following processors failed [SubtermCriterion (4ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 4
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U32(X)) | → | U32#(mark(X)) |
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(U21(tt)) | → | mark#(tt) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U52(tt, M, N)) | → | plus#(N, M) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(U52(tt, M, N)) | → | s#(plus(N, M)) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, s(M))) | → | U71#(isNat(M), M, N) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 5
Dependency Pairs
| U52#(X1, mark(X2), X3) | → | U52#(X1, X2, X3) | | U52#(X1, X2, mark(X3)) | → | U52#(X1, X2, X3) |
| U52#(X1, X2, active(X3)) | → | U52#(X1, X2, X3) | | U52#(X1, active(X2), X3) | → | U52#(X1, X2, X3) |
| U52#(active(X1), X2, X3) | → | U52#(X1, X2, X3) | | U52#(mark(X1), X2, X3) | → | U52#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 6
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U32(X)) | → | U32#(mark(X)) |
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | U32#(mark(X)) | → | U32#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(isNat(X)) | → | isNat#(X) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | U32#(active(X)) | → | U32#(X) |
| mark#(U12(X)) | → | mark#(X) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 7
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | active#(U72(tt, M, N)) | → | plus#(x(N, M), N) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 8
Dependency Pairs
| plus#(X1, mark(X2)) | → | plus#(X1, X2) | | plus#(X1, active(X2)) | → | plus#(X1, X2) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | plus#(active(X1), X2) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 9
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(plus(N, s(M))) | → | U51#(isNat(M), M, N) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | active#(isNat(x(V1, V2))) | → | U31#(isNat(V1), V2) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | active#(U11(tt, V2)) | → | U12#(isNat(V2)) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | active#(x(N, 0)) | → | isNat#(N) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| active#(U72(tt, M, N)) | → | plus#(x(N, M), N) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | active#(x(N, s(M))) | → | isNat#(M) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 10
Dependency Pairs
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U11(X1, X2)) | → | mark#(X1) |
| mark#(U32(X)) | → | U32#(mark(X)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, s(M))) | → | U71#(isNat(M), M, N) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| active#(U12(tt)) | → | mark#(tt) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | active#(x(N, 0)) | → | isNat#(N) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| U32#(mark(X)) | → | U32#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | U32#(active(X)) | → | U32#(X) |
| mark#(U12(X)) | → | mark#(X) | | mark#(0) | → | active#(0) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| active#(x(N, s(M))) | → | isNat#(M) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 12
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(U61(tt)) | → | mark#(0) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | active#(U72(tt, M, N)) | → | plus#(x(N, M), N) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 13
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U32(X)) | → | U32#(mark(X)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(U21(tt)) | → | mark#(tt) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, s(M))) | → | U71#(isNat(M), M, N) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | active#(U11(tt, V2)) | → | U12#(isNat(V2)) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| U32#(mark(X)) | → | U32#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | U32#(active(X)) | → | U32#(X) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 14
Dependency Pairs
| U41#(X1, mark(X2)) | → | U41#(X1, X2) | | U41#(X1, active(X2)) | → | U41#(X1, X2) |
| U41#(mark(X1), X2) | → | U41#(X1, X2) | | U41#(active(X1), X2) | → | U41#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 17
Dependency Pairs
| U72#(X1, active(X2), X3) | → | U72#(X1, X2, X3) | | U72#(active(X1), X2, X3) | → | U72#(X1, X2, X3) |
| U72#(X1, X2, active(X3)) | → | U72#(X1, X2, X3) | | U72#(mark(X1), X2, X3) | → | U72#(X1, X2, X3) |
| U72#(X1, X2, mark(X3)) | → | U72#(X1, X2, X3) | | U72#(X1, mark(X2), X3) | → | U72#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 16
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| active#(U12(tt)) | → | mark#(tt) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(U12(X)) | → | mark#(X) |
| mark#(0) | → | active#(0) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 19
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U32(X)) | → | U32#(mark(X)) |
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| U32#(mark(X)) | → | U32#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| mark#(isNat(X)) | → | isNat#(X) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | U32#(active(X)) | → | U32#(X) |
| mark#(U12(X)) | → | mark#(X) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 21
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | U12#(isNat(V2)) |
| active#(U12(tt)) | → | mark#(tt) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) | → | mark#(X) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(U12(X)) | → | mark#(X) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 20
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| mark#(isNat(X)) | → | isNat#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| active#(U72(tt, M, N)) | → | plus#(x(N, M), N) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) | → | mark#(X) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(U12(X)) | → | mark#(X) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(U51(tt, M, N)) | → | isNat#(N) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 23
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(isNat(X)) | → | isNat#(X) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| active#(U72(tt, M, N)) | → | plus#(x(N, M), N) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) | → | mark#(X) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(U12(X)) | → | mark#(X) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(U51(tt, M, N)) | → | isNat#(N) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 25
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| mark#(isNat(X)) | → | isNat#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 24
Dependency Pairs
| U71#(X1, X2, active(X3)) | → | U71#(X1, X2, X3) | | U71#(active(X1), X2, X3) | → | U71#(X1, X2, X3) |
| U71#(X1, active(X2), X3) | → | U71#(X1, X2, X3) | | U71#(mark(X1), X2, X3) | → | U71#(X1, X2, X3) |
| U71#(X1, mark(X2), X3) | → | U71#(X1, X2, X3) | | U71#(X1, X2, mark(X3)) | → | U71#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 27
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U32(X)) | → | U32#(mark(X)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(isNat(X)) | → | isNat#(X) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(U12(X)) | → | mark#(X) |
