TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60003 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (12965ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (2ms).
| – Problem 5 was processed with processor SubtermCriterion (3ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 was processed with processor SubtermCriterion (4ms).
| – Problem 9 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (8675ms), DependencyGraph (48ms), PolynomialLinearRange8NegiUR (30000ms), DependencyGraph (timeout), ReductionPairSAT (timeout)].
| – Problem 10 was processed with processor SubtermCriterion (2ms).
| – Problem 11 was processed with processor SubtermCriterion (1ms).
| – Problem 12 was processed with processor SubtermCriterion (4ms).
| – Problem 13 was processed with processor SubtermCriterion (1ms).
| – Problem 14 was processed with processor SubtermCriterion (1ms).
| – Problem 15 was processed with processor SubtermCriterion (2ms).
| – Problem 16 was processed with processor SubtermCriterion (3ms).
| – Problem 17 was processed with processor SubtermCriterion (3ms).
| – Problem 18 was processed with processor SubtermCriterion (4ms).
| – Problem 19 was processed with processor SubtermCriterion (2ms).
The following open problems remain:
Open Dependency Pair Problem 9
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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, active, U11, ok, proper, U31, U21, top, nil, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| proper#(U41(X1, X2)) | → | proper#(X2) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(length(X)) | → | length#(proper(X)) | | proper#(U41(X1, X2)) | → | proper#(X1) |
| active#(U51(X1, X2)) | → | active#(X1) | | proper#(U62(X1, X2)) | → | U62#(proper(X1), proper(X2)) |
| active#(U41(tt, V2)) | → | isNatIList#(V2) | | proper#(U52(X)) | → | U52#(proper(X)) |
| active#(U21(X)) | → | U21#(active(X)) | | active#(U51(tt, V2)) | → | U52#(isNatList(V2)) |
| active#(U51(tt, V2)) | → | isNatList#(V2) | | U51#(mark(X1), X2) | → | U51#(X1, X2) |
| active#(isNat(length(V1))) | → | isNatList#(V1) | | active#(U31(X)) | → | U31#(active(X)) |
| proper#(U42(X)) | → | U42#(proper(X)) | | active#(U42(X)) | → | U42#(active(X)) |
| top#(mark(X)) | → | proper#(X) | | active#(U62(tt, L)) | → | length#(L) |
| length#(mark(X)) | → | length#(X) | | active#(U11(X)) | → | U11#(active(X)) |
| proper#(U61(X1, X2, X3)) | → | U61#(proper(X1), proper(X2), proper(X3)) | | active#(length(X)) | → | active#(X) |
| isNat#(ok(X)) | → | isNat#(X) | | isNatIList#(ok(X)) | → | isNatIList#(X) |
| active#(isNatIList(cons(V1, V2))) | → | isNat#(V1) | | active#(U41(X1, X2)) | → | active#(X1) |
| proper#(U21(X)) | → | U21#(proper(X)) | | active#(U61(tt, L, N)) | → | U62#(isNat(N), L) |
| proper#(U61(X1, X2, X3)) | → | proper#(X1) | | active#(U62(X1, X2)) | → | U62#(active(X1), X2) |
| active#(U61(X1, X2, X3)) | → | active#(X1) | | proper#(U62(X1, X2)) | → | proper#(X2) |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | proper#(U62(X1, X2)) | → | proper#(X1) |
| proper#(U11(X)) | → | proper#(X) | | top#(ok(X)) | → | active#(X) |
| active#(isNat(length(V1))) | → | U11#(isNatList(V1)) | | U62#(ok(X1), ok(X2)) | → | U62#(X1, X2) |
| active#(U41(tt, V2)) | → | U42#(isNatIList(V2)) | | active#(U52(X)) | → | active#(X) |
| proper#(isNatList(X)) | → | proper#(X) | | proper#(isNat(X)) | → | isNat#(proper(X)) |
| proper#(U51(X1, X2)) | → | proper#(X2) | | U61#(mark(X1), X2, X3) | → | U61#(X1, X2, X3) |
| U42#(ok(X)) | → | U42#(X) | | proper#(U21(X)) | → | proper#(X) |
| active#(isNat(s(V1))) | → | isNat#(V1) | | U11#(mark(X)) | → | U11#(X) |
