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 (56964ms).
| – Problem 2 was processed with processor SubtermCriterion (3ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (13ms)].
| – Problem 4 was processed with processor SubtermCriterion (4ms).
| – Problem 5 was processed with processor SubtermCriterion (3ms).
| – Problem 6 was processed with processor SubtermCriterion (5ms).
| – Problem 7 was processed with processor SubtermCriterion (3ms).
| – Problem 8 was processed with processor SubtermCriterion (4ms).
| – Problem 9 was processed with processor SubtermCriterion (2ms).
| – Problem 10 was processed with processor SubtermCriterion (4ms).
| – Problem 11 was processed with processor SubtermCriterion (3ms).
| – Problem 12 was processed with processor SubtermCriterion (4ms).
| – Problem 13 was processed with processor SubtermCriterion (3ms).
| | – Problem 29 remains open; application of the following processors failed [].
| – Problem 14 remains open; application of the following processors failed [SubtermCriterion (3ms)].
| – Problem 15 was processed with processor SubtermCriterion (4ms).
| – Problem 16 was processed with processor SubtermCriterion (4ms).
| – Problem 17 was processed with processor SubtermCriterion (2ms).
| – Problem 18 was processed with processor SubtermCriterion (4ms).
| – Problem 19 was processed with processor SubtermCriterion (2ms).
| – Problem 20 was processed with processor SubtermCriterion (4ms).
| – Problem 21 was processed with processor SubtermCriterion (3ms).
| – Problem 22 remains open; application of the following processors failed [SubtermCriterion (3ms), DependencyGraph (timeout)].
| – Problem 23 was processed with processor SubtermCriterion (3ms).
| – Problem 24 was processed with processor SubtermCriterion (4ms).
| – Problem 25 was processed with processor SubtermCriterion (7ms).
| – Problem 26 was processed with processor SubtermCriterion (2ms).
| – Problem 27 was processed with processor SubtermCriterion (3ms).
| – Problem 28 was processed with processor SubtermCriterion (4ms).
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U93, U42, U41, U92, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U52, U81, U11, active, U31, nil
Open Dependency Pair Problem 13
Dependency Pairs
| take#(mark(X1), X2) | → | take#(X1, X2) | | take#(X1, mark(X2)) | → | take#(X1, X2) |
| take#(ok(X1), ok(X2)) | → | take#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U93, U42, U41, U92, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U52, U81, U11, active, U31, nil
Open Dependency Pair Problem 14
Dependency Pairs
| active#(U91(X1, X2, X3, X4)) | → | U91#(active(X1), X2, X3, X4) | | active#(U71(X1, X2, X3)) | → | U71#(active(X1), X2, X3) |
| active#(take(s(M), cons(N, IL))) | → | U91#(isNatIList(IL), IL, M, N) | | active#(U72(tt, L)) | → | length#(L) |
| active#(U51(X1, X2)) | → | active#(X1) | | active#(isNatIList(cons(V1, V2))) | → | U41#(isNat(V1), V2) |
| active#(take(s(M), cons(N, IL))) | → | isNatIList#(IL) | | active#(U81(X)) | → | U81#(active(X)) |
| active#(U51(tt, V2)) | → | isNatList#(V2) | | active#(cons(X1, X2)) | → | cons#(active(X1), X2) |
| active#(U41(X1, X2)) | → | U41#(active(X1), X2) | | active#(U93(X1, X2, X3, X4)) | → | active#(X1) |
| active#(take(X1, X2)) | → | take#(active(X1), X2) | | active#(U61(X1, X2)) | → | U61#(active(X1), X2) |
| active#(isNat(length(V1))) | → | isNatList#(V1) | | active#(U31(X)) | → | U31#(active(X)) |
| active#(U51(X1, X2)) | → | U51#(active(X1), X2) | | active#(length(cons(N, L))) | → | U71#(isNatList(L), L, N) |
| active#(U62(X)) | → | U62#(active(X)) | | active#(U31(X)) | → | active#(X) |
| active#(U11(X)) | → | U11#(active(X)) | | active#(take(X1, X2)) | → | active#(X2) |
| active#(U71(X1, X2, X3)) | → | active#(X1) | | active#(U61(X1, X2)) | → | active#(X1) |
| active#(length(X)) | → | active#(X) | | active#(U92(X1, X2, X3, X4)) | → | U92#(active(X1), X2, X3, X4) |
| active#(U62(X)) | → | active#(X) | | active#(U91(X1, X2, X3, X4)) | → | active#(X1) |
| active#(U81(X)) | → | active#(X) | | active#(isNatList(take(V1, V2))) | → | U61#(isNat(V1), V2) |
| active#(isNatIList(cons(V1, V2))) | → | isNat#(V1) | | active#(U41(X1, X2)) | → | active#(X1) |
| active#(U72(X1, X2)) | → | active#(X1) | | active#(cons(X1, X2)) | → | active#(X1) |
| active#(U91(tt, IL, M, N)) | → | U92#(isNat(M), IL, M, N) | | active#(U11(X)) | → | active#(X) |
| active#(U42(X)) | → | active#(X) | | active#(U21(X)) | → | active#(X) |
| active#(isNat(length(V1))) | → | U11#(isNatList(V1)) | | active#(U41(tt, V2)) | → | U42#(isNatIList(V2)) |
| active#(U93(tt, IL, M, N)) | → | take#(M, IL) | | active#(U92(tt, IL, M, N)) | → | isNat#(N) |
| active#(U61(tt, V2)) | → | U62#(isNatIList(V2)) | | active#(U52(X)) | → | active#(X) |
| active#(isNat(s(V1))) | → | isNat#(V1) | | active#(U71(tt, L, N)) | → | isNat#(N) |
| active#(U93(X1, X2, X3, X4)) | → | U93#(active(X1), X2, X3, X4) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| active#(U61(tt, V2)) | → | isNatIList#(V2) | | active#(take(X1, X2)) | → | active#(X1) |
| active#(s(X)) | → | s#(active(X)) | | active#(take(0, IL)) | → | U81#(isNatIList(IL)) |
