26.13/8.00 YES 26.57/8.02 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 26.57/8.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 26.57/8.02 26.57/8.02 26.57/8.02 Termination w.r.t. Q of the given QTRS could be proven: 26.57/8.02 26.57/8.02 (0) QTRS 26.57/8.02 (1) DependencyPairsProof [EQUIVALENT, 215 ms] 26.57/8.02 (2) QDP 26.57/8.02 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 26.57/8.02 (4) AND 26.57/8.02 (5) QDP 26.57/8.02 (6) UsableRulesProof [EQUIVALENT, 0 ms] 26.57/8.02 (7) QDP 26.57/8.02 (8) QReductionProof [EQUIVALENT, 0 ms] 26.57/8.02 (9) QDP 26.57/8.02 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 26.57/8.02 (11) YES 26.57/8.02 (12) QDP 26.57/8.02 (13) UsableRulesProof [EQUIVALENT, 0 ms] 26.57/8.02 (14) QDP 26.57/8.02 (15) QReductionProof [EQUIVALENT, 0 ms] 26.57/8.02 (16) QDP 26.57/8.02 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 26.57/8.02 (18) YES 26.57/8.02 (19) QDP 26.57/8.02 (20) UsableRulesProof [EQUIVALENT, 0 ms] 26.57/8.02 (21) QDP 26.57/8.02 (22) QReductionProof [EQUIVALENT, 0 ms] 26.57/8.02 (23) QDP 26.57/8.02 (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] 26.57/8.02 (25) YES 26.57/8.02 (26) QDP 26.57/8.02 (27) QDPOrderProof [EQUIVALENT, 530 ms] 26.57/8.02 (28) QDP 26.57/8.02 (29) QDPOrderProof [EQUIVALENT, 437 ms] 26.57/8.02 (30) QDP 26.57/8.02 (31) DependencyGraphProof [EQUIVALENT, 0 ms] 26.57/8.02 (32) AND 26.57/8.02 (33) QDP 26.57/8.02 (34) QDPOrderProof [EQUIVALENT, 277 ms] 26.57/8.02 (35) QDP 26.57/8.02 (36) DependencyGraphProof [EQUIVALENT, 0 ms] 26.57/8.02 (37) TRUE 26.57/8.02 (38) QDP 26.57/8.02 (39) UsableRulesProof [EQUIVALENT, 0 ms] 26.57/8.02 (40) QDP 26.57/8.02 (41) QReductionProof [EQUIVALENT, 0 ms] 26.57/8.02 (42) QDP 26.57/8.02 (43) QDPSizeChangeProof [EQUIVALENT, 0 ms] 26.57/8.02 (44) YES 26.57/8.02 26.57/8.02 26.57/8.02 ---------------------------------------- 26.57/8.02 26.57/8.02 (0) 26.57/8.02 Obligation: 26.57/8.02 Q restricted rewrite system: 26.57/8.02 The TRS R consists of the following rules: 26.57/8.02 26.57/8.02 a__zeros -> cons(0, zeros) 26.57/8.02 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.02 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.02 a__U13(tt) -> tt 26.57/8.02 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.02 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.02 a__U23(tt) -> tt 26.57/8.02 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.02 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.02 a__U33(tt) -> tt 26.57/8.02 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.02 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.02 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.02 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.02 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.02 a__U46(tt) -> tt 26.57/8.02 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.02 a__U52(tt) -> tt 26.57/8.02 a__U61(tt) -> tt 26.57/8.02 a__U71(tt) -> tt 26.57/8.02 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.02 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.02 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.02 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.02 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.02 a__U86(tt) -> tt 26.57/8.02 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.02 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.02 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.02 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.02 a__isNat(0) -> tt 26.57/8.02 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.02 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.02 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.02 a__isNatIList(zeros) -> tt 26.57/8.02 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.02 a__isNatIListKind(nil) -> tt 26.57/8.02 a__isNatIListKind(zeros) -> tt 26.57/8.02 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.02 a__isNatKind(0) -> tt 26.57/8.02 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.02 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.02 a__isNatList(nil) -> tt 26.57/8.02 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.02 a__length(nil) -> 0 26.57/8.02 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.02 mark(zeros) -> a__zeros 26.57/8.02 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.02 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.02 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.02 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.02 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.02 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.02 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.02 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.02 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.02 mark(isNat(X)) -> a__isNat(X) 26.57/8.02 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.02 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.02 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.02 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.02 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.02 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.02 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.02 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.02 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.02 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.02 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.02 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.02 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.02 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.02 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.02 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.02 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.02 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.02 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.02 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.02 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.02 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.02 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.02 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.02 mark(length(X)) -> a__length(mark(X)) 26.57/8.02 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.02 mark(0) -> 0 26.57/8.02 mark(tt) -> tt 26.57/8.02 mark(s(X)) -> s(mark(X)) 26.57/8.02 mark(nil) -> nil 26.57/8.02 a__zeros -> zeros 26.57/8.02 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.02 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.02 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.02 a__U13(X) -> U13(X) 26.57/8.02 a__isNatList(X) -> isNatList(X) 26.57/8.02 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.02 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.02 a__isNatKind(X) -> isNatKind(X) 26.57/8.02 a__U23(X) -> U23(X) 26.57/8.02 a__isNat(X) -> isNat(X) 26.57/8.02 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.02 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.02 a__U33(X) -> U33(X) 26.57/8.02 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.02 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.02 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.02 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.02 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.02 a__U46(X) -> U46(X) 26.57/8.02 a__isNatIList(X) -> isNatIList(X) 26.57/8.02 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.02 a__U52(X) -> U52(X) 26.57/8.02 a__U61(X) -> U61(X) 26.57/8.02 a__U71(X) -> U71(X) 26.57/8.02 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.02 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.02 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.02 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.02 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.02 a__U86(X) -> U86(X) 26.57/8.02 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.02 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.02 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.02 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.02 a__length(X) -> length(X) 26.57/8.02 26.57/8.02 The set Q consists of the following terms: 26.57/8.02 26.57/8.02 a__zeros 26.57/8.02 a__isNatIList(x0) 26.57/8.02 mark(zeros) 26.57/8.02 mark(U11(x0, x1)) 26.57/8.02 mark(U12(x0, x1)) 26.57/8.02 mark(isNatIListKind(x0)) 26.57/8.02 mark(U13(x0)) 26.57/8.02 mark(isNatList(x0)) 26.57/8.02 mark(U21(x0, x1)) 26.57/8.02 mark(U22(x0, x1)) 26.57/8.02 mark(isNatKind(x0)) 26.57/8.02 mark(U23(x0)) 26.57/8.02 mark(isNat(x0)) 26.57/8.02 mark(U31(x0, x1)) 26.57/8.02 mark(U32(x0, x1)) 26.57/8.02 mark(U33(x0)) 26.57/8.02 mark(U41(x0, x1, x2)) 26.57/8.02 mark(U42(x0, x1, x2)) 26.57/8.02 mark(U43(x0, x1, x2)) 26.57/8.02 mark(U44(x0, x1, x2)) 26.57/8.02 mark(U45(x0, x1)) 26.57/8.02 mark(U46(x0)) 26.57/8.02 mark(isNatIList(x0)) 26.57/8.02 mark(U51(x0, x1)) 26.57/8.02 mark(U52(x0)) 26.57/8.02 mark(U61(x0)) 26.57/8.02 mark(U71(x0)) 26.57/8.02 mark(U81(x0, x1, x2)) 26.57/8.02 mark(U82(x0, x1, x2)) 26.57/8.02 mark(U83(x0, x1, x2)) 26.57/8.02 mark(U84(x0, x1, x2)) 26.57/8.02 mark(U85(x0, x1)) 26.57/8.02 mark(U86(x0)) 26.57/8.02 mark(U91(x0, x1, x2)) 26.57/8.02 mark(U92(x0, x1, x2)) 26.57/8.02 mark(U93(x0, x1, x2)) 26.57/8.02 mark(U94(x0, x1)) 26.57/8.02 mark(length(x0)) 26.57/8.02 mark(cons(x0, x1)) 26.57/8.02 mark(0) 26.57/8.02 mark(tt) 26.57/8.02 mark(s(x0)) 26.57/8.02 mark(nil) 26.57/8.02 a__U11(x0, x1) 26.57/8.02 a__U12(x0, x1) 26.57/8.02 a__isNatIListKind(x0) 26.57/8.02 a__U13(x0) 26.57/8.02 a__isNatList(x0) 26.57/8.02 a__U21(x0, x1) 26.57/8.02 a__U22(x0, x1) 26.57/8.02 a__isNatKind(x0) 26.57/8.02 a__U23(x0) 26.57/8.02 a__isNat(x0) 26.57/8.02 a__U31(x0, x1) 26.57/8.02 a__U32(x0, x1) 26.57/8.02 a__U33(x0) 26.57/8.02 a__U41(x0, x1, x2) 26.57/8.02 a__U42(x0, x1, x2) 26.57/8.02 a__U43(x0, x1, x2) 26.57/8.02 a__U44(x0, x1, x2) 26.57/8.02 a__U45(x0, x1) 26.57/8.02 a__U46(x0) 26.57/8.02 a__U51(x0, x1) 26.57/8.02 a__U52(x0) 26.57/8.02 a__U61(x0) 26.57/8.02 a__U71(x0) 26.57/8.02 a__U81(x0, x1, x2) 26.57/8.02 a__U82(x0, x1, x2) 26.57/8.02 a__U83(x0, x1, x2) 26.57/8.02 a__U84(x0, x1, x2) 26.57/8.02 a__U85(x0, x1) 26.57/8.02 a__U86(x0) 26.57/8.02 a__U91(x0, x1, x2) 26.57/8.02 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (1) DependencyPairsProof (EQUIVALENT) 26.57/8.03 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (2) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 26.57/8.03 A__U11(tt, V1) -> A__ISNATILISTKIND(V1) 26.57/8.03 A__U12(tt, V1) -> A__U13(a__isNatList(V1)) 26.57/8.03 A__U12(tt, V1) -> A__ISNATLIST(V1) 26.57/8.03 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 26.57/8.03 A__U21(tt, V1) -> A__ISNATKIND(V1) 26.57/8.03 A__U22(tt, V1) -> A__U23(a__isNat(V1)) 26.57/8.03 A__U22(tt, V1) -> A__ISNAT(V1) 26.57/8.03 A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) 26.57/8.03 A__U31(tt, V) -> A__ISNATILISTKIND(V) 26.57/8.03 A__U32(tt, V) -> A__U33(a__isNatList(V)) 26.57/8.03 A__U32(tt, V) -> A__ISNATLIST(V) 26.57/8.03 A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U41(tt, V1, V2) -> A__ISNATKIND(V1) 26.57/8.03 A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U42(tt, V1, V2) -> A__ISNATILISTKIND(V2) 26.57/8.03 A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U43(tt, V1, V2) -> A__ISNATILISTKIND(V2) 26.57/8.03 A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 26.57/8.03 A__U44(tt, V1, V2) -> A__ISNAT(V1) 26.57/8.03 A__U45(tt, V2) -> A__U46(a__isNatIList(V2)) 26.57/8.03 A__U45(tt, V2) -> A__ISNATILIST(V2) 26.57/8.03 A__U51(tt, V2) -> A__U52(a__isNatIListKind(V2)) 26.57/8.03 A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 26.57/8.03 A__U81(tt, V1, V2) -> A__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U81(tt, V1, V2) -> A__ISNATKIND(V1) 26.57/8.03 A__U82(tt, V1, V2) -> A__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U82(tt, V1, V2) -> A__ISNATILISTKIND(V2) 26.57/8.03 A__U83(tt, V1, V2) -> A__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U83(tt, V1, V2) -> A__ISNATILISTKIND(V2) 26.57/8.03 A__U84(tt, V1, V2) -> A__U85(a__isNat(V1), V2) 26.57/8.03 A__U84(tt, V1, V2) -> A__ISNAT(V1) 26.57/8.03 A__U85(tt, V2) -> A__U86(a__isNatList(V2)) 26.57/8.03 A__U85(tt, V2) -> A__ISNATLIST(V2) 26.57/8.03 A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) 26.57/8.03 A__U91(tt, L, N) -> A__ISNATILISTKIND(L) 26.57/8.03 A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) 26.57/8.03 A__U92(tt, L, N) -> A__ISNAT(N) 26.57/8.03 A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) 26.57/8.03 A__U93(tt, L, N) -> A__ISNATKIND(N) 26.57/8.03 A__U94(tt, L) -> A__LENGTH(mark(L)) 26.57/8.03 A__U94(tt, L) -> MARK(L) 26.57/8.03 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 26.57/8.03 A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) 26.57/8.03 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 26.57/8.03 A__ISNAT(s(V1)) -> A__ISNATKIND(V1) 26.57/8.03 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) 26.57/8.03 A__ISNATILIST(V) -> A__ISNATILISTKIND(V) 26.57/8.03 A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 26.57/8.03 A__ISNATILIST(cons(V1, V2)) -> A__ISNATKIND(V1) 26.57/8.03 A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 26.57/8.03 A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 26.57/8.03 A__ISNATKIND(length(V1)) -> A__U61(a__isNatIListKind(V1)) 26.57/8.03 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 26.57/8.03 A__ISNATKIND(s(V1)) -> A__U71(a__isNatKind(V1)) 26.57/8.03 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 26.57/8.03 A__ISNATLIST(cons(V1, V2)) -> A__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) 26.57/8.03 A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) 26.57/8.03 A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) 26.57/8.03 MARK(zeros) -> A__ZEROS 26.57/8.03 MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) 26.57/8.03 MARK(U11(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(U12(X1, X2)) -> A__U12(mark(X1), X2) 26.57/8.03 MARK(U12(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) 26.57/8.03 MARK(U13(X)) -> A__U13(mark(X)) 26.57/8.03 MARK(U13(X)) -> MARK(X) 26.57/8.03 MARK(isNatList(X)) -> A__ISNATLIST(X) 26.57/8.03 MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) 26.57/8.03 MARK(U21(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(U22(X1, X2)) -> A__U22(mark(X1), X2) 26.57/8.03 MARK(U22(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(isNatKind(X)) -> A__ISNATKIND(X) 26.57/8.03 MARK(U23(X)) -> A__U23(mark(X)) 26.57/8.03 MARK(U23(X)) -> MARK(X) 26.57/8.03 MARK(isNat(X)) -> A__ISNAT(X) 26.57/8.03 MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) 26.57/8.03 MARK(U31(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(U32(X1, X2)) -> A__U32(mark(X1), X2) 26.57/8.03 MARK(U32(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(U33(X)) -> A__U33(mark(X)) 26.57/8.03 MARK(U33(X)) -> MARK(X) 26.57/8.03 MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) 26.57/8.03 MARK(U41(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U42(X1, X2, X3)) -> A__U42(mark(X1), X2, X3) 26.57/8.03 MARK(U42(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U43(X1, X2, X3)) -> A__U43(mark(X1), X2, X3) 26.57/8.03 MARK(U43(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U44(X1, X2, X3)) -> A__U44(mark(X1), X2, X3) 26.57/8.03 MARK(U44(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U45(X1, X2)) -> A__U45(mark(X1), X2) 26.57/8.03 MARK(U45(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(U46(X)) -> A__U46(mark(X)) 26.57/8.03 MARK(U46(X)) -> MARK(X) 26.57/8.03 MARK(isNatIList(X)) -> A__ISNATILIST(X) 26.57/8.03 MARK(U51(X1, X2)) -> A__U51(mark(X1), X2) 26.57/8.03 MARK(U51(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(U52(X)) -> A__U52(mark(X)) 26.57/8.03 MARK(U52(X)) -> MARK(X) 26.57/8.03 MARK(U61(X)) -> A__U61(mark(X)) 26.57/8.03 MARK(U61(X)) -> MARK(X) 26.57/8.03 MARK(U71(X)) -> A__U71(mark(X)) 26.57/8.03 MARK(U71(X)) -> MARK(X) 26.57/8.03 MARK(U81(X1, X2, X3)) -> A__U81(mark(X1), X2, X3) 26.57/8.03 MARK(U81(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U82(X1, X2, X3)) -> A__U82(mark(X1), X2, X3) 26.57/8.03 MARK(U82(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U83(X1, X2, X3)) -> A__U83(mark(X1), X2, X3) 26.57/8.03 MARK(U83(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U84(X1, X2, X3)) -> A__U84(mark(X1), X2, X3) 26.57/8.03 MARK(U84(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U85(X1, X2)) -> A__U85(mark(X1), X2) 26.57/8.03 MARK(U85(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(U86(X)) -> A__U86(mark(X)) 26.57/8.03 MARK(U86(X)) -> MARK(X) 26.57/8.03 MARK(U91(X1, X2, X3)) -> A__U91(mark(X1), X2, X3) 26.57/8.03 MARK(U91(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U92(X1, X2, X3)) -> A__U92(mark(X1), X2, X3) 26.57/8.03 MARK(U92(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U93(X1, X2, X3)) -> A__U93(mark(X1), X2, X3) 26.57/8.03 MARK(U93(X1, X2, X3)) -> MARK(X1) 26.57/8.03 MARK(U94(X1, X2)) -> A__U94(mark(X1), X2) 26.57/8.03 MARK(U94(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(length(X)) -> A__LENGTH(mark(X)) 