TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60043 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (48624ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| – Problem 4 was processed with processor SubtermCriterion (3ms).
| – Problem 5 was processed with processor SubtermCriterion (2ms).
| – Problem 6 was processed with processor DependencyGraph (5991ms).
| | – Problem 27 remains open; application of the following processors failed [].
| – Problem 7 was processed with processor SubtermCriterion (2ms).
| – Problem 8 was processed with processor SubtermCriterion (2ms).
| – Problem 9 was processed with processor SubtermCriterion (3ms).
| – Problem 10 was processed with processor SubtermCriterion (3ms).
| – Problem 11 was processed with processor SubtermCriterion (47ms).
| | – Problem 25 remains open; application of the following processors failed [DependencyGraph (4ms)].
| – Problem 12 was processed with processor SubtermCriterion (2ms).
| – Problem 13 was processed with processor SubtermCriterion (3ms).
| – Problem 14 was processed with processor SubtermCriterion (3ms).
| – Problem 15 was processed with processor SubtermCriterion (3ms).
| – Problem 16 was processed with processor SubtermCriterion (3ms).
| – Problem 17 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (10ms)].
| – Problem 18 was processed with processor SubtermCriterion (4ms).
| – Problem 19 was processed with processor SubtermCriterion (3ms).
| – Problem 20 was processed with processor SubtermCriterion (6ms).
| – Problem 21 was processed with processor SubtermCriterion (2ms).
| | – Problem 26 remains open; application of the following processors failed [].
| – Problem 22 was processed with processor SubtermCriterion (4ms).
| – Problem 23 was processed with processor SubtermCriterion (2ms).
| – Problem 24 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (timeout)].
The following open problems remain:
Open Dependency Pair Problem 6
Dependency Pairs
| active#(x(N, s(M))) | → | U71#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | | active#(U71(X1, X2, X3)) | → | U71#(active(X1), X2, X3) |
| active#(isNat(x(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | | active#(U32(tt, V2)) | → | U33#(isNat(V2)) |
| active#(isNat(plus(V1, V2))) | → | U11#(and(isNatKind(V1), isNatKind(V2)), V1, V2) | | active#(isNat(x(V1, V2))) | → | isNatKind#(V2) |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V1) | | active#(x(N, s(M))) | → | isNatKind#(N) |
| active#(x(N, s(M))) | → | and#(isNat(N), isNatKind(N)) | | active#(U51(X1, X2, X3)) | → | active#(X1) |
| active#(plus(N, s(M))) | → | and#(isNat(N), isNatKind(N)) | | active#(U71(X1, X2, X3)) | → | active#(X1) |
| active#(U61(X)) | → | U61#(active(X)) | | active#(x(X1, X2)) | → | x#(X1, active(X2)) |
| active#(U12(tt, V2)) | → | U13#(isNat(V2)) | | active#(U21(tt, V1)) | → | U22#(isNat(V1)) |
| active#(U11(X1, X2, X3)) | → | active#(X1) | | active#(U41(X1, X2)) | → | active#(X1) |
| active#(plus(X1, X2)) | → | active#(X1) | | active#(plus(N, s(M))) | → | isNatKind#(M) |
| active#(U31(X1, X2, X3)) | → | active#(X1) | | active#(U51(tt, M, N)) | → | s#(plus(N, M)) |
| active#(plus(X1, X2)) | → | active#(X2) | | active#(x(N, 0)) | → | U61#(and(isNat(N), isNatKind(N))) |
| active#(U33(X)) | → | active#(X) | | active#(x(N, 0)) | → | isNat#(N) |
| active#(plus(N, s(M))) | → | isNatKind#(N) | | active#(x(N, s(M))) | → | and#(isNat(M), isNatKind(M)) |
| active#(U11(X1, X2, X3)) | → | U11#(active(X1), X2, X3) | | active#(x(X1, X2)) | → | x#(active(X1), X2) |
| active#(isNat(plus(V1, V2))) | → | isNatKind#(V1) | | active#(plus(N, 0)) | → | isNatKind#(N) |
| active#(U11(tt, V1, V2)) | → | isNat#(V1) | | active#(isNat(x(V1, V2))) | → | isNatKind#(V1) |
| active#(plus(N, s(M))) | → | U51#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | | active#(isNatKind(x(V1, V2))) | → | isNatKind#(V1) |
| active#(U13(X)) | → | active#(X) | | active#(isNat(plus(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) |
| active#(x(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | | active#(isNatKind(s(V1))) | → | isNatKind#(V1) |
| active#(U21(X1, X2)) | → | U21#(active(X1), X2) | | active#(x(N, 0)) | → | isNatKind#(N) |
| active#(U12(X1, X2)) | → | active#(X1) | | active#(x(X1, X2)) | → | active#(X1) |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V2) | | active#(plus(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) |
| active#(plus(X1, X2)) | → | plus#(X1, active(X2)) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(x(N, s(M))) | → | isNat#(M) | | active#(U51(tt, M, N)) | → | plus#(N, M) |
| active#(U32(X1, X2)) | → | active#(X1) | | active#(U22(X)) | → | active#(X) |
| active#(U32(X1, X2)) | → | U32#(active(X1), X2) | | active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) |
| active#(x(N, s(M))) | → | isNat#(N) | | active#(s(X)) | → | active#(X) |
| active#(x(N, 0)) | → | and#(isNat(N), isNatKind(N)) | | active#(x(X1, X2)) | → | active#(X2) |
| active#(U61(X)) | → | active#(X) | | active#(and(X1, X2)) | → | active#(X1) |
| active#(U31(tt, V1, V2)) | → | U32#(isNat(V1), V2) | | active#(plus(N, s(M))) | → | and#(isNat(M), isNatKind(M)) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, U22, top, plus, isNatKind, mark, U71, and, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Open Dependency Pair Problem 11