| active#(U91(tt, IL, M, N)) | → | isNat#(M) | | active#(s(X)) | → | active#(X) |
| active#(take(X1, X2)) | → | take#(X1, active(X2)) | | active#(zeros) | → | cons#(0, zeros) |
| active#(U93(tt, IL, M, N)) | → | cons#(N, take(M, IL)) | | active#(take(0, IL)) | → | isNatIList#(IL) |
| active#(U92(X1, X2, X3, X4)) | → | active#(X1) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U93, U42, U41, U92, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U52, U81, U11, active, U31, nil
Open Dependency Pair Problem 22
Dependency Pairs
| active#(U91(X1, X2, X3, X4)) | → | U91#(active(X1), X2, X3, X4) | | active#(U71(X1, X2, X3)) | → | U71#(active(X1), X2, X3) |
| active#(take(s(M), cons(N, IL))) | → | U91#(isNatIList(IL), IL, M, N) | | active#(U72(tt, L)) | → | length#(L) |
| active#(U51(X1, X2)) | → | active#(X1) | | active#(isNatIList(cons(V1, V2))) | → | U41#(isNat(V1), V2) |
| active#(take(s(M), cons(N, IL))) | → | isNatIList#(IL) | | active#(U81(X)) | → | U81#(active(X)) |
| active#(U51(tt, V2)) | → | isNatList#(V2) | | active#(cons(X1, X2)) | → | cons#(active(X1), X2) |
| active#(U41(X1, X2)) | → | U41#(active(X1), X2) | | active#(U93(X1, X2, X3, X4)) | → | active#(X1) |
| active#(take(X1, X2)) | → | take#(active(X1), X2) | | active#(U61(X1, X2)) | → | U61#(active(X1), X2) |
| active#(isNat(length(V1))) | → | isNatList#(V1) | | active#(U31(X)) | → | U31#(active(X)) |
| active#(U51(X1, X2)) | → | U51#(active(X1), X2) | | active#(length(cons(N, L))) | → | U71#(isNatList(L), L, N) |
| active#(U62(X)) | → | U62#(active(X)) | | active#(U31(X)) | → | active#(X) |
| active#(U11(X)) | → | U11#(active(X)) | | active#(take(X1, X2)) | → | active#(X2) |
| active#(U71(X1, X2, X3)) | → | active#(X1) | | active#(U61(X1, X2)) | → | active#(X1) |
| active#(length(X)) | → | active#(X) | | active#(U92(X1, X2, X3, X4)) | → | U92#(active(X1), X2, X3, X4) |
| active#(U62(X)) | → | active#(X) | | active#(U91(X1, X2, X3, X4)) | → | active#(X1) |
| active#(U81(X)) | → | active#(X) | | active#(isNatList(take(V1, V2))) | → | U61#(isNat(V1), V2) |
| active#(isNatIList(cons(V1, V2))) | → | isNat#(V1) | | active#(U41(X1, X2)) | → | active#(X1) |
| active#(U72(X1, X2)) | → | active#(X1) | | active#(cons(X1, X2)) | → | active#(X1) |
| active#(U91(tt, IL, M, N)) | → | U92#(isNat(M), IL, M, N) | | active#(U11(X)) | → | active#(X) |
| active#(U42(X)) | → | active#(X) | | active#(U21(X)) | → | active#(X) |
| active#(U41(tt, V2)) | → | U42#(isNatIList(V2)) | | active#(U93(tt, IL, M, N)) | → | take#(M, IL) |
| active#(U92(tt, IL, M, N)) | → | isNat#(N) | | active#(U61(tt, V2)) | → | U62#(isNatIList(V2)) |
| active#(U52(X)) | → | active#(X) | | active#(isNat(s(V1))) | → | isNat#(V1) |
| active#(U71(tt, L, N)) | → | isNat#(N) | | active#(U93(X1, X2, X3, X4)) | → | U93#(active(X1), X2, X3, X4) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(U61(tt, V2)) | → | isNatIList#(V2) |
| active#(take(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | s#(active(X)) |
| active#(take(0, IL)) | → | U81#(isNatIList(IL)) | | active#(U91(tt, IL, M, N)) | → | isNat#(M) |
| active#(s(X)) | → | active#(X) | | active#(take(X1, X2)) | → | take#(X1, active(X2)) |
| active#(zeros) | → | cons#(0, zeros) | | active#(U93(tt, IL, M, N)) | → | cons#(N, take(M, IL)) |
| active#(take(0, IL)) | → | isNatIList#(IL) | | active#(U92(X1, X2, X3, X4)) | → | active#(X1) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U93, U42, U41, U92, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U52, U81, U11, active, U31, nil
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#(cons(X1, X2)) | → | proper#(X1) | | active#(take(s(M), cons(N, IL))) | → | U91#(isNatIList(IL), IL, M, N) |
| active#(U72(tt, L)) | → | length#(L) | | proper#(U41(X1, X2)) | → | proper#(X1) |
| proper#(length(X)) | → | length#(proper(X)) | | active#(U92(tt, IL, M, N)) | → | U93#(isNat(N), IL, M, N) |
| active#(U51(X1, X2)) | → | active#(X1) | | U92#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U92#(X1, X2, X3, X4) |
| active#(U41(tt, V2)) | → | isNatIList#(V2) | | active#(U72(tt, L)) | → | s#(length(L)) |
| proper#(U52(X)) | → | U52#(proper(X)) | | active#(U21(X)) | → | U21#(active(X)) |
| active#(U51(tt, V2)) | → | isNatList#(V2) | | active#(U51(tt, V2)) | → | U52#(isNatList(V2)) |
| proper#(U93(X1, X2, X3, X4)) | → | U93#(proper(X1), proper(X2), proper(X3), proper(X4)) | | active#(take(X1, X2)) | → | take#(active(X1), X2) |
| U51#(mark(X1), X2) | → | U51#(X1, X2) | | proper#(U93(X1, X2, X3, X4)) | → | proper#(X3) |
| proper#(U71(X1, X2, X3)) | → | proper#(X1) | | proper#(U71(X1, X2, X3)) | → | U71#(proper(X1), proper(X2), proper(X3)) |
| proper#(U62(X)) | → | proper#(X) | | active#(isNat(length(V1))) | → | isNatList#(V1) |
| U72#(mark(X1), X2) | → | U72#(X1, X2) | | active#(U31(X)) | → | U31#(active(X)) |
| proper#(U42(X)) | → | U42#(proper(X)) | | proper#(U93(X1, X2, X3, X4)) | → | proper#(X1) |
| active#(U42(X)) | → | U42#(active(X)) | | top#(mark(X)) | → | proper#(X) |
| proper#(U93(X1, X2, X3, X4)) | → | proper#(X2) | | proper#(U72(X1, X2)) | → | proper#(X2) |
| length#(mark(X)) | → | length#(X) | | U81#(mark(X)) | → | U81#(X) |
