TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60152 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (582ms).
| – Problem 2 was processed with processor PolynomialLinearRange4 (842ms).
| | – Problem 5 was processed with processor DependencyGraph (40ms).
| | | – Problem 6 was processed with processor PolynomialLinearRange4 (189ms).
| | | | – Problem 8 was processed with processor DependencyGraph (10ms).
| | | | | – Problem 10 was processed with processor PolynomialLinearRange4 (7ms).
| | | | | – Problem 11 was processed with processor PolynomialLinearRange4 (148ms).
| | | | | | – Problem 14 remains open; application of the following processors failed [DependencyGraph (4ms), PolynomialLinearRange4 (149ms), DependencyGraph (3ms), ReductionPairSAT (304ms), DependencyGraph (3ms)].
| | | – Problem 7 was processed with processor PolynomialLinearRange4 (200ms).
| | | | – Problem 9 was processed with processor DependencyGraph (4ms).
| | | | | – Problem 12 remains open; application of the following processors failed [PolynomialLinearRange4 (195ms), DependencyGraph (1ms), PolynomialLinearRange4 (226ms), DependencyGraph (1ms), ReductionPairSAT (451ms), DependencyGraph (1ms)].
| | | | | – Problem 13 remains open; application of the following processors failed [PolynomialLinearRange4 (170ms), DependencyGraph (4ms), PolynomialLinearRange4 (141ms), DependencyGraph (1ms), ReductionPairSAT (437ms), DependencyGraph (1ms)].
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (0ms), DependencyGraph (19ms), PolynomialOrderingProcessor (0ms), DependencyGraph (1ms), PolynomialLinearRange4 (241ms), DependencyGraph (1ms), PolynomialLinearRange4 (201ms), DependencyGraph (1ms), PolynomialLinearRange4 (194ms), DependencyGraph (1ms), PolynomialLinearRange4 (188ms), DependencyGraph (5ms), ReductionPairSAT (517ms), DependencyGraph (1ms), SizeChangePrinciple (timeout)].
| – Problem 4 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (0ms), DependencyGraph (1ms), PolynomialOrderingProcessor (0ms), DependencyGraph (1ms), PolynomialLinearRange4 (222ms), DependencyGraph (1ms), PolynomialLinearRange4 (208ms), DependencyGraph (1ms), PolynomialLinearRange4 (178ms), DependencyGraph (1ms), PolynomialLinearRange4 (154ms), DependencyGraph (3ms), ReductionPairSAT (410ms), DependencyGraph (1ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| length#(cons(N, L)) | → | U61#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | | U61#(tt, L) | → | length#(L) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, U21, U22, cons, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Open Dependency Pair Problem 4
Dependency Pairs
| U42#(tt, V2) | → | isNatIList#(V2) | | U41#(tt, V1, V2) | → | U42#(isNat(V1), V2) |
| isNatIList#(cons(V1, V2)) | → | U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, U21, U22, cons, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Open Dependency Pair Problem 12
Dependency Pairs
| isNat#(s(V1)) | → | U21#(isNatKind(V1), V1) | | U21#(tt, V1) | → | isNat#(V1) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, U21, U22, cons, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Open Dependency Pair Problem 13
Dependency Pairs
| isNatList#(cons(V1, V2)) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | U52#(tt, V2) | → | isNatList#(V2) |
| U51#(tt, V1, V2) | → | U52#(isNat(V1), V2) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, U21, U22, cons, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Open Dependency Pair Problem 14
Dependency Pairs