| mark#(0) | → | active#(0) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 26
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U32(X)) | → | U32#(mark(X)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| U32#(mark(X)) | → | U32#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(isNat(X)) | → | isNat#(X) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | U32#(active(X)) | → | U32#(X) |
| mark#(U12(X)) | → | mark#(X) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 29
Dependency Pairs
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U11(X1, X2)) | → | mark#(X1) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| active#(x(N, 0)) | → | isNat#(N) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | active#(x(N, s(M))) | → | isNat#(M) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 28
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U32(X)) | → | U32#(mark(X)) |
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(U21(tt)) | → | mark#(tt) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| U32#(mark(X)) | → | U32#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(isNat(X)) | → | isNat#(X) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) | → | mark#(X) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| U32#(active(X)) | → | U32#(X) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 31
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | active#(x(N, 0)) | → | isNat#(N) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| active#(x(N, s(M))) | → | isNat#(M) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 30
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U32(X)) | → | U32#(mark(X)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| U32#(mark(X)) | → | U32#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| mark#(isNat(X)) | → | isNat#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) | → | mark#(X) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| U32#(active(X)) | → | U32#(X) | | mark#(U12(X)) | → | mark#(X) |
| mark#(0) | → | active#(0) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 34
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| active#(U72(tt, M, N)) | → | plus#(x(N, M), N) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 35
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(isNat(X)) | → | isNat#(X) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(U12(X)) | → | mark#(X) |
| mark#(0) | → | active#(0) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 32
Dependency Pairs
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U11(X1, X2)) | → | mark#(X1) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(U61(tt)) | → | mark#(0) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| active#(U52(tt, M, N)) | → | plus#(N, M) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | active#(U52(tt, M, N)) | → | s#(plus(N, M)) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| mark#(isNat(X)) | → | isNat#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 33
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) | → | mark#(X) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 38
Dependency Pairs
| U11#(X1, active(X2)) | → | U11#(X1, X2) | | U11#(active(X1), X2) | → | U11#(X1, X2) |
| U11#(mark(X1), X2) | → | U11#(X1, X2) | | U11#(X1, mark(X2)) | → | U11#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 39
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| active#(U11(tt, V2)) | → | U12#(isNat(V2)) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 36
Dependency Pairs
| U32#(active(X)) | → | U32#(X) | | mark#(U32(X)) | → | U32#(mark(X)) |
| U32#(mark(X)) | → | U32#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 37
Dependency Pairs
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U11(X1, X2)) | → | mark#(X1) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| active#(x(N, 0)) | → | isNat#(N) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| active#(x(N, s(M))) | → | isNat#(M) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 42
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U32(X)) | → | U32#(mark(X)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, s(M))) | → | U71#(isNat(M), M, N) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | active#(U11(tt, V2)) | → | U12#(isNat(V2)) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 43
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, s(M))) | → | U71#(isNat(M), M, N) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | U12#(isNat(V2)) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 40
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| active#(U52(tt, M, N)) | → | plus#(N, M) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| active#(U52(tt, M, N)) | → | s#(plus(N, M)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| mark#(isNat(X)) | → | isNat#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(U12(X)) | → | mark#(X) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 41
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U32(X)) | → | U32#(mark(X)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| active#(U11(tt, V2)) | → | U12#(isNat(V2)) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| active#(U12(tt)) | → | mark#(tt) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(isNat(X)) | → | isNat#(X) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 46
Dependency Pairs
| U31#(active(X1), X2) | → | U31#(X1, X2) | | U31#(X1, active(X2)) | → | U31#(X1, X2) |
| U31#(X1, mark(X2)) | → | U31#(X1, X2) | | U31#(mark(X1), X2) | → | U31#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 47
Dependency Pairs
| U51#(X1, active(X2), X3) | → | U51#(X1, X2, X3) | | U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) |
| U51#(X1, mark(X2), X3) | → | U51#(X1, X2, X3) | | U51#(X1, X2, active(X3)) | → | U51#(X1, X2, X3) |
| U51#(X1, X2, mark(X3)) | → | U51#(X1, X2, X3) | | U51#(active(X1), X2, X3) | → | U51#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 44
Dependency Pairs
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(X1, active(X2)) | → | x#(X1, X2) |
| x#(X1, mark(X2)) | → | x#(X1, X2) | | x#(active(X1), X2) | → | x#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 50
Dependency Pairs
| mark#(U32(X)) | → | U32#(mark(X)) | | mark#(U11(X1, X2)) | → | mark#(X1) |
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(U61(tt)) | → | mark#(0) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | active#(U52(tt, M, N)) | → | plus#(N, M) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(U32(tt)) | → | mark#(tt) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| active#(x(N, s(M))) | → | U71#(isNat(M), M, N) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | active#(x(N, 0)) | → | isNat#(N) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| active#(x(N, s(M))) | → | isNat#(M) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 49
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| active#(U52(tt, M, N)) | → | plus#(N, M) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X2) |
| active#(U12(tt)) | → | mark#(tt) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(isNat(X)) | → | isNat#(X) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Open Dependency Pair Problem 48
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U32(X)) | → | U32#(mark(X)) |