26.57/8.03 MARK(length(X)) -> MARK(X) 26.57/8.03 MARK(cons(X1, X2)) -> MARK(X1) 26.57/8.03 MARK(s(X)) -> MARK(X) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__zeros -> cons(0, zeros) 26.57/8.03 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.03 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.03 a__U13(tt) -> tt 26.57/8.03 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.03 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.03 a__U23(tt) -> tt 26.57/8.03 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.03 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.03 a__U33(tt) -> tt 26.57/8.03 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.03 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.03 a__U46(tt) -> tt 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.03 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.03 a__U86(tt) -> tt 26.57/8.03 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.03 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.03 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.03 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.03 a__isNat(0) -> tt 26.57/8.03 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.03 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.03 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.03 a__isNatIList(zeros) -> tt 26.57/8.03 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatList(nil) -> tt 26.57/8.03 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 a__length(nil) -> 0 26.57/8.03 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.03 mark(zeros) -> a__zeros 26.57/8.03 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.03 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.03 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.03 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.03 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.03 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.03 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.03 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.03 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.03 mark(isNat(X)) -> a__isNat(X) 26.57/8.03 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.03 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.03 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.03 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.03 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.03 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.03 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.03 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.03 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.03 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.03 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.03 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.03 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.03 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.03 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.03 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.03 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.03 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.03 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.03 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.03 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.03 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.03 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.03 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.03 mark(length(X)) -> a__length(mark(X)) 26.57/8.03 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.03 mark(0) -> 0 26.57/8.03 mark(tt) -> tt 26.57/8.03 mark(s(X)) -> s(mark(X)) 26.57/8.03 mark(nil) -> nil 26.57/8.03 a__zeros -> zeros 26.57/8.03 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.03 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__U13(X) -> U13(X) 26.57/8.03 a__isNatList(X) -> isNatList(X) 26.57/8.03 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.03 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U23(X) -> U23(X) 26.57/8.03 a__isNat(X) -> isNat(X) 26.57/8.03 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.03 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.03 a__U33(X) -> U33(X) 26.57/8.03 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.03 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.03 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.03 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.03 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.03 a__U46(X) -> U46(X) 26.57/8.03 a__isNatIList(X) -> isNatIList(X) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.03 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.03 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.03 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.03 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.03 a__U86(X) -> U86(X) 26.57/8.03 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.03 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.03 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.03 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.03 a__length(X) -> length(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (3) DependencyGraphProof (EQUIVALENT) 26.57/8.03 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 61 less nodes. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (4) 26.57/8.03 Complex Obligation (AND) 26.57/8.03 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (5) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 26.57/8.03 A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 26.57/8.03 A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 26.57/8.03 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 26.57/8.03 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__zeros -> cons(0, zeros) 26.57/8.03 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.03 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.03 a__U13(tt) -> tt 26.57/8.03 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.03 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.03 a__U23(tt) -> tt 26.57/8.03 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.03 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.03 a__U33(tt) -> tt 26.57/8.03 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.03 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.03 a__U46(tt) -> tt 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.03 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.03 a__U86(tt) -> tt 26.57/8.03 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.03 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.03 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.03 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.03 a__isNat(0) -> tt 26.57/8.03 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.03 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.03 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.03 a__isNatIList(zeros) -> tt 26.57/8.03 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatList(nil) -> tt 26.57/8.03 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 a__length(nil) -> 0 26.57/8.03 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.03 mark(zeros) -> a__zeros 26.57/8.03 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.03 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.03 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.03 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.03 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.03 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.03 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.03 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.03 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.03 mark(isNat(X)) -> a__isNat(X) 26.57/8.03 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.03 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.03 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.03 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.03 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.03 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.03 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.03 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.03 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.03 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.03 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.03 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.03 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.03 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.03 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.03 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.03 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.03 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.03 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.03 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.03 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.03 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.03 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.03 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.03 mark(length(X)) -> a__length(mark(X)) 26.57/8.03 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.03 mark(0) -> 0 26.57/8.03 mark(tt) -> tt 26.57/8.03 mark(s(X)) -> s(mark(X)) 26.57/8.03 mark(nil) -> nil 26.57/8.03 a__zeros -> zeros 26.57/8.03 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.03 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__U13(X) -> U13(X) 26.57/8.03 a__isNatList(X) -> isNatList(X) 26.57/8.03 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.03 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U23(X) -> U23(X) 26.57/8.03 a__isNat(X) -> isNat(X) 26.57/8.03 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.03 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.03 a__U33(X) -> U33(X) 26.57/8.03 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.03 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.03 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.03 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.03 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.03 a__U46(X) -> U46(X) 26.57/8.03 a__isNatIList(X) -> isNatIList(X) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.03 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.03 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.03 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.03 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.03 a__U86(X) -> U86(X) 26.57/8.03 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.03 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.03 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.03 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.03 a__length(X) -> length(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (6) UsableRulesProof (EQUIVALENT) 26.57/8.03 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (7) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 26.57/8.03 A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 26.57/8.03 A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 26.57/8.03 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 26.57/8.03 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (8) QReductionProof (EQUIVALENT) 26.57/8.03 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (9) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 26.57/8.03 A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 26.57/8.03 A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 26.57/8.03 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 26.57/8.03 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (10) QDPSizeChangeProof (EQUIVALENT) 26.57/8.03 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 26.57/8.03 26.57/8.03 From the DPs we obtained the following set of size-change graphs: 26.57/8.03 *A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 26.57/8.03 The graph contains the following edges 1 > 2 26.57/8.03 26.57/8.03 26.57/8.03 *A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 26.57/8.03 The graph contains the following edges 1 > 1 26.57/8.03 26.57/8.03 26.57/8.03 *A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 26.57/8.03 The graph contains the following edges 2 >= 1 26.57/8.03 26.57/8.03 26.57/8.03 *A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 26.57/8.03 The graph contains the following edges 1 > 1 26.57/8.03 26.57/8.03 26.57/8.03 *A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 26.57/8.03 The graph contains the following edges 1 > 1 26.57/8.03 26.57/8.03 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (11) 26.57/8.03 YES 26.57/8.03 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (12) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U12(tt, V1) -> A__ISNATLIST(V1) 26.57/8.03 A__ISNATLIST(cons(V1, V2)) -> A__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U81(tt, V1, V2) -> A__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U82(tt, V1, V2) -> A__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U83(tt, V1, V2) -> A__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U84(tt, V1, V2) -> A__U85(a__isNat(V1), V2) 26.57/8.03 A__U85(tt, V2) -> A__ISNATLIST(V2) 26.57/8.03 A__U84(tt, V1, V2) -> A__ISNAT(V1) 26.57/8.03 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 26.57/8.03 A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 26.57/8.03 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 26.57/8.03 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 26.57/8.03 A__U22(tt, V1) -> A__ISNAT(V1) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__zeros -> cons(0, zeros) 26.57/8.03 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.03 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.03 a__U13(tt) -> tt 26.57/8.03 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.03 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.03 a__U23(tt) -> tt 26.57/8.03 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.03 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.03 a__U33(tt) -> tt 26.57/8.03 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.03 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.03 a__U46(tt) -> tt 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.03 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.03 a__U86(tt) -> tt 26.57/8.03 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.03 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.03 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.03 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.03 a__isNat(0) -> tt 26.57/8.03 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.03 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.03 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.03 a__isNatIList(zeros) -> tt 26.57/8.03 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatList(nil) -> tt 26.57/8.03 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 a__length(nil) -> 0 26.57/8.03 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.03 mark(zeros) -> a__zeros 26.57/8.03 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.03 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.03 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.03 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.03 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.03 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.03 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.03 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.03 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.03 mark(isNat(X)) -> a__isNat(X) 26.57/8.03 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.03 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.03 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.03 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.03 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.03 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.03 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.03 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.03 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.03 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.03 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.03 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.03 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.03 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.03 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.03 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.03 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.03 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.03 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.03 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.03 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.03 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.03 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.03 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.03 