| U51#(ok(X1), ok(X2)) | → | U51#(X1, X2) | | active#(U52(X)) | → | U52#(active(X)) |
| proper#(isNat(X)) | → | proper#(X) | | active#(U61(X1, X2, X3)) | → | U61#(active(X1), X2, X3) |
| active#(isNat(s(V1))) | → | U21#(isNat(V1)) | | active#(s(X)) | → | s#(active(X)) |
| active#(isNatList(cons(V1, V2))) | → | U51#(isNat(V1), V2) | | U31#(ok(X)) | → | U31#(X) |
| s#(ok(X)) | → | s#(X) | | proper#(length(X)) | → | proper#(X) |
| proper#(s(X)) | → | s#(proper(X)) | | active#(zeros) | → | cons#(0, zeros) |
| active#(isNatIList(V)) | → | U31#(isNatList(V)) | | top#(ok(X)) | → | top#(active(X)) |
| active#(isNatIList(cons(V1, V2))) | → | U41#(isNat(V1), V2) | | proper#(isNatIList(X)) | → | isNatIList#(proper(X)) |
| cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) | | active#(cons(X1, X2)) | → | cons#(active(X1), X2) |
| U42#(mark(X)) | → | U42#(X) | | proper#(U51(X1, X2)) | → | U51#(proper(X1), proper(X2)) |
| active#(U62(X1, X2)) | → | active#(X1) | | active#(U41(X1, X2)) | → | U41#(active(X1), X2) |
| active#(length(cons(N, L))) | → | U61#(isNatList(L), L, N) | | active#(U51(X1, X2)) | → | U51#(active(X1), X2) |
| length#(ok(X)) | → | length#(X) | | U62#(mark(X1), X2) | → | U62#(X1, X2) |
| U21#(ok(X)) | → | U21#(X) | | top#(mark(X)) | → | top#(proper(X)) |
| active#(U31(X)) | → | active#(X) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(isNatIList(X)) | → | proper#(X) | | active#(U61(tt, L, N)) | → | isNat#(N) |
| U52#(ok(X)) | → | U52#(X) | | proper#(U61(X1, X2, X3)) | → | proper#(X2) |
| isNatList#(ok(X)) | → | isNatList#(X) | | proper#(s(X)) | → | proper#(X) |
| proper#(isNatList(X)) | → | isNatList#(proper(X)) | | active#(isNatList(cons(V1, V2))) | → | isNat#(V1) |
| U31#(mark(X)) | → | U31#(X) | | active#(cons(X1, X2)) | → | active#(X1) |
| active#(U11(X)) | → | active#(X) | | active#(U42(X)) | → | active#(X) |
| proper#(U61(X1, X2, X3)) | → | proper#(X3) | | proper#(U31(X)) | → | U31#(proper(X)) |
| proper#(U42(X)) | → | proper#(X) | | proper#(U51(X1, X2)) | → | proper#(X1) |
| active#(U21(X)) | → | active#(X) | | active#(length(cons(N, L))) | → | isNatList#(L) |
| U41#(ok(X1), ok(X2)) | → | U41#(X1, X2) | | active#(U62(tt, L)) | → | s#(length(L)) |
| proper#(U11(X)) | → | U11#(proper(X)) | | U41#(mark(X1), X2) | → | U41#(X1, X2) |
| active#(length(X)) | → | length#(active(X)) | | U21#(mark(X)) | → | U21#(X) |
| U61#(ok(X1), ok(X2), ok(X3)) | → | U61#(X1, X2, X3) | | proper#(U41(X1, X2)) | → | U41#(proper(X1), proper(X2)) |
| active#(isNatIList(V)) | → | isNatList#(V) | | s#(mark(X)) | → | s#(X) |
| proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) | | active#(s(X)) | → | active#(X) |
| U11#(ok(X)) | → | U11#(X) | | U52#(mark(X)) | → | U52#(X) |
| 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
The following SCCs where found
| U41#(ok(X1), ok(X2)) → U41#(X1, X2) | U41#(mark(X1), X2) → U41#(X1, X2) |
| U61#(ok(X1), ok(X2), ok(X3)) → U61#(X1, X2, X3) | U61#(mark(X1), X2, X3) → U61#(X1, X2, X3) |
| U42#(mark(X)) → U42#(X) | U42#(ok(X)) → U42#(X) |
| U31#(ok(X)) → U31#(X) | U31#(mark(X)) → U31#(X) |
| isNat#(ok(X)) → isNat#(X) |
| U21#(ok(X)) → U21#(X) | U21#(mark(X)) → U21#(X) |
| isNatList#(ok(X)) → isNatList#(X) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| isNatIList#(ok(X)) → isNatIList#(X) |
| cons#(mark(X1), X2) → cons#(X1, X2) | cons#(ok(X1), ok(X2)) → cons#(X1, X2) |
| length#(mark(X)) → length#(X) | length#(ok(X)) → length#(X) |
| proper#(U41(X1, X2)) → proper#(X2) | proper#(cons(X1, X2)) → proper#(X1) |
| proper#(U62(X1, X2)) → proper#(X2) | proper#(isNat(X)) → proper#(X) |
| proper#(U61(X1, X2, X3)) → proper#(X3) | proper#(U41(X1, X2)) → proper#(X1) |
