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 (49735ms).
| – Problem 2 was processed with processor SubtermCriterion (46ms).
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| – Problem 4 was processed with processor SubtermCriterion (12ms).
| – Problem 5 was processed with processor SubtermCriterion (2ms).
| – Problem 6 was processed with processor SubtermCriterion (3ms).
| – Problem 7 was processed with processor SubtermCriterion (3ms).
| – Problem 8 was processed with processor DependencyGraph (4628ms).
| | – Problem 30 remains open; application of the following processors failed [].
| – Problem 9 was processed with processor SubtermCriterion (4ms).
| – Problem 10 was processed with processor SubtermCriterion (2ms).
| – Problem 11 was processed with processor SubtermCriterion (2ms).
| – Problem 12 was processed with processor SubtermCriterion (4ms).
| – Problem 13 was processed with processor SubtermCriterion (2ms).
| – Problem 14 remains open; application of the following processors failed [SubtermCriterion (3ms), DependencyGraph (9ms), PolynomialLinearRange4iUR (timeout)].
| – Problem 15 was processed with processor SubtermCriterion (2ms).
| – Problem 16 was processed with processor SubtermCriterion (4ms).
| – Problem 17 was processed with processor SubtermCriterion (3ms).
| – Problem 18 was processed with processor DependencyGraph (4529ms).
| | – Problem 29 was processed with processor PolynomialLinearRange4iUR (220ms).
| | | – Problem 31 remains open; application of the following processors failed [].
| – Problem 19 was processed with processor SubtermCriterion (2ms).
| – Problem 20 was processed with processor SubtermCriterion (4ms).
| – Problem 21 was processed with processor SubtermCriterion (2ms).
| – Problem 22 was processed with processor SubtermCriterion (3ms).
| – Problem 23 was processed with processor SubtermCriterion (2ms).
| – Problem 24 was processed with processor SubtermCriterion (2ms).
| – Problem 25 was processed with processor SubtermCriterion (4ms).
| – Problem 26 was processed with processor SubtermCriterion (2ms).
| – Problem 27 was processed with processor SubtermCriterion (6ms).
| – Problem 28 was processed with processor SubtermCriterion (4ms).
The following open problems remain:
Open Dependency Pair Problem 29
Dependency Pairs
| active#(U51(X1, X2, X3)) | → | active#(X1) | | active#(U32(X)) | → | active#(X) |
| active#(U61(X1, X2)) | → | active#(X1) | | active#(U42(X1, X2)) | → | active#(X1) |
| active#(U12(X)) | → | active#(X) | | active#(length(X)) | → | active#(X) |
| active#(U43(X)) | → | active#(X) | | active#(U52(X1, X2)) | → | active#(X1) |
| active#(U41(X1, X2, X3)) | → | active#(X1) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(U53(X)) | → | active#(X) | | active#(U22(X)) | → | active#(X) |
| active#(s(X)) | → | active#(X) | | active#(U31(X1, X2)) | → | active#(X1) |
| active#(and(X1, X2)) | → | active#(X1) | | active#(U11(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, ok, proper, U21, top, U22, cons, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, active, U12, U31, U32, nil
Open Dependency Pair Problem 14
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, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, ok, proper, U21, top, U22, cons, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, active, U12, U31, U32, nil
Open Dependency Pair Problem 30
Dependency Pairs
| active#(U51(X1, X2, X3)) | → | active#(X1) | | active#(U32(X)) | → | active#(X) |
| active#(U61(X1, X2)) | → | active#(X1) | | active#(U42(X1, X2)) | → | active#(X1) |
| active#(U12(X)) | → | active#(X) | | active#(length(X)) | → | active#(X) |
| active#(U43(X)) | → | active#(X) | | active#(U52(X1, X2)) | → | active#(X1) |
| active#(U41(X1, X2, X3)) | → | active#(X1) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(U53(X)) | → | active#(X) | | active#(U22(X)) | → | active#(X) |
| active#(s(X)) | → | active#(X) | | active#(U31(X1, X2)) | → | active#(X1) |
| active#(and(X1, X2)) | → | active#(X1) | | active#(U11(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, ok, proper, U21, top, U22, cons, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, active, U12, U31, U32, nil
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| proper#(cons(X1, X2)) | → | proper#(X1) | | proper#(length(X)) | → | length#(proper(X)) |
| proper#(U11(X1, X2)) | → | proper#(X1) | | active#(isNatIList(V)) | → | isNatIListKind#(V) |
| active#(U52(tt, V2)) | → | isNatList#(V2) | | active#(U12(X)) | → | active#(X) |
| U11#(mark(X1), X2) | → | U11#(X1, X2) | | proper#(U41(X1, X2, X3)) | → | proper#(X2) |
| proper#(isNatKind(X)) | → | proper#(X) | | proper#(U11(X1, X2)) | → | proper#(X2) |
| active#(isNatIListKind(cons(V1, V2))) | → | isNatKind#(V1) | | U43#(ok(X)) | → | U43#(X) |
| top#(mark(X)) | → | proper#(X) | | length#(mark(X)) | → | length#(X) |
| isNatKind#(ok(X)) | → | isNatKind#(X) | | active#(U51(tt, V1, V2)) | → | U52#(isNat(V1), V2) |
| active#(U32(X)) | → | active#(X) | | active#(U43(X)) | → | U43#(active(X)) |
| active#(U61(X1, X2)) | → | active#(X1) | | active#(U52(tt, V2)) | → | U53#(isNatList(V2)) |
| active#(U61(tt, L)) | → | length#(L) | | active#(length(X)) | → | active#(X) |
| proper#(U12(X)) | → | U12#(proper(X)) | | isNat#(ok(X)) | → | isNat#(X) |