| active#(U11(X)) | → | U11#(active(X)) | | active#(take(X1, X2)) | → | active#(X2) |
| active#(U71(X1, X2, X3)) | → | active#(X1) | | active#(U61(X1, X2)) | → | active#(X1) |
| U71#(mark(X1), X2, X3) | → | U71#(X1, X2, X3) | | active#(U92(X1, X2, X3, X4)) | → | U92#(active(X1), X2, X3, X4) |
| active#(length(X)) | → | active#(X) | | active#(U62(X)) | → | active#(X) |
| proper#(U71(X1, X2, X3)) | → | proper#(X3) | | isNat#(ok(X)) | → | isNat#(X) |
| active#(isNatIList(cons(V1, V2))) | → | isNat#(V1) | | isNatIList#(ok(X)) | → | isNatIList#(X) |
| active#(U41(X1, X2)) | → | active#(X1) | | proper#(U21(X)) | → | U21#(proper(X)) |
| active#(U72(X1, X2)) | → | active#(X1) | | proper#(U62(X)) | → | U62#(proper(X)) |
| active#(U91(tt, IL, M, N)) | → | U92#(isNat(M), IL, M, N) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| proper#(U11(X)) | → | proper#(X) | | top#(ok(X)) | → | active#(X) |
| active#(isNat(length(V1))) | → | U11#(isNatList(V1)) | | active#(U41(tt, V2)) | → | U42#(isNatIList(V2)) |
| active#(U93(tt, IL, M, N)) | → | take#(M, IL) | | U62#(mark(X)) | → | U62#(X) |
| active#(U61(tt, V2)) | → | U62#(isNatIList(V2)) | | active#(U52(X)) | → | active#(X) |
| proper#(isNatList(X)) | → | proper#(X) | | proper#(isNat(X)) | → | isNat#(proper(X)) |
| proper#(U51(X1, X2)) | → | proper#(X2) | | take#(ok(X1), ok(X2)) | → | take#(X1, X2) |
| U42#(ok(X)) | → | U42#(X) | | proper#(U92(X1, X2, X3, X4)) | → | proper#(X4) |
| proper#(U21(X)) | → | proper#(X) | | active#(isNat(s(V1))) | → | isNat#(V1) |
| U11#(mark(X)) | → | U11#(X) | | proper#(U81(X)) | → | U81#(proper(X)) |
| U51#(ok(X1), ok(X2)) | → | U51#(X1, X2) | | proper#(isNat(X)) | → | proper#(X) |
| active#(U52(X)) | → | U52#(active(X)) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| active#(U61(tt, V2)) | → | isNatIList#(V2) | | proper#(U91(X1, X2, X3, X4)) | → | proper#(X3) |
| proper#(U72(X1, X2)) | → | proper#(X1) | | active#(s(X)) | → | s#(active(X)) |
| U31#(ok(X)) | → | U31#(X) | | active#(isNatList(cons(V1, V2))) | → | U51#(isNat(V1), V2) |
| s#(ok(X)) | → | s#(X) | | proper#(U92(X1, X2, X3, X4)) | → | U92#(proper(X1), proper(X2), proper(X3), proper(X4)) |
| U91#(mark(X1), X2, X3, X4) | → | U91#(X1, X2, X3, X4) | | proper#(length(X)) | → | proper#(X) |
| active#(U91(tt, IL, M, N)) | → | isNat#(M) | | active#(U72(X1, X2)) | → | U72#(active(X1), X2) |
| active#(take(X1, X2)) | → | take#(X1, active(X2)) | | proper#(s(X)) | → | s#(proper(X)) |
| active#(zeros) | → | cons#(0, zeros) | | active#(isNatIList(V)) | → | U31#(isNatList(V)) |
| U81#(ok(X)) | → | U81#(X) | | proper#(U91(X1, X2, X3, X4)) | → | proper#(X1) |
| active#(U92(X1, X2, X3, X4)) | → | active#(X1) | | U92#(mark(X1), X2, X3, X4) | → | U92#(X1, X2, X3, X4) |
| active#(U91(X1, X2, X3, X4)) | → | U91#(active(X1), X2, X3, X4) | | top#(ok(X)) | → | top#(active(X)) |
| proper#(U91(X1, X2, X3, X4)) | → | proper#(X2) | | active#(isNatIList(cons(V1, V2))) | → | U41#(isNat(V1), V2) |
| active#(take(s(M), cons(N, IL))) | → | isNatIList#(IL) | | proper#(U93(X1, X2, X3, X4)) | → | proper#(X4) |
| active#(U81(X)) | → | U81#(active(X)) | | proper#(isNatIList(X)) | → | isNatIList#(proper(X)) |
| cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) | | proper#(U72(X1, X2)) | → | U72#(proper(X1), proper(X2)) |
| proper#(U92(X1, X2, X3, X4)) | → | proper#(X2) | | active#(cons(X1, X2)) | → | cons#(active(X1), X2) |
| U42#(mark(X)) | → | U42#(X) | | proper#(U51(X1, X2)) | → | U51#(proper(X1), proper(X2)) |
| active#(U93(X1, X2, X3, X4)) | → | active#(X1) | | active#(U41(X1, X2)) | → | U41#(active(X1), X2) |
| active#(U61(X1, X2)) | → | U61#(active(X1), X2) | | U91#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U91#(X1, X2, X3, X4) |
| proper#(U61(X1, X2)) | → | proper#(X2) | | U61#(mark(X1), X2) | → | U61#(X1, X2) |
| active#(U51(X1, X2)) | → | U51#(active(X1), X2) | | active#(length(cons(N, L))) | → | U71#(isNatList(L), L, N) |
| length#(ok(X)) | → | length#(X) | | active#(U62(X)) | → | U62#(active(X)) |
| U21#(ok(X)) | → | U21#(X) | | U61#(ok(X1), ok(X2)) | → | U61#(X1, X2) |
| U71#(ok(X1), ok(X2), ok(X3)) | → | U71#(X1, X2, X3) | | active#(U31(X)) | → | active#(X) |
| top#(mark(X)) | → | top#(proper(X)) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(isNatIList(X)) | → | proper#(X) | | take#(X1, mark(X2)) | → | take#(X1, X2) |
| active#(U91(X1, X2, X3, X4)) | → | active#(X1) | | U52#(ok(X)) | → | U52#(X) |
| active#(U81(X)) | → | active#(X) | | isNatList#(ok(X)) | → | isNatList#(X) |
| U62#(ok(X)) | → | U62#(X) | | active#(isNatList(take(V1, V2))) | → | U61#(isNat(V1), V2) |
| proper#(s(X)) | → | proper#(X) | | proper#(U92(X1, X2, X3, X4)) | → | proper#(X3) |
| proper#(isNatList(X)) | → | isNatList#(proper(X)) | | proper#(U91(X1, X2, X3, X4)) | → | proper#(X4) |
| proper#(take(X1, X2)) | → | take#(proper(X1), proper(X2)) | | active#(isNatList(cons(V1, V2))) | → | isNat#(V1) |
| U31#(mark(X)) | → | U31#(X) | | proper#(U61(X1, X2)) | → | U61#(proper(X1), proper(X2)) |
| active#(cons(X1, X2)) | → | active#(X1) | | take#(mark(X1), X2) | → | take#(X1, X2) |
| active#(U11(X)) | → | active#(X) | | active#(U42(X)) | → | active#(X) |
| proper#(U31(X)) | → | U31#(proper(X)) | | U93#(mark(X1), X2, X3, X4) | → | U93#(X1, X2, X3, X4) |