Dependency Pairs
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(ok(X1), ok(X2)) | → | x#(X1, X2) |
| x#(X1, mark(X2)) | → | x#(X1, X2) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, U22, top, plus, isNatKind, mark, U71, and, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Open Dependency Pair Problem 17
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, U22, top, plus, isNatKind, mark, U71, and, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Open Dependency Pair Problem 21
Dependency Pairs
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, U22, top, plus, isNatKind, mark, U71, and, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Open Dependency Pair Problem 24
Dependency Pairs
| active#(x(N, s(M))) | → | U71#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | | active#(U71(X1, X2, X3)) | → | U71#(active(X1), X2, X3) |
| active#(isNat(x(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | | active#(U32(tt, V2)) | → | U33#(isNat(V2)) |
| active#(isNat(plus(V1, V2))) | → | U11#(and(isNatKind(V1), isNatKind(V2)), V1, V2) | | active#(isNat(x(V1, V2))) | → | isNatKind#(V2) |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V1) | | active#(x(N, s(M))) | → | and#(isNat(N), isNatKind(N)) |
| active#(U51(X1, X2, X3)) | → | active#(X1) | | active#(plus(N, s(M))) | → | and#(isNat(N), isNatKind(N)) |
| active#(U71(X1, X2, X3)) | → | active#(X1) | | active#(U61(X)) | → | U61#(active(X)) |
| active#(x(X1, X2)) | → | x#(X1, active(X2)) | | active#(U12(tt, V2)) | → | U13#(isNat(V2)) |
| active#(U21(tt, V1)) | → | U22#(isNat(V1)) | | active#(U11(X1, X2, X3)) | → | active#(X1) |
| active#(U41(X1, X2)) | → | active#(X1) | | active#(plus(X1, X2)) | → | active#(X1) |
| active#(plus(N, s(M))) | → | isNatKind#(M) | | active#(U31(X1, X2, X3)) | → | active#(X1) |
| active#(U51(tt, M, N)) | → | s#(plus(N, M)) | | active#(plus(X1, X2)) | → | active#(X2) |
| active#(x(N, 0)) | → | U61#(and(isNat(N), isNatKind(N))) | | active#(U33(X)) | → | active#(X) |
| active#(x(N, 0)) | → | isNat#(N) | | active#(plus(N, s(M))) | → | isNatKind#(N) |
| active#(x(N, s(M))) | → | and#(isNat(M), isNatKind(M)) | | active#(U11(X1, X2, X3)) | → | U11#(active(X1), X2, X3) |
| active#(x(X1, X2)) | → | x#(active(X1), X2) | | active#(isNat(plus(V1, V2))) | → | isNatKind#(V1) |
| active#(plus(N, 0)) | → | isNatKind#(N) | | active#(U11(tt, V1, V2)) | → | isNat#(V1) |
| active#(isNat(x(V1, V2))) | → | isNatKind#(V1) | | active#(plus(N, s(M))) | → | U51#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) |
| active#(isNatKind(x(V1, V2))) | → | isNatKind#(V1) | | active#(U13(X)) | → | active#(X) |
| active#(isNat(plus(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | | active#(x(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) |
| active#(isNatKind(s(V1))) | → | isNatKind#(V1) | | active#(U21(X1, X2)) | → | U21#(active(X1), X2) |
| active#(x(N, 0)) | → | isNatKind#(N) | | active#(U12(X1, X2)) | → | active#(X1) |
| active#(x(X1, X2)) | → | active#(X1) | | active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V2) |
| active#(plus(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | | active#(plus(X1, X2)) | → | plus#(X1, active(X2)) |
| active#(U21(X1, X2)) | → | active#(X1) | | active#(x(N, s(M))) | → | isNat#(M) |
| active#(U51(tt, M, N)) | → | plus#(N, M) | | active#(U32(X1, X2)) | → | active#(X1) |
| active#(U22(X)) | → | active#(X) | | active#(U32(X1, X2)) | → | U32#(active(X1), X2) |
| active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | | active#(x(N, s(M))) | → | isNat#(N) |
| active#(s(X)) | → | active#(X) | | active#(x(N, 0)) | → | and#(isNat(N), isNatKind(N)) |
| active#(x(X1, X2)) | → | active#(X2) | | active#(U61(X)) | → | active#(X) |
| active#(and(X1, X2)) | → | active#(X1) | | active#(U31(tt, V1, V2)) | → | U32#(isNat(V1), V2) |
| active#(plus(N, s(M))) | → | and#(isNat(M), isNatKind(M)) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, U22, top, plus, isNatKind, mark, U71, and, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(U71(X1, X2, X3)) | → | U71#(active(X1), X2, X3) | | proper#(U41(X1, X2)) | → | proper#(X2) |
| proper#(U41(X1, X2)) | → | proper#(X1) | | proper#(U11(X1, X2, X3)) | → | proper#(X3) |
| active#(isNat(x(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | | active#(U32(tt, V2)) | → | U33#(isNat(V2)) |
| active#(isNat(plus(V1, V2))) | → | U11#(and(isNatKind(V1), isNatKind(V2)), V1, V2) | | active#(U71(tt, M, N)) | → | x#(N, M) |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V1) | | proper#(U32(X1, X2)) | → | proper#(X2) |
| active#(x(N, s(M))) | → | isNatKind#(N) | | proper#(U31(X1, X2, X3)) | → | proper#(X2) |
| proper#(isNatKind(X)) | → | proper#(X) | | proper#(U71(X1, X2, X3)) | → | proper#(X1) |
| proper#(U71(X1, X2, X3)) | → | U71#(proper(X1), proper(X2), proper(X3)) | | active#(plus(N, 0)) | → | and#(isNat(N), isNatKind(N)) |
| active#(isNatKind(x(V1, V2))) | → | isNatKind#(V2) | | proper#(U11(X1, X2, X3)) | → | proper#(X2) |
| active#(U12(tt, V2)) | → | isNat#(V2) | | x#(ok(X1), ok(X2)) | → | x#(X1, X2) |