| and#(mark(X1), X2) | → | and#(X1, X2) | | active#(U21(tt, V1)) | → | isNat#(V1) |
| U42#(mark(X1), X2) | → | U42#(X1, X2) | | isNatIList#(ok(X)) | → | isNatIList#(X) |
| active#(U41(X1, X2, X3)) | → | active#(X1) | | active#(U42(X1, X2)) | → | U42#(active(X1), X2) |
| active#(isNat(s(V1))) | → | isNatKind#(V1) | | active#(isNatKind(length(V1))) | → | isNatIListKind#(V1) |
| active#(isNatList(cons(V1, V2))) | → | isNatIListKind#(V2) | | proper#(U51(X1, X2, X3)) | → | proper#(X1) |
| U53#(mark(X)) | → | U53#(X) | | proper#(U32(X)) | → | U32#(proper(X)) |
| active#(U31(tt, V)) | → | U32#(isNatList(V)) | | proper#(U22(X)) | → | proper#(X) |
| active#(U53(X)) | → | U53#(active(X)) | | active#(isNatList(cons(V1, V2))) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| proper#(U51(X1, X2, X3)) | → | proper#(X2) | | proper#(U52(X1, X2)) | → | proper#(X1) |
| proper#(isNatKind(X)) | → | isNatKind#(proper(X)) | | U51#(ok(X1), ok(X2), ok(X3)) | → | U51#(X1, X2, X3) |
| proper#(isNatIListKind(X)) | → | proper#(X) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| active#(U52(X1, X2)) | → | U52#(active(X1), X2) | | active#(U41(X1, X2, X3)) | → | U41#(active(X1), X2, X3) |
| U32#(mark(X)) | → | U32#(X) | | top#(ok(X)) | → | active#(X) |
| proper#(U42(X1, X2)) | → | proper#(X1) | | active#(isNatList(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| active#(U52(X1, X2)) | → | active#(X1) | | active#(U53(X)) | → | active#(X) |
| active#(and(X1, X2)) | → | and#(active(X1), X2) | | U52#(ok(X1), ok(X2)) | → | U52#(X1, X2) |
| proper#(U41(X1, X2, X3)) | → | proper#(X1) | | active#(isNatIListKind(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| active#(length(cons(N, L))) | → | U61#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | | proper#(isNatList(X)) | → | proper#(X) |
| active#(U31(X1, X2)) | → | active#(X1) | | proper#(isNat(X)) | → | isNat#(proper(X)) |
| active#(U41(tt, V1, V2)) | → | U42#(isNat(V1), V2) | | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) |
| active#(isNatIList(cons(V1, V2))) | → | isNatIListKind#(V2) | | active#(isNat(length(V1))) | → | isNatIListKind#(V1) |
| active#(U21(X1, X2)) | → | U21#(active(X1), X2) | | U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) |
| proper#(isNat(X)) | → | proper#(X) | | proper#(U12(X)) | → | proper#(X) |
| active#(length(cons(N, L))) | → | and#(isNatList(L), isNatIListKind(L)) | | proper#(U22(X)) | → | U22#(proper(X)) |
| proper#(U42(X1, X2)) | → | U42#(proper(X1), proper(X2)) | | active#(U11(X1, X2)) | → | U11#(active(X1), X2) |
| active#(s(X)) | → | s#(active(X)) | | s#(ok(X)) | → | s#(X) |
| proper#(length(X)) | → | proper#(X) | | active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) |
| active#(U51(X1, X2, X3)) | → | U51#(active(X1), X2, X3) | | proper#(s(X)) | → | s#(proper(X)) |
| active#(zeros) | → | cons#(0, zeros) | | proper#(U31(X1, X2)) | → | U31#(proper(X1), proper(X2)) |
| proper#(U21(X1, X2)) | → | proper#(X1) | | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) |
| active#(U11(tt, V1)) | → | isNatList#(V1) | | U52#(mark(X1), X2) | → | U52#(X1, X2) |
| top#(ok(X)) | → | top#(active(X)) | | proper#(U42(X1, X2)) | → | proper#(X2) |
| active#(U51(tt, V1, V2)) | → | isNat#(V1) | | proper#(U31(X1, X2)) | → | proper#(X1) |
| U32#(ok(X)) | → | U32#(X) | | U11#(ok(X1), ok(X2)) | → | U11#(X1, X2) |
| isNatIListKind#(ok(X)) | → | isNatIListKind#(X) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
| proper#(isNatIList(X)) | → | isNatIList#(proper(X)) | | active#(U42(tt, V2)) | → | isNatIList#(V2) |
| proper#(U11(X1, X2)) | → | U11#(proper(X1), proper(X2)) | | proper#(U52(X1, X2)) | → | U52#(proper(X1), proper(X2)) |
| proper#(and(X1, X2)) | → | and#(proper(X1), proper(X2)) | | U12#(mark(X)) | → | U12#(X) |
| active#(cons(X1, X2)) | → | cons#(active(X1), X2) | | active#(U42(tt, V2)) | → | U43#(isNatIList(V2)) |
| active#(U61(X1, X2)) | → | U61#(active(X1), X2) | | proper#(and(X1, X2)) | → | proper#(X2) |
| proper#(U61(X1, X2)) | → | proper#(X2) | | U61#(mark(X1), X2) | → | U61#(X1, X2) |
| active#(U11(tt, V1)) | → | U12#(isNatList(V1)) | | U22#(mark(X)) | → | U22#(X) |
| length#(ok(X)) | → | length#(X) | | active#(U61(tt, L)) | → | s#(length(L)) |
| active#(U11(X1, X2)) | → | active#(X1) | | proper#(U41(X1, X2, X3)) | → | U41#(proper(X1), proper(X2), proper(X3)) |
| proper#(U43(X)) | → | proper#(X) | | U61#(ok(X1), ok(X2)) | → | U61#(X1, X2) |
| active#(U51(X1, X2, X3)) | → | active#(X1) | | top#(mark(X)) | → | top#(proper(X)) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(isNatIList(X)) | → | proper#(X) |
| active#(isNatIListKind(cons(V1, V2))) | → | isNatIListKind#(V2) | | active#(length(cons(N, L))) | → | and#(isNat(N), isNatKind(N)) |
| U22#(ok(X)) | → | U22#(X) | | active#(isNatIList(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| isNatList#(ok(X)) | → | isNatList#(X) | | active#(U21(tt, V1)) | → | U22#(isNat(V1)) |
| active#(U31(tt, V)) | → | isNatList#(V) | | proper#(s(X)) | → | proper#(X) |
| active#(length(cons(N, L))) | → | isNatIListKind#(L) | | proper#(isNatList(X)) | → | isNatList#(proper(X)) |
| active#(isNat(length(V1))) | → | U11#(isNatIListKind(V1), V1) | | proper#(U43(X)) | → | U43#(proper(X)) |