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(U21(tt)) | → | mark#(tt) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(U32(tt)) | → | mark#(tt) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(U12(tt)) | → | mark#(tt) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) | → | U32#(isNat(V2)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| U32#(mark(X)) | → | U32#(X) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | U32#(active(X)) | → | U32#(X) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, active, U11, U12, U31, U32, U21, x
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(U32(X)) | → | U32#(mark(X)) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | U61#(mark(X)) |
| U11#(mark(X1), X2) | → | U11#(X1, X2) | | U72#(mark(X1), X2, X3) | → | U72#(X1, X2, X3) |
| U72#(X1, X2, mark(X3)) | → | U72#(X1, X2, X3) | | x#(active(X1), X2) | → | x#(X1, X2) |
| x#(X1, active(X2)) | → | x#(X1, X2) | | active#(U61(tt)) | → | mark#(0) |
| mark#(U52(X1, X2, X3)) | → | U52#(mark(X1), X2, X3) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| isNat#(active(X)) | → | isNat#(X) | | U71#(X1, mark(X2), X3) | → | U71#(X1, X2, X3) |
| mark#(s(X)) | → | mark#(X) | | x#(X1, mark(X2)) | → | x#(X1, X2) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) | → | isNat#(N) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| U51#(X1, X2, active(X3)) | → | U51#(X1, X2, X3) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| U11#(active(X1), X2) | → | U11#(X1, X2) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| U71#(mark(X1), X2, X3) | → | U71#(X1, X2, X3) | | U41#(active(X1), X2) | → | U41#(X1, X2) |
| active#(isNat(x(V1, V2))) | → | U31#(isNat(V1), V2) | | active#(x(N, s(M))) | → | U71#(isNat(M), M, N) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | U11#(X1, mark(X2)) | → | U11#(X1, X2) |
| U11#(X1, active(X2)) | → | U11#(X1, X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| U72#(X1, X2, active(X3)) | → | U72#(X1, X2, X3) | | U61#(active(X)) | → | U61#(X) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | active#(plus(N, 0)) | → | U41#(isNat(N), N) |
| plus#(active(X1), X2) | → | plus#(X1, X2) | | active#(x(N, 0)) | → | isNat#(N) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| U31#(X1, active(X2)) | → | U31#(X1, X2) | | U32#(mark(X)) | → | U32#(X) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| x#(mark(X1), X2) | → | x#(X1, X2) | | U71#(active(X1), X2, X3) | → | U71#(X1, X2, X3) |
| U41#(X1, active(X2)) | → | U41#(X1, X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(isNat(s(V1))) | → | isNat#(V1) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| active#(U72(tt, M, N)) | → | x#(N, M) | | U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) |
| mark#(0) | → | active#(0) | | U52#(X1, X2, mark(X3)) | → | U52#(X1, X2, X3) |
| mark#(s(X)) | → | active#(s(mark(X))) | | U61#(mark(X)) | → | U61#(X) |
| U52#(X1, active(X2), X3) | → | U52#(X1, X2, X3) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | U52#(active(X1), X2, X3) | → | U52#(X1, X2, X3) |
| U71#(X1, X2, mark(X3)) | → | U71#(X1, X2, X3) | | active#(U51(tt, M, N)) | → | U52#(isNat(N), M, N) |
| mark#(U71(X1, X2, X3)) | → | U71#(mark(X1), X2, X3) | | active#(isNat(plus(V1, V2))) | → | U11#(isNat(V1), V2) |
| active#(x(N, s(M))) | → | isNat#(M) | | active#(plus(N, s(M))) | → | isNat#(M) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U31(tt, V2)) | → | isNat#(V2) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | x#(mark(X1), mark(X2)) | | mark#(U21(X)) | → | U21#(mark(X)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | U12#(mark(X)) | → | U12#(X) |
| mark#(s(X)) | → | s#(mark(X)) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| U52#(X1, mark(X2), X3) | → | U52#(X1, X2, X3) | | U51#(X1, active(X2), X3) | → | U51#(X1, X2, X3) |
| U52#(X1, X2, active(X3)) | → | U52#(X1, X2, X3) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(plus(N, s(M))) | → | U51#(isNat(M), M, N) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| U31#(active(X1), X2) | → | U31#(X1, X2) | | active#(U52(tt, M, N)) | → | plus#(N, M) |
| U71#(X1, active(X2), X3) | → | U71#(X1, X2, X3) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| active#(U32(tt)) | → | mark#(tt) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| active#(U52(tt, M, N)) | → | s#(plus(N, M)) | | U51#(X1, mark(X2), X3) | → | U51#(X1, X2, X3) |
| mark#(U12(X)) | → | U12#(mark(X)) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| U41#(X1, mark(X2)) | → | U41#(X1, X2) | | active#(isNat(0)) | → | mark#(tt) |
| U72#(X1, active(X2), X3) | → | U72#(X1, X2, X3) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | active#(U71(tt, M, N)) | → | isNat#(N) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(U41(X1, X2)) | → | U41#(mark(X1), X2) |
| active#(U11(tt, V2)) | → | U12#(isNat(V2)) | | mark#(x(X1, X2)) | → | mark#(X2) |
| U12#(active(X)) | → | U12#(X) | | plus#(X1, active(X2)) | → | plus#(X1, X2) |
| U51#(active(X1), X2, X3) | → | U51#(X1, X2, X3) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| U71#(X1, X2, active(X3)) | → | U71#(X1, X2, X3) | | U72#(active(X1), X2, X3) | → | U72#(X1, X2, X3) |
| mark#(isNat(X)) | → | isNat#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | isNat#(mark(X)) | → | isNat#(X) |
| U31#(X1, mark(X2)) | → | U31#(X1, X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| active#(U72(tt, M, N)) | → | plus#(x(N, M), N) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| U41#(mark(X1), X2) | → | U41#(X1, X2) | | U32#(active(X)) | → | U32#(X) |
| mark#(U12(X)) | → | mark#(X) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| active#(U11(tt, V2)) | → | isNat#(V2) | | U21#(mark(X)) | → | U21#(X) |
| U21#(active(X)) | → | U21#(X) | | active#(U71(tt, M, N)) | → | U72#(isNat(N), M, N) |
| U51#(X1, X2, mark(X3)) | → | U51#(X1, X2, X3) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(U51(tt, M, N)) | → | isNat#(N) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| U31#(mark(X1), X2) | → | U31#(X1, X2) | | s#(mark(X)) | → | s#(X) |
| mark#(U31(X1, X2)) | → | U31#(mark(X1), X2) | | mark#(U51(X1, X2, X3)) | → | U51#(mark(X1), X2, X3) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | s#(active(X)) | → | s#(X) |
| U52#(mark(X1), X2, X3) | → | U52#(X1, X2, X3) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
| U72#(X1, mark(X2), X3) | → | U72#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
The following SCCs where found
| U41#(X1, mark(X2)) → U41#(X1, X2) | U41#(mark(X1), X2) → U41#(X1, X2) |
| U41#(X1, active(X2)) → U41#(X1, X2) | U41#(active(X1), X2) → U41#(X1, X2) |
| x#(X1, active(X2)) → x#(X1, X2) | x#(mark(X1), X2) → x#(X1, X2) |
| x#(X1, mark(X2)) → x#(X1, X2) | x#(active(X1), X2) → x#(X1, X2) |