| proper#(U42(X)) | → | proper#(X) | | proper#(U51(X1, X2)) | → | proper#(X1) |
| active#(U21(X)) | → | active#(X) | | active#(length(cons(N, L))) | → | isNatList#(L) |
| active#(U92(tt, IL, M, N)) | → | isNat#(N) | | U41#(ok(X1), ok(X2)) | → | U41#(X1, X2) |
| proper#(take(X1, X2)) | → | proper#(X1) | | active#(isNatList(take(V1, V2))) | → | isNat#(V1) |
| proper#(U11(X)) | → | U11#(proper(X)) | | U41#(mark(X1), X2) | → | U41#(X1, X2) |
| active#(U71(tt, L, N)) | → | isNat#(N) | | active#(U93(X1, X2, X3, X4)) | → | U93#(active(X1), X2, X3, X4) |
| active#(length(X)) | → | length#(active(X)) | | U21#(mark(X)) | → | U21#(X) |
| active#(take(X1, X2)) | → | active#(X1) | | proper#(U92(X1, X2, X3, X4)) | → | proper#(X1) |
| proper#(take(X1, X2)) | → | proper#(X2) | | proper#(U91(X1, X2, X3, X4)) | → | U91#(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper#(U81(X)) | → | proper#(X) | | proper#(U41(X1, X2)) | → | U41#(proper(X1), proper(X2)) |
| active#(isNatIList(V)) | → | isNatList#(V) | | proper#(U61(X1, X2)) | → | proper#(X1) |
| U72#(ok(X1), ok(X2)) | → | U72#(X1, X2) | | s#(mark(X)) | → | s#(X) |
| active#(take(0, IL)) | → | U81#(isNatIList(IL)) | | proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) |
| active#(s(X)) | → | active#(X) | | U93#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U93#(X1, X2, X3, X4) |
| U52#(mark(X)) | → | U52#(X) | | active#(U93(tt, IL, M, N)) | → | cons#(N, take(M, IL)) |
| U11#(ok(X)) | → | U11#(X) | | proper#(U71(X1, X2, X3)) | → | proper#(X2) |
| active#(U71(tt, L, N)) | → | U72#(isNat(N), L) | | active#(take(0, IL)) | → | isNatIList#(IL) |
| proper#(U52(X)) | → | proper#(X) | | proper#(U31(X)) | → | proper#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
The following SCCs where found
| U31#(ok(X)) → U31#(X) | U31#(mark(X)) → U31#(X) |
| take#(mark(X1), X2) → take#(X1, X2) | take#(X1, mark(X2)) → take#(X1, X2) |
| take#(ok(X1), ok(X2)) → take#(X1, X2) |
| U92#(ok(X1), ok(X2), ok(X3), ok(X4)) → U92#(X1, X2, X3, X4) | U92#(mark(X1), X2, X3, X4) → U92#(X1, X2, X3, X4) |
| U21#(ok(X)) → U21#(X) | U21#(mark(X)) → U21#(X) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| U62#(mark(X)) → U62#(X) | U62#(ok(X)) → U62#(X) |
| proper#(U41(X1, X2)) → proper#(X2) | proper#(cons(X1, X2)) → proper#(X1) |
| proper#(U41(X1, X2)) → proper#(X1) | proper#(U91(X1, X2, X3, X4)) → proper#(X2) |
| proper#(U93(X1, X2, X3, X4)) → proper#(X4) | proper#(U11(X)) → proper#(X) |
| proper#(U51(X1, X2)) → proper#(X1) | proper#(U42(X)) → proper#(X) |
| proper#(U92(X1, X2, X3, X4)) → proper#(X2) | proper#(U93(X1, X2, X3, X4)) → proper#(X3) |
| proper#(U71(X1, X2, X3)) → proper#(X1) | proper#(U62(X)) → proper#(X) |
| proper#(U93(X1, X2, X3, X4)) → proper#(X1) | proper#(U61(X1, X2)) → proper#(X2) |
| proper#(take(X1, X2)) → proper#(X1) | proper#(isNatList(X)) → proper#(X) |
| proper#(U51(X1, X2)) → proper#(X2) | proper#(U92(X1, X2, X3, X4)) → proper#(X4) |
| proper#(U21(X)) → proper#(X) | proper#(U93(X1, X2, X3, X4)) → proper#(X2) |
| proper#(U72(X1, X2)) → proper#(X2) | proper#(isNat(X)) → proper#(X) |
| proper#(cons(X1, X2)) → proper#(X2) | proper#(isNatIList(X)) → proper#(X) |
| proper#(U92(X1, X2, X3, X4)) → proper#(X1) | proper#(U91(X1, X2, X3, X4)) → proper#(X3) |
| proper#(take(X1, X2)) → proper#(X2) | proper#(U71(X1, X2, X3)) → proper#(X3) |
| proper#(U72(X1, X2)) → proper#(X1) | proper#(U81(X)) → proper#(X) |
| proper#(U61(X1, X2)) → proper#(X1) | proper#(U92(X1, X2, X3, X4)) → proper#(X3) |
| proper#(s(X)) → proper#(X) | proper#(length(X)) → proper#(X) |
| proper#(U91(X1, X2, X3, X4)) → proper#(X4) | proper#(U71(X1, X2, X3)) → proper#(X2) |
| proper#(U52(X)) → proper#(X) | proper#(U91(X1, X2, X3, X4)) → proper#(X1) |
| proper#(U31(X)) → proper#(X) |
| U11#(ok(X)) → U11#(X) | U11#(mark(X)) → U11#(X) |
| U71#(ok(X1), ok(X2), ok(X3)) → U71#(X1, X2, X3) | U71#(mark(X1), X2, X3) → U71#(X1, X2, X3) |
| active#(U71(X1, X2, X3)) → U71#(active(X1), X2, X3) | active#(U91(X1, X2, X3, X4)) → U91#(active(X1), X2, X3, X4) |
| active#(take(s(M), cons(N, IL))) → U91#(isNatIList(IL), IL, M, N) | active#(U72(tt, L)) → length#(L) |
| active#(U51(X1, X2)) → active#(X1) | active#(take(s(M), cons(N, IL))) → isNatIList#(IL) |
| active#(isNatIList(cons(V1, V2))) → U41#(isNat(V1), V2) | active#(U81(X)) → U81#(active(X)) |
| active#(U51(tt, V2)) → isNatList#(V2) | active#(cons(X1, X2)) → cons#(active(X1), X2) |
| active#(take(X1, X2)) → take#(active(X1), X2) | active#(U93(X1, X2, X3, X4)) → active#(X1) |
| active#(U41(X1, X2)) → U41#(active(X1), X2) | active#(U61(X1, X2)) → U61#(active(X1), X2) |
| active#(isNat(length(V1))) → isNatList#(V1) | active#(U31(X)) → U31#(active(X)) |
| active#(U51(X1, X2)) → U51#(active(X1), X2) | active#(length(cons(N, L))) → U71#(isNatList(L), L, N) |
| active#(U62(X)) → U62#(active(X)) | active#(U31(X)) → active#(X) |
| active#(U11(X)) → U11#(active(X)) | active#(U71(X1, X2, X3)) → active#(X1) |
| active#(take(X1, X2)) → active#(X2) | active#(U61(X1, X2)) → active#(X1) |
| active#(U92(X1, X2, X3, X4)) → U92#(active(X1), X2, X3, X4) | active#(length(X)) → active#(X) |