mark(length(X)) -> a__length(mark(X)) 26.57/8.03 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.03 mark(0) -> 0 26.57/8.03 mark(tt) -> tt 26.57/8.03 mark(s(X)) -> s(mark(X)) 26.57/8.03 mark(nil) -> nil 26.57/8.03 a__zeros -> zeros 26.57/8.03 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.03 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__U13(X) -> U13(X) 26.57/8.03 a__isNatList(X) -> isNatList(X) 26.57/8.03 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.03 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U23(X) -> U23(X) 26.57/8.03 a__isNat(X) -> isNat(X) 26.57/8.03 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.03 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.03 a__U33(X) -> U33(X) 26.57/8.03 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.03 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.03 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.03 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.03 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.03 a__U46(X) -> U46(X) 26.57/8.03 a__isNatIList(X) -> isNatIList(X) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.03 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.03 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.03 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.03 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.03 a__U86(X) -> U86(X) 26.57/8.03 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.03 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.03 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.03 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.03 a__length(X) -> length(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (13) UsableRulesProof (EQUIVALENT) 26.57/8.03 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (14) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U12(tt, V1) -> A__ISNATLIST(V1) 26.57/8.03 A__ISNATLIST(cons(V1, V2)) -> A__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U81(tt, V1, V2) -> A__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U82(tt, V1, V2) -> A__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U83(tt, V1, V2) -> A__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U84(tt, V1, V2) -> A__U85(a__isNat(V1), V2) 26.57/8.03 A__U85(tt, V2) -> A__ISNATLIST(V2) 26.57/8.03 A__U84(tt, V1, V2) -> A__ISNAT(V1) 26.57/8.03 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 26.57/8.03 A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 26.57/8.03 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 26.57/8.03 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 26.57/8.03 A__U22(tt, V1) -> A__ISNAT(V1) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 a__isNat(0) -> tt 26.57/8.03 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.03 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.03 a__isNat(X) -> isNat(X) 26.57/8.03 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.03 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.03 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.03 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.03 a__U23(tt) -> tt 26.57/8.03 a__U23(X) -> U23(X) 26.57/8.03 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.03 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.03 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.03 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.03 a__isNatList(nil) -> tt 26.57/8.03 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 a__isNatList(X) -> isNatList(X) 26.57/8.03 a__U13(tt) -> tt 26.57/8.03 a__U13(X) -> U13(X) 26.57/8.03 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.03 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.03 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.03 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.03 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.03 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.03 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.03 a__U86(tt) -> tt 26.57/8.03 a__U86(X) -> U86(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (15) QReductionProof (EQUIVALENT) 26.57/8.03 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (16) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U12(tt, V1) -> A__ISNATLIST(V1) 26.57/8.03 A__ISNATLIST(cons(V1, V2)) -> A__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U81(tt, V1, V2) -> A__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U82(tt, V1, V2) -> A__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U83(tt, V1, V2) -> A__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U84(tt, V1, V2) -> A__U85(a__isNat(V1), V2) 26.57/8.03 A__U85(tt, V2) -> A__ISNATLIST(V2) 26.57/8.03 A__U84(tt, V1, V2) -> A__ISNAT(V1) 26.57/8.03 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 26.57/8.03 A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 26.57/8.03 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 26.57/8.03 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 26.57/8.03 A__U22(tt, V1) -> A__ISNAT(V1) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 a__isNat(0) -> tt 26.57/8.03 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.03 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.03 a__isNat(X) -> isNat(X) 26.57/8.03 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.03 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.03 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.03 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.03 a__U23(tt) -> tt 26.57/8.03 a__U23(X) -> U23(X) 26.57/8.03 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.03 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.03 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.03 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.03 a__isNatList(nil) -> tt 26.57/8.03 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 a__isNatList(X) -> isNatList(X) 26.57/8.03 a__U13(tt) -> tt 26.57/8.03 a__U13(X) -> U13(X) 26.57/8.03 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.03 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.03 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.03 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.03 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.03 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.03 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.03 a__U86(tt) -> tt 26.57/8.03 a__U86(X) -> U86(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (17) QDPSizeChangeProof (EQUIVALENT) 26.57/8.03 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 26.57/8.03 26.57/8.03 From the DPs we obtained the following set of size-change graphs: 26.57/8.03 *A__ISNATLIST(cons(V1, V2)) -> A__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 The graph contains the following edges 1 > 2, 1 > 3 26.57/8.03 26.57/8.03 26.57/8.03 *A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 26.57/8.03 The graph contains the following edges 2 >= 2 26.57/8.03 26.57/8.03 26.57/8.03 *A__U81(tt, V1, V2) -> A__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 The graph contains the following edges 2 >= 2, 3 >= 3 26.57/8.03 26.57/8.03 26.57/8.03 *A__U82(tt, V1, V2) -> A__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 The graph contains the following edges 2 >= 2, 3 >= 3 26.57/8.03 26.57/8.03 26.57/8.03 *A__U83(tt, V1, V2) -> A__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 The graph contains the following edges 2 >= 2, 3 >= 3 26.57/8.03 26.57/8.03 26.57/8.03 *A__U85(tt, V2) -> A__ISNATLIST(V2) 26.57/8.03 The graph contains the following edges 2 >= 1 26.57/8.03 26.57/8.03 26.57/8.03 *A__U12(tt, V1) -> A__ISNATLIST(V1) 26.57/8.03 The graph contains the following edges 2 >= 1 26.57/8.03 26.57/8.03 26.57/8.03 *A__U84(tt, V1, V2) -> A__U85(a__isNat(V1), V2) 26.57/8.03 The graph contains the following edges 3 >= 2 26.57/8.03 26.57/8.03 26.57/8.03 *A__U84(tt, V1, V2) -> A__ISNAT(V1) 26.57/8.03 The graph contains the following edges 2 >= 1 26.57/8.03 26.57/8.03 26.57/8.03 *A__U22(tt, V1) -> A__ISNAT(V1) 26.57/8.03 The graph contains the following edges 2 >= 1 26.57/8.03 26.57/8.03 26.57/8.03 *A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 26.57/8.03 The graph contains the following edges 2 >= 2 26.57/8.03 26.57/8.03 26.57/8.03 *A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 26.57/8.03 The graph contains the following edges 1 > 2 26.57/8.03 26.57/8.03 26.57/8.03 *A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 26.57/8.03 The graph contains the following edges 1 > 2 26.57/8.03 26.57/8.03 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (18) 26.57/8.03 YES 26.57/8.03 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (19) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 26.57/8.03 A__U45(tt, V2) -> A__ISNATILIST(V2) 26.57/8.03 A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__zeros -> cons(0, zeros) 26.57/8.03 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.03 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.03 a__U13(tt) -> tt 26.57/8.03 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.03 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.03 a__U23(tt) -> tt 26.57/8.03 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.03 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.03 a__U33(tt) -> tt 26.57/8.03 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.03 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.03 a__U46(tt) -> tt 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.03 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.03 a__U86(tt) -> tt 26.57/8.03 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.03 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.03 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.03 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.03 a__isNat(0) -> tt 26.57/8.03 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.03 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.03 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.03 a__isNatIList(zeros) -> tt 26.57/8.03 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatList(nil) -> tt 26.57/8.03 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 a__length(nil) -> 0 26.57/8.03 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.03 mark(zeros) -> a__zeros 26.57/8.03 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.03 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.03 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.03 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.03 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.03 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.03 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.03 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.03 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.03 mark(isNat(X)) -> a__isNat(X) 26.57/8.03 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.03 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.03 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.03 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.03 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.03 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.03 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.03 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.03 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.03 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.03 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.03 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.03 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.03 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.03 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.03 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.03 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.03 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.03 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.03 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.03 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.03 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.03 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.03 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.03 mark(length(X)) -> a__length(mark(X)) 26.57/8.03 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.03 mark(0) -> 0 26.57/8.03 mark(tt) -> tt 26.57/8.03 mark(s(X)) -> s(mark(X)) 26.57/8.03 mark(nil) -> nil 26.57/8.03 a__zeros -> zeros 26.57/8.03 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.03 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__U13(X) -> U13(X) 26.57/8.03 a__isNatList(X) -> isNatList(X) 26.57/8.03 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.03 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U23(X) -> U23(X) 26.57/8.03 a__isNat(X) -> isNat(X) 26.57/8.03 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.03 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.03 a__U33(X) -> U33(X) 26.57/8.03 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.03 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.03 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.03 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.03 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.03 a__U46(X) -> U46(X) 26.57/8.03 a__isNatIList(X) -> isNatIList(X) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.03 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.03 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.03 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.03 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.03 a__U86(X) -> U86(X) 26.57/8.03 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.03 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.03 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.03 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.03 a__length(X) -> length(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (20) UsableRulesProof (EQUIVALENT) 26.57/8.03 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (21) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 26.57/8.03 A__U45(tt, V2) -> A__ISNATILIST(V2) 26.57/8.03 A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__isNat(0) -> tt 26.57/8.03 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.03 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.03 a__isNat(X) -> isNat(X) 26.57/8.03 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.03 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.03 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.03 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.03 a__U23(tt) -> tt 26.57/8.03 a__U23(X) -> U23(X) 26.57/8.03 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.03 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.03 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.03 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.03 a__isNatList(nil) -> tt 26.57/8.03 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 a__isNatList(X) -> isNatList(X) 26.57/8.03 a__U13(tt) -> tt 26.57/8.03 a__U13(X) -> U13(X) 26.57/8.03 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.03 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.03 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.03 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.03 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.03 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.03 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.03 a__U86(tt) -> tt 26.57/8.03 a__U86(X) -> U86(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (22) QReductionProof (EQUIVALENT) 26.57/8.03 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 26.57/8.03 26.57/8.03 a__zeros 26.57/8.03 a__isNatIList(x0) 26.57/8.03 mark(zeros) 26.57/8.03 mark(U11(x0, x1)) 26.57/8.03 mark(U12(x0, x1)) 26.57/8.03 mark(isNatIListKind(x0)) 26.57/8.03 mark(U13(x0)) 26.57/8.03 mark(isNatList(x0)) 26.57/8.03 mark(U21(x0, x1)) 26.57/8.03 mark(U22(x0, x1)) 26.57/8.03 mark(isNatKind(x0)) 26.57/8.03 mark(U23(x0)) 26.57/8.03 mark(isNat(x0)) 26.57/8.03 mark(U31(x0, x1)) 26.57/8.03 mark(U32(x0, x1)) 26.57/8.03 mark(U33(x0)) 26.57/8.03 mark(U41(x0, x1, x2)) 26.57/8.03 mark(U42(x0, x1, x2)) 26.57/8.03 mark(U43(x0, x1, x2)) 26.57/8.03 mark(U44(x0, x1, x2)) 26.57/8.03 mark(U45(x0, x1)) 26.57/8.03 mark(U46(x0)) 26.57/8.03 mark(isNatIList(x0)) 26.57/8.03 mark(U51(x0, x1)) 26.57/8.03 mark(U52(x0)) 26.57/8.03 mark(U61(x0)) 26.57/8.03 mark(U71(x0)) 26.57/8.03 mark(U81(x0, x1, x2)) 26.57/8.03 mark(U82(x0, x1, x2)) 26.57/8.03 mark(U83(x0, x1, x2)) 26.57/8.03 mark(U84(x0, x1, x2)) 26.57/8.03 mark(U85(x0, x1)) 26.57/8.03 mark(U86(x0)) 26.57/8.03 mark(U91(x0, x1, x2)) 26.57/8.03 mark(U92(x0, x1, x2)) 26.57/8.03 mark(U93(x0, x1, x2)) 26.57/8.03 mark(U94(x0, x1)) 26.57/8.03 mark(length(x0)) 26.57/8.03 mark(cons(x0, x1)) 26.57/8.03 mark(0) 26.57/8.03 mark(tt) 26.57/8.03 mark(s(x0)) 26.57/8.03 mark(nil) 26.57/8.03 a__U31(x0, x1) 26.57/8.03 a__U32(x0, x1) 26.57/8.03 a__U33(x0) 26.57/8.03 a__U41(x0, x1, x2) 26.57/8.03 a__U42(x0, x1, x2) 26.57/8.03 a__U43(x0, x1, x2) 26.57/8.03 a__U44(x0, x1, x2) 