| x#(X1, mark(X2)) | → | x#(X1, X2) | | top#(mark(X)) | → | proper#(X) |
| active#(plus(N, 0)) | → | isNat#(N) | | isNatKind#(ok(X)) | → | isNatKind#(X) |
| active#(U71(X1, X2, X3)) | → | active#(X1) | | active#(plus(N, s(M))) | → | and#(isNat(N), isNatKind(N)) |
| U71#(mark(X1), X2, X3) | → | U71#(X1, X2, X3) | | proper#(U71(X1, X2, X3)) | → | proper#(X3) |
| isNat#(ok(X)) | → | isNat#(X) | | and#(mark(X1), X2) | → | and#(X1, X2) |
| active#(U21(tt, V1)) | → | isNat#(V1) | | active#(U12(X1, X2)) | → | U12#(active(X1), X2) |
| active#(isNat(s(V1))) | → | isNatKind#(V1) | | active#(U41(X1, X2)) | → | active#(X1) |
| proper#(U13(X)) | → | U13#(proper(X)) | | proper#(U51(X1, X2, X3)) | → | proper#(X1) |
| active#(plus(N, s(M))) | → | isNatKind#(M) | | active#(U31(X1, X2, X3)) | → | active#(X1) |
| active#(U51(tt, M, N)) | → | s#(plus(N, M)) | | active#(plus(X1, X2)) | → | active#(X2) |
| proper#(U22(X)) | → | proper#(X) | | U11#(ok(X1), ok(X2), ok(X3)) | → | U11#(X1, X2, X3) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | proper#(U51(X1, X2, X3)) | → | proper#(X2) |
| proper#(isNatKind(X)) | → | isNatKind#(proper(X)) | | U51#(ok(X1), ok(X2), ok(X3)) | → | U51#(X1, X2, X3) |
| active#(U33(X)) | → | active#(X) | | active#(x(N, 0)) | → | isNat#(N) |
| active#(plus(N, 0)) | → | U41#(and(isNat(N), isNatKind(N)), N) | | active#(x(N, s(M))) | → | and#(isNat(M), isNatKind(M)) |
| top#(ok(X)) | → | active#(X) | | active#(x(N, s(M))) | → | isNatKind#(M) |
| active#(U71(tt, M, N)) | → | plus#(x(N, M), N) | | active#(U11(tt, V1, V2)) | → | isNat#(V1) |
| active#(isNat(x(V1, V2))) | → | isNatKind#(V1) | | x#(mark(X1), X2) | → | x#(X1, X2) |
| active#(and(X1, X2)) | → | and#(active(X1), X2) | | active#(plus(N, s(M))) | → | U51#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) |
| active#(U31(X1, X2, X3)) | → | U31#(active(X1), X2, X3) | | U13#(mark(X)) | → | U13#(X) |
| proper#(isNat(X)) | → | isNat#(proper(X)) | | active#(U32(tt, V2)) | → | isNat#(V2) |
| active#(U13(X)) | → | active#(X) | | active#(U11(tt, V1, V2)) | → | U12#(isNat(V1), V2) |
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | active#(U21(X1, X2)) | → | U21#(active(X1), X2) |
| U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | | active#(x(N, 0)) | → | isNatKind#(N) |
| proper#(isNat(X)) | → | proper#(X) | | U61#(mark(X)) | → | U61#(X) |
| active#(U12(X1, X2)) | → | active#(X1) | | active#(isNat(x(V1, V2))) | → | U31#(and(isNatKind(V1), isNatKind(V2)), V1, V2) |
| proper#(U22(X)) | → | U22#(proper(X)) | | U33#(mark(X)) | → | U33#(X) |
| proper#(U31(X1, X2, X3)) | → | proper#(X1) | | active#(U31(tt, V1, V2)) | → | isNat#(V1) |
| active#(s(X)) | → | s#(active(X)) | | s#(ok(X)) | → | s#(X) |
| active#(plus(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | | proper#(U11(X1, X2, X3)) | → | proper#(X1) |
| active#(U51(tt, M, N)) | → | plus#(N, M) | | active#(x(N, s(M))) | → | isNat#(M) |
| active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | | active#(U32(X1, X2)) | → | U32#(active(X1), X2) |
| active#(U32(X1, X2)) | → | active#(X1) | | active#(U51(X1, X2, X3)) | → | U51#(active(X1), X2, X3) |
| active#(plus(N, s(M))) | → | isNat#(M) | | active#(x(N, s(M))) | → | isNat#(N) |
| proper#(s(X)) | → | s#(proper(X)) | | active#(x(N, 0)) | → | and#(isNat(N), isNatKind(N)) |
| active#(x(X1, X2)) | → | active#(X2) | | proper#(U21(X1, X2)) | → | proper#(X1) |
| active#(U31(tt, V1, V2)) | → | U32#(isNat(V1), V2) | | active#(plus(N, s(M))) | → | and#(isNat(M), isNatKind(M)) |
| active#(x(N, s(M))) | → | U71#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | | top#(ok(X)) | → | top#(active(X)) |
| active#(isNatKind(x(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | | U61#(ok(X)) | → | U61#(X) |
| U12#(ok(X1), ok(X2)) | → | U12#(X1, X2) | | active#(isNat(x(V1, V2))) | → | isNatKind#(V2) |
| proper#(U12(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | and#(proper(X1), proper(X2)) |
| active#(U41(X1, X2)) | → | U41#(active(X1), X2) | | proper#(and(X1, X2)) | → | proper#(X2) |
| plus#(X1, mark(X2)) | → | plus#(X1, X2) | | U22#(mark(X)) | → | U22#(X) |
| U12#(mark(X1), X2) | → | U12#(X1, X2) | | proper#(plus(X1, X2)) | → | proper#(X1) |
| active#(isNatKind(plus(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | | active#(x(N, s(M))) | → | and#(isNat(N), isNatKind(N)) |
| proper#(U32(X1, X2)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | plus#(proper(X1), proper(X2)) |
| U71#(ok(X1), ok(X2), ok(X3)) | → | U71#(X1, X2, X3) | | active#(U51(X1, X2, X3)) | → | active#(X1) |
| top#(mark(X)) | → | top#(proper(X)) | | U13#(ok(X)) | → | U13#(X) |
| active#(U61(X)) | → | U61#(active(X)) | | U22#(ok(X)) | → | U22#(X) |
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | | active#(x(X1, X2)) | → | x#(X1, active(X2)) |
| active#(U12(tt, V2)) | → | U13#(isNat(V2)) | | active#(U21(tt, V1)) | → | U22#(isNat(V1)) |
| active#(U11(X1, X2, X3)) | → | active#(X1) | | proper#(s(X)) | → | proper#(X) |
| active#(plus(X1, X2)) | → | active#(X1) | | U32#(ok(X1), ok(X2)) | → | U32#(X1, X2) |
| active#(U13(X)) | → | U13#(active(X)) | | active#(U33(X)) | → | U33#(active(X)) |
| proper#(U51(X1, X2, X3)) | → | U51#(proper(X1), proper(X2), proper(X3)) | | active#(x(N, 0)) | → | U61#(and(isNat(N), isNatKind(N))) |