| active#(U62(X)) → active#(X) | active#(U91(X1, X2, X3, X4)) → active#(X1) |
| active#(U81(X)) → active#(X) | active#(isNatIList(cons(V1, V2))) → isNat#(V1) |
| active#(isNatList(take(V1, V2))) → U61#(isNat(V1), V2) | active#(U41(X1, X2)) → active#(X1) |
| active#(U72(X1, X2)) → active#(X1) | active#(cons(X1, X2)) → active#(X1) |
| active#(U91(tt, IL, M, N)) → U92#(isNat(M), IL, M, N) | active#(U11(X)) → active#(X) |
| active#(U42(X)) → active#(X) | active#(U21(X)) → active#(X) |
| active#(isNat(length(V1))) → U11#(isNatList(V1)) | active#(U93(tt, IL, M, N)) → take#(M, IL) |
| active#(U41(tt, V2)) → U42#(isNatIList(V2)) | active#(U92(tt, IL, M, N)) → isNat#(N) |
| active#(U52(X)) → active#(X) | active#(U61(tt, V2)) → U62#(isNatIList(V2)) |
| active#(isNat(s(V1))) → isNat#(V1) | active#(U71(tt, L, N)) → isNat#(N) |
| active#(U93(X1, X2, X3, X4)) → U93#(active(X1), X2, X3, X4) | active#(isNat(s(V1))) → U21#(isNat(V1)) |
| active#(U61(tt, V2)) → isNatIList#(V2) | active#(take(X1, X2)) → active#(X1) |
| active#(s(X)) → s#(active(X)) | active#(take(0, IL)) → U81#(isNatIList(IL)) |
| active#(U91(tt, IL, M, N)) → isNat#(M) | active#(take(X1, X2)) → take#(X1, active(X2)) |
| active#(s(X)) → active#(X) | active#(zeros) → cons#(0, zeros) |
| active#(U93(tt, IL, M, N)) → cons#(N, take(M, IL)) | active#(U92(X1, X2, X3, X4)) → active#(X1) |
| active#(take(0, IL)) → isNatIList#(IL) |
| active#(U71(X1, X2, X3)) → U71#(active(X1), X2, X3) | active#(U91(X1, X2, X3, X4)) → U91#(active(X1), X2, X3, X4) |
| active#(take(s(M), cons(N, IL))) → U91#(isNatIList(IL), IL, M, N) | active#(U72(tt, L)) → length#(L) |
| active#(U51(X1, X2)) → active#(X1) | active#(take(s(M), cons(N, IL))) → isNatIList#(IL) |
| active#(isNatIList(cons(V1, V2))) → U41#(isNat(V1), V2) | active#(U81(X)) → U81#(active(X)) |
| active#(U51(tt, V2)) → isNatList#(V2) | active#(cons(X1, X2)) → cons#(active(X1), X2) |
| active#(take(X1, X2)) → take#(active(X1), X2) | active#(U93(X1, X2, X3, X4)) → active#(X1) |
| active#(U41(X1, X2)) → U41#(active(X1), X2) | active#(U61(X1, X2)) → U61#(active(X1), X2) |
| active#(isNat(length(V1))) → isNatList#(V1) | active#(U31(X)) → U31#(active(X)) |
| active#(U51(X1, X2)) → U51#(active(X1), X2) | active#(length(cons(N, L))) → U71#(isNatList(L), L, N) |
| active#(U62(X)) → U62#(active(X)) | active#(U31(X)) → active#(X) |
| active#(U11(X)) → U11#(active(X)) | active#(U71(X1, X2, X3)) → active#(X1) |
| active#(take(X1, X2)) → active#(X2) | active#(U61(X1, X2)) → active#(X1) |
| active#(U92(X1, X2, X3, X4)) → U92#(active(X1), X2, X3, X4) | active#(length(X)) → active#(X) |
| active#(U62(X)) → active#(X) | active#(U91(X1, X2, X3, X4)) → active#(X1) |
| active#(U81(X)) → active#(X) | active#(isNatIList(cons(V1, V2))) → isNat#(V1) |
| active#(isNatList(take(V1, V2))) → U61#(isNat(V1), V2) | active#(U41(X1, X2)) → active#(X1) |
| active#(U72(X1, X2)) → active#(X1) | active#(cons(X1, X2)) → active#(X1) |
| active#(U91(tt, IL, M, N)) → U92#(isNat(M), IL, M, N) | active#(U11(X)) → active#(X) |
| active#(U42(X)) → active#(X) | active#(U21(X)) → active#(X) |
| active#(U93(tt, IL, M, N)) → take#(M, IL) | active#(U41(tt, V2)) → U42#(isNatIList(V2)) |
| active#(U92(tt, IL, M, N)) → isNat#(N) | active#(U52(X)) → active#(X) |
| active#(U61(tt, V2)) → U62#(isNatIList(V2)) | active#(isNat(s(V1))) → isNat#(V1) |
| active#(U71(tt, L, N)) → isNat#(N) | active#(U93(X1, X2, X3, X4)) → U93#(active(X1), X2, X3, X4) |
| active#(isNat(s(V1))) → U21#(isNat(V1)) | active#(U61(tt, V2)) → isNatIList#(V2) |
| active#(take(X1, X2)) → active#(X1) | active#(s(X)) → s#(active(X)) |
| active#(take(0, IL)) → U81#(isNatIList(IL)) | active#(U91(tt, IL, M, N)) → isNat#(M) |
| active#(take(X1, X2)) → take#(X1, active(X2)) | active#(s(X)) → active#(X) |
| active#(zeros) → cons#(0, zeros) | active#(U93(tt, IL, M, N)) → cons#(N, take(M, IL)) |
| active#(U92(X1, X2, X3, X4)) → active#(X1) | active#(take(0, IL)) → isNatIList#(IL) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| U51#(mark(X1), X2) → U51#(X1, X2) | U51#(ok(X1), ok(X2)) → U51#(X1, X2) |
| U81#(mark(X)) → U81#(X) | U81#(ok(X)) → U81#(X) |
| U41#(ok(X1), ok(X2)) → U41#(X1, X2) | U41#(mark(X1), X2) → U41#(X1, X2) |
| U42#(mark(X)) → U42#(X) | U42#(ok(X)) → U42#(X) |
| U72#(mark(X1), X2) → U72#(X1, X2) | U72#(ok(X1), ok(X2)) → U72#(X1, X2) |
| isNat#(ok(X)) → isNat#(X) |
| isNatList#(ok(X)) → isNatList#(X) |
| isNatIList#(ok(X)) → isNatIList#(X) |
| length#(mark(X)) → length#(X) | length#(ok(X)) → length#(X) |
| cons#(mark(X1), X2) → cons#(X1, X2) | cons#(ok(X1), ok(X2)) → cons#(X1, X2) |
| U91#(ok(X1), ok(X2), ok(X3), ok(X4)) → U91#(X1, X2, X3, X4) | U91#(mark(X1), X2, X3, X4) → U91#(X1, X2, X3, X4) |
| U52#(mark(X)) → U52#(X) | U52#(ok(X)) → U52#(X) |
| active#(U11(X)) → active#(X) | active#(U31(X)) → active#(X) |
| active#(U42(X)) → active#(X) | active#(U51(X1, X2)) → active#(X1) |
| active#(U71(X1, X2, X3)) → active#(X1) | active#(take(X1, X2)) → active#(X2) |
| active#(U61(X1, X2)) → active#(X1) | active#(take(X1, X2)) → active#(X1) |
| active#(length(X)) → active#(X) | active#(U62(X)) → active#(X) |
| active#(U91(X1, X2, X3, X4)) → active#(X1) | active#(U81(X)) → active#(X) |