| proper#(U51(X1, X2, X3)) | → | U51#(proper(X1), proper(X2), proper(X3)) | | proper#(U61(X1, X2)) | → | U61#(proper(X1), proper(X2)) |
| active#(cons(X1, X2)) | → | active#(X1) | | proper#(U53(X)) | → | U53#(proper(X)) |
| active#(length(cons(N, L))) | → | isNatKind#(N) | | proper#(U41(X1, X2, X3)) | → | proper#(X3) |
| active#(U42(X1, X2)) | → | active#(X1) | | proper#(U31(X1, X2)) | → | proper#(X2) |
| and#(ok(X1), ok(X2)) | → | and#(X1, X2) | | active#(U43(X)) | → | active#(X) |
| proper#(and(X1, X2)) | → | proper#(X1) | | active#(length(cons(N, L))) | → | and#(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))) |
| U53#(ok(X)) | → | U53#(X) | | active#(length(cons(N, L))) | → | isNatList#(L) |
| active#(U41(tt, V1, V2)) | → | isNat#(V1) | | proper#(U21(X1, X2)) | → | proper#(X2) |
| proper#(U32(X)) | → | proper#(X) | | active#(U31(X1, X2)) | → | U31#(active(X1), X2) |
| active#(U12(X)) | → | U12#(active(X)) | | active#(length(cons(N, L))) | → | isNat#(N) |
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | | active#(U22(X)) | → | U22#(active(X)) |
| proper#(isNatIListKind(X)) | → | isNatIListKind#(proper(X)) | | proper#(U51(X1, X2, X3)) | → | proper#(X3) |
| active#(isNatKind(s(V1))) | → | isNatKind#(V1) | | proper#(U21(X1, X2)) | → | U21#(proper(X1), proper(X2)) |
| proper#(U53(X)) | → | proper#(X) | | U43#(mark(X)) | → | U43#(X) |
| active#(length(X)) | → | length#(active(X)) | | active#(U32(X)) | → | U32#(active(X)) |
| U42#(ok(X1), ok(X2)) | → | U42#(X1, X2) | | active#(isNatIList(cons(V1, V2))) | → | isNatKind#(V1) |
| U31#(mark(X1), X2) | → | U31#(X1, X2) | | proper#(U61(X1, X2)) | → | proper#(X1) |
| s#(mark(X)) | → | s#(X) | | proper#(U52(X1, X2)) | → | proper#(X2) |
| active#(U21(X1, X2)) | → | active#(X1) | | active#(isNatIList(cons(V1, V2))) | → | U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) | | active#(U22(X)) | → | active#(X) |
| U12#(ok(X)) | → | U12#(X) | | U21#(mark(X1), X2) | → | U21#(X1, X2) |
| active#(s(X)) | → | active#(X) | | active#(isNatList(cons(V1, V2))) | → | isNatKind#(V1) |
| active#(and(X1, X2)) | → | active#(X1) | | active#(isNatIList(V)) | → | U31#(isNatIListKind(V), V) |
| U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
The following SCCs where found
| active#(U52(tt, V2)) → isNatList#(V2) | active#(isNatIList(V)) → isNatIListKind#(V) |
| active#(U12(X)) → active#(X) | active#(U61(X1, X2)) → U61#(active(X1), X2) |
| active#(isNatIListKind(cons(V1, V2))) → isNatKind#(V1) | active#(U11(tt, V1)) → U12#(isNatList(V1)) |
| active#(U11(X1, X2)) → active#(X1) | active#(U61(tt, L)) → s#(length(L)) |
| active#(U51(X1, X2, X3)) → active#(X1) | active#(U51(tt, V1, V2)) → U52#(isNat(V1), V2) |
| active#(U32(X)) → active#(X) | active#(U61(X1, X2)) → active#(X1) |
| active#(length(cons(N, L))) → and#(isNat(N), isNatKind(N)) | active#(length(X)) → active#(X) |
| active#(U52(tt, V2)) → U53#(isNatList(V2)) | active#(U21(tt, V1)) → isNat#(V1) |
| active#(U41(X1, X2, X3)) → active#(X1) | active#(U31(tt, V)) → isNatList#(V) |
| active#(isNatList(cons(V1, V2))) → isNatIListKind#(V2) | active#(length(cons(N, L))) → isNatIListKind#(L) |
| active#(isNat(length(V1))) → U11#(isNatIListKind(V1), V1) | active#(U31(tt, V)) → U32#(isNatList(V)) |
| active#(isNatList(cons(V1, V2))) → U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | active#(cons(X1, X2)) → active#(X1) |
| active#(length(cons(N, L))) → isNatKind#(N) | active#(U42(X1, X2)) → active#(X1) |
| active#(U52(X1, X2)) → U52#(active(X1), X2) | active#(U41(X1, X2, X3)) → U41#(active(X1), X2, X3) |
| active#(U43(X)) → active#(X) | active#(isNatList(cons(V1, V2))) → and#(isNatKind(V1), isNatIListKind(V2)) |
| active#(U52(X1, X2)) → active#(X1) | active#(U41(tt, V1, V2)) → isNat#(V1) |
| active#(U53(X)) → active#(X) | active#(and(X1, X2)) → and#(active(X1), X2) |
| active#(U12(X)) → U12#(active(X)) | active#(U31(X1, X2)) → U31#(active(X1), X2) |
| active#(length(cons(N, L))) → isNat#(N) | active#(length(cons(N, L))) → U61#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
| active#(U31(X1, X2)) → active#(X1) | active#(length(X)) → length#(active(X)) |
| active#(U32(X)) → U32#(active(X)) | active#(length(cons(N, L))) → and#(isNatList(L), isNatIListKind(L)) |
| active#(isNatIList(cons(V1, V2))) → isNatKind#(V1) | active#(U11(X1, X2)) → U11#(active(X1), X2) |
| active#(isNatIList(cons(V1, V2))) → U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | active#(U21(X1, X2)) → active#(X1) |
| active#(U22(X)) → active#(X) | active#(U51(X1, X2, X3)) → U51#(active(X1), X2, X3) |
| active#(isNat(s(V1))) → U21#(isNatKind(V1), V1) | active#(s(X)) → active#(X) |
| active#(zeros) → cons#(0, zeros) | active#(isNatList(cons(V1, V2))) → isNatKind#(V1) |
| active#(isNatIList(V)) → U31#(isNatIListKind(V), V) | active#(and(X1, X2)) → active#(X1) |
| isNatIListKind#(ok(X)) → isNatIListKind#(X) |
| U32#(ok(X)) → U32#(X) | U32#(mark(X)) → U32#(X) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| active#(U51(X1, X2, X3)) → active#(X1) | active#(U32(X)) → active#(X) |
| active#(U42(X1, X2)) → active#(X1) | active#(U12(X)) → active#(X) |
| active#(U61(X1, X2)) → active#(X1) | active#(length(X)) → active#(X) |
| active#(U43(X)) → active#(X) | active#(U52(X1, X2)) → active#(X1) |
| active#(U41(X1, X2, X3)) → active#(X1) | active#(U53(X)) → active#(X) |