26.57/8.03 a__U45(x0, x1) 26.57/8.03 a__U46(x0) 26.57/8.03 a__U91(x0, x1, x2) 26.57/8.03 a__U92(x0, x1, x2) 26.57/8.03 a__U93(x0, x1, x2) 26.57/8.03 a__U94(x0, x1) 26.57/8.03 a__length(x0) 26.57/8.03 26.57/8.03 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (23) 26.57/8.03 Obligation: 26.57/8.03 Q DP problem: 26.57/8.03 The TRS P consists of the following rules: 26.57/8.03 26.57/8.03 A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 26.57/8.03 A__U45(tt, V2) -> A__ISNATILIST(V2) 26.57/8.03 A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 26.57/8.03 A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.03 A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.03 26.57/8.03 The TRS R consists of the following rules: 26.57/8.03 26.57/8.03 a__isNatIListKind(nil) -> tt 26.57/8.03 a__isNatIListKind(zeros) -> tt 26.57/8.03 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.03 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.03 a__isNatKind(0) -> tt 26.57/8.03 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.03 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.03 a__isNatKind(X) -> isNatKind(X) 26.57/8.03 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.03 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.03 a__U52(tt) -> tt 26.57/8.03 a__U52(X) -> U52(X) 26.57/8.03 a__U71(tt) -> tt 26.57/8.03 a__U71(X) -> U71(X) 26.57/8.03 a__U61(tt) -> tt 26.57/8.03 a__U61(X) -> U61(X) 26.57/8.03 a__isNat(0) -> tt 26.57/8.03 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.03 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.03 a__isNat(X) -> isNat(X) 26.57/8.03 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.03 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.03 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.03 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.03 a__U23(tt) -> tt 26.57/8.03 a__U23(X) -> U23(X) 26.57/8.03 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.03 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.03 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.03 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.03 a__isNatList(nil) -> tt 26.57/8.03 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.03 a__isNatList(X) -> isNatList(X) 26.57/8.03 a__U13(tt) -> tt 26.57/8.03 a__U13(X) -> U13(X) 26.57/8.03 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.03 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.03 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.03 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.03 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.03 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.03 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.03 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.03 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.03 a__U86(tt) -> tt 26.57/8.03 a__U86(X) -> U86(X) 26.57/8.03 26.57/8.03 The set Q consists of the following terms: 26.57/8.03 26.57/8.03 a__U11(x0, x1) 26.57/8.03 a__U12(x0, x1) 26.57/8.03 a__isNatIListKind(x0) 26.57/8.03 a__U13(x0) 26.57/8.03 a__isNatList(x0) 26.57/8.03 a__U21(x0, x1) 26.57/8.03 a__U22(x0, x1) 26.57/8.03 a__isNatKind(x0) 26.57/8.03 a__U23(x0) 26.57/8.03 a__isNat(x0) 26.57/8.03 a__U51(x0, x1) 26.57/8.03 a__U52(x0) 26.57/8.03 a__U61(x0) 26.57/8.03 a__U71(x0) 26.57/8.03 a__U81(x0, x1, x2) 26.57/8.03 a__U82(x0, x1, x2) 26.57/8.03 a__U83(x0, x1, x2) 26.57/8.03 a__U84(x0, x1, x2) 26.57/8.03 a__U85(x0, x1) 26.57/8.03 a__U86(x0) 26.57/8.03 26.57/8.03 We have to consider all minimal (P,Q,R)-chains. 26.57/8.03 ---------------------------------------- 26.57/8.03 26.57/8.03 (24) QDPSizeChangeProof (EQUIVALENT) 26.57/8.03 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 26.57/8.04 26.57/8.04 From the DPs we obtained the following set of size-change graphs: 26.57/8.04 *A__U45(tt, V2) -> A__ISNATILIST(V2) 26.57/8.04 The graph contains the following edges 2 >= 1 26.57/8.04 26.57/8.04 26.57/8.04 *A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 The graph contains the following edges 2 >= 2, 3 >= 3 26.57/8.04 26.57/8.04 26.57/8.04 *A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 26.57/8.04 The graph contains the following edges 1 > 2, 1 > 3 26.57/8.04 26.57/8.04 26.57/8.04 *A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 26.57/8.04 The graph contains the following edges 3 >= 2 26.57/8.04 26.57/8.04 26.57/8.04 *A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 The graph contains the following edges 2 >= 2, 3 >= 3 26.57/8.04 26.57/8.04 26.57/8.04 *A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 The graph contains the following edges 2 >= 2, 3 >= 3 26.57/8.04 26.57/8.04 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (25) 26.57/8.04 YES 26.57/8.04 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (26) 26.57/8.04 Obligation: 26.57/8.04 Q DP problem: 26.57/8.04 The TRS P consists of the following rules: 26.57/8.04 26.57/8.04 MARK(U11(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U12(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U13(X)) -> MARK(X) 26.57/8.04 MARK(U21(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U22(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U23(X)) -> MARK(X) 26.57/8.04 MARK(U31(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U32(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U33(X)) -> MARK(X) 26.57/8.04 MARK(U41(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U42(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U43(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U44(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U45(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U46(X)) -> MARK(X) 26.57/8.04 MARK(U51(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U52(X)) -> MARK(X) 26.57/8.04 MARK(U61(X)) -> MARK(X) 26.57/8.04 MARK(U71(X)) -> MARK(X) 26.57/8.04 MARK(U81(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U82(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U83(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U84(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U85(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U86(X)) -> MARK(X) 26.57/8.04 MARK(U91(X1, X2, X3)) -> A__U91(mark(X1), X2, X3) 26.57/8.04 A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) 26.57/8.04 A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) 26.57/8.04 A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) 26.57/8.04 A__U94(tt, L) -> A__LENGTH(mark(L)) 26.57/8.04 A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) 26.57/8.04 A__U94(tt, L) -> MARK(L) 26.57/8.04 MARK(U91(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U92(X1, X2, X3)) -> A__U92(mark(X1), X2, X3) 26.57/8.04 MARK(U92(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U93(X1, X2, X3)) -> A__U93(mark(X1), X2, X3) 26.57/8.04 MARK(U93(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U94(X1, X2)) -> A__U94(mark(X1), X2) 26.57/8.04 MARK(U94(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(length(X)) -> A__LENGTH(mark(X)) 26.57/8.04 MARK(length(X)) -> MARK(X) 26.57/8.04 MARK(cons(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(s(X)) -> MARK(X) 26.57/8.04 26.57/8.04 The TRS R consists of the following rules: 26.57/8.04 26.57/8.04 a__zeros -> cons(0, zeros) 26.57/8.04 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.04 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.04 a__U13(tt) -> tt 26.57/8.04 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.04 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.04 a__U23(tt) -> tt 26.57/8.04 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.04 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.04 a__U33(tt) -> tt 26.57/8.04 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.04 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.04 a__U46(tt) -> tt 26.57/8.04 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.04 a__U52(tt) -> tt 26.57/8.04 a__U61(tt) -> tt 26.57/8.04 a__U71(tt) -> tt 26.57/8.04 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.04 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.04 a__U86(tt) -> tt 26.57/8.04 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.04 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.04 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.04 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.04 a__isNat(0) -> tt 26.57/8.04 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.04 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.04 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.04 a__isNatIList(zeros) -> tt 26.57/8.04 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatIListKind(nil) -> tt 26.57/8.04 a__isNatIListKind(zeros) -> tt 26.57/8.04 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.04 a__isNatKind(0) -> tt 26.57/8.04 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.04 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.04 a__isNatList(nil) -> tt 26.57/8.04 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.04 a__length(nil) -> 0 26.57/8.04 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.04 mark(zeros) -> a__zeros 26.57/8.04 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.04 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.04 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.04 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.04 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.04 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.04 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.04 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.04 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.04 mark(isNat(X)) -> a__isNat(X) 26.57/8.04 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.04 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.04 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.04 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.04 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.04 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.04 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.04 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.04 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.04 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.04 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.04 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.04 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.04 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.04 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.04 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.04 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.04 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.04 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.04 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.04 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.04 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.04 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.04 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.04 mark(length(X)) -> a__length(mark(X)) 26.57/8.04 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.04 mark(0) -> 0 26.57/8.04 mark(tt) -> tt 26.57/8.04 mark(s(X)) -> s(mark(X)) 26.57/8.04 mark(nil) -> nil 26.57/8.04 a__zeros -> zeros 26.57/8.04 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.04 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.04 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.04 a__U13(X) -> U13(X) 26.57/8.04 a__isNatList(X) -> isNatList(X) 26.57/8.04 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.04 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.04 a__isNatKind(X) -> isNatKind(X) 26.57/8.04 a__U23(X) -> U23(X) 26.57/8.04 a__isNat(X) -> isNat(X) 26.57/8.04 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.04 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.04 a__U33(X) -> U33(X) 26.57/8.04 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.04 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.04 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.04 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.04 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.04 a__U46(X) -> U46(X) 26.57/8.04 a__isNatIList(X) -> isNatIList(X) 26.57/8.04 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.04 a__U52(X) -> U52(X) 26.57/8.04 a__U61(X) -> U61(X) 26.57/8.04 a__U71(X) -> U71(X) 26.57/8.04 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.04 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.04 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.04 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.04 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.04 a__U86(X) -> U86(X) 26.57/8.04 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.04 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.04 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.04 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.04 a__length(X) -> length(X) 26.57/8.04 26.57/8.04 The set Q consists of the following terms: 26.57/8.04 26.57/8.04 a__zeros 26.57/8.04 a__isNatIList(x0) 26.57/8.04 mark(zeros) 26.57/8.04 mark(U11(x0, x1)) 26.57/8.04 mark(U12(x0, x1)) 26.57/8.04 mark(isNatIListKind(x0)) 26.57/8.04 mark(U13(x0)) 26.57/8.04 mark(isNatList(x0)) 26.57/8.04 mark(U21(x0, x1)) 26.57/8.04 mark(U22(x0, x1)) 26.57/8.04 mark(isNatKind(x0)) 26.57/8.04 mark(U23(x0)) 26.57/8.04 mark(isNat(x0)) 26.57/8.04 mark(U31(x0, x1)) 26.57/8.04 mark(U32(x0, x1)) 26.57/8.04 mark(U33(x0)) 26.57/8.04 mark(U41(x0, x1, x2)) 26.57/8.04 mark(U42(x0, x1, x2)) 26.57/8.04 mark(U43(x0, x1, x2)) 26.57/8.04 mark(U44(x0, x1, x2)) 26.57/8.04 mark(U45(x0, x1)) 26.57/8.04 mark(U46(x0)) 26.57/8.04 mark(isNatIList(x0)) 26.57/8.04 mark(U51(x0, x1)) 26.57/8.04 mark(U52(x0)) 26.57/8.04 mark(U61(x0)) 26.57/8.04 mark(U71(x0)) 26.57/8.04 mark(U81(x0, x1, x2)) 26.57/8.04 mark(U82(x0, x1, x2)) 26.57/8.04 mark(U83(x0, x1, x2)) 26.57/8.04 mark(U84(x0, x1, x2)) 26.57/8.04 mark(U85(x0, x1)) 26.57/8.04 mark(U86(x0)) 26.57/8.04 mark(U91(x0, x1, x2)) 26.57/8.04 mark(U92(x0, x1, x2)) 26.57/8.04 mark(U93(x0, x1, x2)) 26.57/8.04 mark(U94(x0, x1)) 26.57/8.04 mark(length(x0)) 26.57/8.04 mark(cons(x0, x1)) 26.57/8.04 mark(0) 26.57/8.04 mark(tt) 26.57/8.04 mark(s(x0)) 26.57/8.04 mark(nil) 26.57/8.04 a__U11(x0, x1) 26.57/8.04 a__U12(x0, x1) 26.57/8.04 a__isNatIListKind(x0) 26.57/8.04 a__U13(x0) 26.57/8.04 a__isNatList(x0) 26.57/8.04 a__U21(x0, x1) 26.57/8.04 a__U22(x0, x1) 26.57/8.04 a__isNatKind(x0) 26.57/8.04 a__U23(x0) 26.57/8.04 a__isNat(x0) 26.57/8.04 a__U31(x0, x1) 26.57/8.04 a__U32(x0, x1) 26.57/8.04 a__U33(x0) 26.57/8.04 a__U41(x0, x1, x2) 26.57/8.04 a__U42(x0, x1, x2) 26.57/8.04 a__U43(x0, x1, x2) 26.57/8.04 a__U44(x0, x1, x2) 26.57/8.04 a__U45(x0, x1) 26.57/8.04 a__U46(x0) 26.57/8.04 a__U51(x0, x1) 26.57/8.04 a__U52(x0) 26.57/8.04 a__U61(x0) 26.57/8.04 a__U71(x0) 26.57/8.04 a__U81(x0, x1, x2) 26.57/8.04 a__U82(x0, x1, x2) 26.57/8.04 a__U83(x0, x1, x2) 26.57/8.04 a__U84(x0, x1, x2) 26.57/8.04 a__U85(x0, x1) 26.57/8.04 a__U86(x0) 26.57/8.04 a__U91(x0, x1, x2) 26.57/8.04 a__U92(x0, x1, x2) 26.57/8.04 a__U93(x0, x1, x2) 26.57/8.04 a__U94(x0, x1) 26.57/8.04 a__length(x0) 26.57/8.04 26.57/8.04 We have to consider all minimal (P,Q,R)-chains. 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (27) QDPOrderProof (EQUIVALENT) 26.57/8.04 We use the reduction pair processor [LPAR04,JAR06]. 26.57/8.04 26.57/8.04 26.57/8.04 The following pairs can be oriented strictly and are deleted. 26.57/8.04 26.57/8.04 MARK(U91(X1, X2, X3)) -> A__U91(mark(X1), X2, X3) 26.57/8.04 MARK(U91(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U92(X1, X2, X3)) -> A__U92(mark(X1), X2, X3) 26.57/8.04 MARK(U92(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U93(X1, X2, X3)) -> A__U93(mark(X1), X2, X3) 26.57/8.04 MARK(U93(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U94(X1, X2)) -> A__U94(mark(X1), X2) 26.57/8.04 MARK(U94(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(cons(X1, X2)) -> MARK(X1) 26.57/8.04 The remaining pairs can at least be oriented weakly. 