| active#(U21(X)) → active#(X) | active#(U93(X1, X2, X3, X4)) → active#(X1) |
| active#(U41(X1, X2)) → active#(X1) | active#(s(X)) → active#(X) |
| active#(U52(X)) → active#(X) | active#(U72(X1, X2)) → active#(X1) |
| active#(U92(X1, X2, X3, X4)) → active#(X1) | active#(cons(X1, X2)) → active#(X1) |
| U61#(ok(X1), ok(X2)) → U61#(X1, X2) | U61#(mark(X1), X2) → U61#(X1, X2) |
| U93#(ok(X1), ok(X2), ok(X3), ok(X4)) → U93#(X1, X2, X3, X4) | U93#(mark(X1), X2, X3, X4) → U93#(X1, X2, X3, X4) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U42#(mark(X)) | → | U42#(X) | | U42#(ok(X)) | → | U42#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U42#(mark(X)) | → | U42#(X) | | U42#(ok(X)) | → | U42#(X) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNat#(ok(X)) | → | isNat#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNat#(ok(X)) | → | isNat#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(U11(X)) | → | active#(X) | | active#(U42(X)) | → | active#(X) |
| active#(U31(X)) | → | active#(X) | | active#(U51(X1, X2)) | → | active#(X1) |
| active#(take(X1, X2)) | → | active#(X2) | | active#(U71(X1, X2, X3)) | → | active#(X1) |
| active#(U61(X1, X2)) | → | active#(X1) | | active#(length(X)) | → | active#(X) |
| active#(take(X1, X2)) | → | active#(X1) | | active#(U62(X)) | → | active#(X) |
| active#(U91(X1, X2, X3, X4)) | → | active#(X1) | | active#(U81(X)) | → | active#(X) |
| active#(U21(X)) | → | active#(X) | | active#(U93(X1, X2, X3, X4)) | → | active#(X1) |
| active#(U41(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(U52(X)) | → | active#(X) | | active#(U72(X1, X2)) | → | active#(X1) |
| active#(U92(X1, X2, X3, X4)) | → | active#(X1) | | active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(U11(X)) | → | active#(X) | | active#(U42(X)) | → | active#(X) |
| active#(U31(X)) | → | active#(X) | | active#(U51(X1, X2)) | → | active#(X1) |
| active#(take(X1, X2)) | → | active#(X2) | | active#(U71(X1, X2, X3)) | → | active#(X1) |
| active#(U61(X1, X2)) | → | active#(X1) | | active#(take(X1, X2)) | → | active#(X1) |
| active#(length(X)) | → | active#(X) | | active#(U62(X)) | → | active#(X) |
| active#(U91(X1, X2, X3, X4)) | → | active#(X1) | | active#(U81(X)) | → | active#(X) |
| active#(U21(X)) | → | active#(X) | | active#(U93(X1, X2, X3, X4)) | → | active#(X1) |
| active#(U41(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(U52(X)) | → | active#(X) | | active#(U72(X1, X2)) | → | active#(X1) |
| active#(U92(X1, X2, X3, X4)) | → | active#(X1) | | active#(cons(X1, X2)) | → | active#(X1) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U93#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U93#(X1, X2, X3, X4) | | U93#(mark(X1), X2, X3, X4) | → | U93#(X1, X2, X3, X4) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U93#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U93#(X1, X2, X3, X4) | | U93#(mark(X1), X2, X3, X4) | → | U93#(X1, X2, X3, X4) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U51#(mark(X1), X2) | → | U51#(X1, X2) | | U51#(ok(X1), ok(X2)) | → | U51#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U51#(ok(X1), ok(X2)) | → | U51#(X1, X2) | | U51#(mark(X1), X2) | → | U51#(X1, X2) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNatIList#(ok(X)) | → | isNatIList#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatIList#(ok(X)) | → | isNatIList#(X) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| length#(mark(X)) | → | length#(X) | | length#(ok(X)) | → | length#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| length#(mark(X)) | → | length#(X) | | length#(ok(X)) | → | length#(X) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNatList#(ok(X)) | → | isNatList#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatList#(ok(X)) | → | isNatList#(X) |
Problem 12: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U91#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U91#(X1, X2, X3, X4) | | U91#(mark(X1), X2, X3, X4) | → | U91#(X1, X2, X3, X4) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U91#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U91#(X1, X2, X3, X4) | | U91#(mark(X1), X2, X3, X4) | → | U91#(X1, X2, X3, X4) |
Problem 13: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| take#(mark(X1), X2) | → | take#(X1, X2) | | take#(X1, mark(X2)) | → | take#(X1, X2) |
| take#(ok(X1), ok(X2)) | → | take#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| take#(mark(X1), X2) | → | take#(X1, X2) | | take#(ok(X1), ok(X2)) | → | take#(X1, X2) |
Problem 15: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U81#(mark(X)) | → | U81#(X) | | U81#(ok(X)) | → | U81#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U81#(mark(X)) | → | U81#(X) | | U81#(ok(X)) | → | U81#(X) |
Problem 16: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
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 17: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U31#(ok(X)) | → | U31#(X) | | U31#(mark(X)) | → | U31#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U31#(ok(X)) | → | U31#(X) | | U31#(mark(X)) | → | U31#(X) |