| active#(U21(X1, X2)) → active#(X1) | active#(U22(X)) → active#(X) |
| active#(s(X)) → active#(X) | active#(U31(X1, X2)) → active#(X1) |
| active#(and(X1, X2)) → active#(X1) | active#(U11(X1, X2)) → active#(X1) |
| active#(cons(X1, X2)) → active#(X1) |
| proper#(U52(X1, X2)) → proper#(X1) | proper#(U51(X1, X2, X3)) → proper#(X2) |
| proper#(cons(X1, X2)) → proper#(X1) | proper#(U11(X1, X2)) → proper#(X1) |
| proper#(isNatIListKind(X)) → proper#(X) | proper#(U42(X1, X2)) → proper#(X2) |
| proper#(U41(X1, X2, X3)) → proper#(X3) | proper#(U31(X1, X2)) → proper#(X1) |
| proper#(U31(X1, X2)) → proper#(X2) | proper#(and(X1, X2)) → proper#(X1) |
| proper#(U41(X1, X2, X3)) → proper#(X2) | proper#(U42(X1, X2)) → proper#(X1) |
| proper#(U21(X1, X2)) → proper#(X2) | proper#(U32(X)) → proper#(X) |
| proper#(isNatKind(X)) → proper#(X) | proper#(U11(X1, X2)) → proper#(X2) |
| proper#(and(X1, X2)) → proper#(X2) | proper#(U61(X1, X2)) → proper#(X2) |
| proper#(U41(X1, X2, X3)) → proper#(X1) | proper#(isNatList(X)) → proper#(X) |
| proper#(U51(X1, X2, X3)) → proper#(X3) | proper#(U53(X)) → proper#(X) |
| proper#(U43(X)) → proper#(X) | proper#(isNat(X)) → proper#(X) |
| proper#(cons(X1, X2)) → proper#(X2) | proper#(U12(X)) → proper#(X) |
| proper#(isNatIList(X)) → proper#(X) | proper#(U61(X1, X2)) → proper#(X1) |
| proper#(U52(X1, X2)) → proper#(X2) | proper#(length(X)) → proper#(X) |
| proper#(s(X)) → proper#(X) | proper#(U51(X1, X2, X3)) → proper#(X1) |
| proper#(U21(X1, X2)) → proper#(X1) | proper#(U22(X)) → proper#(X) |
| U43#(mark(X)) → U43#(X) | U43#(ok(X)) → U43#(X) |
| U42#(ok(X1), ok(X2)) → U42#(X1, X2) | U42#(mark(X1), X2) → U42#(X1, X2) |
| U51#(mark(X1), X2, X3) → U51#(X1, X2, X3) | U51#(ok(X1), ok(X2), ok(X3)) → U51#(X1, X2, X3) |
| active#(U52(tt, V2)) → isNatList#(V2) | active#(isNatIList(V)) → isNatIListKind#(V) |
| active#(U12(X)) → active#(X) | active#(cons(X1, X2)) → cons#(active(X1), X2) |
| active#(U61(X1, X2)) → U61#(active(X1), X2) | active#(isNatIListKind(cons(V1, V2))) → isNatKind#(V1) |
| active#(U11(tt, V1)) → U12#(isNatList(V1)) | active#(U11(X1, X2)) → active#(X1) |
| active#(U61(tt, L)) → s#(length(L)) | active#(U51(X1, X2, X3)) → active#(X1) |
| active#(U51(tt, V1, V2)) → U52#(isNat(V1), V2) | active#(U32(X)) → active#(X) |
| active#(U61(X1, X2)) → active#(X1) | active#(length(cons(N, L))) → and#(isNat(N), isNatKind(N)) |
| active#(length(X)) → active#(X) | active#(U52(tt, V2)) → U53#(isNatList(V2)) |
| active#(U21(tt, V1)) → isNat#(V1) | active#(U41(X1, X2, X3)) → active#(X1) |
| active#(U31(tt, V)) → isNatList#(V) | active#(isNatList(cons(V1, V2))) → isNatIListKind#(V2) |
| active#(length(cons(N, L))) → isNatIListKind#(L) | active#(isNat(length(V1))) → U11#(isNatIListKind(V1), V1) |
| active#(U31(tt, V)) → U32#(isNatList(V)) | active#(isNatList(cons(V1, V2))) → U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| active#(cons(X1, X2)) → active#(X1) | active#(length(cons(N, L))) → isNatKind#(N) |
| active#(U42(X1, X2)) → active#(X1) | active#(U52(X1, X2)) → U52#(active(X1), X2) |
| active#(U41(X1, X2, X3)) → U41#(active(X1), X2, X3) | active#(U43(X)) → active#(X) |
| active#(isNatList(cons(V1, V2))) → and#(isNatKind(V1), isNatIListKind(V2)) | active#(U52(X1, X2)) → active#(X1) |
| active#(U41(tt, V1, V2)) → isNat#(V1) | active#(U53(X)) → active#(X) |
| active#(and(X1, X2)) → and#(active(X1), X2) | active#(U12(X)) → U12#(active(X)) |
| active#(U31(X1, X2)) → U31#(active(X1), X2) | active#(length(cons(N, L))) → isNat#(N) |
| active#(length(cons(N, L))) → U61#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | active#(U31(X1, X2)) → active#(X1) |
| active#(length(X)) → length#(active(X)) | active#(U32(X)) → U32#(active(X)) |
| active#(length(cons(N, L))) → and#(isNatList(L), isNatIListKind(L)) | active#(isNatIList(cons(V1, V2))) → isNatKind#(V1) |
| active#(U11(X1, X2)) → U11#(active(X1), X2) | active#(isNatIList(cons(V1, V2))) → U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| active#(U21(X1, X2)) → active#(X1) | active#(U22(X)) → active#(X) |
| active#(U51(X1, X2, X3)) → U51#(active(X1), X2, X3) | active#(isNat(s(V1))) → U21#(isNatKind(V1), V1) |
| active#(s(X)) → active#(X) | active#(zeros) → cons#(0, zeros) |
| active#(isNatList(cons(V1, V2))) → isNatKind#(V1) | active#(isNatIList(V)) → U31#(isNatIListKind(V), V) |
| active#(and(X1, X2)) → active#(X1) |
| U22#(ok(X)) → U22#(X) | U22#(mark(X)) → U22#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| U52#(mark(X1), X2) → U52#(X1, X2) | U52#(ok(X1), ok(X2)) → U52#(X1, X2) |
| isNatKind#(ok(X)) → isNatKind#(X) |
| 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) |
| U12#(ok(X)) → U12#(X) | U12#(mark(X)) → U12#(X) |
| U53#(mark(X)) → U53#(X) | U53#(ok(X)) → U53#(X) |
| U21#(mark(X1), X2) → U21#(X1, X2) | U21#(ok(X1), ok(X2)) → U21#(X1, X2) |
| U11#(ok(X1), ok(X2)) → U11#(X1, X2) | U11#(mark(X1), X2) → U11#(X1, X2) |
| U61#(ok(X1), ok(X2)) → U61#(X1, X2) | U61#(mark(X1), X2) → U61#(X1, X2) |
| and#(ok(X1), ok(X2)) → and#(X1, X2) | and#(mark(X1), X2) → and#(X1, X2) |
| U41#(mark(X1), X2, X3) → U41#(X1, X2, X3) | U41#(ok(X1), ok(X2), ok(X3)) → U41#(X1, X2, X3) |