26.57/8.04 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 26.57/8.04 26.57/8.04 POL( A__LENGTH_1(x_1) ) = max{0, x_1 - 1} 26.57/8.04 POL( A__U91_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 1 26.57/8.04 POL( A__U92_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 1 26.57/8.04 POL( A__U93_3(x_1, ..., x_3) ) = 2x_2 + 1 26.57/8.04 POL( A__U94_2(x_1, x_2) ) = 2x_2 + 1 26.57/8.04 POL( mark_1(x_1) ) = x_1 + 2 26.57/8.04 POL( zeros ) = 0 26.57/8.04 POL( a__zeros ) = 2 26.57/8.04 POL( U11_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U11_2(x_1, x_2) ) = x_1 26.57/8.04 POL( U12_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U12_2(x_1, x_2) ) = x_1 26.57/8.04 POL( isNatIListKind_1(x_1) ) = 0 26.57/8.04 POL( a__isNatIListKind_1(x_1) ) = 0 26.57/8.04 POL( U13_1(x_1) ) = x_1 26.57/8.04 POL( a__U13_1(x_1) ) = x_1 26.57/8.04 POL( isNatList_1(x_1) ) = 0 26.57/8.04 POL( a__isNatList_1(x_1) ) = 0 26.57/8.04 POL( U21_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U21_2(x_1, x_2) ) = x_1 26.57/8.04 POL( U22_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U22_2(x_1, x_2) ) = x_1 26.57/8.04 POL( isNatKind_1(x_1) ) = 0 26.57/8.04 POL( a__isNatKind_1(x_1) ) = 0 26.57/8.04 POL( U23_1(x_1) ) = x_1 26.57/8.04 POL( a__U23_1(x_1) ) = x_1 26.57/8.04 POL( isNat_1(x_1) ) = 0 26.57/8.04 POL( a__isNat_1(x_1) ) = 0 26.57/8.04 POL( U31_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U31_2(x_1, x_2) ) = x_1 26.57/8.04 POL( U32_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U32_2(x_1, x_2) ) = x_1 26.57/8.04 POL( U33_1(x_1) ) = x_1 26.57/8.04 POL( a__U33_1(x_1) ) = x_1 26.57/8.04 POL( U41_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U41_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U42_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U42_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U43_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U43_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U44_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U44_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U45_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U45_2(x_1, x_2) ) = x_1 26.57/8.04 POL( U46_1(x_1) ) = x_1 26.57/8.04 POL( a__U46_1(x_1) ) = x_1 26.57/8.04 POL( isNatIList_1(x_1) ) = 0 26.57/8.04 POL( a__isNatIList_1(x_1) ) = 0 26.57/8.04 POL( U51_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U51_2(x_1, x_2) ) = x_1 26.57/8.04 POL( U52_1(x_1) ) = x_1 26.57/8.04 POL( a__U52_1(x_1) ) = x_1 26.57/8.04 POL( U61_1(x_1) ) = x_1 26.57/8.04 POL( a__U61_1(x_1) ) = x_1 26.57/8.04 POL( U71_1(x_1) ) = x_1 26.57/8.04 POL( a__U71_1(x_1) ) = x_1 26.57/8.04 POL( U81_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U81_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U82_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U82_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U83_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U83_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U84_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U84_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U85_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U85_2(x_1, x_2) ) = x_1 26.57/8.04 POL( U86_1(x_1) ) = x_1 26.57/8.04 POL( a__U86_1(x_1) ) = x_1 26.57/8.04 POL( U91_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2 26.57/8.04 POL( a__U91_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2 26.57/8.04 POL( U92_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 26.57/8.04 POL( a__U92_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 26.57/8.04 POL( U93_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 26.57/8.04 POL( a__U93_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 26.57/8.04 POL( U94_2(x_1, x_2) ) = x_1 + 2x_2 + 2 26.57/8.04 POL( a__U94_2(x_1, x_2) ) = x_1 + 2x_2 + 2 26.57/8.04 POL( length_1(x_1) ) = x_1 26.57/8.04 POL( a__length_1(x_1) ) = x_1 26.57/8.04 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 + 2 26.57/8.04 POL( 0 ) = 0 26.57/8.04 POL( tt ) = 0 26.57/8.04 POL( s_1(x_1) ) = x_1 26.57/8.04 POL( nil ) = 2 26.57/8.04 POL( MARK_1(x_1) ) = 2x_1 + 1 26.57/8.04 26.57/8.04 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.57/8.04 26.57/8.04 mark(zeros) -> a__zeros 26.57/8.04 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.04 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.04 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.04 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.04 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.04 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.04 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.04 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.04 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.04 mark(isNat(X)) -> a__isNat(X) 26.57/8.04 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.04 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.04 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.04 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.04 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.04 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.04 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.04 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.04 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.04 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.04 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.04 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.04 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.04 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.04 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.04 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.04 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.04 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.04 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.04 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.04 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.04 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.04 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.04 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.04 mark(length(X)) -> a__length(mark(X)) 26.57/8.04 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.04 mark(0) -> 0 26.57/8.04 mark(tt) -> tt 26.57/8.04 mark(s(X)) -> s(mark(X)) 26.57/8.04 mark(nil) -> nil 26.57/8.04 a__isNatIListKind(nil) -> tt 26.57/8.04 a__isNatIListKind(zeros) -> tt 26.57/8.04 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.04 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.04 a__isNat(0) -> tt 26.57/8.04 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.04 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.04 a__isNat(X) -> isNat(X) 26.57/8.04 a__isNatKind(0) -> tt 26.57/8.04 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.04 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.04 a__isNatKind(X) -> isNatKind(X) 26.57/8.04 a__isNatList(nil) -> tt 26.57/8.04 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatList(X) -> isNatList(X) 26.57/8.04 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.04 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.04 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.04 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.04 a__U13(tt) -> tt 26.57/8.04 a__U13(X) -> U13(X) 26.57/8.04 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.04 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.04 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.04 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.04 a__U23(tt) -> tt 26.57/8.04 a__U23(X) -> U23(X) 26.57/8.04 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.04 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.04 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.04 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.04 a__U33(tt) -> tt 26.57/8.04 a__U33(X) -> U33(X) 26.57/8.04 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.04 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.04 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.04 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.04 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.04 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.04 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.04 a__U46(tt) -> tt 26.57/8.04 a__U46(X) -> U46(X) 26.57/8.04 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.04 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.04 a__U52(tt) -> tt 26.57/8.04 a__U52(X) -> U52(X) 26.57/8.04 a__U61(tt) -> tt 26.57/8.04 a__U61(X) -> U61(X) 26.57/8.04 a__U71(tt) -> tt 26.57/8.04 a__U71(X) -> U71(X) 26.57/8.04 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.04 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.04 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.04 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.04 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.04 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.04 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.04 a__U86(tt) -> tt 26.57/8.04 a__U86(X) -> U86(X) 26.57/8.04 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.04 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.04 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.04 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.04 a__length(nil) -> 0 26.57/8.04 a__length(X) -> length(X) 26.57/8.04 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.04 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.04 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.04 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.04 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.04 a__zeros -> cons(0, zeros) 26.57/8.04 a__zeros -> zeros 26.57/8.04 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.04 a__isNatIList(zeros) -> tt 26.57/8.04 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatIList(X) -> isNatIList(X) 26.57/8.04 26.57/8.04 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (28) 26.57/8.04 Obligation: 26.57/8.04 Q DP problem: 26.57/8.04 The TRS P consists of the following rules: 26.57/8.04 26.57/8.04 MARK(U11(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U12(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U13(X)) -> MARK(X) 26.57/8.04 MARK(U21(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U22(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U23(X)) -> MARK(X) 26.57/8.04 MARK(U31(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U32(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U33(X)) -> MARK(X) 26.57/8.04 MARK(U41(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U42(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U43(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U44(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U45(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U46(X)) -> MARK(X) 26.57/8.04 MARK(U51(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U52(X)) -> MARK(X) 26.57/8.04 MARK(U61(X)) -> MARK(X) 26.57/8.04 MARK(U71(X)) -> MARK(X) 26.57/8.04 MARK(U81(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U82(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U83(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U84(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U85(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U86(X)) -> MARK(X) 26.57/8.04 A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) 26.57/8.04 A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) 26.57/8.04 A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) 26.57/8.04 A__U94(tt, L) -> A__LENGTH(mark(L)) 26.57/8.04 A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) 26.57/8.04 A__U94(tt, L) -> MARK(L) 26.57/8.04 MARK(length(X)) -> A__LENGTH(mark(X)) 26.57/8.04 MARK(length(X)) -> MARK(X) 26.57/8.04 MARK(s(X)) -> MARK(X) 26.57/8.04 26.57/8.04 The TRS R consists of the following rules: 26.57/8.04 26.57/8.04 a__zeros -> cons(0, zeros) 26.57/8.04 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.04 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.04 a__U13(tt) -> tt 26.57/8.04 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.04 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.04 a__U23(tt) -> tt 26.57/8.04 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.04 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.04 a__U33(tt) -> tt 26.57/8.04 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.04 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.04 a__U46(tt) -> tt 26.57/8.04 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.04 a__U52(tt) -> tt 26.57/8.04 a__U61(tt) -> tt 26.57/8.04 a__U71(tt) -> tt 26.57/8.04 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.04 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.04 a__U86(tt) -> tt 26.57/8.04 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.04 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.04 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.04 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.04 a__isNat(0) -> tt 26.57/8.04 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.04 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.04 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.04 a__isNatIList(zeros) -> tt 26.57/8.04 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatIListKind(nil) -> tt 26.57/8.04 a__isNatIListKind(zeros) -> tt 26.57/8.04 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.04 a__isNatKind(0) -> tt 26.57/8.04 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.04 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.04 a__isNatList(nil) -> tt 26.57/8.04 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.04 a__length(nil) -> 0 26.57/8.04 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.04 mark(zeros) -> a__zeros 26.57/8.04 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.04 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.04 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.04 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.04 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.04 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.04 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.04 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.04 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.04 mark(isNat(X)) -> a__isNat(X) 26.57/8.04 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.04 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.04 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.04 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.04 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.04 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.04 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.04 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.04 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.04 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.04 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.04 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.04 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.04 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.04 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.04 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.04 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.04 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.04 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.04 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.04 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.04 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.04 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.04 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.04 mark(length(X)) -> a__length(mark(X)) 26.57/8.04 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.04 mark(0) -> 0 26.57/8.04 mark(tt) -> tt 26.57/8.04 mark(s(X)) -> s(mark(X)) 26.57/8.04 mark(nil) -> nil 26.57/8.04 a__zeros -> zeros 26.57/8.04 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.04 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.04 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.04 a__U13(X) -> U13(X) 26.57/8.04 a__isNatList(X) -> isNatList(X) 26.57/8.04 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.04 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.04 a__isNatKind(X) -> isNatKind(X) 26.57/8.04 a__U23(X) -> U23(X) 26.57/8.04 a__isNat(X) -> isNat(X) 26.57/8.04 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.04 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.04 a__U33(X) -> U33(X) 26.57/8.04 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.04 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.04 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.04 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.04 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.04 a__U46(X) -> U46(X) 26.57/8.04 a__isNatIList(X) -> isNatIList(X) 26.57/8.04 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.04 a__U52(X) -> U52(X) 26.57/8.04 a__U61(X) -> U61(X) 26.57/8.04 a__U71(X) -> U71(X) 26.57/8.04 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.04 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.04 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.04 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.04 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.04 a__U86(X) -> U86(X) 26.57/8.04 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.04 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.04 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.04 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.04 a__length(X) -> length(X) 26.57/8.04 26.57/8.04 The set Q consists of the following terms: 26.57/8.04 26.57/8.04 a__zeros 26.57/8.04 a__isNatIList(x0) 26.57/8.04 mark(zeros) 26.57/8.04 mark(U11(x0, x1)) 26.57/8.04 mark(U12(x0, x1)) 26.57/8.04 mark(isNatIListKind(x0)) 26.57/8.04 mark(U13(x0)) 26.57/8.04 mark(isNatList(x0)) 26.57/8.04 mark(U21(x0, x1)) 26.57/8.04 mark(U22(x0, x1)) 26.57/8.04 mark(isNatKind(x0)) 26.57/8.04 mark(U23(x0)) 26.57/8.04 mark(isNat(x0)) 26.57/8.04 mark(U31(x0, x1)) 26.57/8.04 mark(U32(x0, x1)) 26.57/8.04 mark(U33(x0)) 26.57/8.04 mark(U41(x0, x1, x2)) 26.57/8.04 mark(U42(x0, x1, x2)) 26.57/8.04 mark(U43(x0, x1, x2)) 26.57/8.04 mark(U44(x0, x1, x2)) 26.57/8.04 mark(U45(x0, x1)) 26.57/8.04 mark(U46(x0)) 26.57/8.04 mark(isNatIList(x0)) 26.57/8.04 mark(U51(x0, x1)) 26.57/8.04 mark(U52(x0)) 26.57/8.04 mark(U61(x0)) 26.57/8.04 mark(U71(x0)) 26.57/8.04 mark(U81(x0, x1, x2)) 26.57/8.04 mark(U82(x0, x1, x2)) 26.57/8.04 mark(U83(x0, x1, x2)) 26.57/8.04 mark(U84(x0, x1, x2)) 26.57/8.04 mark(U85(x0, x1)) 26.57/8.04 mark(U86(x0)) 26.57/8.04 mark(U91(x0, x1, x2)) 26.57/8.04 mark(U92(x0, x1, x2)) 26.57/8.04 mark(U93(x0, x1, x2)) 26.57/8.04 mark(U94(x0, x1)) 26.57/8.04 mark(length(x0)) 26.57/8.04 mark(cons(x0, x1)) 26.57/8.04 mark(0) 26.57/8.04 mark(tt) 26.57/8.04 mark(s(x0)) 26.57/8.04 mark(nil) 26.57/8.04 a__U11(x0, x1) 26.57/8.04 a__U12(x0, x1) 26.57/8.04 a__isNatIListKind(x0) 26.57/8.04 a__U13(x0) 26.57/8.04 a__isNatList(x0) 26.57/8.04 a__U21(x0, x1) 26.57/8.04 a__U22(x0, x1) 26.57/8.04 a__isNatKind(x0) 26.57/8.04 a__U23(x0) 26.57/8.04 a__isNat(x0) 26.57/8.04 a__U31(x0, x1) 26.57/8.04 a__U32(x0, x1) 26.57/8.04 a__U33(x0) 26.57/8.04 a__U41(x0, x1, x2) 26.57/8.04 a__U42(x0, x1, x2) 26.57/8.04 a__U43(x0, x1, x2) 26.57/8.04 a__U44(x0, x1, x2) 26.57/8.04 a__U45(x0, x1) 26.57/8.04 a__U46(x0) 26.57/8.04 a__U51(x0, x1) 26.57/8.04 a__U52(x0) 26.57/8.04 a__U61(x0) 26.57/8.04 a__U71(x0) 26.57/8.04 a__U81(x0, x1, x2) 26.57/8.04 a__U82(x0, x1, x2) 26.57/8.04 a__U83(x0, x1, x2) 26.57/8.04 a__U84(x0, x1, x2) 26.57/8.04 a__U85(x0, x1) 26.57/8.04 a__U86(x0) 26.57/8.04 a__U91(x0, x1, x2) 26.57/8.04 a__U92(x0, x1, x2) 26.57/8.04 a__U93(x0, x1, x2) 26.57/8.04 a__U94(x0, x1) 26.57/8.04 a__length(x0) 26.57/8.04 26.57/8.04 We have to consider all minimal (P,Q,R)-chains. 