TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60001 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (12291ms).
| – Problem 2 was processed with processor SubtermCriterion (5ms).
| | – Problem 17 was processed with processor PolynomialLinearRange4iUR (117ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (2ms).
| – Problem 5 was processed with processor SubtermCriterion (3ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| – Problem 9 was processed with processor SubtermCriterion (3ms).
| – Problem 10 was processed with processor SubtermCriterion (2ms).
| – Problem 11 was processed with processor SubtermCriterion (1ms).
| – Problem 12 was processed with processor SubtermCriterion (1ms).
| – Problem 13 was processed with processor SubtermCriterion (7ms).
| – Problem 14 was processed with processor SubtermCriterion (1ms).
| – Problem 15 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (5016ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (timeout), DependencyGraph (6ms), PolynomialLinearRange8NegiUR (30034ms), DependencyGraph (timeout), ReductionPairSAT (timeout)].
| – Problem 16 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 15
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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, proper, U13, U31, U21, top, U22
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| proper#(U11(X1, X2, X3)) | → | proper#(X3) | | active#(isNat(plus(V1, V2))) | → | U11#(and(isNatKind(V1), isNatKind(V2)), V1, V2) |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V1) | | proper#(U41(X1, X2, X3)) | → | proper#(X2) |
| proper#(isNatKind(X)) | → | proper#(X) | | active#(plus(N, 0)) | → | and#(isNat(N), isNatKind(N)) |
| proper#(U11(X1, X2, X3)) | → | proper#(X2) | | active#(U12(tt, V2)) | → | isNat#(V2) |
| active#(plus(N, 0)) | → | isNat#(N) | | top#(mark(X)) | → | proper#(X) |
| isNatKind#(ok(X)) | → | isNatKind#(X) | | active#(plus(N, s(M))) | → | and#(isNat(N), isNatKind(N)) |
| isNat#(ok(X)) | → | isNat#(X) | | and#(mark(X1), X2) | → | and#(X1, X2) |
| active#(U21(tt, V1)) | → | isNat#(V1) | | active#(U41(X1, X2, X3)) | → | active#(X1) |
| active#(U12(X1, X2)) | → | U12#(active(X1), X2) | | active#(isNat(s(V1))) | → | isNatKind#(V1) |
| proper#(U13(X)) | → | U13#(proper(X)) | | active#(plus(N, s(M))) | → | isNatKind#(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#(isNatKind(X)) | → | isNatKind#(proper(X)) | | active#(U41(X1, X2, X3)) | → | U41#(active(X1), X2, X3) |
| top#(ok(X)) | → | active#(X) | | active#(U11(tt, V1, V2)) | → | isNat#(V1) |
| active#(and(X1, X2)) | → | and#(active(X1), X2) | | proper#(U41(X1, X2, X3)) | → | proper#(X1) |
| U13#(mark(X)) | → | U13#(X) | | active#(U31(X1, X2)) | → | active#(X1) |
| proper#(isNat(X)) | → | isNat#(proper(X)) | | active#(U13(X)) | → | active#(X) |
| active#(U11(tt, V1, V2)) | → | U12#(isNat(V1), V2) | | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) |
| active#(U21(X1, X2)) | → | U21#(active(X1), X2) | | plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) |
| proper#(isNat(X)) | → | proper#(X) | | active#(U12(X1, X2)) | → | active#(X1) |
| proper#(U22(X)) | → | U22#(proper(X)) | | active#(U41(tt, M, N)) | → | s#(plus(N, M)) |
| 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#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | | active#(plus(N, s(M))) | → | isNat#(M) |
| proper#(s(X)) | → | s#(proper(X)) | | proper#(U31(X1, X2)) | → | U31#(proper(X1), proper(X2)) |
| proper#(U21(X1, X2)) | → | proper#(X1) | | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) |
| active#(plus(N, s(M))) | → | and#(isNat(M), isNatKind(M)) | | top#(ok(X)) | → | top#(active(X)) |
| proper#(U31(X1, X2)) | → | proper#(X1) | | U12#(ok(X1), ok(X2)) | → | U12#(X1, X2) |
| proper#(U12(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | and#(proper(X1), proper(X2)) |
| proper#(and(X1, X2)) | → | proper#(X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| U12#(mark(X1), X2) | → | U12#(X1, X2) | | U22#(mark(X)) | → | U22#(X) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | active#(isNatKind(plus(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) |
| proper#(U41(X1, X2, X3)) | → | U41#(proper(X1), proper(X2), proper(X3)) | | proper#(plus(X1, X2)) | → | plus#(proper(X1), proper(X2)) |
| active#(U41(tt, M, N)) | → | plus#(N, M) | | top#(mark(X)) | → | top#(proper(X)) |
| U13#(ok(X)) | → | U13#(X) | | U22#(ok(X)) | → | U22#(X) |
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | | 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) |
| active#(U13(X)) | → | U13#(active(X)) | | active#(plus(N, s(M))) | → | U41#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) |
| proper#(U12(X1, X2)) | → | U12#(proper(X1), proper(X2)) | | active#(plus(N, s(M))) | → | isNat#(N) |