| and#(tt, X) | → | T(X) | | isNatIListKind#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| T(isNatIListKind(V2)) | → | isNatIListKind#(V2) | | T(isNatKind(x_1)) | → | T(x_1) |
| T(and(x_1, x_2)) | → | T(x_1) | | T(and(x_1, x_2)) | → | T(x_2) |
| T(isNatIListKind(L)) | → | isNatIListKind#(L) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, U21, cons, U22, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| isNatIList#(cons(V1, V2)) | → | isNatKind#(V1) | | isNat#(s(V1)) | → | isNatKind#(V1) |
| isNat#(s(V1)) | → | U21#(isNatKind(V1), V1) | | U31#(tt, V) | → | isNatList#(V) |
| isNatKind#(length(V1)) | → | isNatIListKind#(V1) | | isNatList#(cons(V1, V2)) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| U42#(tt, V2) | → | isNatIList#(V2) | | T(isNat(x_1)) | → | T(x_1) |
| T(and(x_1, x_2)) | → | T(x_1) | | U52#(tt, V2) | → | U53#(isNatList(V2)) |
| isNatList#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) | | U52#(tt, V2) | → | isNatList#(V2) |
| isNatKind#(s(V1)) | → | isNatKind#(V1) | | isNatIListKind#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| T(isNatIListKind(V2)) | → | isNatIListKind#(V2) | | T(zeros) | → | zeros# |
| isNatIList#(V) | → | U31#(isNatIListKind(V), V) | | U41#(tt, V1, V2) | → | U42#(isNat(V1), V2) |
| isNatIList#(cons(V1, V2)) | → | U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatList#(cons(V1, V2)) | → | isNatKind#(V1) |
| isNatIList#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) | | U21#(tt, V1) | → | isNat#(V1) |
| U51#(tt, V1, V2) | → | U52#(isNat(V1), V2) | | and#(tt, X) | → | T(X) |
| U11#(tt, V1) | → | U12#(isNatList(V1)) | | U21#(tt, V1) | → | U22#(isNat(V1)) |
| T(isNatIListKind(x_1)) | → | T(x_1) | | T(and(isNat(N), isNatKind(N))) | → | and#(isNat(N), isNatKind(N)) |
| U61#(tt, L) | → | length#(L) | | U51#(tt, V1, V2) | → | isNat#(V1) |
| T(isNatKind(N)) | → | isNatKind#(N) | | isNat#(length(V1)) | → | isNatIListKind#(V1) |
| length#(cons(N, L)) | → | and#(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))) | | U61#(tt, L) | → | T(L) |
| T(isNatIListKind(L)) | → | isNatIListKind#(L) | | isNat#(length(V1)) | → | U11#(isNatIListKind(V1), V1) |
| U41#(tt, V1, V2) | → | isNat#(V1) | | length#(cons(N, L)) | → | U61#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
| U11#(tt, V1) | → | isNatList#(V1) | | U31#(tt, V) | → | U32#(isNatList(V)) |
| length#(cons(N, L)) | → | and#(isNatList(L), isNatIListKind(L)) | | T(isNat(N)) | → | isNat#(N) |
| T(isNatKind(x_1)) | → | T(x_1) | | T(and(x_1, x_2)) | → | T(x_2) |
| isNatIListKind#(cons(V1, V2)) | → | isNatKind#(V1) | | isNatIList#(V) | → | isNatIListKind#(V) |
| length#(cons(N, L)) | → | isNatList#(L) | | U42#(tt, V2) | → | U43#(isNatIList(V2)) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, U21, cons, U22, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Strategy
Context-sensitive strategy:
μ(isNatList#) = μ(isNatIListKind#) = μ(isNatIListKind) = μ(zeros#) = μ(isNat) = μ(T) = μ(isNatIList) = μ(isNatKind) = μ(isNatIList#) = μ(0) = μ(isNatList) = μ(isNatKind#) = μ(tt) = μ(zeros) = μ(isNat#) = μ(nil) = ∅
μ(U11#) = μ(U31#) = μ(length#) = μ(U21#) = μ(and#) = μ(U52#) = μ(U43#) = μ(U41#) = μ(U43) = μ(U61) = μ(U42) = μ(U41) = μ(length) = μ(U21) = μ(U22) = μ(cons) = μ(U12#) = μ(U22#) = μ(U61#) = μ(U53#) = μ(U51#) = μ(and) = μ(U42#) = μ(U51) = μ(s) = μ(U53) = μ(U52) = μ(U32#) = μ(U11) = μ(U12) = μ(U31) = μ(U32) = {1}
The following SCCs where found
| isNat#(s(V1)) → isNatKind#(V1) | isNat#(s(V1)) → U21#(isNatKind(V1), V1) |
| isNatKind#(length(V1)) → isNatIListKind#(V1) | isNatList#(cons(V1, V2)) → U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| T(isNat(x_1)) → T(x_1) | T(and(x_1, x_2)) → T(x_1) |