| proper#(cons(X1, X2)) → proper#(X2) | proper#(isNatIList(X)) → proper#(X) |
| proper#(U62(X1, X2)) → proper#(X1) | proper#(U11(X)) → proper#(X) |
| proper#(U61(X1, X2, X3)) → proper#(X2) | proper#(U51(X1, X2)) → proper#(X1) |
| proper#(U42(X)) → proper#(X) | proper#(length(X)) → proper#(X) |
| proper#(s(X)) → proper#(X) | proper#(isNatList(X)) → proper#(X) |
| proper#(U61(X1, X2, X3)) → proper#(X1) | proper#(U51(X1, X2)) → proper#(X2) |
| proper#(U21(X)) → proper#(X) | proper#(U52(X)) → proper#(X) |
| proper#(U31(X)) → proper#(X) |
| U11#(ok(X)) → U11#(X) | U11#(mark(X)) → U11#(X) |
| U52#(mark(X)) → U52#(X) | U52#(ok(X)) → U52#(X) |
| active#(U62(X1, X2)) → active#(X1) | active#(U61(X1, X2, X3)) → active#(X1) |
| active#(U11(X)) → active#(X) | active#(U41(X1, X2)) → active#(X1) |
| active#(U42(X)) → active#(X) | active#(U31(X)) → active#(X) |
| active#(U51(X1, X2)) → active#(X1) | active#(s(X)) → active#(X) |
| active#(length(X)) → active#(X) | active#(U52(X)) → active#(X) |
| active#(U21(X)) → active#(X) | active#(cons(X1, X2)) → active#(X1) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| U62#(ok(X1), ok(X2)) → U62#(X1, X2) | U62#(mark(X1), X2) → U62#(X1, X2) |
| U51#(mark(X1), X2) → U51#(X1, X2) | U51#(ok(X1), ok(X2)) → U51#(X1, X2) |
Problem 2: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
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 3: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U52#(mark(X)) | → | U52#(X) | | U52#(ok(X)) | → | U52#(X) |
Problem 4: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
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 5: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatIList#(ok(X)) | → | isNatIList#(X) |
Problem 6: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNat#(ok(X)) | → | isNat#(X) |
Problem 7: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
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 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U62#(ok(X1), ok(X2)) | → | U62#(X1, X2) | | U62#(mark(X1), X2) | → | U62#(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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U62#(ok(X1), ok(X2)) | → | U62#(X1, X2) | | U62#(mark(X1), X2) | → | U62#(X1, X2) |
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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
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
| 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
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 12: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
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 13: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
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 14: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U61#(ok(X1), ok(X2), ok(X3)) | → | U61#(X1, X2, X3) | | U61#(mark(X1), X2, X3) | → | U61#(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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U61#(ok(X1), ok(X2), ok(X3)) | → | U61#(X1, X2, X3) | | U61#(mark(X1), X2, X3) | → | U61#(X1, X2, X3) |
Problem 15: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatList#(ok(X)) | → | isNatList#(X) |
Problem 16: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(U41(X1, X2)) | → | proper#(X2) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(U62(X1, X2)) | → | proper#(X2) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2)) | → | proper#(X1) | | proper#(U61(X1, X2, X3)) | → | proper#(X3) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(isNatIList(X)) | → | proper#(X) |
| proper#(U62(X1, X2)) | → | proper#(X1) | | proper#(U11(X)) | → | proper#(X) |