| U32#(active(X)) → U32#(X) | U32#(mark(X)) → U32#(X) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U32(X)) → U32#(mark(X)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(U21(tt)) → mark#(tt) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) → U61#(isNat(N)) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| active#(U12(tt)) → mark#(tt) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | active#(U31(tt, V2)) → U32#(isNat(V2)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | U32#(mark(X)) → U32#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(isNat(X)) → isNat#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| mark#(U61(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| U32#(active(X)) → U32#(X) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(0) → active#(0) | mark#(U12(X)) → mark#(X) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | active#(U21(tt)) → mark#(tt) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(x(N, 0)) → U61#(isNat(N)) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(x(N, s(M))) → U71#(isNat(M), M, N) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | mark#(U61(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U32(X)) → U32#(mark(X)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | active#(x(N, 0)) → U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| U32#(mark(X)) → U32#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| mark#(U61(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| U32#(active(X)) → U32#(X) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| mark#(U32(X)) → U32#(mark(X)) | active#(plus(N, 0)) → mark#(U41(isNat(N), N)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(U21(tt)) → mark#(tt) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(x(N, 0)) → U61#(isNat(N)) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(x(N, s(M))) → U71#(isNat(M), M, N) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | mark#(isNat(X)) → isNat#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U41(X1, X2)) → mark#(X1) |
| mark#(U61(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(0) → active#(0) |
| mark#(U12(X)) → mark#(X) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | active#(x(N, 0)) → U61#(isNat(N)) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | active#(x(N, s(M))) → U71#(isNat(M), M, N) |
| mark#(U12(X)) → active#(U12(mark(X))) | active#(U11(tt, V2)) → U12#(isNat(V2)) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | active#(U21(tt)) → mark#(tt) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| active#(U12(tt)) → mark#(tt) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | active#(U31(tt, V2)) → U32#(isNat(V2)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| mark#(isNat(X)) → isNat#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(x(N, 0)) → U61#(isNat(N)) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(x(N, s(M))) → U71#(isNat(M), M, N) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(isNat(X)) → isNat#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| mark#(U61(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(0) → active#(0) |
| mark#(U12(X)) → mark#(X) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| U71#(X1, X2, active(X3)) → U71#(X1, X2, X3) | U71#(active(X1), X2, X3) → U71#(X1, X2, X3) |
| U71#(mark(X1), X2, X3) → U71#(X1, X2, X3) | U71#(X1, active(X2), X3) → U71#(X1, X2, X3) |
| U71#(X1, mark(X2), X3) → U71#(X1, X2, X3) | U71#(X1, X2, mark(X3)) → U71#(X1, X2, X3) |
| mark#(U32(X)) → U32#(mark(X)) | active#(plus(N, 0)) → mark#(U41(isNat(N), N)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(U21(tt)) → mark#(tt) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | active#(x(N, 0)) → U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | active#(x(N, s(M))) → U71#(isNat(M), M, N) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| active#(U11(tt, V2)) → U12#(isNat(V2)) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| U31#(active(X1), X2) → U31#(X1, X2) | U31#(X1, active(X2)) → U31#(X1, X2) |
| U31#(mark(X1), X2) → U31#(X1, X2) | U31#(X1, mark(X2)) → U31#(X1, X2) |
| mark#(U32(X)) → U32#(mark(X)) | mark#(U11(X1, X2)) → mark#(X1) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(U21(tt)) → mark#(tt) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | active#(x(N, 0)) → U61#(isNat(N)) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | active#(x(N, s(M))) → U71#(isNat(M), M, N) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U12(tt)) → mark#(tt) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | active#(U31(tt, V2)) → U32#(isNat(V2)) |
| mark#(isNat(X)) → active#(isNat(X)) | active#(x(N, 0)) → isNat#(N) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| U32#(mark(X)) → U32#(X) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| mark#(U61(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| U32#(active(X)) → U32#(X) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(0) → active#(0) | mark#(U12(X)) → mark#(X) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| active#(x(N, s(M))) → isNat#(M) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U52(tt, M, N)) → plus#(N, M) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(isNat(X)) → isNat#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| mark#(isNat(X)) → isNat#(X) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| active#(U72(tt, M, N)) → plus#(x(N, M), N) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | mark#(U61(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(0) → active#(0) | mark#(U12(X)) → mark#(X) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | active#(U51(tt, M, N)) → isNat#(N) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U61(tt)) → mark#(0) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | active#(x(N, 0)) → U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | active#(U72(tt, M, N)) → plus#(x(N, M), N) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| U12#(active(X)) → U12#(X) | U12#(mark(X)) → U12#(X) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| mark#(isNat(X)) → isNat#(X) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| active#(U72(tt, M, N)) → plus#(x(N, M), N) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | mark#(U61(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(0) → active#(0) | mark#(U12(X)) → mark#(X) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(U51(tt, M, N)) → isNat#(N) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | active#(U21(tt)) → mark#(tt) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(x(N, 0)) → U61#(isNat(N)) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(x(N, s(M))) → U71#(isNat(M), M, N) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) → active#(isNat(X)) | active#(x(N, 0)) → isNat#(N) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| active#(x(N, s(M))) → isNat#(M) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| mark#(U32(X)) → U32#(mark(X)) | active#(plus(N, 0)) → mark#(U41(isNat(N), N)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(U21(tt)) → mark#(tt) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | active#(x(N, 0)) → U61#(isNat(N)) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| U32#(mark(X)) → U32#(X) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| mark#(isNat(X)) → isNat#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | mark#(U61(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | U32#(active(X)) → U32#(X) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(0) → active#(0) |
| mark#(U12(X)) → mark#(X) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| isNat#(active(X)) → isNat#(X) | isNat#(mark(X)) → isNat#(X) |