| isNatList#(cons(V1, V2)) → and#(isNatKind(V1), isNatIListKind(V2)) | U52#(tt, V2) → isNatList#(V2) |
| isNatKind#(s(V1)) → isNatKind#(V1) | isNatIListKind#(cons(V1, V2)) → and#(isNatKind(V1), isNatIListKind(V2)) |
| T(isNatIListKind(V2)) → isNatIListKind#(V2) | isNatList#(cons(V1, V2)) → isNatKind#(V1) |
| U21#(tt, V1) → isNat#(V1) | U51#(tt, V1, V2) → U52#(isNat(V1), V2) |
| and#(tt, X) → T(X) | T(isNatIListKind(x_1)) → T(x_1) |
| T(and(isNat(N), isNatKind(N))) → and#(isNat(N), isNatKind(N)) | U51#(tt, V1, V2) → isNat#(V1) |
| T(isNatKind(N)) → isNatKind#(N) | isNat#(length(V1)) → isNatIListKind#(V1) |
| T(isNatIListKind(L)) → isNatIListKind#(L) | isNat#(length(V1)) → U11#(isNatIListKind(V1), V1) |
| U11#(tt, V1) → isNatList#(V1) | T(isNat(N)) → isNat#(N) |
| T(isNatKind(x_1)) → T(x_1) | T(and(x_1, x_2)) → T(x_2) |
| isNatIListKind#(cons(V1, V2)) → isNatKind#(V1) |
| length#(cons(N, L)) → U61#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | U61#(tt, L) → length#(L) |
| U42#(tt, V2) → isNatIList#(V2) | U41#(tt, V1, V2) → U42#(isNat(V1), V2) |
| isNatIList#(cons(V1, V2)) → U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
Problem 2: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| isNat#(s(V1)) | → | isNatKind#(V1) | | isNat#(s(V1)) | → | U21#(isNatKind(V1), V1) |
| isNatKind#(length(V1)) | → | isNatIListKind#(V1) | | isNatList#(cons(V1, V2)) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| T(isNat(x_1)) | → | T(x_1) | | T(and(x_1, x_2)) | → | T(x_1) |
| isNatList#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) | | U52#(tt, V2) | → | isNatList#(V2) |
| isNatKind#(s(V1)) | → | isNatKind#(V1) | | isNatIListKind#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| T(isNatIListKind(V2)) | → | isNatIListKind#(V2) | | isNatList#(cons(V1, V2)) | → | isNatKind#(V1) |
| U21#(tt, V1) | → | isNat#(V1) | | U51#(tt, V1, V2) | → | U52#(isNat(V1), V2) |
| and#(tt, X) | → | T(X) | | T(isNatIListKind(x_1)) | → | T(x_1) |
| T(and(isNat(N), isNatKind(N))) | → | and#(isNat(N), isNatKind(N)) | | U51#(tt, V1, V2) | → | isNat#(V1) |
| T(isNatKind(N)) | → | isNatKind#(N) | | isNat#(length(V1)) | → | isNatIListKind#(V1) |
| T(isNatIListKind(L)) | → | isNatIListKind#(L) | | isNat#(length(V1)) | → | U11#(isNatIListKind(V1), V1) |
| U11#(tt, V1) | → | isNatList#(V1) | | T(isNat(N)) | → | isNat#(N) |
| T(isNatKind(x_1)) | → | T(x_1) | | T(and(x_1, x_2)) | → | T(x_2) |
| isNatIListKind#(cons(V1, V2)) | → | isNatKind#(V1) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, U21, cons, U22, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Strategy
Context-sensitive strategy:
μ(isNatList#) = μ(isNatIListKind) = μ(isNatIListKind#) = μ(zeros#) = μ(isNat) = μ(T) = μ(isNatIList) = μ(isNatKind) = μ(isNatIList#) = μ(0) = μ(isNatList) = μ(zeros) = μ(tt) = μ(isNatKind#) = μ(isNat#) = μ(nil) = ∅
μ(U11#) = μ(U31#) = μ(length#) = μ(U21#) = μ(and#) = μ(U52#) = μ(U43#) = μ(U41#) = μ(U43) = μ(U61) = μ(U42) = μ(U41) = μ(length) = μ(U21) = μ(U22) = μ(cons) = μ(U12#) = μ(U22#) = μ(U61#) = μ(U53#) = μ(U51#) = μ(and) = μ(U42#) = μ(s) = μ(U51) = μ(U53) = μ(U52) = μ(U32#) = μ(U11) = μ(U12) = μ(U31) = μ(U32) = {1}
Polynomial Interpretation
- 0: 0
- T(x): 2x
- U11(x,y): y
- U11#(x,y): 2y + 2
- U12(x): 0
- U21(x,y): 1
- U21#(x,y): y + 2
- U22(x): 0
- U31(x,y): 0
- U32(x): 0
- U41(x,y,z): 0
- U42(x,y): 0
- U43(x): 0
- U51(x,y,z): 0
- U51#(x,y,z): 2z + y + 2
- U52(x,y): 0
- U52#(x,y): 2y + 2
- U53(x): 0
- U61(x,y): 2y
- and(x,y): 2y + x
- and#(x,y): 2y
- cons(x,y): 2y + x
- isNat(x): x + 1
- isNat#(x): x + 2