Problem 18: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U11#(ok(X)) | → | U11#(X) | | U11#(mark(X)) | → | U11#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(ok(X)) | → | U11#(X) | | U11#(mark(X)) | → | U11#(X) |
Problem 19: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U92#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U92#(X1, X2, X3, X4) | | U92#(mark(X1), X2, X3, X4) | → | U92#(X1, X2, X3, X4) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U92#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U92#(X1, X2, X3, X4) | | U92#(mark(X1), X2, X3, X4) | → | U92#(X1, X2, X3, X4) |
Problem 20: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U21#(ok(X)) | → | U21#(X) | | U21#(mark(X)) | → | U21#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U21#(ok(X)) | → | U21#(X) | | U21#(mark(X)) | → | U21#(X) |
Problem 21: 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(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
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 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(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
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) |
Problem 24: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U61#(ok(X1), ok(X2)) | → | U61#(X1, X2) | | U61#(mark(X1), X2) | → | U61#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U61#(ok(X1), ok(X2)) | → | U61#(X1, X2) | | U61#(mark(X1), X2) | → | U61#(X1, X2) |
Problem 25: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(U41(X1, X2)) | → | proper#(X2) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2)) | → | proper#(X1) | | proper#(U91(X1, X2, X3, X4)) | → | proper#(X2) |
| proper#(U93(X1, X2, X3, X4)) | → | proper#(X4) | | proper#(U11(X)) | → | proper#(X) |
| proper#(U51(X1, X2)) | → | proper#(X1) | | proper#(U42(X)) | → | proper#(X) |
| proper#(U92(X1, X2, X3, X4)) | → | proper#(X2) | | proper#(U93(X1, X2, X3, X4)) | → | proper#(X3) |
| proper#(U71(X1, X2, X3)) | → | proper#(X1) | | proper#(U62(X)) | → | proper#(X) |
| proper#(U93(X1, X2, X3, X4)) | → | proper#(X1) | | proper#(take(X1, X2)) | → | proper#(X1) |
| proper#(U61(X1, X2)) | → | proper#(X2) | | proper#(isNatList(X)) | → | proper#(X) |
| proper#(U51(X1, X2)) | → | proper#(X2) | | proper#(U21(X)) | → | proper#(X) |
| proper#(U92(X1, X2, X3, X4)) | → | proper#(X4) | | proper#(U93(X1, X2, X3, X4)) | → | proper#(X2) |
| proper#(U72(X1, X2)) | → | proper#(X2) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(isNatIList(X)) | → | proper#(X) |
| proper#(U92(X1, X2, X3, X4)) | → | proper#(X1) | | proper#(U91(X1, X2, X3, X4)) | → | proper#(X3) |
| proper#(take(X1, X2)) | → | proper#(X2) | | proper#(U71(X1, X2, X3)) | → | proper#(X3) |
| proper#(U72(X1, X2)) | → | proper#(X1) | | proper#(U81(X)) | → | proper#(X) |
| proper#(U61(X1, X2)) | → | proper#(X1) | | proper#(U92(X1, X2, X3, X4)) | → | proper#(X3) |
| proper#(s(X)) | → | proper#(X) | | proper#(length(X)) | → | proper#(X) |
| proper#(U91(X1, X2, X3, X4)) | → | proper#(X4) | | proper#(U71(X1, X2, X3)) | → | proper#(X2) |
| proper#(U52(X)) | → | proper#(X) | | proper#(U91(X1, X2, X3, X4)) | → | proper#(X1) |
| proper#(U31(X)) | → | proper#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(U41(X1, X2)) | → | proper#(X2) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2)) | → | proper#(X1) | | proper#(U91(X1, X2, X3, X4)) | → | proper#(X2) |
| proper#(U93(X1, X2, X3, X4)) | → | proper#(X4) | | proper#(U11(X)) | → | proper#(X) |
| proper#(U92(X1, X2, X3, X4)) | → | proper#(X2) | | proper#(U51(X1, X2)) | → | proper#(X1) |
| proper#(U42(X)) | → | proper#(X) | | proper#(U93(X1, X2, X3, X4)) | → | proper#(X3) |
| proper#(U62(X)) | → | proper#(X) | | proper#(U71(X1, X2, X3)) | → | proper#(X1) |
| proper#(U93(X1, X2, X3, X4)) | → | proper#(X1) | | proper#(take(X1, X2)) | → | proper#(X1) |
| proper#(U61(X1, X2)) | → | proper#(X2) | | proper#(isNatList(X)) | → | proper#(X) |
| proper#(U51(X1, X2)) | → | proper#(X2) | | proper#(U92(X1, X2, X3, X4)) | → | proper#(X4) |
| proper#(U21(X)) | → | proper#(X) | | proper#(U93(X1, X2, X3, X4)) | → | proper#(X2) |
| proper#(U72(X1, X2)) | → | proper#(X2) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(isNatIList(X)) | → | proper#(X) |
| proper#(U92(X1, X2, X3, X4)) | → | proper#(X1) | | proper#(U91(X1, X2, X3, X4)) | → | proper#(X3) |
| proper#(take(X1, X2)) | → | proper#(X2) | | proper#(U71(X1, X2, X3)) | → | proper#(X3) |
| proper#(U72(X1, X2)) | → | proper#(X1) | | proper#(U81(X)) | → | proper#(X) |
| proper#(U61(X1, X2)) | → | proper#(X1) | | proper#(U92(X1, X2, X3, X4)) | → | proper#(X3) |
| proper#(s(X)) | → | proper#(X) | | proper#(length(X)) | → | proper#(X) |
| proper#(U91(X1, X2, X3, X4)) | → | proper#(X4) | | proper#(U71(X1, X2, X3)) | → | proper#(X2) |
| proper#(U91(X1, X2, X3, X4)) | → | proper#(X1) | | proper#(U52(X)) | → | proper#(X) |
| proper#(U31(X)) | → | proper#(X) |
Problem 26: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U72#(mark(X1), X2) | → | U72#(X1, X2) | | U72#(ok(X1), ok(X2)) | → | U72#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U72#(mark(X1), X2) | → | U72#(X1, X2) | | U72#(ok(X1), ok(X2)) | → | U72#(X1, X2) |