| mark#(U11(X1, X2)) → mark#(X1) | active#(plus(N, 0)) → mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) → U61#(isNat(N)) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(x(N, s(M))) → U71#(isNat(M), M, N) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| active#(x(N, 0)) → isNat#(N) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | active#(x(N, s(M))) → isNat#(M) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | active#(U21(tt)) → mark#(tt) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(x(N, 0)) → U61#(isNat(N)) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(x(N, s(M))) → U71#(isNat(M), M, N) | mark#(U12(X)) → active#(U12(mark(X))) |
| active#(U11(tt, V2)) → U12#(isNat(V2)) | mark#(plus(X1, X2)) → mark#(X1) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U12(tt)) → mark#(tt) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | active#(U31(tt, V2)) → U32#(isNat(V2)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U41(X1, X2)) → mark#(X1) | mark#(U61(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(0) → active#(0) | mark#(U12(X)) → mark#(X) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U32(X)) → U32#(mark(X)) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(U21(tt)) → mark#(tt) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U61(tt)) → mark#(0) |
| active#(U52(tt, M, N)) → plus#(N, M) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) → U61#(isNat(N)) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(x(N, s(M))) → U71#(isNat(M), M, N) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) → active#(isNat(X)) | active#(x(N, 0)) → isNat#(N) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| active#(x(N, s(M))) → isNat#(M) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) → U61#(isNat(N)) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| active#(U12(tt)) → mark#(tt) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | active#(U31(tt, V2)) → U32#(isNat(V2)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U52(tt, M, N)) → plus#(N, M) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(U52(tt, M, N)) → s#(plus(N, M)) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| active#(U12(tt)) → mark#(tt) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | active#(U31(tt, V2)) → U32#(isNat(V2)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| mark#(isNat(X)) → isNat#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(0) → active#(0) | mark#(U12(X)) → mark#(X) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
| U51#(X1, active(X2), X3) → U51#(X1, X2, X3) | U51#(mark(X1), X2, X3) → U51#(X1, X2, X3) |
| U51#(X1, mark(X2), X3) → U51#(X1, X2, X3) | U51#(X1, X2, active(X3)) → U51#(X1, X2, X3) |
| U51#(X1, X2, mark(X3)) → U51#(X1, X2, X3) | U51#(active(X1), X2, X3) → U51#(X1, X2, X3) |
| U32#(active(X)) → U32#(X) | mark#(U32(X)) → U32#(mark(X)) |
| U32#(mark(X)) → U32#(X) |
| mark#(U32(X)) → U32#(mark(X)) | active#(plus(N, 0)) → mark#(U41(isNat(N), N)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | active#(x(N, 0)) → U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | active#(x(N, s(M))) → U71#(isNat(M), M, N) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| active#(U11(tt, V2)) → U12#(isNat(V2)) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| U32#(mark(X)) → U32#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| mark#(U61(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| U32#(active(X)) → U32#(X) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| U11#(X1, active(X2)) → U11#(X1, X2) | U11#(active(X1), X2) → U11#(X1, X2) |
| U11#(mark(X1), X2) → U11#(X1, X2) | U11#(X1, mark(X2)) → U11#(X1, X2) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | active#(x(N, 0)) → U61#(isNat(N)) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | active#(U72(tt, M, N)) → plus#(x(N, M), N) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U32(X)) → U32#(mark(X)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(U21(tt)) → mark#(tt) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | active#(x(N, 0)) → U61#(isNat(N)) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| U32#(mark(X)) → U32#(X) | mark#(isNat(X)) → isNat#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| mark#(U61(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| U32#(active(X)) → U32#(X) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(0) → active#(0) | mark#(U12(X)) → mark#(X) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| mark#(U11(X1, X2)) → mark#(X1) | active#(plus(N, 0)) → mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) → U61#(isNat(N)) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(x(N, s(M))) → U71#(isNat(M), M, N) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| active#(x(N, 0)) → isNat#(N) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U41(X1, X2)) → mark#(X1) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| active#(x(N, s(M))) → isNat#(M) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | active#(x(N, 0)) → U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| active#(U72(tt, M, N)) → plus#(x(N, M), N) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| mark#(U32(X)) → U32#(mark(X)) | active#(plus(N, 0)) → mark#(U41(isNat(N), N)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(U21(tt)) → mark#(tt) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| U32#(mark(X)) → U32#(X) | mark#(isNat(X)) → isNat#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U41(X1, X2)) → mark#(X1) |
| mark#(U61(X)) → mark#(X) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| U32#(active(X)) → U32#(X) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(0) → active#(0) | mark#(U12(X)) → mark#(X) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| U72#(active(X1), X2, X3) → U72#(X1, X2, X3) | U72#(X1, active(X2), X3) → U72#(X1, X2, X3) |
| U72#(X1, X2, active(X3)) → U72#(X1, X2, X3) | U72#(X1, X2, mark(X3)) → U72#(X1, X2, X3) |
| U72#(mark(X1), X2, X3) → U72#(X1, X2, X3) | U72#(X1, mark(X2), X3) → U72#(X1, X2, X3) |
| plus#(X1, active(X2)) → plus#(X1, X2) | plus#(X1, mark(X2)) → plus#(X1, X2) |
| plus#(mark(X1), X2) → plus#(X1, X2) | plus#(active(X1), X2) → plus#(X1, X2) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | active#(U21(tt)) → mark#(tt) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(x(N, 0)) → U61#(isNat(N)) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(x(N, s(M))) → U71#(isNat(M), M, N) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U11(tt, V2)) → U12#(isNat(V2)) |
| active#(U12(tt)) → mark#(tt) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| U61#(mark(X)) → U61#(X) | U61#(active(X)) → U61#(X) |
| mark#(U32(X)) → U32#(mark(X)) | active#(plus(N, 0)) → mark#(U41(isNat(N), N)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(U21(tt)) → mark#(tt) |