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (29) QDPOrderProof (EQUIVALENT) 26.57/8.04 We use the reduction pair processor [LPAR04,JAR06]. 26.57/8.04 26.57/8.04 26.57/8.04 The following pairs can be oriented strictly and are deleted. 26.57/8.04 26.57/8.04 A__U94(tt, L) -> MARK(L) 26.57/8.04 MARK(length(X)) -> A__LENGTH(mark(X)) 26.57/8.04 MARK(length(X)) -> MARK(X) 26.57/8.04 The remaining pairs can at least be oriented weakly. 26.57/8.04 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 26.57/8.04 26.57/8.04 POL( A__LENGTH_1(x_1) ) = x_1 + 1 26.57/8.04 POL( A__U92_3(x_1, ..., x_3) ) = 2x_2 + 1 26.57/8.04 POL( A__U91_3(x_1, ..., x_3) ) = 2x_2 + 1 26.57/8.04 POL( a__U11_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( a__U12_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( a__U83_3(x_1, ..., x_3) ) = 2x_1 26.57/8.04 POL( a__U84_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( A__U93_3(x_1, ..., x_3) ) = 2x_2 + 1 26.57/8.04 POL( A__U94_2(x_1, x_2) ) = 2x_2 + 1 26.57/8.04 POL( a__U52_1(x_1) ) = x_1 26.57/8.04 POL( a__U61_1(x_1) ) = 2x_1 26.57/8.04 POL( a__U21_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( a__U22_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( a__U81_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U82_3(x_1, ..., x_3) ) = 2x_1 26.57/8.04 POL( a__isNatIListKind_1(x_1) ) = 0 26.57/8.04 POL( nil ) = 0 26.57/8.04 POL( tt ) = 0 26.57/8.04 POL( zeros ) = 0 26.57/8.04 POL( cons_2(x_1, x_2) ) = 2x_2 26.57/8.04 POL( a__U51_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( a__isNatKind_1(x_1) ) = 0 26.57/8.04 POL( isNatIListKind_1(x_1) ) = 0 26.57/8.04 POL( a__U23_1(x_1) ) = x_1 26.57/8.04 POL( a__U85_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( a__isNat_1(x_1) ) = 0 26.57/8.04 POL( 0 ) = 0 26.57/8.04 POL( length_1(x_1) ) = x_1 + 2 26.57/8.04 POL( s_1(x_1) ) = x_1 26.57/8.04 POL( isNat_1(x_1) ) = 0 26.57/8.04 POL( a__U71_1(x_1) ) = x_1 26.57/8.04 POL( isNatKind_1(x_1) ) = 0 26.57/8.04 POL( mark_1(x_1) ) = x_1 26.57/8.04 POL( a__zeros ) = 0 26.57/8.04 POL( U11_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( U12_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( U13_1(x_1) ) = 2x_1 26.57/8.04 POL( a__U13_1(x_1) ) = 2x_1 26.57/8.04 POL( isNatList_1(x_1) ) = 0 26.57/8.04 POL( a__isNatList_1(x_1) ) = 0 26.57/8.04 POL( U21_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( U22_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( U23_1(x_1) ) = x_1 26.57/8.04 POL( U31_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U31_2(x_1, x_2) ) = x_1 26.57/8.04 POL( U32_2(x_1, x_2) ) = x_1 26.57/8.04 POL( a__U32_2(x_1, x_2) ) = x_1 26.57/8.04 POL( U33_1(x_1) ) = x_1 26.57/8.04 POL( a__U33_1(x_1) ) = x_1 26.57/8.04 POL( U41_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U41_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U42_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U42_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U43_3(x_1, ..., x_3) ) = 2x_1 26.57/8.04 POL( a__U43_3(x_1, ..., x_3) ) = 2x_1 26.57/8.04 POL( U44_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( a__U44_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U45_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( a__U45_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( U46_1(x_1) ) = 2x_1 26.57/8.04 POL( a__U46_1(x_1) ) = 2x_1 26.57/8.04 POL( isNatIList_1(x_1) ) = 0 26.57/8.04 POL( a__isNatIList_1(x_1) ) = 0 26.57/8.04 POL( U51_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( U52_1(x_1) ) = x_1 26.57/8.04 POL( U61_1(x_1) ) = 2x_1 26.57/8.04 POL( U71_1(x_1) ) = x_1 26.57/8.04 POL( U81_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U82_3(x_1, ..., x_3) ) = 2x_1 26.57/8.04 POL( U83_3(x_1, ..., x_3) ) = 2x_1 26.57/8.04 POL( U84_3(x_1, ..., x_3) ) = x_1 26.57/8.04 POL( U85_2(x_1, x_2) ) = 2x_1 26.57/8.04 POL( U86_1(x_1) ) = 2x_1 26.57/8.04 POL( a__U86_1(x_1) ) = 2x_1 26.57/8.04 POL( U91_3(x_1, ..., x_3) ) = 2x_2 + 2 26.57/8.04 POL( a__U91_3(x_1, ..., x_3) ) = 2x_2 + 2 26.57/8.04 POL( U92_3(x_1, ..., x_3) ) = 2x_2 + 2 26.57/8.04 POL( a__U92_3(x_1, ..., x_3) ) = 2x_2 + 2 26.57/8.04 POL( U93_3(x_1, ..., x_3) ) = 2x_2 + 2 26.57/8.04 POL( a__U93_3(x_1, ..., x_3) ) = 2x_2 + 2 26.57/8.04 POL( U94_2(x_1, x_2) ) = 2x_2 + 2 26.57/8.04 POL( a__U94_2(x_1, x_2) ) = 2x_2 + 2 26.57/8.04 POL( a__length_1(x_1) ) = x_1 + 2 26.57/8.04 POL( MARK_1(x_1) ) = 2x_1 26.57/8.04 26.57/8.04 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.57/8.04 26.57/8.04 a__isNatIListKind(nil) -> tt 26.57/8.04 a__isNatIListKind(zeros) -> tt 26.57/8.04 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.04 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.04 a__isNat(0) -> tt 26.57/8.04 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.04 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.04 a__isNat(X) -> isNat(X) 26.57/8.04 a__isNatKind(0) -> tt 26.57/8.04 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.04 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.04 a__isNatKind(X) -> isNatKind(X) 26.57/8.04 mark(zeros) -> a__zeros 26.57/8.04 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.04 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.04 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.04 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.04 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.04 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.04 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.04 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.04 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.04 mark(isNat(X)) -> a__isNat(X) 26.57/8.04 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.04 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.04 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.04 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.04 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.04 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.04 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.04 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.04 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.04 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.04 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.04 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.04 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.04 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.04 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.04 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.04 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.04 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.04 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.04 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.04 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.04 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.04 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.04 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.04 mark(length(X)) -> a__length(mark(X)) 26.57/8.04 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.04 mark(0) -> 0 26.57/8.04 mark(tt) -> tt 26.57/8.04 mark(s(X)) -> s(mark(X)) 26.57/8.04 mark(nil) -> nil 26.57/8.04 a__isNatList(nil) -> tt 26.57/8.04 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatList(X) -> isNatList(X) 26.57/8.04 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.04 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.04 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.04 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.04 a__U13(tt) -> tt 26.57/8.04 a__U13(X) -> U13(X) 26.57/8.04 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.04 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.04 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.04 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.04 a__U23(tt) -> tt 26.57/8.04 a__U23(X) -> U23(X) 26.57/8.04 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.04 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.04 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.04 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.04 a__U33(tt) -> tt 26.57/8.04 a__U33(X) -> U33(X) 26.57/8.04 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.04 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.04 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.04 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.04 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.04 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.04 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.04 a__U46(tt) -> tt 26.57/8.04 a__U46(X) -> U46(X) 26.57/8.04 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.04 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.04 a__U52(tt) -> tt 26.57/8.04 a__U52(X) -> U52(X) 26.57/8.04 a__U61(tt) -> tt 26.57/8.04 a__U61(X) -> U61(X) 26.57/8.04 a__U71(tt) -> tt 26.57/8.04 a__U71(X) -> U71(X) 26.57/8.04 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.04 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.04 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.04 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.04 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.04 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.04 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.04 a__U86(tt) -> tt 26.57/8.04 a__U86(X) -> U86(X) 26.57/8.04 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.04 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.04 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.04 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.04 a__length(nil) -> 0 26.57/8.04 a__length(X) -> length(X) 26.57/8.04 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.04 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.04 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.04 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.04 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.04 a__zeros -> cons(0, zeros) 26.57/8.04 a__zeros -> zeros 26.57/8.04 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.04 a__isNatIList(zeros) -> tt 26.57/8.04 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatIList(X) -> isNatIList(X) 26.57/8.04 26.57/8.04 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (30) 26.57/8.04 Obligation: 26.57/8.04 Q DP problem: 26.57/8.04 The TRS P consists of the following rules: 26.57/8.04 26.57/8.04 MARK(U11(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U12(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U13(X)) -> MARK(X) 26.57/8.04 MARK(U21(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U22(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U23(X)) -> MARK(X) 26.57/8.04 MARK(U31(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U32(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U33(X)) -> MARK(X) 26.57/8.04 MARK(U41(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U42(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U43(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U44(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U45(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U46(X)) -> MARK(X) 26.57/8.04 MARK(U51(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U52(X)) -> MARK(X) 26.57/8.04 MARK(U61(X)) -> MARK(X) 26.57/8.04 MARK(U71(X)) -> MARK(X) 26.57/8.04 MARK(U81(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U82(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U83(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U84(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U85(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U86(X)) -> MARK(X) 26.57/8.04 A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) 26.57/8.04 A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) 26.57/8.04 A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) 26.57/8.04 A__U94(tt, L) -> A__LENGTH(mark(L)) 26.57/8.04 A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) 26.57/8.04 MARK(s(X)) -> MARK(X) 26.57/8.04 26.57/8.04 The TRS R consists of the following rules: 26.57/8.04 26.57/8.04 a__zeros -> cons(0, zeros) 26.57/8.04 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.04 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.04 a__U13(tt) -> tt 26.57/8.04 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.04 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.04 a__U23(tt) -> tt 26.57/8.04 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.04 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.04 a__U33(tt) -> tt 26.57/8.04 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.04 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.04 a__U46(tt) -> tt 26.57/8.04 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.04 a__U52(tt) -> tt 26.57/8.04 a__U61(tt) -> tt 26.57/8.04 a__U71(tt) -> tt 26.57/8.04 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.04 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.04 a__U86(tt) -> tt 26.57/8.04 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.04 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.04 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.04 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.04 a__isNat(0) -> tt 26.57/8.04 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.04 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.04 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.04 a__isNatIList(zeros) -> tt 26.57/8.04 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatIListKind(nil) -> tt 26.57/8.04 a__isNatIListKind(zeros) -> tt 26.57/8.04 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.04 a__isNatKind(0) -> tt 26.57/8.04 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.04 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.04 a__isNatList(nil) -> tt 26.57/8.04 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.04 a__length(nil) -> 0 26.57/8.04 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.04 mark(zeros) -> a__zeros 26.57/8.04 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.04 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.04 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.04 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.04 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.04 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.04 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.04 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.04 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.04 mark(isNat(X)) -> a__isNat(X) 26.57/8.04 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.04 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.04 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.04 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.04 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.04 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.04 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.04 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.04 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.04 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.04 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.04 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.04 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.04 