| proper#(U32(X1, X2)) | → | U32#(proper(X1), proper(X2)) | | proper#(U12(X1, X2)) | → | U12#(proper(X1), proper(X2)) |
| U33#(ok(X)) | → | U33#(X) | | active#(plus(N, s(M))) | → | isNat#(N) |
| active#(plus(N, s(M))) | → | isNatKind#(N) | | active#(U11(X1, X2, X3)) | → | U11#(active(X1), X2, X3) |
| proper#(x(X1, X2)) | → | proper#(X1) | | and#(ok(X1), ok(X2)) | → | and#(X1, X2) |
| active#(x(X1, X2)) | → | x#(active(X1), X2) | | proper#(U61(X)) | → | U61#(proper(X)) |
| U31#(ok(X1), ok(X2), ok(X3)) | → | U31#(X1, X2, X3) | | proper#(and(X1, X2)) | → | proper#(X1) |
| active#(isNat(plus(V1, V2))) | → | isNatKind#(V2) | | active#(isNat(plus(V1, V2))) | → | isNatKind#(V1) |
| active#(plus(N, 0)) | → | isNatKind#(N) | | proper#(U13(X)) | → | proper#(X) |
| proper#(U21(X1, X2)) | → | proper#(X2) | | U41#(ok(X1), ok(X2)) | → | U41#(X1, X2) |
| proper#(U31(X1, X2, X3)) | → | U31#(proper(X1), proper(X2), proper(X3)) | | active#(U22(X)) | → | U22#(active(X)) |
| active#(isNatKind(x(V1, V2))) | → | isNatKind#(V1) | | proper#(U51(X1, X2, X3)) | → | proper#(X3) |
| U41#(mark(X1), X2) | → | U41#(X1, X2) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(x(X1, X2)) | → | x#(proper(X1), proper(X2)) | | active#(isNat(plus(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) |
| proper#(U31(X1, X2, X3)) | → | proper#(X3) | | active#(x(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) |
| active#(isNatKind(s(V1))) | → | isNatKind#(V1) | | proper#(U21(X1, X2)) | → | U21#(proper(X1), proper(X2)) |
| proper#(U12(X1, X2)) | → | proper#(X1) | | proper#(U61(X)) | → | proper#(X) |
| active#(x(X1, X2)) | → | active#(X1) | | active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V2) |
| proper#(U33(X)) | → | U33#(proper(X)) | | proper#(U41(X1, X2)) | → | U41#(proper(X1), proper(X2)) |
| proper#(x(X1, X2)) | → | proper#(X2) | | proper#(U11(X1, X2, X3)) | → | U11#(proper(X1), proper(X2), proper(X3)) |
| s#(mark(X)) | → | s#(X) | | active#(plus(X1, X2)) | → | plus#(X1, active(X2)) |
| active#(U21(X1, X2)) | → | active#(X1) | | active#(U22(X)) | → | active#(X) |
| U21#(mark(X1), X2) | → | U21#(X1, X2) | | active#(plus(X1, X2)) | → | plus#(active(X1), X2) |
| active#(s(X)) | → | active#(X) | | U32#(mark(X1), X2) | → | U32#(X1, X2) |
| active#(U61(X)) | → | active#(X) | | proper#(U71(X1, X2, X3)) | → | proper#(X2) |
| active#(and(X1, X2)) | → | active#(X1) | | U31#(mark(X1), X2, X3) | → | U31#(X1, X2, X3) |
| U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) | | proper#(U33(X)) | → | proper#(X) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
The following SCCs where found
| active#(U71(X1, X2, X3)) → U71#(active(X1), X2, X3) | active#(x(N, s(M))) → U71#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) |
| active#(isNat(x(V1, V2))) → and#(isNatKind(V1), isNatKind(V2)) | active#(U32(tt, V2)) → U33#(isNat(V2)) |
| active#(isNat(plus(V1, V2))) → U11#(and(isNatKind(V1), isNatKind(V2)), V1, V2) | active#(isNat(x(V1, V2))) → isNatKind#(V2) |
| active#(isNatKind(plus(V1, V2))) → isNatKind#(V1) | active#(x(N, s(M))) → isNatKind#(N) |
| active#(x(N, s(M))) → and#(isNat(N), isNatKind(N)) | active#(U51(X1, X2, X3)) → active#(X1) |
| active#(U71(X1, X2, X3)) → active#(X1) | active#(plus(N, s(M))) → and#(isNat(N), isNatKind(N)) |
| active#(U61(X)) → U61#(active(X)) | active#(U12(tt, V2)) → U13#(isNat(V2)) |
| active#(x(X1, X2)) → x#(X1, active(X2)) | active#(U21(tt, V1)) → U22#(isNat(V1)) |
| active#(U11(X1, X2, X3)) → active#(X1) | active#(U41(X1, X2)) → active#(X1) |
| active#(plus(N, s(M))) → isNatKind#(M) | active#(plus(X1, X2)) → active#(X1) |
| active#(U51(tt, M, N)) → s#(plus(N, M)) | active#(U31(X1, X2, X3)) → active#(X1) |
| active#(plus(X1, X2)) → active#(X2) | active#(x(N, 0)) → U61#(and(isNat(N), isNatKind(N))) |
| active#(U33(X)) → active#(X) | active#(plus(N, s(M))) → isNatKind#(N) |
| active#(x(N, 0)) → isNat#(N) | active#(U11(X1, X2, X3)) → U11#(active(X1), X2, X3) |
| active#(x(N, s(M))) → and#(isNat(M), isNatKind(M)) | active#(x(X1, X2)) → x#(active(X1), X2) |
| active#(plus(N, 0)) → isNatKind#(N) | active#(isNat(plus(V1, V2))) → isNatKind#(V1) |
| active#(isNat(x(V1, V2))) → isNatKind#(V1) | active#(U11(tt, V1, V2)) → isNat#(V1) |
| active#(plus(N, s(M))) → U51#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | active#(isNatKind(x(V1, V2))) → isNatKind#(V1) |
| active#(U13(X)) → active#(X) | active#(isNat(plus(V1, V2))) → and#(isNatKind(V1), isNatKind(V2)) |
| active#(x(N, s(M))) → and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | active#(isNatKind(s(V1))) → isNatKind#(V1) |
| active#(U21(X1, X2)) → U21#(active(X1), X2) | active#(x(N, 0)) → isNatKind#(N) |
| active#(U12(X1, X2)) → active#(X1) | active#(x(X1, X2)) → active#(X1) |
| active#(isNatKind(plus(V1, V2))) → isNatKind#(V2) | active#(plus(N, s(M))) → and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) |
| active#(plus(X1, X2)) → plus#(X1, active(X2)) | active#(U21(X1, X2)) → active#(X1) |
| active#(U51(tt, M, N)) → plus#(N, M) | active#(x(N, s(M))) → isNat#(M) |
| active#(isNat(s(V1))) → U21#(isNatKind(V1), V1) | active#(U32(X1, X2)) → U32#(active(X1), X2) |