| active#(plus(N, s(M))) | → | isNatKind#(N) | | proper#(U41(X1, X2, X3)) | → | proper#(X3) |
| proper#(U31(X1, X2)) | → | proper#(X2) | | and#(ok(X1), ok(X2)) | → | and#(X1, X2) |
| active#(U11(X1, X2, X3)) | → | U11#(active(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) | | active#(U31(X1, X2)) | → | U31#(active(X1), X2) |
| active#(U22(X)) | → | U22#(active(X)) | | U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) |
| proper#(plus(X1, X2)) | → | proper#(X2) | | active#(plus(N, 0)) | → | U31#(and(isNat(N), isNatKind(N)), N) |
| active#(isNat(plus(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | | active#(isNatKind(s(V1))) | → | isNatKind#(V1) |
| proper#(U21(X1, X2)) | → | U21#(proper(X1), proper(X2)) | | proper#(U12(X1, X2)) | → | proper#(X1) |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V2) | | U31#(mark(X1), X2) | → | U31#(X1, 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#(s(X)) | → | active#(X) | | active#(plus(X1, X2)) | → | plus#(active(X1), X2) |
| active#(and(X1, X2)) | → | active#(X1) | | 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
Strategy
The following SCCs where found
| isNatKind#(ok(X)) → isNatKind#(X) |
| active#(U11(X1, X2, X3)) → active#(X1) | active#(U21(X1, X2)) → active#(X1) |
| active#(plus(X1, X2)) → active#(X1) | active#(U22(X)) → active#(X) |
| active#(U12(X1, X2)) → active#(X1) | active#(s(X)) → active#(X) |
| active#(U31(X1, X2)) → active#(X1) | active#(plus(X1, X2)) → active#(X2) |
| active#(U13(X)) → active#(X) | active#(and(X1, X2)) → active#(X1) |
| active#(U41(X1, X2, X3)) → active#(X1) |
| proper#(isNat(X)) → proper#(X) | proper#(U12(X1, X2)) → proper#(X1) |
| proper#(U11(X1, X2, X3)) → proper#(X3) | proper#(U41(X1, X2, X3)) → proper#(X3) |
| proper#(U31(X1, X2)) → proper#(X1) | proper#(U31(X1, X2)) → proper#(X2) |
| proper#(U12(X1, X2)) → proper#(X2) | proper#(and(X1, X2)) → proper#(X1) |
| proper#(U41(X1, X2, X3)) → proper#(X2) | proper#(U13(X)) → proper#(X) |
| proper#(U11(X1, X2, X3)) → proper#(X1) | proper#(U21(X1, X2)) → proper#(X2) |
| proper#(s(X)) → proper#(X) | proper#(isNatKind(X)) → proper#(X) |
| proper#(and(X1, X2)) → proper#(X2) | proper#(U11(X1, X2, X3)) → proper#(X2) |
| proper#(U41(X1, X2, X3)) → proper#(X1) | proper#(plus(X1, X2)) → proper#(X1) |
| proper#(U21(X1, X2)) → proper#(X1) | proper#(plus(X1, X2)) → proper#(X2) |
| proper#(U22(X)) → proper#(X) |
| isNat#(ok(X)) → isNat#(X) |
| U11#(mark(X1), X2, X3) → U11#(X1, X2, X3) | U11#(ok(X1), ok(X2), ok(X3)) → U11#(X1, X2, X3) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| 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) |
| U13#(ok(X)) → U13#(X) | U13#(mark(X)) → U13#(X) |
| U12#(ok(X1), ok(X2)) → U12#(X1, X2) | U12#(mark(X1), X2) → U12#(X1, X2) |
| U22#(ok(X)) → U22#(X) | U22#(mark(X)) → U22#(X) |
| and#(ok(X1), ok(X2)) → and#(X1, X2) | and#(mark(X1), X2) → and#(X1, X2) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| U41#(mark(X1), X2, X3) → U41#(X1, X2, X3) | U41#(ok(X1), ok(X2), ok(X3)) → U41#(X1, X2, X3) |
| U31#(mark(X1), X2) → U31#(X1, X2) | U31#(ok(X1), ok(X2)) → U31#(X1, X2) |
Problem 2: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
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 17: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| plus#(X1, mark(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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, proper, U13, U31, U21, top, U22
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): 0
- U12(x,y): 0
- U13(x): 0
- U21(x,y): 0
- U22(x): 0
- U31(x,y): 0
- U41(x,y,z): 0
- active(x): 0
- and(x,y): 0
- isNat(x): 0
- isNatKind(x): 0
- mark(x): x + 2
- ok(x): 0
- plus(x,y): 0
- plus#(x,y): y + x + 1
- proper(x): 0
- s(x): 0
- top(x): 0
- tt: 0
There are no usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| plus#(X1, mark(X2)) | → | plus#(X1, X2) |
Problem 3: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
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 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(U11(X1, X2, X3)) | → | active#(X1) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(plus(X1, X2)) | → | active#(X1) | | active#(U22(X)) | → | active#(X) |
| active#(U12(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(U31(X1, X2)) | → | active#(X1) | | active#(plus(X1, X2)) | → | active#(X2) |
| active#(U13(X)) | → | active#(X) | | active#(and(X1, X2)) | → | active#(X1) |
| active#(U41(X1, X2, X3)) | → | 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(U11(X1, X2, X3)) | → | active#(X1) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(plus(X1, X2)) | → | active#(X1) | | active#(U22(X)) | → | active#(X) |
| active#(U12(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(U31(X1, X2)) | → | active#(X1) | | active#(plus(X1, X2)) | → | active#(X2) |