- isNatIList(x): 0
- isNatIListKind(x): 2x
- isNatIListKind#(x): 2x
- isNatKind(x): 2x
- isNatKind#(x): x
- isNatList(x): 0
- isNatList#(x): 2x + 2
- length(x): 2x
- nil: 0
- s(x): x
- tt: 0
- zeros: 0
Standard Usable rules
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNatIListKind(nil) | → | tt |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | U21(tt, V1) | → | U22(isNat(V1)) |
| isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) | | U53(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | U32(tt) | → | tt |
| U11(tt, V1) | → | U12(isNatList(V1)) | | isNatKind(0) | → | tt |
| U22(tt) | → | tt | | isNatIList(zeros) | → | tt |
| length(nil) | → | 0 | | U43(tt) | → | tt |
| U61(tt, L) | → | s(length(L)) | | isNat(0) | → | tt |
| isNatList(nil) | → | tt | | U42(tt, V2) | → | U43(isNatIList(V2)) |
| U31(tt, V) | → | U32(isNatList(V)) | | zeros | → | cons(0, zeros) |
| U12(tt) | → | tt | | isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| U41(tt, V1, V2) | → | U42(isNat(V1), V2) | | and(tt, X) | → | X |
| U52(tt, V2) | → | U53(isNatList(V2)) | | isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| isNatIListKind(zeros) | → | tt |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| isNat#(s(V1)) | → | isNatKind#(V1) | | T(isNat(x_1)) | → | T(x_1) |
| isNatList#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) | | isNatList#(cons(V1, V2)) | → | isNatKind#(V1) |
| T(and(isNat(N), isNatKind(N))) | → | and#(isNat(N), isNatKind(N)) | | isNat#(length(V1)) | → | isNatIListKind#(V1) |
Problem 5: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| and#(tt, X) | → | T(X) | | T(isNatIListKind(x_1)) | → | T(x_1) |
| isNat#(s(V1)) | → | U21#(isNatKind(V1), V1) | | U51#(tt, V1, V2) | → | isNat#(V1) |
| T(isNatKind(N)) | → | isNatKind#(N) | | isNatKind#(length(V1)) | → | isNatIListKind#(V1) |
| isNatList#(cons(V1, V2)) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | T(and(x_1, x_2)) | → | T(x_1) |
| T(isNatIListKind(L)) | → | isNatIListKind#(L) | | U52#(tt, V2) | → | isNatList#(V2) |
| isNatKind#(s(V1)) | → | isNatKind#(V1) | | isNat#(length(V1)) | → | U11#(isNatIListKind(V1), V1) |
| U11#(tt, V1) | → | isNatList#(V1) | | isNatIListKind#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| T(isNatIListKind(V2)) | → | isNatIListKind#(V2) | | T(isNat(N)) | → | isNat#(N) |
| T(isNatKind(x_1)) | → | T(x_1) | | T(and(x_1, x_2)) | → | T(x_2) |
| isNatIListKind#(cons(V1, V2)) | → | isNatKind#(V1) | | U21#(tt, V1) | → | isNat#(V1) |
| U51#(tt, V1, V2) | → | U52#(isNat(V1), V2) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, U21, U22, cons, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Strategy
Context-sensitive strategy:
μ(isNatList#) = μ(isNatIListKind#) = μ(isNatIListKind) = μ(zeros#) = μ(isNat) = μ(T) = μ(isNatIList) = μ(isNatKind) = μ(isNatIList#) = μ(0) = μ(isNatList) = μ(isNatKind#) = μ(tt) = μ(zeros) = μ(isNat#) = μ(nil) = ∅
μ(U11#) = μ(U31#) = μ(length#) = μ(U21#) = μ(and#) = μ(U52#) = μ(U43#) = μ(U41#) = μ(U43) = μ(U61) = μ(U42) = μ(U41) = μ(length) = μ(U21) = μ(U22) = μ(cons) = μ(U12#) = μ(U22#) = μ(U61#) = μ(U53#) = μ(U51#) = μ(and) = μ(U42#) = μ(U51) = μ(s) = μ(U53) = μ(U52) = μ(U32#) = μ(U11) = μ(U12) = μ(U31) = μ(U32) = {1}
The following SCCs where found
| isNatKind#(s(V1)) → isNatKind#(V1) | and#(tt, X) → T(X) |
| T(isNatIListKind(x_1)) → T(x_1) | isNatIListKind#(cons(V1, V2)) → and#(isNatKind(V1), isNatIListKind(V2)) |
| T(isNatIListKind(V2)) → isNatIListKind#(V2) | T(isNatKind(N)) → isNatKind#(N) |
| isNatKind#(length(V1)) → isNatIListKind#(V1) | T(and(x_1, x_2)) → T(x_1) |