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.04 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.04 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.04 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.04 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.04 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.04 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.04 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.04 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.04 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.04 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.04 mark(length(X)) -> a__length(mark(X)) 26.57/8.04 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.04 mark(0) -> 0 26.57/8.04 mark(tt) -> tt 26.57/8.04 mark(s(X)) -> s(mark(X)) 26.57/8.04 mark(nil) -> nil 26.57/8.04 a__zeros -> zeros 26.57/8.04 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.04 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.04 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.04 a__U13(X) -> U13(X) 26.57/8.04 a__isNatList(X) -> isNatList(X) 26.57/8.04 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.04 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.04 a__isNatKind(X) -> isNatKind(X) 26.57/8.04 a__U23(X) -> U23(X) 26.57/8.04 a__isNat(X) -> isNat(X) 26.57/8.04 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.04 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.04 a__U33(X) -> U33(X) 26.57/8.04 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.04 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.04 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.04 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.04 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.04 a__U46(X) -> U46(X) 26.57/8.04 a__isNatIList(X) -> isNatIList(X) 26.57/8.04 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.04 a__U52(X) -> U52(X) 26.57/8.04 a__U61(X) -> U61(X) 26.57/8.04 a__U71(X) -> U71(X) 26.57/8.04 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.04 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.04 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.04 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.04 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.04 a__U86(X) -> U86(X) 26.57/8.04 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.04 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.04 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.04 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.04 a__length(X) -> length(X) 26.57/8.04 26.57/8.04 The set Q consists of the following terms: 26.57/8.04 26.57/8.04 a__zeros 26.57/8.04 a__isNatIList(x0) 26.57/8.04 mark(zeros) 26.57/8.04 mark(U11(x0, x1)) 26.57/8.04 mark(U12(x0, x1)) 26.57/8.04 mark(isNatIListKind(x0)) 26.57/8.04 mark(U13(x0)) 26.57/8.04 mark(isNatList(x0)) 26.57/8.04 mark(U21(x0, x1)) 26.57/8.04 mark(U22(x0, x1)) 26.57/8.04 mark(isNatKind(x0)) 26.57/8.04 mark(U23(x0)) 26.57/8.04 mark(isNat(x0)) 26.57/8.04 mark(U31(x0, x1)) 26.57/8.04 mark(U32(x0, x1)) 26.57/8.04 mark(U33(x0)) 26.57/8.04 mark(U41(x0, x1, x2)) 26.57/8.04 mark(U42(x0, x1, x2)) 26.57/8.04 mark(U43(x0, x1, x2)) 26.57/8.04 mark(U44(x0, x1, x2)) 26.57/8.04 mark(U45(x0, x1)) 26.57/8.04 mark(U46(x0)) 26.57/8.04 mark(isNatIList(x0)) 26.57/8.04 mark(U51(x0, x1)) 26.57/8.04 mark(U52(x0)) 26.57/8.04 mark(U61(x0)) 26.57/8.04 mark(U71(x0)) 26.57/8.04 mark(U81(x0, x1, x2)) 26.57/8.04 mark(U82(x0, x1, x2)) 26.57/8.04 mark(U83(x0, x1, x2)) 26.57/8.04 mark(U84(x0, x1, x2)) 26.57/8.04 mark(U85(x0, x1)) 26.57/8.04 mark(U86(x0)) 26.57/8.04 mark(U91(x0, x1, x2)) 26.57/8.04 mark(U92(x0, x1, x2)) 26.57/8.04 mark(U93(x0, x1, x2)) 26.57/8.04 mark(U94(x0, x1)) 26.57/8.04 mark(length(x0)) 26.57/8.04 mark(cons(x0, x1)) 26.57/8.04 mark(0) 26.57/8.04 mark(tt) 26.57/8.04 mark(s(x0)) 26.57/8.04 mark(nil) 26.57/8.04 a__U11(x0, x1) 26.57/8.04 a__U12(x0, x1) 26.57/8.04 a__isNatIListKind(x0) 26.57/8.04 a__U13(x0) 26.57/8.04 a__isNatList(x0) 26.57/8.04 a__U21(x0, x1) 26.57/8.04 a__U22(x0, x1) 26.57/8.04 a__isNatKind(x0) 26.57/8.04 a__U23(x0) 26.57/8.04 a__isNat(x0) 26.57/8.04 a__U31(x0, x1) 26.57/8.04 a__U32(x0, x1) 26.57/8.04 a__U33(x0) 26.57/8.04 a__U41(x0, x1, x2) 26.57/8.04 a__U42(x0, x1, x2) 26.57/8.04 a__U43(x0, x1, x2) 26.57/8.04 a__U44(x0, x1, x2) 26.57/8.04 a__U45(x0, x1) 26.57/8.04 a__U46(x0) 26.57/8.04 a__U51(x0, x1) 26.57/8.04 a__U52(x0) 26.57/8.04 a__U61(x0) 26.57/8.04 a__U71(x0) 26.57/8.04 a__U81(x0, x1, x2) 26.57/8.04 a__U82(x0, x1, x2) 26.57/8.04 a__U83(x0, x1, x2) 26.57/8.04 a__U84(x0, x1, x2) 26.57/8.04 a__U85(x0, x1) 26.57/8.04 a__U86(x0) 26.57/8.04 a__U91(x0, x1, x2) 26.57/8.04 a__U92(x0, x1, x2) 26.57/8.04 a__U93(x0, x1, x2) 26.57/8.04 a__U94(x0, x1) 26.57/8.04 a__length(x0) 26.57/8.04 26.57/8.04 We have to consider all minimal (P,Q,R)-chains. 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (31) DependencyGraphProof (EQUIVALENT) 26.57/8.04 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (32) 26.57/8.04 Complex Obligation (AND) 26.57/8.04 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (33) 26.57/8.04 Obligation: 26.57/8.04 Q DP problem: 26.57/8.04 The TRS P consists of the following rules: 26.57/8.04 26.57/8.04 A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) 26.57/8.04 A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) 26.57/8.04 A__U94(tt, L) -> A__LENGTH(mark(L)) 26.57/8.04 A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) 26.57/8.04 A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) 26.57/8.04 26.57/8.04 The TRS R consists of the following rules: 26.57/8.04 26.57/8.04 a__zeros -> cons(0, zeros) 26.57/8.04 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.04 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.04 a__U13(tt) -> tt 26.57/8.04 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.04 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.04 a__U23(tt) -> tt 26.57/8.04 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.04 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.04 a__U33(tt) -> tt 26.57/8.04 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.04 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.04 a__U46(tt) -> tt 26.57/8.04 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.04 a__U52(tt) -> tt 26.57/8.04 a__U61(tt) -> tt 26.57/8.04 a__U71(tt) -> tt 26.57/8.04 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.04 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.04 a__U86(tt) -> tt 26.57/8.04 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.04 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.04 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.04 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.04 a__isNat(0) -> tt 26.57/8.04 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.04 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.04 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.04 a__isNatIList(zeros) -> tt 26.57/8.04 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatIListKind(nil) -> tt 26.57/8.04 a__isNatIListKind(zeros) -> tt 26.57/8.04 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.04 a__isNatKind(0) -> tt 26.57/8.04 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.04 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.04 a__isNatList(nil) -> tt 26.57/8.04 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.04 a__length(nil) -> 0 26.57/8.04 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.04 mark(zeros) -> a__zeros 26.57/8.04 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.04 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.04 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.04 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.04 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.04 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.04 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.04 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.04 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.04 mark(isNat(X)) -> a__isNat(X) 26.57/8.04 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.04 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.04 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.04 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.04 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.04 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.04 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.04 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.04 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.04 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.04 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.04 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.04 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.04 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.04 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.04 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.04 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.04 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.04 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.04 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.04 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.04 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.04 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.04 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.04 mark(length(X)) -> a__length(mark(X)) 26.57/8.04 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.04 mark(0) -> 0 26.57/8.04 mark(tt) -> tt 26.57/8.04 mark(s(X)) -> s(mark(X)) 26.57/8.04 mark(nil) -> nil 26.57/8.04 a__zeros -> zeros 26.57/8.04 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.04 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.04 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.04 a__U13(X) -> U13(X) 26.57/8.04 a__isNatList(X) -> isNatList(X) 26.57/8.04 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.04 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.04 a__isNatKind(X) -> isNatKind(X) 26.57/8.04 a__U23(X) -> U23(X) 26.57/8.04 a__isNat(X) -> isNat(X) 26.57/8.04 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.04 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.04 a__U33(X) -> U33(X) 26.57/8.04 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.04 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.04 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.04 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.04 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.04 a__U46(X) -> U46(X) 26.57/8.04 a__isNatIList(X) -> isNatIList(X) 26.57/8.04 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.04 a__U52(X) -> U52(X) 26.57/8.04 a__U61(X) -> U61(X) 26.57/8.04 a__U71(X) -> U71(X) 26.57/8.04 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.04 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.04 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.04 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.04 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.04 a__U86(X) -> U86(X) 26.57/8.04 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.04 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.04 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.04 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.04 a__length(X) -> length(X) 26.57/8.04 26.57/8.04 The set Q consists of the following terms: 26.57/8.04 26.57/8.04 a__zeros 26.57/8.04 a__isNatIList(x0) 26.57/8.04 mark(zeros) 26.57/8.04 mark(U11(x0, x1)) 26.57/8.04 mark(U12(x0, x1)) 26.57/8.04 mark(isNatIListKind(x0)) 26.57/8.04 mark(U13(x0)) 26.57/8.04 mark(isNatList(x0)) 26.57/8.04 mark(U21(x0, x1)) 26.57/8.04 mark(U22(x0, x1)) 26.57/8.04 mark(isNatKind(x0)) 26.57/8.04 mark(U23(x0)) 26.57/8.04 mark(isNat(x0)) 26.57/8.04 mark(U31(x0, x1)) 26.57/8.04 mark(U32(x0, x1)) 26.57/8.04 mark(U33(x0)) 26.57/8.04 mark(U41(x0, x1, x2)) 26.57/8.04 mark(U42(x0, x1, x2)) 26.57/8.04 mark(U43(x0, x1, x2)) 26.57/8.04 mark(U44(x0, x1, x2)) 26.57/8.04 mark(U45(x0, x1)) 26.57/8.04 mark(U46(x0)) 26.57/8.04 mark(isNatIList(x0)) 26.57/8.04 mark(U51(x0, x1)) 26.57/8.04 mark(U52(x0)) 26.57/8.04 mark(U61(x0)) 26.57/8.04 mark(U71(x0)) 26.57/8.04 mark(U81(x0, x1, x2)) 26.57/8.04 mark(U82(x0, x1, x2)) 26.57/8.04 mark(U83(x0, x1, x2)) 26.57/8.04 mark(U84(x0, x1, x2)) 26.57/8.04 mark(U85(x0, x1)) 26.57/8.04 mark(U86(x0)) 26.57/8.04 mark(U91(x0, x1, x2)) 26.57/8.04 mark(U92(x0, x1, x2)) 26.57/8.04 mark(U93(x0, x1, x2)) 26.57/8.04 mark(U94(x0, x1)) 26.57/8.04 mark(length(x0)) 26.57/8.04 mark(cons(x0, x1)) 26.57/8.04 mark(0) 26.57/8.04 mark(tt) 26.57/8.04 mark(s(x0)) 26.57/8.04 mark(nil) 26.57/8.04 a__U11(x0, x1) 26.57/8.04 a__U12(x0, x1) 26.57/8.04 a__isNatIListKind(x0) 26.57/8.04 a__U13(x0) 26.57/8.04 a__isNatList(x0) 26.57/8.04 a__U21(x0, x1) 26.57/8.04 a__U22(x0, x1) 26.57/8.04 a__isNatKind(x0) 26.57/8.04 a__U23(x0) 26.57/8.04 a__isNat(x0) 26.57/8.04 a__U31(x0, x1) 26.57/8.04 a__U32(x0, x1) 26.57/8.04 a__U33(x0) 26.57/8.04 a__U41(x0, x1, x2) 26.57/8.04 a__U42(x0, x1, x2) 26.57/8.04 a__U43(x0, x1, x2) 26.57/8.04 a__U44(x0, x1, x2) 26.57/8.04 a__U45(x0, x1) 26.57/8.04 a__U46(x0) 26.57/8.04 a__U51(x0, x1) 26.57/8.04 a__U52(x0) 26.57/8.04 a__U61(x0) 26.57/8.04 a__U71(x0) 26.57/8.04 a__U81(x0, x1, x2) 26.57/8.04 a__U82(x0, x1, x2) 26.57/8.04 a__U83(x0, x1, x2) 26.57/8.04 a__U84(x0, x1, x2) 26.57/8.04 a__U85(x0, x1) 26.57/8.04 a__U86(x0) 26.57/8.04 a__U91(x0, x1, x2) 26.57/8.04 a__U92(x0, x1, x2) 26.57/8.04 a__U93(x0, x1, x2) 26.57/8.04 a__U94(x0, x1) 26.57/8.04 a__length(x0) 26.57/8.04 26.57/8.04 We have to consider all minimal (P,Q,R)-chains. 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (34) QDPOrderProof (EQUIVALENT) 26.57/8.04 We use the reduction pair processor [LPAR04,JAR06]. 26.57/8.04 26.57/8.04 26.57/8.04 The following pairs can be oriented strictly and are deleted. 26.57/8.04 26.57/8.04 A__U93(tt, L, N) -> A__U94(a__isNatKind(N), L) 26.57/8.04 A__LENGTH(cons(N, L)) -> A__U91(a__isNatList(L), L, N) 26.57/8.04 The remaining pairs can at least be oriented weakly. 26.57/8.04 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 26.57/8.04 26.57/8.04 POL( A__LENGTH_1(x_1) ) = 2x_1 + 1 26.57/8.04 POL( A__U93_3(x_1, ..., x_3) ) = 2x_2 + 2 26.57/8.04 POL( A__U91_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 26.57/8.04 POL( a__U23_1(x_1) ) = 1 26.57/8.04 POL( a__U85_2(x_1, x_2) ) = x_2 26.57/8.04 POL( a__isNat_1(x_1) ) = x_1 26.57/8.04 POL( 0 ) = 1 26.57/8.04 POL( tt ) = 1 26.57/8.04 POL( length_1(x_1) ) = 2 26.57/8.04 POL( a__U11_2(x_1, x_2) ) = 1 26.57/8.04 POL( a__isNatIListKind_1(x_1) ) = 2 26.57/8.04 POL( s_1(x_1) ) = 1 26.57/8.04 POL( a__U21_2(x_1, x_2) ) = 1 26.57/8.04 POL( a__isNatKind_1(x_1) ) = 2x_1 + 2 26.57/8.04 POL( isNat_1(x_1) ) = x_1 26.57/8.04 POL( A__U94_2(x_1, x_2) ) = 2x_2 + 1 26.57/8.04 POL( a__U22_2(x_1, x_2) ) = 1 26.57/8.04 POL( a__U81_3(x_1, ..., x_3) ) = 2x_3 26.57/8.04 POL( a__U82_3(x_1, ..., x_3) ) = 2x_3 26.57/8.04 POL( A__U92_3(x_1, ..., x_3) ) = 2x_2 + 2 26.57/8.04 POL( a__U51_2(x_1, x_2) ) = 2 26.57/8.04 POL( a__U71_1(x_1) ) = 1 26.57/8.04 POL( a__U12_2(x_1, x_2) ) = 1 26.57/8.04 POL( a__U83_3(x_1, ..., x_3) ) = 2x_3 26.57/8.04 POL( a__U84_3(x_1, ..., x_3) ) = x_3 26.57/8.04 POL( a__U61_1(x_1) ) = 2x_1 + 2 26.57/8.04 POL( isNatKind_1(x_1) ) = 2x_1 + 2 26.57/8.04 POL( mark_1(x_1) ) = x_1 26.57/8.04 POL( zeros ) = 0 26.57/8.04 POL( a__zeros ) = 0 26.57/8.04 POL( U11_2(x_1, x_2) ) = 1 26.57/8.04 POL( U12_2(x_1, x_2) ) = 1 26.57/8.04 POL( isNatIListKind_1(x_1) ) = 2 26.57/8.04 POL( U13_1(x_1) ) = 1 26.57/8.04 POL( a__U13_1(x_1) ) = 1 26.57/8.04 POL( isNatList_1(x_1) ) = x_1 26.57/8.04 POL( a__isNatList_1(x_1) ) = x_1 26.57/8.04 POL( U21_2(x_1, x_2) ) = 1 26.57/8.04 POL( U22_2(x_1, x_2) ) = 1 26.57/8.04 POL( U23_1(x_1) ) = 1 26.57/8.04 POL( U31_2(x_1, x_2) ) = 2 26.57/8.04 POL( a__U31_2(x_1, x_2) ) = 2 26.57/8.04 POL( U32_2(x_1, x_2) ) = 1 26.57/8.04 POL( a__U32_2(x_1, x_2) ) = 1 26.57/8.04 POL( U33_1(x_1) ) = 1 26.57/8.04 POL( a__U33_1(x_1) ) = 1 26.57/8.04 POL( U41_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( a__U41_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( U42_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( a__U42_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( U43_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( a__U43_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( U44_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( a__U44_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( U45_2(x_1, x_2) ) = 1 26.57/8.04 POL( a__U45_2(x_1, x_2) ) = 1 26.57/8.04 POL( U46_1(x_1) ) = 1 26.57/8.04 POL( a__U46_1(x_1) ) = 1 26.57/8.04 POL( isNatIList_1(x_1) ) = 2 26.57/8.04 POL( a__isNatIList_1(x_1) ) = 2 26.57/8.04 POL( U51_2(x_1, x_2) ) = 2 26.57/8.04 POL( U52_1(x_1) ) = 2 26.57/8.04 POL( a__U52_1(x_1) ) = 2 26.57/8.04 POL( U61_1(x_1) ) = 2x_1 + 2 26.57/8.04 POL( U71_1(x_1) ) = 1 26.57/8.04 POL( U81_3(x_1, ..., x_3) ) = 2x_3 26.57/8.04 POL( U82_3(x_1, ..., x_3) ) = 2x_3 26.57/8.04 POL( U83_3(x_1, ..., x_3) ) = 2x_3 26.57/8.04 POL( U84_3(x_1, ..., x_3) ) = x_3 