| active#(U22(X)) → active#(X) | active#(U32(X1, X2)) → active#(X1) |
| active#(s(X)) → active#(X) | active#(x(N, s(M))) → isNat#(N) |
| active#(x(X1, X2)) → active#(X2) | active#(x(N, 0)) → and#(isNat(N), isNatKind(N)) |
| active#(U61(X)) → active#(X) | active#(and(X1, X2)) → active#(X1) |
| active#(U31(tt, V1, V2)) → U32#(isNat(V1), V2) | active#(plus(N, s(M))) → and#(isNat(M), isNatKind(M)) |
| U33#(mark(X)) → U33#(X) | U33#(ok(X)) → U33#(X) |
| isNatKind#(ok(X)) → isNatKind#(X) |
| U41#(ok(X1), ok(X2)) → U41#(X1, X2) | U41#(mark(X1), X2) → U41#(X1, X2) |
| isNat#(ok(X)) → isNat#(X) |
| active#(U71(X1, X2, X3)) → U71#(active(X1), X2, X3) | active#(x(N, s(M))) → U71#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) |
| active#(isNat(x(V1, V2))) → and#(isNatKind(V1), isNatKind(V2)) | active#(U32(tt, V2)) → U33#(isNat(V2)) |
| active#(isNat(plus(V1, V2))) → U11#(and(isNatKind(V1), isNatKind(V2)), V1, V2) | active#(isNat(x(V1, V2))) → isNatKind#(V2) |
| active#(isNatKind(plus(V1, V2))) → isNatKind#(V1) | active#(x(N, s(M))) → and#(isNat(N), isNatKind(N)) |
| active#(U51(X1, X2, X3)) → active#(X1) | active#(U71(X1, X2, X3)) → active#(X1) |
| active#(plus(N, s(M))) → and#(isNat(N), isNatKind(N)) | active#(U61(X)) → U61#(active(X)) |
| active#(U12(tt, V2)) → U13#(isNat(V2)) | active#(x(X1, X2)) → x#(X1, active(X2)) |
| active#(U21(tt, V1)) → U22#(isNat(V1)) | active#(U11(X1, X2, X3)) → active#(X1) |
| active#(U41(X1, X2)) → active#(X1) | active#(plus(N, s(M))) → isNatKind#(M) |
| active#(plus(X1, X2)) → active#(X1) | active#(U51(tt, M, N)) → s#(plus(N, M)) |
| active#(U31(X1, X2, X3)) → active#(X1) | active#(plus(X1, X2)) → active#(X2) |
| active#(x(N, 0)) → U61#(and(isNat(N), isNatKind(N))) | active#(U33(X)) → active#(X) |
| active#(plus(N, s(M))) → isNatKind#(N) | active#(x(N, 0)) → isNat#(N) |
| active#(U11(X1, X2, X3)) → U11#(active(X1), X2, X3) | active#(x(N, s(M))) → and#(isNat(M), isNatKind(M)) |
| active#(x(X1, X2)) → x#(active(X1), X2) | active#(plus(N, 0)) → isNatKind#(N) |
| active#(isNat(plus(V1, V2))) → isNatKind#(V1) | active#(isNat(x(V1, V2))) → isNatKind#(V1) |
| active#(U11(tt, V1, V2)) → isNat#(V1) | active#(plus(N, s(M))) → U51#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) |
| active#(isNatKind(x(V1, V2))) → isNatKind#(V1) | active#(U13(X)) → active#(X) |
| active#(isNat(plus(V1, V2))) → and#(isNatKind(V1), isNatKind(V2)) | active#(x(N, s(M))) → and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) |
| active#(isNatKind(s(V1))) → isNatKind#(V1) | active#(U21(X1, X2)) → U21#(active(X1), X2) |
| active#(x(N, 0)) → isNatKind#(N) | active#(U12(X1, X2)) → active#(X1) |
| active#(x(X1, X2)) → active#(X1) | active#(isNatKind(plus(V1, V2))) → isNatKind#(V2) |
| active#(plus(N, s(M))) → and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | active#(plus(X1, X2)) → plus#(X1, active(X2)) |
| active#(U21(X1, X2)) → active#(X1) | active#(U51(tt, M, N)) → plus#(N, M) |
| active#(x(N, s(M))) → isNat#(M) | active#(isNat(s(V1))) → U21#(isNatKind(V1), V1) |
| active#(U32(X1, X2)) → U32#(active(X1), X2) | active#(U22(X)) → active#(X) |
| active#(U32(X1, X2)) → active#(X1) | active#(s(X)) → active#(X) |
| active#(x(N, s(M))) → isNat#(N) | active#(x(X1, X2)) → active#(X2) |
| active#(x(N, 0)) → and#(isNat(N), isNatKind(N)) | active#(U61(X)) → active#(X) |
| active#(and(X1, X2)) → active#(X1) | active#(U31(tt, V1, V2)) → U32#(isNat(V1), V2) |
| active#(plus(N, s(M))) → and#(isNat(M), isNatKind(M)) |
| U11#(mark(X1), X2, X3) → U11#(X1, X2, X3) | U11#(ok(X1), ok(X2), ok(X3)) → U11#(X1, X2, X3) |
| U61#(mark(X)) → U61#(X) | U61#(ok(X)) → U61#(X) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| active#(U51(X1, X2, X3)) → active#(X1) | active#(U33(X)) → active#(X) |
| active#(U12(X1, X2)) → active#(X1) | active#(U71(X1, X2, X3)) → active#(X1) |
| active#(x(X1, X2)) → active#(X1) | active#(U11(X1, X2, X3)) → active#(X1) |
| active#(U21(X1, X2)) → active#(X1) | active#(U41(X1, X2)) → active#(X1) |
| active#(plus(X1, X2)) → active#(X1) | active#(U22(X)) → active#(X) |
| active#(U32(X1, X2)) → active#(X1) | active#(s(X)) → active#(X) |
| active#(U31(X1, X2, X3)) → active#(X1) | active#(x(X1, X2)) → active#(X2) |
| active#(U61(X)) → active#(X) | active#(U13(X)) → active#(X) |
| active#(plus(X1, X2)) → active#(X2) | active#(and(X1, X2)) → active#(X1) |
| U32#(ok(X1), ok(X2)) → U32#(X1, X2) | U32#(mark(X1), X2) → U32#(X1, X2) |
| plus#(ok(X1), ok(X2)) → plus#(X1, X2) | plus#(X1, mark(X2)) → plus#(X1, X2) |
| plus#(mark(X1), X2) → plus#(X1, X2) |
| U21#(mark(X1), X2) → U21#(X1, X2) | U21#(ok(X1), ok(X2)) → U21#(X1, X2) |
| U71#(ok(X1), ok(X2), ok(X3)) → U71#(X1, X2, X3) | U71#(mark(X1), X2, X3) → U71#(X1, X2, X3) |
| U31#(ok(X1), ok(X2), ok(X3)) → U31#(X1, X2, X3) | U31#(mark(X1), X2, X3) → U31#(X1, X2, X3) |
| U12#(ok(X1), ok(X2)) → U12#(X1, X2) | U12#(mark(X1), X2) → U12#(X1, X2) |
| U13#(ok(X)) → U13#(X) | U13#(mark(X)) → U13#(X) |
| x#(mark(X1), X2) → x#(X1, X2) | x#(ok(X1), ok(X2)) → x#(X1, X2) |
| x#(X1, mark(X2)) → x#(X1, X2) |
| proper#(U51(X1, X2, X3)) → proper#(X2) | proper#(U41(X1, X2)) → proper#(X2) |
| proper#(U41(X1, X2)) → proper#(X1) | proper#(U11(X1, X2, X3)) → proper#(X3) |