| T(isNatKind(x_1)) → T(x_1) | T(and(x_1, x_2)) → T(x_2) |
| isNatIListKind#(cons(V1, V2)) → isNatKind#(V1) | T(isNatIListKind(L)) → isNatIListKind#(L) |
| isNat#(length(V1)) → U11#(isNatIListKind(V1), V1) | U11#(tt, V1) → isNatList#(V1) |
| isNat#(s(V1)) → U21#(isNatKind(V1), V1) | U51#(tt, V1, V2) → isNat#(V1) |
| isNatList#(cons(V1, V2)) → U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | U52#(tt, V2) → isNatList#(V2) |
| U21#(tt, V1) → isNat#(V1) | U51#(tt, V1, V2) → U52#(isNat(V1), V2) |
Problem 6: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| isNatKind#(s(V1)) | → | isNatKind#(V1) | | and#(tt, X) | → | T(X) |
| T(isNatIListKind(x_1)) | → | T(x_1) | | isNatIListKind#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| T(isNatIListKind(V2)) | → | isNatIListKind#(V2) | | T(isNatKind(N)) | → | isNatKind#(N) |
| isNatKind#(length(V1)) | → | isNatIListKind#(V1) | | T(isNatKind(x_1)) | → | T(x_1) |
| T(and(x_1, x_2)) | → | T(x_1) | | T(and(x_1, x_2)) | → | T(x_2) |
| isNatIListKind#(cons(V1, V2)) | → | isNatKind#(V1) | | T(isNatIListKind(L)) | → | isNatIListKind#(L) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, U21, U22, cons, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Strategy
Context-sensitive strategy:
μ(isNatList#) = μ(isNatIListKind) = μ(isNatIListKind#) = μ(zeros#) = μ(isNat) = μ(T) = μ(isNatIList) = μ(isNatKind) = μ(isNatIList#) = μ(0) = μ(isNatList) = μ(zeros) = μ(tt) = μ(isNatKind#) = μ(isNat#) = μ(nil) = ∅
μ(U11#) = μ(U31#) = μ(length#) = μ(U21#) = μ(and#) = μ(U52#) = μ(U43#) = μ(U41#) = μ(U43) = μ(U61) = μ(U42) = μ(U41) = μ(length) = μ(U21) = μ(U22) = μ(cons) = μ(U12#) = μ(U22#) = μ(U61#) = μ(U53#) = μ(U51#) = μ(and) = μ(U42#) = μ(s) = μ(U51) = μ(U53) = μ(U52) = μ(U32#) = μ(U11) = μ(U12) = μ(U31) = μ(U32) = {1}
Polynomial Interpretation
- 0: 0
- T(x): x
- U11(x,y): 2y + 2
- U12(x): 1
- U21(x,y): 0
- U22(x): 0
- U31(x,y): 2
- U32(x): x
- U41(x,y,z): z + 3
- U42(x,y): y
- U43(x): 0
- U51(x,y,z): 2
- U52(x,y): 0
- U53(x): 0
- U61(x,y): y + 1
- and(x,y): y + x
- and#(x,y): y
- cons(x,y): y + x
- isNat(x): 2x
- isNatIList(x): x + 3
- isNatIListKind(x): 2x
- isNatIListKind#(x): 2x
- isNatKind(x): 2x
- isNatKind#(x): 2x
- isNatList(x): 2
- length(x): x + 1
- nil: 1
- s(x): x
- tt: 0
- zeros: 0
Standard Usable rules
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNatIListKind(nil) | → | tt |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | U21(tt, V1) | → | U22(isNat(V1)) |
| isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) | | U53(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | U32(tt) | → | tt |
| U11(tt, V1) | → | U12(isNatList(V1)) | | isNatKind(0) | → | tt |
| U22(tt) | → | tt | | isNatIList(zeros) | → | tt |
| length(nil) | → | 0 | | U43(tt) | → | tt |
| U61(tt, L) | → | s(length(L)) | | isNat(0) | → | tt |
| isNatList(nil) | → | tt | | U42(tt, V2) | → | U43(isNatIList(V2)) |
| U31(tt, V) | → | U32(isNatList(V)) | | zeros | → | cons(0, zeros) |
| U12(tt) | → | tt | | isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| U41(tt, V1, V2) | → | U42(isNat(V1), V2) | | and(tt, X) | → | X |
| U52(tt, V2) | → | U53(isNatList(V2)) | | isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| isNatIListKind(zeros) | → | tt |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| isNatKind#(length(V1)) | → | isNatIListKind#(V1) |
Problem 8: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| isNatKind#(s(V1)) | → | isNatKind#(V1) | | and#(tt, X) | → | T(X) |
| T(isNatIListKind(x_1)) | → | T(x_1) | | isNatIListKind#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) |