| proper#(U61(X1, X2, X3)) | → | proper#(X2) | | proper#(U42(X)) | → | proper#(X) |
| proper#(U51(X1, X2)) | → | proper#(X1) | | proper#(length(X)) | → | proper#(X) |
| proper#(s(X)) | → | proper#(X) | | proper#(isNatList(X)) | → | proper#(X) |
| proper#(U61(X1, X2, X3)) | → | proper#(X1) | | proper#(U51(X1, X2)) | → | proper#(X2) |
| proper#(U21(X)) | → | proper#(X) | | 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(U41(X1, X2)) | → | proper#(X2) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(U62(X1, X2)) | → | proper#(X2) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2)) | → | proper#(X1) | | proper#(U61(X1, X2, X3)) | → | proper#(X3) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(isNatIList(X)) | → | proper#(X) |
| proper#(U62(X1, X2)) | → | proper#(X1) | | proper#(U11(X)) | → | proper#(X) |
| proper#(U61(X1, X2, X3)) | → | proper#(X2) | | proper#(U42(X)) | → | proper#(X) |
| proper#(U51(X1, X2)) | → | proper#(X1) | | proper#(length(X)) | → | proper#(X) |
| proper#(s(X)) | → | proper#(X) | | proper#(isNatList(X)) | → | proper#(X) |
| proper#(U61(X1, X2, X3)) | → | proper#(X1) | | proper#(U51(X1, X2)) | → | proper#(X2) |
| proper#(U21(X)) | → | proper#(X) | | proper#(U52(X)) | → | proper#(X) |
| proper#(U31(X)) | → | proper#(X) |
Problem 17: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Problem 18: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(U62(X1, X2)) | → | active#(X1) | | active#(U61(X1, X2, X3)) | → | active#(X1) |
| active#(U41(X1, X2)) | → | active#(X1) | | active#(U11(X)) | → | active#(X) |
| active#(U42(X)) | → | active#(X) | | active#(U31(X)) | → | active#(X) |
| active#(U51(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(U52(X)) | → | active#(X) | | active#(length(X)) | → | active#(X) |
| active#(U21(X)) | → | active#(X) | | 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(U61(X1, X2, X3)) | → | active#(X1) | | active#(U62(X1, X2)) | → | active#(X1) |
| active#(U41(X1, X2)) | → | active#(X1) | | active#(U11(X)) | → | active#(X) |
| active#(U31(X)) | → | active#(X) | | active#(U42(X)) | → | active#(X) |
| active#(U51(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(U52(X)) | → | active#(X) | | active#(length(X)) | → | active#(X) |
| active#(U21(X)) | → | active#(X) | | active#(cons(X1, X2)) | → | active#(X1) |
Problem 19: 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, L, N)) | → | mark(U62(isNat(N), L)) | | active(U62(tt, L)) | → | mark(s(length(L))) |
| 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(length(nil)) | → | mark(0) | | active(length(cons(N, L))) | → | mark(U61(isNatList(L), L, 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, X3)) | → | U61(active(X1), X2, X3) | | active(U62(X1, X2)) | → | U62(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(length(X)) | → | length(active(X)) |
| 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, X3) | → | mark(U61(X1, X2, X3)) | | U62(mark(X1), X2) | → | mark(U62(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| 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, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | | proper(isNat(X)) | → | isNat(proper(X)) |
| proper(s(X)) | → | s(proper(X)) | | proper(length(X)) | → | length(proper(X)) |
| proper(nil) | → | ok(nil) | | 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(X3)) | → | ok(U61(X1, X2, X3)) |
| U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIList, mark, isNat, U62, U61, 0, s, U51, U42, isNatList, zeros, tt, U41, U52, length, U11, active, ok, U31, proper, U21, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U21#(ok(X)) | → | U21#(X) | | U21#(mark(X)) | → | U21#(X) |