| proper#(x(X1, X2)) → proper#(X1) | proper#(U12(X1, X2)) → proper#(X2) |
| proper#(and(X1, X2)) → proper#(X1) | proper#(U13(X)) → proper#(X) |
| proper#(U21(X1, X2)) → proper#(X2) | proper#(U32(X1, X2)) → proper#(X2) |
| proper#(U31(X1, X2, X3)) → proper#(X2) | proper#(isNatKind(X)) → proper#(X) |
| proper#(U71(X1, X2, X3)) → proper#(X1) | proper#(and(X1, X2)) → proper#(X2) |
| proper#(U11(X1, X2, X3)) → proper#(X2) | proper#(plus(X1, X2)) → proper#(X1) |
| proper#(U51(X1, X2, X3)) → proper#(X3) | proper#(plus(X1, X2)) → proper#(X2) |
| proper#(U31(X1, X2, X3)) → proper#(X3) | proper#(U32(X1, X2)) → proper#(X1) |
| proper#(U12(X1, X2)) → proper#(X1) | proper#(isNat(X)) → proper#(X) |
| proper#(U61(X)) → proper#(X) | proper#(U71(X1, X2, X3)) → proper#(X3) |
| proper#(U31(X1, X2, X3)) → proper#(X1) | proper#(x(X1, X2)) → proper#(X2) |
| proper#(U11(X1, X2, X3)) → proper#(X1) | proper#(s(X)) → proper#(X) |
| proper#(U51(X1, X2, X3)) → proper#(X1) | proper#(U71(X1, X2, X3)) → proper#(X2) |
| proper#(U21(X1, X2)) → proper#(X1) | proper#(U22(X)) → proper#(X) |
| proper#(U33(X)) → proper#(X) |
| U51#(mark(X1), X2, X3) → U51#(X1, X2, X3) | U51#(ok(X1), ok(X2), ok(X3)) → U51#(X1, X2, X3) |
| and#(ok(X1), ok(X2)) → and#(X1, X2) | and#(mark(X1), X2) → and#(X1, X2) |
| U22#(ok(X)) → U22#(X) | U22#(mark(X)) → U22#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U71#(ok(X1), ok(X2), ok(X3)) | → | U71#(X1, X2, X3) | | U71#(mark(X1), X2, X3) | → | U71#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U71#(ok(X1), ok(X2), ok(X3)) | → | U71#(X1, X2, X3) | | U71#(mark(X1), X2, X3) | → | U71#(X1, X2, X3) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U33#(mark(X)) | → | U33#(X) | | U33#(ok(X)) | → | U33#(X) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U33#(mark(X)) | → | U33#(X) | | U33#(ok(X)) | → | U33#(X) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U61#(mark(X)) | → | U61#(X) | | U61#(ok(X)) | → | U61#(X) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U61#(mark(X)) | → | U61#(X) | | U61#(ok(X)) | → | U61#(X) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U22#(ok(X)) | → | U22#(X) | | U22#(mark(X)) | → | U22#(X) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U22#(ok(X)) | → | U22#(X) | | U22#(mark(X)) | → | U22#(X) |
Problem 6: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(x(N, s(M))) | → | U71#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | | active#(U71(X1, X2, X3)) | → | U71#(active(X1), X2, X3) |
| active#(isNat(x(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | | active#(U32(tt, V2)) | → | U33#(isNat(V2)) |
| active#(isNat(plus(V1, V2))) | → | U11#(and(isNatKind(V1), isNatKind(V2)), V1, V2) | | active#(isNat(x(V1, V2))) | → | isNatKind#(V2) |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V1) | | active#(x(N, s(M))) | → | isNatKind#(N) |
| active#(x(N, s(M))) | → | and#(isNat(N), isNatKind(N)) | | active#(U51(X1, X2, X3)) | → | active#(X1) |
| active#(plus(N, s(M))) | → | and#(isNat(N), isNatKind(N)) | | active#(U71(X1, X2, X3)) | → | active#(X1) |
| active#(U61(X)) | → | U61#(active(X)) | | active#(U12(tt, V2)) | → | U13#(isNat(V2)) |
| active#(x(X1, X2)) | → | x#(X1, active(X2)) | | active#(U21(tt, V1)) | → | U22#(isNat(V1)) |
| active#(U11(X1, X2, X3)) | → | active#(X1) | | active#(U41(X1, X2)) | → | active#(X1) |
| active#(plus(N, s(M))) | → | isNatKind#(M) | | active#(plus(X1, X2)) | → | active#(X1) |
| active#(U51(tt, M, N)) | → | s#(plus(N, M)) | | active#(U31(X1, X2, X3)) | → | active#(X1) |
| active#(plus(X1, X2)) | → | active#(X2) | | active#(x(N, 0)) | → | U61#(and(isNat(N), isNatKind(N))) |
| active#(U33(X)) | → | active#(X) | | active#(x(N, 0)) | → | isNat#(N) |
| active#(plus(N, s(M))) | → | isNatKind#(N) | | active#(x(N, s(M))) | → | and#(isNat(M), isNatKind(M)) |
| active#(U11(X1, X2, X3)) | → | U11#(active(X1), X2, X3) | | active#(x(X1, X2)) | → | x#(active(X1), X2) |
| active#(isNat(plus(V1, V2))) | → | isNatKind#(V1) | | active#(plus(N, 0)) | → | isNatKind#(N) |
| active#(U11(tt, V1, V2)) | → | isNat#(V1) | | active#(isNat(x(V1, V2))) | → | isNatKind#(V1) |
| active#(plus(N, s(M))) | → | U51#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | | active#(isNatKind(x(V1, V2))) | → | isNatKind#(V1) |
| active#(U13(X)) | → | active#(X) | | active#(isNat(plus(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) |
| active#(x(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | | active#(isNatKind(s(V1))) | → | isNatKind#(V1) |
| active#(U21(X1, X2)) | → | U21#(active(X1), X2) | | active#(x(N, 0)) | → | isNatKind#(N) |
| active#(U12(X1, X2)) | → | active#(X1) | | active#(x(X1, X2)) | → | active#(X1) |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V2) | | active#(plus(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) |
| active#(plus(X1, X2)) | → | plus#(X1, active(X2)) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(U51(tt, M, N)) | → | plus#(N, M) | | active#(x(N, s(M))) | → | isNat#(M) |
| active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | | active#(U32(X1, X2)) | → | U32#(active(X1), X2) |
| active#(U22(X)) | → | active#(X) | | active#(U32(X1, X2)) | → | active#(X1) |
| active#(s(X)) | → | active#(X) | | active#(x(N, s(M))) | → | isNat#(N) |