| T(isNatIListKind(V2)) | → | isNatIListKind#(V2) | | T(isNatKind(N)) | → | isNatKind#(N) |
| T(and(x_1, x_2)) | → | T(x_1) | | T(isNatKind(x_1)) | → | T(x_1) |
| T(and(x_1, x_2)) | → | T(x_2) | | isNatIListKind#(cons(V1, V2)) | → | isNatKind#(V1) |
| T(isNatIListKind(L)) | → | isNatIListKind#(L) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, U21, cons, U22, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Strategy
Context-sensitive strategy:
μ(isNatList#) = μ(isNatIListKind#) = μ(isNatIListKind) = μ(zeros#) = μ(isNat) = μ(T) = μ(isNatIList) = μ(isNatKind) = μ(isNatIList#) = μ(0) = μ(isNatList) = μ(isNatKind#) = μ(tt) = μ(zeros) = μ(isNat#) = μ(nil) = ∅
μ(U11#) = μ(U31#) = μ(length#) = μ(U21#) = μ(and#) = μ(U52#) = μ(U43#) = μ(U41#) = μ(U43) = μ(U61) = μ(U42) = μ(U41) = μ(length) = μ(U21) = μ(U22) = μ(cons) = μ(U12#) = μ(U22#) = μ(U61#) = μ(U53#) = μ(U51#) = μ(and) = μ(U42#) = μ(U51) = μ(s) = μ(U53) = μ(U52) = μ(U32#) = μ(U11) = μ(U12) = μ(U31) = μ(U32) = {1}
The following SCCs where found
| isNatKind#(s(V1)) → isNatKind#(V1) |
| and#(tt, X) → T(X) | T(isNatIListKind(x_1)) → T(x_1) |
| isNatIListKind#(cons(V1, V2)) → and#(isNatKind(V1), isNatIListKind(V2)) | T(isNatIListKind(V2)) → isNatIListKind#(V2) |
| T(isNatKind(x_1)) → T(x_1) | T(and(x_1, x_2)) → T(x_1) |
| T(and(x_1, x_2)) → T(x_2) | T(isNatIListKind(L)) → isNatIListKind#(L) |
Problem 10: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| isNatKind#(s(V1)) | → | isNatKind#(V1) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, U21, cons, U22, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Strategy
Context-sensitive strategy:
μ(isNatList#) = μ(isNatIListKind) = μ(isNatIListKind#) = μ(zeros#) = μ(isNat) = μ(T) = μ(isNatIList) = μ(isNatKind) = μ(isNatIList#) = μ(0) = μ(isNatList) = μ(zeros) = μ(tt) = μ(isNatKind#) = μ(isNat#) = μ(nil) = ∅
μ(U11#) = μ(U31#) = μ(length#) = μ(U21#) = μ(and#) = μ(U52#) = μ(U43#) = μ(U41#) = μ(U43) = μ(U61) = μ(U42) = μ(U41) = μ(length) = μ(U21) = μ(U22) = μ(cons) = μ(U12#) = μ(U22#) = μ(U61#) = μ(U53#) = μ(U51#) = μ(and) = μ(U42#) = μ(s) = μ(U51) = μ(U53) = μ(U52) = μ(U32#) = μ(U11) = μ(U12) = μ(U31) = μ(U32) = {1}
Polynomial Interpretation
- 0: 0
- U11(x,y): 0
- U12(x): 0
- U21(x,y): 0
- U22(x): 0
- U31(x,y): 0
- U32(x): 0
- U41(x,y,z): 0
- U42(x,y): 0
- U43(x): 0
- U51(x,y,z): 0
- U52(x,y): 0
- U53(x): 0
- U61(x,y): 0
- and(x,y): 0
- cons(x,y): 0
- isNat(x): 0
- isNatIList(x): 0
- isNatIListKind(x): 0
- isNatKind(x): 0
- isNatKind#(x): 2x + 1
- isNatList(x): 0
- length(x): 0
- nil: 0
- s(x): x + 1
- 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:
| isNatKind#(s(V1)) | → | isNatKind#(V1) |
Problem 11: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| and#(tt, X) | → | T(X) | | T(isNatIListKind(x_1)) | → | T(x_1) |
| isNatIListKind#(cons(V1, V2)) | → | and#(isNatKind(V1), isNatIListKind(V2)) | | T(isNatIListKind(V2)) | → | isNatIListKind#(V2) |
| T(isNatKind(x_1)) | → | T(x_1) | | T(and(x_1, x_2)) | → | T(x_1) |
| T(and(x_1, x_2)) | → | T(x_2) | | T(isNatIListKind(L)) | → | isNatIListKind#(L) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, U21, cons, U22, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Strategy
Context-sensitive strategy:
μ(isNatList#) = μ(isNatIListKind) = μ(isNatIListKind#) = μ(zeros#) = μ(isNat) = μ(T) = μ(isNatIList) = μ(isNatKind) = μ(isNatIList#) = μ(0) = μ(isNatList) = μ(zeros) = μ(tt) = μ(isNatKind#) = μ(isNat#) = μ(nil) = ∅