| U31#(mark(X1), X2) → U31#(X1, X2) | U31#(ok(X1), ok(X2)) → U31#(X1, X2) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U21#(mark(X1), X2) | → | U21#(X1, X2) | | U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U21#(mark(X1), X2) | → | U21#(X1, X2) | | U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U42#(ok(X1), ok(X2)) | → | U42#(X1, X2) | | U42#(mark(X1), X2) | → | U42#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U42#(ok(X1), ok(X2)) | → | U42#(X1, X2) | | U42#(mark(X1), X2) | → | U42#(X1, X2) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U43#(mark(X)) | → | U43#(X) | | U43#(ok(X)) | → | U43#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U43#(mark(X)) | → | U43#(X) | | U43#(ok(X)) | → | U43#(X) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U32#(ok(X)) | → | U32#(X) | | U32#(mark(X)) | → | U32#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U32#(ok(X)) | → | U32#(X) | | U32#(mark(X)) | → | U32#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U31#(mark(X1), X2) | → | U31#(X1, X2) | | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U31#(mark(X1), X2) | → | U31#(X1, X2) | | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNat#(ok(X)) | → | isNat#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNat#(ok(X)) | → | isNat#(X) |
Problem 8: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(isNatIList(V)) | → | isNatIListKind#(V) | | active#(U52(tt, V2)) | → | isNatList#(V2) |
| active#(U12(X)) | → | active#(X) | | active#(U61(X1, X2)) | → | U61#(active(X1), X2) |
| active#(isNatIListKind(cons(V1, V2))) | → | isNatKind#(V1) | | active#(U11(tt, V1)) | → | U12#(isNatList(V1)) |
| active#(U61(tt, L)) | → | s#(length(L)) | | active#(U11(X1, X2)) | → | active#(X1) |
| active#(U51(X1, X2, X3)) | → | active#(X1) | | active#(U51(tt, V1, V2)) | → | U52#(isNat(V1), V2) |
| active#(U32(X)) | → | active#(X) | | active#(U61(X1, X2)) | → | active#(X1) |
| active#(length(cons(N, L))) | → | and#(isNat(N), isNatKind(N)) | | active#(length(X)) | → | active#(X) |
| active#(U52(tt, V2)) | → | U53#(isNatList(V2)) | | active#(U21(tt, V1)) | → | isNat#(V1) |
| active#(U41(X1, X2, X3)) | → | active#(X1) | | active#(U31(tt, V)) | → | isNatList#(V) |
| active#(isNatList(cons(V1, V2))) | → | isNatIListKind#(V2) | | active#(length(cons(N, L))) | → | isNatIListKind#(L) |
| active#(isNat(length(V1))) | → | U11#(isNatIListKind(V1), V1) | | active#(U31(tt, V)) | → | U32#(isNatList(V)) |
| active#(isNatList(cons(V1, V2))) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | active#(cons(X1, X2)) | → | active#(X1) |
| active#(length(cons(N, L))) | → | isNatKind#(N) | | active#(U42(X1, X2)) | → | active#(X1) |
| active#(U41(X1, X2, X3)) | → | U41#(active(X1), X2, X3) | | active#(U52(X1, X2)) | → | U52#(active(X1), X2) |
| active#(U43(X)) | → | active#(X) | | active#(U52(X1, X2)) | → | active#(X1) |
| active#(isNatList(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) | | active#(U41(tt, V1, V2)) | → | isNat#(V1) |
| active#(and(X1, X2)) | → | and#(active(X1), X2) | | active#(U53(X)) | → | active#(X) |
| active#(U31(X1, X2)) | → | U31#(active(X1), X2) | | active#(U12(X)) | → | U12#(active(X)) |
| active#(length(cons(N, L))) | → | isNat#(N) | | active#(length(cons(N, L))) | → | U61#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
| active#(U31(X1, X2)) | → | active#(X1) | | active#(length(X)) | → | length#(active(X)) |
| active#(U32(X)) | → | U32#(active(X)) | | active#(length(cons(N, L))) | → | and#(isNatList(L), isNatIListKind(L)) |
| active#(isNatIList(cons(V1, V2))) | → | isNatKind#(V1) | | active#(U11(X1, X2)) | → | U11#(active(X1), X2) |
| active#(U21(X1, X2)) | → | active#(X1) | | active#(isNatIList(cons(V1, V2))) | → | U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | | active#(U51(X1, X2, X3)) | → | U51#(active(X1), X2, X3) |
| active#(U22(X)) | → | active#(X) | | active#(s(X)) | → | active#(X) |
| active#(zeros) | → | cons#(0, zeros) | | active#(isNatList(cons(V1, V2))) | → | isNatKind#(V1) |
| active#(isNatIList(V)) | → | U31#(isNatIListKind(V), V) | | active#(and(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
The following SCCs where found
| active#(U51(X1, X2, X3)) → active#(X1) | active#(U32(X)) → active#(X) |
| active#(U12(X)) → active#(X) | active#(U42(X1, X2)) → active#(X1) |
| active#(U61(X1, X2)) → active#(X1) | active#(length(X)) → active#(X) |
| active#(U43(X)) → active#(X) | active#(U52(X1, X2)) → active#(X1) |
| active#(U41(X1, X2, X3)) → active#(X1) | active#(U53(X)) → active#(X) |
| active#(U21(X1, X2)) → active#(X1) | active#(U22(X)) → active#(X) |
| active#(s(X)) → active#(X) | active#(U31(X1, X2)) → active#(X1) |
| active#(and(X1, X2)) → active#(X1) | active#(U11(X1, X2)) → active#(X1) |
| active#(cons(X1, X2)) → active#(X1) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U53#(mark(X)) | → | U53#(X) | | U53#(ok(X)) | → | U53#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U53#(mark(X)) | → | U53#(X) | | U53#(ok(X)) | → | U53#(X) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U12#(ok(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(ok(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNatIListKind#(ok(X)) | → | isNatIListKind#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatIListKind#(ok(X)) | → | isNatIListKind#(X) |
Problem 12: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNatKind#(ok(X)) | → | isNatKind#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatKind#(ok(X)) | → | isNatKind#(X) |