| active#(isNat(x(V1, V2))) → isNat#(V1) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | active#(x(N, 0)) → U61#(isNat(N)) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U11(tt, V2)) → U12#(isNat(V2)) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U12(tt)) → mark#(tt) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(isNat(X)) → isNat#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U32(X)) → U32#(mark(X)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | active#(U52(tt, M, N)) → plus#(N, M) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(U52(tt, M, N)) → s#(plus(N, M)) | active#(x(N, 0)) → U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | active#(x(N, s(M))) → U71#(isNat(M), M, N) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| active#(U12(tt)) → mark#(tt) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| U52#(X1, mark(X2), X3) → U52#(X1, X2, X3) | U52#(X1, X2, mark(X3)) → U52#(X1, X2, X3) |
| U52#(X1, X2, active(X3)) → U52#(X1, X2, X3) | U52#(X1, active(X2), X3) → U52#(X1, X2, X3) |
| U52#(active(X1), X2, X3) → U52#(X1, X2, X3) | U52#(mark(X1), X2, X3) → U52#(X1, X2, X3) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(plus(N, s(M))) → U51#(isNat(M), M, N) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | active#(plus(N, 0)) → isNat#(N) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(x(N, 0)) → U61#(isNat(N)) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| active#(isNat(x(V1, V2))) → U31#(isNat(V1), V2) | active#(x(N, s(M))) → U71#(isNat(M), M, N) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| active#(U11(tt, V2)) → U12#(isNat(V2)) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) → active#(isNat(X)) | active#(x(N, 0)) → isNat#(N) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| active#(U72(tt, M, N)) → plus#(x(N, M), N) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | active#(x(N, s(M))) → isNat#(M) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(U61(tt)) → mark#(0) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(U32(tt)) → mark#(tt) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(isNat(0)) → mark#(tt) |
| active#(x(N, 0)) → mark#(U61(isNat(N))) | mark#(U72(X1, X2, X3)) → mark#(X1) |
| mark#(U12(X)) → active#(U12(mark(X))) | mark#(plus(X1, X2)) → mark#(X1) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U12(tt)) → mark#(tt) |
| active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) | active#(U31(tt, V2)) → U32#(isNat(V2)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → mark#(X2) |
| mark#(U61(X)) → active#(U61(mark(X))) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) → U11#(mark(X1), X2) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(0) → active#(0) |
| mark#(U12(X)) → mark#(X) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| mark#(U11(X1, X2)) → mark#(X1) | active#(plus(N, 0)) → mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) → mark#(0) | active#(U52(tt, M, N)) → plus#(N, M) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | active#(U52(tt, M, N)) → s#(plus(N, M)) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | active#(U12(tt)) → mark#(tt) |
| mark#(x(X1, X2)) → mark#(X2) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| active#(U31(tt, V2)) → mark#(U32(isNat(V2))) | active#(isNat(plus(V1, V2))) → isNat#(V1) |
| mark#(isNat(X)) → isNat#(X) | mark#(U21(X)) → mark#(X) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(U61(X)) → active#(U61(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X2) | active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) |
| mark#(U41(X1, X2)) → mark#(X1) | active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | active#(U52(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U12(X)) → mark#(X) | mark#(0) → active#(0) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | mark#(plus(X1, X2)) → plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) → mark#(U21(isNat(V1))) | mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) | mark#(U32(X)) → active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| U21#(mark(X)) → U21#(X) | U21#(active(X)) → U21#(X) |
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U32(X)) → U32#(mark(X)) |
| mark#(U11(X1, X2)) → mark#(X1) | mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) → active#(U21(mark(X))) | active#(isNat(x(V1, V2))) → isNat#(V1) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(U61(tt)) → mark#(0) | active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(U32(tt)) → mark#(tt) |
| mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| active#(plus(N, 0)) → isNat#(N) | mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) |
| mark#(U71(X1, X2, X3)) → mark#(X1) | active#(x(N, 0)) → U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) → mark#(X1) | active#(U11(tt, V2)) → mark#(U12(isNat(V2))) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| active#(U31(tt, V2)) → U32#(isNat(V2)) | mark#(isNat(X)) → active#(isNat(X)) |
| mark#(tt) → active#(tt) | mark#(U32(X)) → mark#(X) |
| U32#(mark(X)) → U32#(X) | mark#(isNat(X)) → isNat#(X) |
| active#(isNat(plus(V1, V2))) → isNat#(V1) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U11(X1, X2)) → U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) → mark#(X1) | mark#(U61(X)) → mark#(X) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) → mark#(N) | U32#(active(X)) → U32#(X) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(U72(X1, X2, X3)) → U72#(mark(X1), X2, X3) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U32#(active(X)) | → | U32#(X) | | U32#(mark(X)) | → | U32#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U32#(active(X)) | → | U32#(X) | | U32#(mark(X)) | → | U32#(X) |
Problem 3: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) |
| mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | | mark#(U21(X)) | → | active#(U21(mark(X))) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | active#(U21(tt)) | → | mark#(tt) |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | | active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) |
| active#(U61(tt)) | → | mark#(0) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| active#(U32(tt)) | → | mark#(tt) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | mark#(U52(X1, X2, X3)) | → | mark#(X1) |
| active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| mark#(U61(X)) | → | mark#(X) | | active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(0) | → | active#(0) |
| mark#(U12(X)) | → | mark#(X) | | mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
The following SCCs where found
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U61(tt)) → mark#(0) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(U32(tt)) → mark#(tt) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
Problem 4: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U32(X)) | → | U32#(mark(X)) |
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(U21(tt)) | → | mark#(tt) |
| active#(isNat(x(V1, V2))) | → | isNat#(V1) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U52(tt, M, N)) | → | plus#(N, M) | | active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| mark#(U31(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| active#(U32(tt)) | → | mark#(tt) | | mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) |
| active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) | | active#(plus(N, 0)) | → | isNat#(N) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(U71(X1, X2, X3)) | → | mark#(X1) |