| active#(x(X1, X2)) | → | active#(X2) | | active#(x(N, 0)) | → | and#(isNat(N), isNatKind(N)) |
| active#(U61(X)) | → | active#(X) | | active#(and(X1, X2)) | → | active#(X1) |
| active#(U31(tt, V1, V2)) | → | U32#(isNat(V1), V2) | | active#(plus(N, s(M))) | → | and#(isNat(M), isNatKind(M)) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
The following SCCs where found
| active#(U51(X1, X2, X3)) → active#(X1) | active#(U33(X)) → active#(X) |
| active#(U12(X1, X2)) → active#(X1) | active#(U71(X1, X2, X3)) → active#(X1) |
| active#(x(X1, X2)) → active#(X1) | active#(U11(X1, X2, X3)) → active#(X1) |
| active#(U41(X1, X2)) → active#(X1) | active#(U21(X1, X2)) → active#(X1) |
| active#(plus(X1, X2)) → active#(X1) | active#(U32(X1, X2)) → active#(X1) |
| active#(U22(X)) → active#(X) | active#(s(X)) → active#(X) |
| active#(U31(X1, X2, X3)) → active#(X1) | active#(x(X1, X2)) → active#(X2) |
| active#(U61(X)) → active#(X) | active#(U13(X)) → active#(X) |
| active#(plus(X1, X2)) → active#(X2) | active#(and(X1, X2)) → active#(X1) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNatKind#(ok(X)) | → | isNatKind#(X) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatKind#(ok(X)) | → | isNatKind#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U32#(ok(X1), ok(X2)) | → | U32#(X1, X2) | | U32#(mark(X1), X2) | → | U32#(X1, X2) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U32#(ok(X1), ok(X2)) | → | U32#(X1, X2) | | U32#(mark(X1), X2) | → | U32#(X1, X2) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U31#(ok(X1), ok(X2), ok(X3)) | → | U31#(X1, X2, X3) | | U31#(mark(X1), X2, X3) | → | U31#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U31#(ok(X1), ok(X2), ok(X3)) | → | U31#(X1, X2, X3) | | U31#(mark(X1), X2, X3) | → | U31#(X1, X2, X3) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | | U11#(ok(X1), ok(X2), ok(X3)) | → | U11#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | | U11#(ok(X1), ok(X2), ok(X3)) | → | U11#(X1, X2, X3) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(ok(X1), ok(X2)) | → | x#(X1, X2) |
| x#(X1, mark(X2)) | → | x#(X1, X2) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(ok(X1), ok(X2)) | → | x#(X1, X2) |
Problem 12: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U13#(ok(X)) | → | U13#(X) | | U13#(mark(X)) | → | U13#(X) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U13#(ok(X)) | → | U13#(X) | | U13#(mark(X)) | → | U13#(X) |
Problem 13: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U12#(ok(X1), ok(X2)) | → | U12#(X1, X2) | | U12#(mark(X1), X2) | → | U12#(X1, X2) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(ok(X1), ok(X2)) | → | U12#(X1, X2) | | U12#(mark(X1), X2) | → | U12#(X1, X2) |
Problem 14: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Problem 15: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| and#(ok(X1), ok(X2)) | → | and#(X1, X2) | | and#(mark(X1), X2) | → | and#(X1, X2) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| and#(ok(X1), ok(X2)) | → | and#(X1, X2) | | and#(mark(X1), X2) | → | and#(X1, X2) |
Problem 16: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U21#(mark(X1), X2) | → | U21#(X1, X2) | | U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U21#(mark(X1), X2) | → | U21#(X1, X2) | | U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) |
Problem 18: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | | U51#(ok(X1), ok(X2), ok(X3)) | → | U51#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, 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#(ok(X1), ok(X2), ok(X3)) | → | U51#(X1, X2, X3) |
Problem 19: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(U51(X1, X2, X3)) | → | active#(X1) | | active#(U33(X)) | → | active#(X) |
| active#(U12(X1, X2)) | → | active#(X1) | | active#(U71(X1, X2, X3)) | → | active#(X1) |
| active#(x(X1, X2)) | → | active#(X1) | | active#(U11(X1, X2, X3)) | → | active#(X1) |
| active#(U41(X1, X2)) | → | active#(X1) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(plus(X1, X2)) | → | active#(X1) | | active#(U32(X1, X2)) | → | active#(X1) |
| active#(U22(X)) | → | active#(X) | | active#(s(X)) | → | active#(X) |
| active#(U31(X1, X2, X3)) | → | active#(X1) | | active#(x(X1, X2)) | → | active#(X2) |
| active#(U61(X)) | → | active#(X) | | active#(U13(X)) | → | active#(X) |
| active#(plus(X1, X2)) | → | active#(X2) | | active#(and(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(U51(X1, X2, X3)) | → | active#(X1) | | active#(U33(X)) | → | active#(X) |
| active#(U12(X1, X2)) | → | active#(X1) | | active#(U71(X1, X2, X3)) | → | active#(X1) |
| active#(x(X1, X2)) | → | active#(X1) | | active#(U11(X1, X2, X3)) | → | active#(X1) |
| active#(U41(X1, X2)) | → | active#(X1) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(plus(X1, X2)) | → | active#(X1) | | active#(U22(X)) | → | active#(X) |
| active#(U32(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(U31(X1, X2, X3)) | → | active#(X1) | | active#(x(X1, X2)) | → | active#(X2) |
| active#(U61(X)) | → | active#(X) | | active#(plus(X1, X2)) | → | active#(X2) |
| active#(U13(X)) | → | active#(X) | | active#(and(X1, X2)) | → | active#(X1) |