Problem 13: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) |
Problem 15: 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, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, 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 16: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U11#(ok(X1), ok(X2)) | → | U11#(X1, X2) | | U11#(mark(X1), X2) | → | U11#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(ok(X1), ok(X2)) | → | U11#(X1, X2) | | U11#(mark(X1), X2) | → | U11#(X1, X2) |
Problem 17: 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, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, 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 18: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(isNatIList(V)) | → | isNatIListKind#(V) | | active#(U52(tt, V2)) | → | isNatList#(V2) |
| active#(U12(X)) | → | active#(X) | | active#(cons(X1, X2)) | → | cons#(active(X1), X2) |
| active#(U61(X1, X2)) | → | U61#(active(X1), X2) | | active#(isNatIListKind(cons(V1, V2))) | → | isNatKind#(V1) |
| active#(U11(tt, V1)) | → | U12#(isNatList(V1)) | | active#(U61(tt, L)) | → | s#(length(L)) |
| active#(U11(X1, X2)) | → | active#(X1) | | active#(U51(X1, X2, X3)) | → | active#(X1) |
| active#(U51(tt, V1, V2)) | → | U52#(isNat(V1), V2) | | active#(U32(X)) | → | active#(X) |
| active#(U61(X1, X2)) | → | active#(X1) | | active#(length(cons(N, L))) | → | and#(isNat(N), isNatKind(N)) |
| active#(length(X)) | → | active#(X) | | active#(U52(tt, V2)) | → | U53#(isNatList(V2)) |
| active#(U21(tt, V1)) | → | isNat#(V1) | | active#(U41(X1, X2, X3)) | → | active#(X1) |
| active#(U31(tt, V)) | → | isNatList#(V) | | active#(isNatList(cons(V1, V2))) | → | isNatIListKind#(V2) |
| active#(length(cons(N, L))) | → | isNatIListKind#(L) | | active#(isNat(length(V1))) | → | U11#(isNatIListKind(V1), V1) |
| active#(U31(tt, V)) | → | U32#(isNatList(V)) | | active#(isNatList(cons(V1, V2))) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| active#(cons(X1, X2)) | → | active#(X1) | | active#(length(cons(N, L))) | → | isNatKind#(N) |
| active#(U42(X1, X2)) | → | active#(X1) | | active#(U41(X1, X2, X3)) | → | U41#(active(X1), X2, X3) |
| active#(U52(X1, X2)) | → | U52#(active(X1), X2) | | active#(U43(X)) | → | active#(X) |
| active#(U52(X1, X2)) | → | active#(X1) | | active#(isNatList(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| active#(U41(tt, V1, V2)) | → | isNat#(V1) | | active#(and(X1, X2)) | → | and#(active(X1), X2) |
| active#(U53(X)) | → | active#(X) | | active#(U31(X1, X2)) | → | U31#(active(X1), X2) |
| active#(U12(X)) | → | U12#(active(X)) | | active#(length(cons(N, L))) | → | isNat#(N) |
| active#(length(cons(N, L))) | → | U61#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | | active#(U31(X1, X2)) | → | active#(X1) |
| active#(length(X)) | → | length#(active(X)) | | active#(U32(X)) | → | U32#(active(X)) |
| active#(length(cons(N, L))) | → | and#(isNatList(L), isNatIListKind(L)) | | active#(isNatIList(cons(V1, V2))) | → | isNatKind#(V1) |
| active#(U11(X1, X2)) | → | U11#(active(X1), X2) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(isNatIList(cons(V1, V2))) | → | U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | active#(U22(X)) | → | active#(X) |
| active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | | active#(U51(X1, X2, X3)) | → | U51#(active(X1), X2, X3) |
| active#(s(X)) | → | active#(X) | | active#(zeros) | → | cons#(0, zeros) |
| active#(isNatList(cons(V1, V2))) | → | isNatKind#(V1) | | active#(isNatIList(V)) | → | U31#(isNatIListKind(V), V) |
| active#(and(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
The following SCCs where found
| active#(U51(X1, X2, X3)) → active#(X1) | active#(U32(X)) → active#(X) |
| active#(U12(X)) → active#(X) | active#(U42(X1, X2)) → active#(X1) |
| active#(U61(X1, X2)) → active#(X1) | active#(length(X)) → active#(X) |
| active#(U43(X)) → active#(X) | active#(U52(X1, X2)) → active#(X1) |
| active#(U41(X1, X2, X3)) → active#(X1) | active#(U53(X)) → active#(X) |
| active#(U21(X1, X2)) → active#(X1) | active#(U22(X)) → active#(X) |
| active#(s(X)) → active#(X) | active#(U31(X1, X2)) → active#(X1) |
| active#(and(X1, X2)) → active#(X1) | active#(U11(X1, X2)) → active#(X1) |
| active#(cons(X1, X2)) → active#(X1) |
Problem 29: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(U51(X1, X2, X3)) | → | active#(X1) | | active#(U32(X)) | → | active#(X) |
| active#(U61(X1, X2)) | → | active#(X1) | | active#(U42(X1, X2)) | → | active#(X1) |
| active#(U12(X)) | → | active#(X) | | active#(length(X)) | → | active#(X) |
| active#(U43(X)) | → | active#(X) | | active#(U52(X1, X2)) | → | active#(X1) |
| active#(U41(X1, X2, X3)) | → | active#(X1) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(U53(X)) | → | active#(X) | | active#(U22(X)) | → | active#(X) |
| active#(s(X)) | → | active#(X) | | active#(U31(X1, X2)) | → | active#(X1) |
| active#(and(X1, X2)) | → | active#(X1) | | active#(U11(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y): 2x
- U12(x): x
- U21(x,y): x
- U22(x): x
- U31(x,y): 2y + x + 1
- U32(x): x
- U41(x,y,z): 2y + x
- U42(x,y): 2y + 2x
- U43(x): 2x
- U51(x,y,z): z + 2x
- U52(x,y): x
- U53(x): 2x
- U61(x,y): 2x
- active(x): 0
- active#(x): 2x + 1