| active#(U52(tt, M, N)) | → | s#(plus(N, M)) | | active#(x(N, 0)) | → | U61#(isNat(N)) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| active#(isNat(0)) | → | mark#(tt) | | active#(x(N, 0)) | → | mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) | → | mark#(X1) | | active#(x(N, s(M))) | → | U71#(isNat(M), M, N) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | mark#(isNat(X)) | → | active#(isNat(X)) |
| mark#(tt) | → | active#(tt) | | mark#(U32(X)) | → | mark#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | active#(isNat(plus(V1, V2))) | → | isNat#(V1) |
| mark#(U21(X)) | → | mark#(X) | | mark#(U51(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U61(X)) | → | active#(U61(mark(X))) |
| active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) | | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) |
| mark#(U41(X1, X2)) | → | mark#(X1) | | mark#(U61(X)) | → | mark#(X) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) |
| active#(U41(tt, N)) | → | mark#(N) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(U12(X)) | → | mark#(X) | | mark#(0) | → | active#(0) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) |
| active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) | | mark#(U32(X)) | → | active#(U32(mark(X))) |
| mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) | | mark#(x(X1, X2)) | → | mark#(X1) |
| mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
The following SCCs where found
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U61(tt)) → mark#(0) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(U32(tt)) → mark#(tt) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U52#(X1, mark(X2), X3) | → | U52#(X1, X2, X3) | | U52#(X1, X2, mark(X3)) | → | U52#(X1, X2, X3) |
| U52#(X1, X2, active(X3)) | → | U52#(X1, X2, X3) | | U52#(X1, active(X2), X3) | → | U52#(X1, X2, X3) |
| U52#(active(X1), X2, X3) | → | U52#(X1, X2, X3) | | U52#(mark(X1), X2, X3) | → | U52#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U52#(active(X1), X2, X3) | → | U52#(X1, X2, X3) | | U52#(mark(X1), X2, X3) | → | U52#(X1, X2, X3) |
Problem 6: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2)) | → | mark#(X1) | | mark#(U32(X)) | → | U32#(mark(X)) |
| active#(plus(N, 0)) | → | mark#(U41(isNat(N), N)) | | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) |
| mark#(U21(X)) | → | active#(U21(mark(X))) | | active#(isNat(x(V1, V2))) | → | isNat#(V1) |
| active#(U21(tt)) | → | mark#(tt) | | mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) | → | mark#(U71(isNat(M), M, N)) | | active#(U61(tt)) | → | mark#(0) |
| active#(U72(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | mark#(U31(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | active#(U32(tt)) | → | mark#(tt) |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | | active#(plus(N, s(M))) | → | mark#(U51(isNat(M), M, N)) |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(x(N, 0)) | → | U61#(isNat(N)) | | active#(U11(tt, V2)) | → | mark#(U12(isNat(V2))) |
| mark#(U52(X1, X2, X3)) | → | mark#(X1) | | active#(isNat(0)) | → | mark#(tt) |
| active#(x(N, 0)) | → | mark#(U61(isNat(N))) | | mark#(U72(X1, X2, X3)) | → | mark#(X1) |
| mark#(U12(X)) | → | active#(U12(mark(X))) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | active#(U12(tt)) | → | mark#(tt) |
| active#(U51(tt, M, N)) | → | mark#(U52(isNat(N), M, N)) | | active#(U31(tt, V2)) | → | U32#(isNat(V2)) |
| mark#(isNat(X)) | → | active#(isNat(X)) | | mark#(tt) | → | active#(tt) |
| mark#(U32(X)) | → | mark#(X) | | U32#(mark(X)) | → | U32#(X) |
| active#(U31(tt, V2)) | → | mark#(U32(isNat(V2))) | | mark#(isNat(X)) | → | isNat#(X) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | mark#(U21(X)) | → | mark#(X) |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| mark#(U61(X)) | → | active#(U61(mark(X))) | | active#(isNat(x(V1, V2))) | → | mark#(U31(isNat(V1), V2)) |
| mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | | mark#(U41(X1, X2)) | → | mark#(X1) |
| active#(isNat(plus(V1, V2))) | → | mark#(U11(isNat(V1), V2)) | | mark#(U61(X)) | → | mark#(X) |
| mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | | active#(U41(tt, N)) | → | mark#(N) |
| U32#(active(X)) | → | U32#(X) | | active#(U52(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(0) | → | active#(0) | | mark#(U12(X)) | → | mark#(X) |
| mark#(U72(X1, X2, X3)) | → | U72#(mark(X1), X2, X3) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | | active#(U71(tt, M, N)) | → | mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) | → | active#(U32(mark(X))) | | mark#(U72(X1, X2, X3)) | → | active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) | → | mark#(X1) | | mark#(U52(X1, X2, X3)) | → | active#(U52(mark(X1), X2, X3)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
The following SCCs where found
| active#(plus(N, 0)) → mark#(U41(isNat(N), N)) | mark#(U11(X1, X2)) → mark#(X1) |
| mark#(U71(X1, X2, X3)) → active#(U71(mark(X1), X2, X3)) | mark#(U21(X)) → active#(U21(mark(X))) |
| active#(U21(tt)) → mark#(tt) | mark#(U41(X1, X2)) → active#(U41(mark(X1), X2)) |
| active#(x(N, s(M))) → mark#(U71(isNat(M), M, N)) | active#(U61(tt)) → mark#(0) |
| active#(U72(tt, M, N)) → mark#(plus(x(N, M), N)) | mark#(U31(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | mark#(U11(X1, X2)) → active#(U11(mark(X1), X2)) |
| active#(U32(tt)) → mark#(tt) | active#(plus(N, s(M))) → mark#(U51(isNat(M), M, N)) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| active#(U11(tt, V2)) → mark#(U12(isNat(V2))) | mark#(U52(X1, X2, X3)) → mark#(X1) |
| active#(isNat(0)) → mark#(tt) | active#(x(N, 0)) → mark#(U61(isNat(N))) |
| mark#(U72(X1, X2, X3)) → mark#(X1) | mark#(U12(X)) → active#(U12(mark(X))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| active#(U12(tt)) → mark#(tt) | active#(U51(tt, M, N)) → mark#(U52(isNat(N), M, N)) |
| mark#(isNat(X)) → active#(isNat(X)) | mark#(tt) → active#(tt) |
| mark#(U32(X)) → mark#(X) | active#(U31(tt, V2)) → mark#(U32(isNat(V2))) |
| mark#(U21(X)) → mark#(X) | mark#(U51(X1, X2, X3)) → mark#(X1) |
| mark#(U61(X)) → active#(U61(mark(X))) | mark#(plus(X1, X2)) → mark#(X2) |
| active#(isNat(x(V1, V2))) → mark#(U31(isNat(V1), V2)) | mark#(U41(X1, X2)) → mark#(X1) |
| active#(isNat(plus(V1, V2))) → mark#(U11(isNat(V1), V2)) | mark#(U61(X)) → mark#(X) |
| mark#(U31(X1, X2)) → active#(U31(mark(X1), X2)) | active#(U41(tt, N)) → mark#(N) |
| active#(U52(tt, M, N)) → mark#(s(plus(N, M))) | mark#(U12(X)) → mark#(X) |
| mark#(0) → active#(0) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) | active#(isNat(s(V1))) → mark#(U21(isNat(V1))) |
| mark#(U51(X1, X2, X3)) → active#(U51(mark(X1), X2, X3)) | active#(U71(tt, M, N)) → mark#(U72(isNat(N), M, N)) |
| mark#(U32(X)) → active#(U32(mark(X))) | mark#(U72(X1, X2, X3)) → active#(U72(mark(X1), X2, X3)) |
| mark#(x(X1, X2)) → mark#(X1) | mark#(U52(X1, X2, X3)) → active#(U52(mark(X1), X2, X3)) |
| U32#(active(X)) → U32#(X) | U32#(mark(X)) → U32#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(X1, active(X2)) | → | plus#(X1, X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | plus#(active(X1), X2) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | plus#(active(X1), X2) | → | plus#(X1, X2) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U12#(active(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(active(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |
Problem 14: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U41#(X1, mark(X2)) | → | U41#(X1, X2) | | U41#(mark(X1), X2) | → | U41#(X1, X2) |
| U41#(X1, active(X2)) | → | U41#(X1, X2) | | U41#(active(X1), X2) | → | U41#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U41#(mark(X1), X2) | → | U41#(X1, X2) | | U41#(active(X1), X2) | → | U41#(X1, X2) |
Problem 15: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNat#(active(X)) | → | isNat#(X) | | isNat#(mark(X)) | → | isNat#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNat#(active(X)) | → | isNat#(X) | | isNat#(mark(X)) | → | isNat#(X) |
Problem 17: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U72#(active(X1), X2, X3) | → | U72#(X1, X2, X3) | | U72#(X1, active(X2), X3) | → | U72#(X1, X2, X3) |
| U72#(X1, X2, active(X3)) | → | U72#(X1, X2, X3) | | U72#(X1, X2, mark(X3)) | → | U72#(X1, X2, X3) |
| U72#(mark(X1), X2, X3) | → | U72#(X1, X2, X3) | | U72#(X1, mark(X2), X3) | → | U72#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U72#(active(X1), X2, X3) | → | U72#(X1, X2, X3) | | U72#(mark(X1), X2, X3) | → | U72#(X1, X2, X3) |
Problem 18: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U21#(mark(X)) | → | U21#(X) | | U21#(active(X)) | → | U21#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U21#(mark(X)) | → | U21#(X) | | U21#(active(X)) | → | U21#(X) |
Problem 22: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Problem 24: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U71#(X1, X2, active(X3)) | → | U71#(X1, X2, X3) | | U71#(active(X1), X2, X3) | → | U71#(X1, X2, X3) |
| U71#(mark(X1), X2, X3) | → | U71#(X1, X2, X3) | | U71#(X1, active(X2), X3) | → | U71#(X1, X2, X3) |
| U71#(X1, mark(X2), X3) | → | U71#(X1, X2, X3) | | U71#(X1, X2, mark(X3)) | → | U71#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U71#(active(X1), X2, X3) | → | U71#(X1, X2, X3) | | U71#(mark(X1), X2, X3) | → | U71#(X1, X2, X3) |
Problem 38: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U11#(X1, active(X2)) | → | U11#(X1, X2) | | U11#(active(X1), X2) | → | U11#(X1, X2) |
| U11#(mark(X1), X2) | → | U11#(X1, X2) | | U11#(X1, mark(X2)) | → | U11#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(active(X1), X2) | → | U11#(X1, X2) | | U11#(mark(X1), X2) | → | U11#(X1, X2) |
Problem 44: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| x#(X1, active(X2)) | → | x#(X1, X2) | | x#(mark(X1), X2) | → | x#(X1, X2) |
| x#(X1, mark(X2)) | → | x#(X1, X2) | | x#(active(X1), X2) | → | x#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(active(X1), X2) | → | x#(X1, X2) |
Problem 45: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U61#(mark(X)) | → | U61#(X) | | U61#(active(X)) | → | U61#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U61#(mark(X)) | → | U61#(X) | | U61#(active(X)) | → | U61#(X) |
Problem 46: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U31#(active(X1), X2) | → | U31#(X1, X2) | | U31#(X1, active(X2)) | → | U31#(X1, X2) |
| U31#(mark(X1), X2) | → | U31#(X1, X2) | | U31#(X1, mark(X2)) | → | U31#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U31#(active(X1), X2) | → | U31#(X1, X2) | | U31#(mark(X1), X2) | → | U31#(X1, X2) |
Problem 47: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U51#(X1, active(X2), X3) | → | U51#(X1, X2, X3) | | U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) |
| U51#(X1, mark(X2), X3) | → | U51#(X1, X2, X3) | | U51#(X1, X2, active(X3)) | → | U51#(X1, X2, X3) |
| U51#(X1, X2, mark(X3)) | → | U51#(X1, X2, X3) | | U51#(active(X1), X2, X3) | → | U51#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, V2)) | → | mark(U32(isNat(V2))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, N)) | → | mark(N) |
| active(U51(tt, M, N)) | → | mark(U52(isNat(N), M, N)) | | active(U52(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(U72(isNat(N), M, N)) |
| active(U72(tt, M, N)) | → | mark(plus(x(N, M), N)) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) | | active(isNat(s(V1))) | → | mark(U21(isNat(V1))) |
| active(isNat(x(V1, V2))) | → | mark(U31(isNat(V1), V2)) | | active(plus(N, 0)) | → | mark(U41(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U51(isNat(M), M, N)) | | active(x(N, 0)) | → | mark(U61(isNat(N))) |
| active(x(N, s(M))) | → | mark(U71(isNat(M), M, N)) | | mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| mark(isNat(X)) | → | active(isNat(X)) | | mark(U21(X)) | → | active(U21(mark(X))) |
| mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | | mark(U32(X)) | → | active(U32(mark(X))) |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) |
| mark(U52(X1, X2, X3)) | → | active(U52(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U61(X)) | → | active(U61(mark(X))) |
| mark(0) | → | active(0) | | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) |
| mark(U72(X1, X2, X3)) | → | active(U72(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| U11(mark(X1), X2) | → | U11(X1, X2) | | U11(X1, mark(X2)) | → | U11(X1, X2) |
| U11(active(X1), X2) | → | U11(X1, X2) | | U11(X1, active(X2)) | → | U11(X1, X2) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNat(mark(X)) | → | isNat(X) | | isNat(active(X)) | → | isNat(X) |
| U21(mark(X)) | → | U21(X) | | U21(active(X)) | → | U21(X) |
| U31(mark(X1), X2) | → | U31(X1, X2) | | U31(X1, mark(X2)) | → | U31(X1, X2) |
| U31(active(X1), X2) | → | U31(X1, X2) | | U31(X1, active(X2)) | → | U31(X1, X2) |
| U32(mark(X)) | → | U32(X) | | U32(active(X)) | → | U32(X) |
| U41(mark(X1), X2) | → | U41(X1, X2) | | U41(X1, mark(X2)) | → | U41(X1, X2) |
| U41(active(X1), X2) | → | U41(X1, X2) | | U41(X1, active(X2)) | → | U41(X1, X2) |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) |
| U52(mark(X1), X2, X3) | → | U52(X1, X2, X3) | | U52(X1, mark(X2), X3) | → | U52(X1, X2, X3) |
| U52(X1, X2, mark(X3)) | → | U52(X1, X2, X3) | | U52(active(X1), X2, X3) | → | U52(X1, X2, X3) |
| U52(X1, active(X2), X3) | → | U52(X1, X2, X3) | | U52(X1, X2, active(X3)) | → | U52(X1, X2, X3) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| plus(mark(X1), X2) | → | plus(X1, X2) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| plus(active(X1), X2) | → | plus(X1, X2) | | plus(X1, active(X2)) | → | plus(X1, X2) |
| U61(mark(X)) | → | U61(X) | | U61(active(X)) | → | U61(X) |
| U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | | U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) |
| U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | | U71(active(X1), X2, X3) | → | U71(X1, X2, X3) |
| U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | | U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) |
| U72(mark(X1), X2, X3) | → | U72(X1, X2, X3) | | U72(X1, mark(X2), X3) | → | U72(X1, X2, X3) |
| U72(X1, X2, mark(X3)) | → | U72(X1, X2, X3) | | U72(active(X1), X2, X3) | → | U72(X1, X2, X3) |
| U72(X1, active(X2), X3) | → | U72(X1, X2, X3) | | U72(X1, X2, active(X3)) | → | U72(X1, X2, X3) |
| x(mark(X1), X2) | → | x(X1, X2) | | x(X1, mark(X2)) | → | x(X1, X2) |
| x(active(X1), X2) | → | x(X1, X2) | | x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, U71, isNat, U72, U61, 0, s, U51, tt, U41, U52, U11, active, U12, U31, U32, U21, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | | U51#(active(X1), X2, X3) | → | U51#(X1, X2, X3) |