μ(U11#) = μ(U31#) = μ(length#) = μ(U21#) = μ(and#) = μ(U52#) = μ(U43#) = μ(U41#) = μ(U43) = μ(U61) = μ(U42) = μ(U41) = μ(length) = μ(U21) = μ(U22) = μ(cons) = μ(U12#) = μ(U22#) = μ(U61#) = μ(U53#) = μ(U51#) = μ(and) = μ(U42#) = μ(s) = μ(U51) = μ(U53) = μ(U52) = μ(U32#) = μ(U11) = μ(U12) = μ(U31) = μ(U32) = {1}
Polynomial Interpretation
- 0: 0
- T(x): 2x
- U11(x,y): y + 1
- U12(x): x
- U21(x,y): 0
- U22(x): 0
- U31(x,y): 1
- U32(x): x
- U41(x,y,z): 1
- U42(x,y): 1
- U43(x): 0
- U51(x,y,z): 1
- U52(x,y): 1
- U53(x): 0
- U61(x,y): y + 2
- and(x,y): y + x
- and#(x,y): 2y + x
- cons(x,y): 2y + x
- isNat(x): x
- isNatIList(x): 1
- isNatIListKind(x): x + 1
- isNatIListKind#(x): x + 2
- isNatKind(x): x
- isNatList(x): 1
- length(x): x + 2
- nil: 1
- s(x): x
- tt: 0
- zeros: 0
Standard Usable rules
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNatIListKind(nil) | → | tt |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | U21(tt, V1) | → | U22(isNat(V1)) |
| isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) | | U53(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | U32(tt) | → | tt |
| U11(tt, V1) | → | U12(isNatList(V1)) | | isNatKind(0) | → | tt |
| U22(tt) | → | tt | | isNatIList(zeros) | → | tt |
| length(nil) | → | 0 | | U43(tt) | → | tt |
| U61(tt, L) | → | s(length(L)) | | isNat(0) | → | tt |
| isNatList(nil) | → | tt | | U42(tt, V2) | → | U43(isNatIList(V2)) |
| U31(tt, V) | → | U32(isNatList(V)) | | zeros | → | cons(0, zeros) |
| U12(tt) | → | tt | | isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| U41(tt, V1, V2) | → | U42(isNat(V1), V2) | | and(tt, X) | → | X |
| U52(tt, V2) | → | U53(isNatList(V2)) | | isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| isNatIListKind(zeros) | → | tt |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| T(isNatIListKind(x_1)) | → | T(x_1) |
Problem 7: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| isNat#(length(V1)) | → | U11#(isNatIListKind(V1), V1) | | U11#(tt, V1) | → | isNatList#(V1) |
| isNat#(s(V1)) | → | U21#(isNatKind(V1), V1) | | U51#(tt, V1, V2) | → | isNat#(V1) |
| isNatList#(cons(V1, V2)) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | U52#(tt, V2) | → | isNatList#(V2) |
| U21#(tt, V1) | → | isNat#(V1) | | U51#(tt, V1, V2) | → | U52#(isNat(V1), V2) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, U21, U22, cons, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Strategy
Context-sensitive strategy:
μ(isNatList#) = μ(isNatIListKind) = μ(isNatIListKind#) = μ(zeros#) = μ(isNat) = μ(T) = μ(isNatIList) = μ(isNatKind) = μ(isNatIList#) = μ(0) = μ(isNatList) = μ(zeros) = μ(tt) = μ(isNatKind#) = μ(isNat#) = μ(nil) = ∅
μ(U11#) = μ(U31#) = μ(length#) = μ(U21#) = μ(and#) = μ(U52#) = μ(U43#) = μ(U41#) = μ(U43) = μ(U61) = μ(U42) = μ(U41) = μ(length) = μ(U21) = μ(U22) = μ(cons) = μ(U12#) = μ(U22#) = μ(U61#) = μ(U53#) = μ(U51#) = μ(and) = μ(U42#) = μ(s) = μ(U51) = μ(U53) = μ(U52) = μ(U32#) = μ(U11) = μ(U12) = μ(U31) = μ(U32) = {1}
Polynomial Interpretation
- 0: 0
- U11(x,y): 0
- U11#(x,y): 2y + 2
- U12(x): 0
- U21(x,y): 0
- U21#(x,y): 2y
- U22(x): 0
- U31(x,y): 0
- U32(x): 0
- U41(x,y,z): 0
- U42(x,y): 0
- U43(x): 0
- U51(x,y,z): 3z + 2y
- U51#(x,y,z): 3z + 2y
- U52(x,y): 2y
- U52#(x,y): 3y
- U53(x): 0
- U61(x,y): 2y + 1
- and(x,y): 2y
- cons(x,y): 2y + x
- isNat(x): 0
- isNat#(x): 2x
- isNatIList(x): 0
- isNatIListKind(x): 0
- isNatKind(x): 0
- isNatList(x): 2x + 1
- isNatList#(x): 2x
- length(x): x + 1
- nil: 1
- s(x): x
- tt: 0
- zeros: 0
Standard Usable rules
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNatIListKind(nil) | → | tt |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | U21(tt, V1) | → | U22(isNat(V1)) |
| isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) | | U53(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | U32(tt) | → | tt |
| U11(tt, V1) | → | U12(isNatList(V1)) | | isNatKind(0) | → | tt |
| U22(tt) | → | tt | | isNatIList(zeros) | → | tt |
| length(nil) | → | 0 | | U43(tt) | → | tt |
| U61(tt, L) | → | s(length(L)) | | isNat(0) | → | tt |
| isNatList(nil) | → | tt | | U42(tt, V2) | → | U43(isNatIList(V2)) |
| U31(tt, V) | → | U32(isNatList(V)) | | zeros | → | cons(0, zeros) |
| U12(tt) | → | tt | | isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| U41(tt, V1, V2) | → | U42(isNat(V1), V2) | | and(tt, X) | → | X |
| U52(tt, V2) | → | U53(isNatList(V2)) | | isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| isNatIListKind(zeros) | → | tt |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| U11#(tt, V1) | → | isNatList#(V1) |
Problem 9: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| isNat#(length(V1)) | → | U11#(isNatIListKind(V1), V1) | | isNat#(s(V1)) | → | U21#(isNatKind(V1), V1) |
| U51#(tt, V1, V2) | → | isNat#(V1) | | isNatList#(cons(V1, V2)) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) |
| U52#(tt, V2) | → | isNatList#(V2) | | U21#(tt, V1) | → | isNat#(V1) |
| U51#(tt, V1, V2) | → | U52#(isNat(V1), V2) |
Rewrite Rules
| zeros | → | cons(0, zeros) | | U11(tt, V1) | → | U12(isNatList(V1)) |
| U12(tt) | → | tt | | U21(tt, V1) | → | U22(isNat(V1)) |
| U22(tt) | → | tt | | U31(tt, V) | → | U32(isNatList(V)) |
| U32(tt) | → | tt | | U41(tt, V1, V2) | → | U42(isNat(V1), V2) |
| U42(tt, V2) | → | U43(isNatIList(V2)) | | U43(tt) | → | tt |
| U51(tt, V1, V2) | → | U52(isNat(V1), V2) | | U52(tt, V2) | → | U53(isNatList(V2)) |
| U53(tt) | → | tt | | U61(tt, L) | → | s(length(L)) |
| and(tt, X) | → | X | | isNat(0) | → | tt |
| isNat(length(V1)) | → | U11(isNatIListKind(V1), V1) | | isNat(s(V1)) | → | U21(isNatKind(V1), V1) |
| isNatIList(V) | → | U31(isNatIListKind(V), V) | | isNatIList(zeros) | → | tt |
| isNatIList(cons(V1, V2)) | → | U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | isNatIListKind(nil) | → | tt |
| isNatIListKind(zeros) | → | tt | | isNatIListKind(cons(V1, V2)) | → | and(isNatKind(V1), isNatIListKind(V2)) |
| isNatKind(0) | → | tt | | isNatKind(length(V1)) | → | isNatIListKind(V1) |
| isNatKind(s(V1)) | → | isNatKind(V1) | | isNatList(nil) | → | tt |
| isNatList(cons(V1, V2)) | → | U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | | length(nil) | → | 0 |
| length(cons(N, L)) | → | U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) |
Original Signature
Termination of terms over the following signature is verified: isNatIListKind, isNat, U61, U43, U42, U41, length, U21, cons, U22, isNatIList, isNatKind, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, U12, U31, U32, nil
Strategy
Context-sensitive strategy:
μ(isNatList#) = μ(isNatIListKind#) = μ(isNatIListKind) = μ(zeros#) = μ(isNat) = μ(T) = μ(isNatIList) = μ(isNatKind) = μ(isNatIList#) = μ(0) = μ(isNatList) = μ(isNatKind#) = μ(tt) = μ(zeros) = μ(isNat#) = μ(nil) = ∅
μ(U11#) = μ(U31#) = μ(length#) = μ(U21#) = μ(and#) = μ(U52#) = μ(U43#) = μ(U41#) = μ(U43) = μ(U61) = μ(U42) = μ(U41) = μ(length) = μ(U21) = μ(U22) = μ(cons) = μ(U12#) = μ(U22#) = μ(U61#) = μ(U53#) = μ(U51#) = μ(and) = μ(U42#) = μ(U51) = μ(s) = μ(U53) = μ(U52) = μ(U32#) = μ(U11) = μ(U12) = μ(U31) = μ(U32) = {1}
The following SCCs where found
| isNatList#(cons(V1, V2)) → U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | U52#(tt, V2) → isNatList#(V2) |
| U51#(tt, V1, V2) → U52#(isNat(V1), V2) |
| isNat#(s(V1)) → U21#(isNatKind(V1), V1) | U21#(tt, V1) → isNat#(V1) |