- and(x,y): x
- cons(x,y): 2y + x
- isNat(x): 0
- isNatIList(x): 0
- isNatIListKind(x): 0
- isNatKind(x): 0
- isNatList(x): 0
- length(x): x
- mark(x): 0
- nil: 0
- ok(x): 0
- proper(x): 0
- s(x): x
- top(x): 0
- tt: 0
- zeros: 0
There are no usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(U31(X1, X2)) | → | active#(X1) |
Problem 19: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U52#(mark(X1), X2) | → | U52#(X1, X2) | | U52#(ok(X1), ok(X2)) | → | U52#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U52#(mark(X1), X2) | → | U52#(X1, X2) | | U52#(ok(X1), ok(X2)) | → | U52#(X1, X2) |
Problem 20: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNatList#(ok(X)) | → | isNatList#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatList#(ok(X)) | → | isNatList#(X) |
Problem 21: 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, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, 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 22: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(U51(X1, X2, X3)) | → | active#(X1) | | active#(U32(X)) | → | active#(X) |
| active#(U61(X1, X2)) | → | active#(X1) | | active#(U12(X)) | → | active#(X) |
| active#(U42(X1, X2)) | → | active#(X1) | | active#(length(X)) | → | active#(X) |
| active#(U43(X)) | → | active#(X) | | active#(U52(X1, X2)) | → | active#(X1) |
| active#(U41(X1, X2, X3)) | → | active#(X1) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(U53(X)) | → | active#(X) | | active#(U22(X)) | → | active#(X) |
| active#(s(X)) | → | active#(X) | | active#(U31(X1, X2)) | → | active#(X1) |
| active#(and(X1, X2)) | → | active#(X1) | | active#(U11(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(U51(X1, X2, X3)) | → | active#(X1) | | active#(U32(X)) | → | active#(X) |
| active#(U61(X1, X2)) | → | active#(X1) | | active#(U12(X)) | → | active#(X) |
| active#(U42(X1, X2)) | → | active#(X1) | | active#(length(X)) | → | active#(X) |
| active#(U43(X)) | → | active#(X) | | active#(U52(X1, X2)) | → | active#(X1) |
| active#(U41(X1, X2, X3)) | → | active#(X1) | | active#(U21(X1, X2)) | → | active#(X1) |
| active#(U53(X)) | → | active#(X) | | active#(U22(X)) | → | active#(X) |
| active#(s(X)) | → | active#(X) | | active#(U31(X1, X2)) | → | active#(X1) |
| active#(and(X1, X2)) | → | active#(X1) | | active#(U11(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Problem 23: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | | U51#(ok(X1), ok(X2), ok(X3)) | → | U51#(X1, X2, X3) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | | U51#(ok(X1), ok(X2), ok(X3)) | → | U51#(X1, X2, X3) |
Problem 24: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| and#(ok(X1), ok(X2)) | → | and#(X1, X2) | | and#(mark(X1), X2) | → | and#(X1, X2) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| and#(ok(X1), ok(X2)) | → | and#(X1, X2) | | and#(mark(X1), X2) | → | and#(X1, X2) |
Problem 25: 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, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, 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 26: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U22#(ok(X)) | → | U22#(X) | | U22#(mark(X)) | → | U22#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U22#(ok(X)) | → | U22#(X) | | U22#(mark(X)) | → | U22#(X) |
Problem 27: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(U52(X1, X2)) | → | proper#(X1) | | proper#(U51(X1, X2, X3)) | → | proper#(X2) |
| proper#(cons(X1, X2)) | → | proper#(X1) | | proper#(U11(X1, X2)) | → | proper#(X1) |
| proper#(isNatIListKind(X)) | → | proper#(X) | | proper#(U42(X1, X2)) | → | proper#(X2) |
| proper#(U41(X1, X2, X3)) | → | proper#(X3) | | proper#(U31(X1, X2)) | → | proper#(X1) |
| proper#(U31(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2, X3)) | → | proper#(X2) | | proper#(U42(X1, X2)) | → | proper#(X1) |
| proper#(U32(X)) | → | proper#(X) | | proper#(U21(X1, X2)) | → | proper#(X2) |
| proper#(isNatKind(X)) | → | proper#(X) | | proper#(U11(X1, X2)) | → | proper#(X2) |
| proper#(and(X1, X2)) | → | proper#(X2) | | proper#(U61(X1, X2)) | → | proper#(X2) |
| proper#(U41(X1, X2, X3)) | → | proper#(X1) | | proper#(U51(X1, X2, X3)) | → | proper#(X3) |
| proper#(isNatList(X)) | → | proper#(X) | | proper#(U53(X)) | → | proper#(X) |
| proper#(U43(X)) | → | proper#(X) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(U12(X)) | → | proper#(X) |
| proper#(isNatIList(X)) | → | proper#(X) | | proper#(U61(X1, X2)) | → | proper#(X1) |
| proper#(U52(X1, X2)) | → | proper#(X2) | | proper#(length(X)) | → | proper#(X) |
| proper#(s(X)) | → | proper#(X) | | proper#(U51(X1, X2, X3)) | → | proper#(X1) |
| proper#(U21(X1, X2)) | → | proper#(X1) | | proper#(U22(X)) | → | proper#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(U51(X1, X2, X3)) | → | proper#(X2) | | proper#(U52(X1, X2)) | → | proper#(X1) |
| proper#(cons(X1, X2)) | → | proper#(X1) | | proper#(U11(X1, X2)) | → | proper#(X1) |
| proper#(isNatIListKind(X)) | → | proper#(X) | | proper#(U42(X1, X2)) | → | proper#(X2) |
| proper#(U31(X1, X2)) | → | proper#(X1) | | proper#(U41(X1, X2, X3)) | → | proper#(X3) |
| proper#(U31(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2, X3)) | → | proper#(X2) | | proper#(U42(X1, X2)) | → | proper#(X1) |
| proper#(U32(X)) | → | proper#(X) | | proper#(U21(X1, X2)) | → | proper#(X2) |