Problem 20: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(U51(X1, X2, X3)) | → | proper#(X2) | | proper#(U41(X1, X2)) | → | proper#(X2) |
| proper#(U41(X1, X2)) | → | proper#(X1) | | proper#(U11(X1, X2, X3)) | → | proper#(X3) |
| proper#(x(X1, X2)) | → | proper#(X1) | | proper#(U12(X1, X2)) | → | proper#(X2) |
| proper#(and(X1, X2)) | → | proper#(X1) | | proper#(U13(X)) | → | proper#(X) |
| proper#(U21(X1, X2)) | → | proper#(X2) | | proper#(U32(X1, X2)) | → | proper#(X2) |
| proper#(U31(X1, X2, X3)) | → | proper#(X2) | | proper#(isNatKind(X)) | → | proper#(X) |
| proper#(U71(X1, X2, X3)) | → | proper#(X1) | | proper#(and(X1, X2)) | → | proper#(X2) |
| proper#(U11(X1, X2, X3)) | → | proper#(X2) | | proper#(U51(X1, X2, X3)) | → | proper#(X3) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(U31(X1, X2, X3)) | → | proper#(X3) | | proper#(U32(X1, X2)) | → | proper#(X1) |
| proper#(isNat(X)) | → | proper#(X) | | proper#(U12(X1, X2)) | → | proper#(X1) |
| proper#(U61(X)) | → | proper#(X) | | proper#(U71(X1, X2, X3)) | → | proper#(X3) |
| proper#(U31(X1, X2, X3)) | → | proper#(X1) | | proper#(x(X1, X2)) | → | proper#(X2) |
| proper#(U11(X1, X2, X3)) | → | proper#(X1) | | proper#(s(X)) | → | proper#(X) |
| proper#(U51(X1, X2, X3)) | → | proper#(X1) | | proper#(U71(X1, X2, X3)) | → | proper#(X2) |
| proper#(U21(X1, X2)) | → | proper#(X1) | | proper#(U22(X)) | → | proper#(X) |
| proper#(U33(X)) | → | proper#(X) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(U41(X1, X2)) | → | proper#(X2) | | proper#(U51(X1, X2, X3)) | → | proper#(X2) |
| proper#(U41(X1, X2)) | → | proper#(X1) | | proper#(U11(X1, X2, X3)) | → | proper#(X3) |
| proper#(U12(X1, X2)) | → | proper#(X2) | | proper#(x(X1, X2)) | → | proper#(X1) |
| proper#(and(X1, X2)) | → | proper#(X1) | | proper#(U13(X)) | → | proper#(X) |
| proper#(U21(X1, X2)) | → | proper#(X2) | | proper#(U32(X1, X2)) | → | proper#(X2) |
| proper#(U31(X1, X2, X3)) | → | proper#(X2) | | proper#(isNatKind(X)) | → | proper#(X) |
| proper#(U71(X1, X2, X3)) | → | proper#(X1) | | proper#(and(X1, X2)) | → | proper#(X2) |
| proper#(U11(X1, X2, X3)) | → | proper#(X2) | | proper#(U51(X1, X2, X3)) | → | proper#(X3) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(U31(X1, X2, X3)) | → | proper#(X3) | | proper#(U32(X1, X2)) | → | proper#(X1) |
| proper#(isNat(X)) | → | proper#(X) | | proper#(U12(X1, X2)) | → | proper#(X1) |
| proper#(U61(X)) | → | proper#(X) | | proper#(U71(X1, X2, X3)) | → | proper#(X3) |
| proper#(U31(X1, X2, X3)) | → | proper#(X1) | | proper#(x(X1, X2)) | → | proper#(X2) |
| proper#(U11(X1, X2, X3)) | → | proper#(X1) | | proper#(s(X)) | → | proper#(X) |
| proper#(U51(X1, X2, X3)) | → | proper#(X1) | | proper#(U71(X1, X2, X3)) | → | proper#(X2) |
| proper#(U21(X1, X2)) | → | proper#(X1) | | proper#(U22(X)) | → | proper#(X) |
| proper#(U33(X)) | → | proper#(X) |
Problem 21: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | plus#(mark(X1), X2) | → | plus#(X1, X2) |
Problem 22: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNat#(ok(X)) | → | isNat#(X) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNat#(ok(X)) | → | isNat#(X) |
Problem 23: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U41#(ok(X1), ok(X2)) | → | U41#(X1, X2) | | U41#(mark(X1), X2) | → | U41#(X1, X2) |
Rewrite Rules
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) |
| active(U13(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | | active(U33(tt)) | → | mark(tt) |
| active(U41(tt, N)) | → | mark(N) | | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U61(tt)) | → | mark(0) | | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | | active(U13(X)) | → | U13(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | | active(U32(X1, X2)) | → | U32(active(X1), X2) |
| active(U33(X)) | → | U33(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(U61(X)) | → | U61(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(x(X1, X2)) | → | x(active(X1), X2) | | active(x(X1, X2)) | → | x(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | | U32(mark(X1), X2) | → | mark(U32(X1, X2)) |
| U33(mark(X)) | → | mark(U33(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U61(mark(X)) | → | mark(U61(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) |
| proper(tt) | → | ok(tt) | | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U13(X)) | → | U13(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) |
| proper(U33(X)) | → | U33(proper(X)) | | proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U61(X)) | → | U61(proper(X)) |
| proper(0) | → | ok(0) | | proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatKind(X)) | → | isNatKind(proper(X)) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U13(ok(X)) | → | ok(U13(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) |
| U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | | U33(ok(X)) | → | ok(U33(X)) |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| U61(ok(X)) | → | ok(U61(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U61, U41, ok, proper, U21, top, U22, plus, isNatKind, mark, and, U71, 0, s, U51, tt, U11, active, U12, U31, U13, U32, U33, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U41#(ok(X1), ok(X2)) | → | U41#(X1, X2) | | U41#(mark(X1), X2) | → | U41#(X1, X2) |