Problem 27: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U62#(mark(X)) | → | U62#(X) | | U62#(ok(X)) | → | U62#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U62#(mark(X)) | → | U62#(X) | | U62#(ok(X)) | → | U62#(X) |
Problem 28: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U52#(mark(X)) | → | U52#(X) | | U52#(ok(X)) | → | U52#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt)) | → | mark(tt) |
| active(U41(tt, V2)) | → | mark(U42(isNatIList(V2))) | | active(U42(tt)) | → | mark(tt) |
| active(U51(tt, V2)) | → | mark(U52(isNatList(V2))) | | active(U52(tt)) | → | mark(tt) |
| active(U61(tt, V2)) | → | mark(U62(isNatIList(V2))) | | active(U62(tt)) | → | mark(tt) |
| active(U71(tt, L, N)) | → | mark(U72(isNat(N), L)) | | active(U72(tt, L)) | → | mark(s(length(L))) |
| active(U81(tt)) | → | mark(nil) | | active(U91(tt, IL, M, N)) | → | mark(U92(isNat(M), IL, M, N)) |
| active(U92(tt, IL, M, N)) | → | mark(U93(isNat(N), IL, M, N)) | | active(U93(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(length(V1))) | → | mark(U11(isNatList(V1))) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(isNatIList(V)) | → | mark(U31(isNatList(V))) |
| active(isNatIList(zeros)) | → | mark(tt) | | active(isNatIList(cons(V1, V2))) | → | mark(U41(isNat(V1), V2)) |
| active(isNatList(nil)) | → | mark(tt) | | active(isNatList(cons(V1, V2))) | → | mark(U51(isNat(V1), V2)) |
| active(isNatList(take(V1, V2))) | → | mark(U61(isNat(V1), V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U71(isNatList(L), L, N)) | | active(take(0, IL)) | → | mark(U81(isNatIList(IL))) |
| active(take(s(M), cons(N, IL))) | → | mark(U91(isNatIList(IL), IL, M, N)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X)) | → | U11(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X)) | → | U31(active(X)) | | active(U41(X1, X2)) | → | U41(active(X1), X2) |
| active(U42(X)) | → | U42(active(X)) | | active(U51(X1, X2)) | → | U51(active(X1), X2) |
| active(U52(X)) | → | U52(active(X)) | | active(U61(X1, X2)) | → | U61(active(X1), X2) |
| active(U62(X)) | → | U62(active(X)) | | active(U71(X1, X2, X3)) | → | U71(active(X1), X2, X3) |
| active(U72(X1, X2)) | → | U72(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(U81(X)) | → | U81(active(X)) |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | | active(U92(X1, X2, X3, X4)) | → | U92(active(X1), X2, X3, X4) |
| active(U93(X1, X2, X3, X4)) | → | U93(active(X1), X2, X3, X4) | | active(take(X1, X2)) | → | take(active(X1), X2) |
| active(take(X1, X2)) | → | take(X1, active(X2)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| U11(mark(X)) | → | mark(U11(X)) | | U21(mark(X)) | → | mark(U21(X)) |
| U31(mark(X)) | → | mark(U31(X)) | | U41(mark(X1), X2) | → | mark(U41(X1, X2)) |
| U42(mark(X)) | → | mark(U42(X)) | | U51(mark(X1), X2) | → | mark(U51(X1, X2)) |
| U52(mark(X)) | → | mark(U52(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| U62(mark(X)) | → | mark(U62(X)) | | U71(mark(X1), X2, X3) | → | mark(U71(X1, X2, X3)) |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| length(mark(X)) | → | mark(length(X)) | | U81(mark(X)) | → | mark(U81(X)) |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | | U92(mark(X1), X2, X3, X4) | → | mark(U92(X1, X2, X3, X4)) |
| U93(mark(X1), X2, X3, X4) | → | mark(U93(X1, X2, X3, X4)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X)) | → | U11(proper(X)) | | proper(tt) | → | ok(tt) |
| proper(U21(X)) | → | U21(proper(X)) | | proper(U31(X)) | → | U31(proper(X)) |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | | proper(U42(X)) | → | U42(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) |
| proper(U52(X)) | → | U52(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(U62(X)) | → | U62(proper(X)) |
| proper(U71(X1, X2, X3)) | → | U71(proper(X1), proper(X2), proper(X3)) | | proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(U81(X)) | → | U81(proper(X)) |
| proper(nil) | → | ok(nil) | | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(U92(X1, X2, X3, X4)) | → | U92(proper(X1), proper(X2), proper(X3), proper(X4)) | | proper(U93(X1, X2, X3, X4)) | → | U93(proper(X1), proper(X2), proper(X3), proper(X4)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X)) | → | ok(U11(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X)) | → | ok(U31(X)) | | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) |
| U42(ok(X)) | → | ok(U42(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | | U52(ok(X)) | → | ok(U52(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| U62(ok(X)) | → | ok(U62(X)) | | U71(ok(X1), ok(X2), ok(X3)) | → | ok(U71(X1, X2, X3)) |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| U81(ok(X)) | → | ok(U81(X)) | | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) |
| U92(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U92(X1, X2, X3, X4)) | | U93(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U93(X1, X2, X3, X4)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNat, U62, U61, U42, U93, U92, U41, U91, length, ok, proper, U21, top, cons, isNatIList, mark, U71, U72, 0, isNatList, U51, s, tt, zeros, take, U81, U52, active, U11, U31, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U52#(mark(X)) | → | U52#(X) | | U52#(ok(X)) | → | U52#(X) |