| proper#(isNatKind(X)) | → | proper#(X) | | proper#(U11(X1, X2)) | → | proper#(X2) |
| proper#(and(X1, X2)) | → | proper#(X2) | | proper#(U61(X1, X2)) | → | proper#(X2) |
| proper#(U41(X1, X2, X3)) | → | proper#(X1) | | proper#(U51(X1, X2, X3)) | → | proper#(X3) |
| proper#(isNatList(X)) | → | proper#(X) | | proper#(U53(X)) | → | proper#(X) |
| proper#(U43(X)) | → | proper#(X) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(U12(X)) | → | proper#(X) |
| proper#(isNatIList(X)) | → | proper#(X) | | proper#(U61(X1, X2)) | → | proper#(X1) |
| proper#(U52(X1, X2)) | → | proper#(X2) | | proper#(s(X)) | → | proper#(X) |
| proper#(length(X)) | → | proper#(X) | | proper#(U51(X1, X2, X3)) | → | proper#(X1) |
| proper#(U21(X1, X2)) | → | proper#(X1) | | proper#(U22(X)) | → | proper#(X) |
Problem 28: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNatIList#(ok(X)) | → | isNatIList#(X) |
Rewrite Rules
| active(zeros) | → | mark(cons(0, zeros)) | | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) |
| active(U12(tt)) | → | mark(tt) | | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) |
| active(U22(tt)) | → | mark(tt) | | active(U31(tt, V)) | → | mark(U32(isNatList(V))) |
| active(U32(tt)) | → | mark(tt) | | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | | active(U43(tt)) | → | mark(tt) |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) |
| active(U53(tt)) | → | mark(tt) | | active(U61(tt, L)) | → | mark(s(length(L))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | | active(isNatIList(zeros)) | → | mark(tt) |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(isNatIListKind(nil)) | → | mark(tt) |
| active(isNatIListKind(zeros)) | → | mark(tt) | | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) |
| active(isNatKind(0)) | → | mark(tt) | | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | | active(isNatList(nil)) | → | mark(tt) |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | | active(length(nil)) | → | mark(0) |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U12(X)) | → | U12(active(X)) |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | | active(U22(X)) | → | U22(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U32(X)) | → | U32(active(X)) |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | | active(U42(X1, X2)) | → | U42(active(X1), X2) |
| active(U43(X)) | → | U43(active(X)) | | active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) |
| active(U52(X1, X2)) | → | U52(active(X1), X2) | | active(U53(X)) | → | U53(active(X)) |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(length(X)) | → | length(active(X)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | U11(mark(X1), X2) | → | mark(U11(X1, X2)) |
| U12(mark(X)) | → | mark(U12(X)) | | U21(mark(X1), X2) | → | mark(U21(X1, X2)) |
| U22(mark(X)) | → | mark(U22(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U32(mark(X)) | → | mark(U32(X)) | | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | | U43(mark(X)) | → | mark(U43(X)) |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | | U52(mark(X1), X2) | → | mark(U52(X1, X2)) |
| U53(mark(X)) | → | mark(U53(X)) | | U61(mark(X1), X2) | → | mark(U61(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | length(mark(X)) | → | mark(length(X)) |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | | proper(zeros) | → | ok(zeros) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U12(X)) | → | U12(proper(X)) | | proper(isNatList(X)) | → | isNatList(proper(X)) |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | | proper(U22(X)) | → | U22(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) |
| proper(U32(X)) | → | U32(proper(X)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | | proper(U43(X)) | → | U43(proper(X)) |
| proper(isNatIList(X)) | → | isNatIList(proper(X)) | | proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) |
| proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | | proper(U53(X)) | → | U53(proper(X)) |
| proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(length(X)) | → | length(proper(X)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | | proper(isNatKind(X)) | → | isNatKind(proper(X)) |
| proper(nil) | → | ok(nil) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNatList(ok(X)) | → | ok(isNatList(X)) | | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) |
| U22(ok(X)) | → | ok(U22(X)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U32(ok(X)) | → | ok(U32(X)) |
| U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | | U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) |
| U43(ok(X)) | → | ok(U43(X)) | | isNatIList(ok(X)) | → | ok(isNatIList(X)) |
| U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | | U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) |
| U53(ok(X)) | → | ok(U53(X)) | | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | length(ok(X)) | → | ok(length(X)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) |
| isNatKind(ok(X)) | → | ok(isNatKind(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, ok, proper, U21, cons, U22, top, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, active, U11, U12, U31, U32, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNatIList#(ok(X)) | → | isNatIList#(X) |