| active#(U13(X)) | → | active#(X) | | active#(and(X1, X2)) | → | active#(X1) |
| active#(U41(X1, X2, X3)) | → | active#(X1) |
Problem 5: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatKind#(ok(X)) | → | isNatKind#(X) |
Problem 6: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
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 7: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
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 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U31#(mark(X1), X2) | → | U31#(X1, X2) | | U31#(ok(X1), ok(X2)) | → | U31#(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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U31#(mark(X1), X2) | → | U31#(X1, X2) | | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) |
Problem 9: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
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 10: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
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 11: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNat#(ok(X)) | → | isNat#(X) |
Problem 12: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
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 13: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(U12(X1, X2)) | → | proper#(X1) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(U11(X1, X2, X3)) | → | proper#(X3) | | proper#(U31(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2, X3)) | → | proper#(X3) | | proper#(U31(X1, X2)) | → | proper#(X2) |
| proper#(U12(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2, X3)) | → | proper#(X2) | | proper#(U13(X)) | → | proper#(X) |
| proper#(U11(X1, X2, X3)) | → | proper#(X1) | | proper#(U21(X1, X2)) | → | proper#(X2) |
| proper#(s(X)) | → | proper#(X) | | proper#(isNatKind(X)) | → | proper#(X) |
| proper#(and(X1, X2)) | → | proper#(X2) | | proper#(U11(X1, X2, X3)) | → | proper#(X2) |
| proper#(U41(X1, X2, X3)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X1) |
| proper#(U21(X1, X2)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(U22(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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(U12(X1, X2)) | → | proper#(X1) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(U11(X1, X2, X3)) | → | proper#(X3) | | proper#(U31(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2, X3)) | → | proper#(X3) | | proper#(U31(X1, X2)) | → | proper#(X2) |
| proper#(U12(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2, X3)) | → | proper#(X2) | | proper#(U13(X)) | → | proper#(X) |
| proper#(U11(X1, X2, X3)) | → | proper#(X1) | | proper#(U21(X1, X2)) | → | proper#(X2) |
| proper#(s(X)) | → | proper#(X) | | proper#(isNatKind(X)) | → | proper#(X) |
| proper#(and(X1, X2)) | → | proper#(X2) | | proper#(U11(X1, X2, X3)) | → | proper#(X2) |
| proper#(U41(X1, X2, X3)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X1) |
| proper#(U21(X1, X2)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(U22(X)) | → | proper#(X) |
Problem 14: 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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
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 16: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(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, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(s(plus(N, M))) | | active(and(tt, X)) | → | mark(X) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | | active(isNatKind(0)) | → | mark(tt) |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) |
| active(plus(N, 0)) | → | mark(U31(and(isNat(N), isNatKind(N)), N)) | | active(plus(N, s(M))) | → | mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2)) | → | U12(active(X1), X2) |
| active(U13(X)) | → | U13(active(X)) | | active(U21(X1, X2)) | → | U21(active(X1), X2) |
| active(U22(X)) | → | U22(active(X)) | | active(U31(X1, X2)) | → | U31(active(X1), X2) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(and(X1, X2)) | → | and(active(X1), X2) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | | U13(mark(X)) | → | mark(U13(X)) |
| U21(mark(X1), X2) | → | mark(U21(X1, X2)) | | U22(mark(X)) | → | mark(U22(X)) |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(U13(X)) | → | U13(proper(X)) | | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) |
| proper(U22(X)) | → | U22(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U13(ok(X)) | → | ok(U13(X)) |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | | U22(ok(X)) | → | ok(U22(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatKind(ok(X)) | → | ok(isNatKind(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, isNat, 0, s, tt, U41, active, U11, ok, U12, U31, U13, proper, U21, U22, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) |