26.57/8.04 POL( U85_2(x_1, x_2) ) = x_2 26.57/8.04 POL( U86_1(x_1) ) = x_1 26.57/8.04 POL( a__U86_1(x_1) ) = x_1 26.57/8.04 POL( U91_3(x_1, ..., x_3) ) = 2 26.57/8.04 POL( a__U91_3(x_1, ..., x_3) ) = 2 26.57/8.04 POL( U92_3(x_1, ..., x_3) ) = 2 26.57/8.04 POL( a__U92_3(x_1, ..., x_3) ) = 2 26.57/8.04 POL( U93_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( a__U93_3(x_1, ..., x_3) ) = 1 26.57/8.04 POL( U94_2(x_1, x_2) ) = 1 26.57/8.04 POL( a__U94_2(x_1, x_2) ) = 1 26.57/8.04 POL( a__length_1(x_1) ) = 2 26.57/8.04 POL( cons_2(x_1, x_2) ) = 2x_2 26.57/8.04 POL( nil ) = 2 26.57/8.04 26.57/8.04 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.57/8.04 26.57/8.04 a__isNat(0) -> tt 26.57/8.04 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.04 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.04 a__isNat(X) -> isNat(X) 26.57/8.04 a__isNatKind(0) -> tt 26.57/8.04 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.04 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.04 a__isNatKind(X) -> isNatKind(X) 26.57/8.04 mark(zeros) -> a__zeros 26.57/8.04 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.04 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.04 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.04 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.04 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.04 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.04 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.04 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.04 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.04 mark(isNat(X)) -> a__isNat(X) 26.57/8.04 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.04 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.04 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.04 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.04 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.04 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.04 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.04 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.04 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.04 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.04 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.04 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.04 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.04 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.04 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.04 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.04 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.04 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.04 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.04 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.04 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.04 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.04 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.04 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.04 mark(length(X)) -> a__length(mark(X)) 26.57/8.04 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.04 mark(0) -> 0 26.57/8.04 mark(tt) -> tt 26.57/8.04 mark(s(X)) -> s(mark(X)) 26.57/8.04 mark(nil) -> nil 26.57/8.04 a__isNatList(nil) -> tt 26.57/8.04 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatList(X) -> isNatList(X) 26.57/8.04 a__isNatIListKind(nil) -> tt 26.57/8.04 a__isNatIListKind(zeros) -> tt 26.57/8.04 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.04 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.04 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.04 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.04 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.04 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.04 a__U13(tt) -> tt 26.57/8.04 a__U13(X) -> U13(X) 26.57/8.04 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.04 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.04 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.04 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.04 a__U23(tt) -> tt 26.57/8.04 a__U23(X) -> U23(X) 26.57/8.04 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.04 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.04 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.04 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.04 a__U33(tt) -> tt 26.57/8.04 a__U33(X) -> U33(X) 26.57/8.04 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.04 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.04 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.04 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.04 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.04 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.04 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.04 a__U46(tt) -> tt 26.57/8.04 a__U46(X) -> U46(X) 26.57/8.04 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.04 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.04 a__U52(tt) -> tt 26.57/8.04 a__U52(X) -> U52(X) 26.57/8.04 a__U61(tt) -> tt 26.57/8.04 a__U61(X) -> U61(X) 26.57/8.04 a__U71(tt) -> tt 26.57/8.04 a__U71(X) -> U71(X) 26.57/8.04 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.04 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.04 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.04 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.04 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.04 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.04 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.04 a__U86(tt) -> tt 26.57/8.04 a__U86(X) -> U86(X) 26.57/8.04 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.04 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.04 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.04 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.04 a__length(nil) -> 0 26.57/8.04 a__length(X) -> length(X) 26.57/8.04 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.04 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.04 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.04 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.04 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.04 a__zeros -> cons(0, zeros) 26.57/8.04 a__zeros -> zeros 26.57/8.04 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.04 a__isNatIList(zeros) -> tt 26.57/8.04 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatIList(X) -> isNatIList(X) 26.57/8.04 26.57/8.04 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (35) 26.57/8.04 Obligation: 26.57/8.04 Q DP problem: 26.57/8.04 The TRS P consists of the following rules: 26.57/8.04 26.57/8.04 A__U92(tt, L, N) -> A__U93(a__isNat(N), L, N) 26.57/8.04 A__U94(tt, L) -> A__LENGTH(mark(L)) 26.57/8.04 A__U91(tt, L, N) -> A__U92(a__isNatIListKind(L), L, N) 26.57/8.04 26.57/8.04 The TRS R consists of the following rules: 26.57/8.04 26.57/8.04 a__zeros -> cons(0, zeros) 26.57/8.04 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.04 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.04 a__U13(tt) -> tt 26.57/8.04 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.04 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.04 a__U23(tt) -> tt 26.57/8.04 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.04 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.04 a__U33(tt) -> tt 26.57/8.04 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.04 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.04 a__U46(tt) -> tt 26.57/8.04 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.04 a__U52(tt) -> tt 26.57/8.04 a__U61(tt) -> tt 26.57/8.04 a__U71(tt) -> tt 26.57/8.04 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.04 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.04 a__U86(tt) -> tt 26.57/8.04 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.04 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.04 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.04 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.04 a__isNat(0) -> tt 26.57/8.04 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.04 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.04 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.04 a__isNatIList(zeros) -> tt 26.57/8.04 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.04 a__isNatIListKind(nil) -> tt 26.57/8.04 a__isNatIListKind(zeros) -> tt 26.57/8.04 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.04 a__isNatKind(0) -> tt 26.57/8.04 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.04 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.04 a__isNatList(nil) -> tt 26.57/8.04 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.04 a__length(nil) -> 0 26.57/8.04 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.04 mark(zeros) -> a__zeros 26.57/8.04 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.04 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.04 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.04 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.04 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.04 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.04 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.04 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.04 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.04 mark(isNat(X)) -> a__isNat(X) 26.57/8.04 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.04 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.04 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.04 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.04 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.04 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.04 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.04 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.04 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.04 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.04 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.04 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.04 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.04 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.04 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.04 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.04 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.04 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.04 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.04 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.04 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.04 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.04 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.04 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.04 mark(length(X)) -> a__length(mark(X)) 26.57/8.04 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.04 mark(0) -> 0 26.57/8.04 mark(tt) -> tt 26.57/8.04 mark(s(X)) -> s(mark(X)) 26.57/8.04 mark(nil) -> nil 26.57/8.04 a__zeros -> zeros 26.57/8.04 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.04 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.04 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.04 a__U13(X) -> U13(X) 26.57/8.04 a__isNatList(X) -> isNatList(X) 26.57/8.04 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.04 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.04 a__isNatKind(X) -> isNatKind(X) 26.57/8.04 a__U23(X) -> U23(X) 26.57/8.04 a__isNat(X) -> isNat(X) 26.57/8.04 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.04 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.04 a__U33(X) -> U33(X) 26.57/8.04 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.04 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.04 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.04 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.04 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.04 a__U46(X) -> U46(X) 26.57/8.04 a__isNatIList(X) -> isNatIList(X) 26.57/8.04 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.04 a__U52(X) -> U52(X) 26.57/8.04 a__U61(X) -> U61(X) 26.57/8.04 a__U71(X) -> U71(X) 26.57/8.04 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.04 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.04 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.04 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.04 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.04 a__U86(X) -> U86(X) 26.57/8.04 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.04 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.04 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.04 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.04 a__length(X) -> length(X) 26.57/8.04 26.57/8.04 The set Q consists of the following terms: 26.57/8.04 26.57/8.04 a__zeros 26.57/8.04 a__isNatIList(x0) 26.57/8.04 mark(zeros) 26.57/8.04 mark(U11(x0, x1)) 26.57/8.04 mark(U12(x0, x1)) 26.57/8.04 mark(isNatIListKind(x0)) 26.57/8.04 mark(U13(x0)) 26.57/8.04 mark(isNatList(x0)) 26.57/8.04 mark(U21(x0, x1)) 26.57/8.04 mark(U22(x0, x1)) 26.57/8.04 mark(isNatKind(x0)) 26.57/8.04 mark(U23(x0)) 26.57/8.04 mark(isNat(x0)) 26.57/8.04 mark(U31(x0, x1)) 26.57/8.04 mark(U32(x0, x1)) 26.57/8.04 mark(U33(x0)) 26.57/8.04 mark(U41(x0, x1, x2)) 26.57/8.04 mark(U42(x0, x1, x2)) 26.57/8.04 mark(U43(x0, x1, x2)) 26.57/8.04 mark(U44(x0, x1, x2)) 26.57/8.04 mark(U45(x0, x1)) 26.57/8.04 mark(U46(x0)) 26.57/8.04 mark(isNatIList(x0)) 26.57/8.04 mark(U51(x0, x1)) 26.57/8.04 mark(U52(x0)) 26.57/8.04 mark(U61(x0)) 26.57/8.04 mark(U71(x0)) 26.57/8.04 mark(U81(x0, x1, x2)) 26.57/8.04 mark(U82(x0, x1, x2)) 26.57/8.04 mark(U83(x0, x1, x2)) 26.57/8.04 mark(U84(x0, x1, x2)) 26.57/8.04 mark(U85(x0, x1)) 26.57/8.04 mark(U86(x0)) 26.57/8.04 mark(U91(x0, x1, x2)) 26.57/8.04 mark(U92(x0, x1, x2)) 26.57/8.04 mark(U93(x0, x1, x2)) 26.57/8.04 mark(U94(x0, x1)) 26.57/8.04 mark(length(x0)) 26.57/8.04 mark(cons(x0, x1)) 26.57/8.04 mark(0) 26.57/8.04 mark(tt) 26.57/8.04 mark(s(x0)) 26.57/8.04 mark(nil) 26.57/8.04 a__U11(x0, x1) 26.57/8.04 a__U12(x0, x1) 26.57/8.04 a__isNatIListKind(x0) 26.57/8.04 a__U13(x0) 26.57/8.04 a__isNatList(x0) 26.57/8.04 a__U21(x0, x1) 26.57/8.04 a__U22(x0, x1) 26.57/8.04 a__isNatKind(x0) 26.57/8.04 a__U23(x0) 26.57/8.04 a__isNat(x0) 26.57/8.04 a__U31(x0, x1) 26.57/8.04 a__U32(x0, x1) 26.57/8.04 a__U33(x0) 26.57/8.04 a__U41(x0, x1, x2) 26.57/8.04 a__U42(x0, x1, x2) 26.57/8.04 a__U43(x0, x1, x2) 26.57/8.04 a__U44(x0, x1, x2) 26.57/8.04 a__U45(x0, x1) 26.57/8.04 a__U46(x0) 26.57/8.04 a__U51(x0, x1) 26.57/8.04 a__U52(x0) 26.57/8.04 a__U61(x0) 26.57/8.04 a__U71(x0) 26.57/8.04 a__U81(x0, x1, x2) 26.57/8.04 a__U82(x0, x1, x2) 26.57/8.04 a__U83(x0, x1, x2) 26.57/8.04 a__U84(x0, x1, x2) 26.57/8.04 a__U85(x0, x1) 26.57/8.04 a__U86(x0) 26.57/8.04 a__U91(x0, x1, x2) 26.57/8.04 a__U92(x0, x1, x2) 26.57/8.04 a__U93(x0, x1, x2) 26.57/8.04 a__U94(x0, x1) 26.57/8.04 a__length(x0) 26.57/8.04 26.57/8.04 We have to consider all minimal (P,Q,R)-chains. 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (36) DependencyGraphProof (EQUIVALENT) 26.57/8.04 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (37) 26.57/8.04 TRUE 26.57/8.04 26.57/8.04 ---------------------------------------- 26.57/8.04 26.57/8.04 (38) 26.57/8.04 Obligation: 26.57/8.04 Q DP problem: 26.57/8.04 The TRS P consists of the following rules: 26.57/8.04 26.57/8.04 MARK(U12(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U11(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U13(X)) -> MARK(X) 26.57/8.04 MARK(U21(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U22(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U23(X)) -> MARK(X) 26.57/8.04 MARK(U31(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U32(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U33(X)) -> MARK(X) 26.57/8.04 MARK(U41(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U42(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U43(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U44(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U45(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U46(X)) -> MARK(X) 26.57/8.04 MARK(U51(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U52(X)) -> MARK(X) 26.57/8.04 MARK(U61(X)) -> MARK(X) 26.57/8.04 MARK(U71(X)) -> MARK(X) 26.57/8.04 MARK(U81(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U82(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U83(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U84(X1, X2, X3)) -> MARK(X1) 26.57/8.04 MARK(U85(X1, X2)) -> MARK(X1) 26.57/8.04 MARK(U86(X)) -> MARK(X) 26.57/8.04 MARK(s(X)) -> MARK(X) 26.57/8.04 26.57/8.04 The TRS R consists of the following rules: 26.57/8.04 26.57/8.04 a__zeros -> cons(0, zeros) 26.57/8.04 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 26.57/8.04 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 26.57/8.04 a__U13(tt) -> tt 26.57/8.04 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 26.57/8.04 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 26.57/8.04 a__U23(tt) -> tt 26.57/8.04 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 26.57/8.04 a__U32(tt, V) -> a__U33(a__isNatList(V)) 26.57/8.04 a__U33(tt) -> tt 26.57/8.04 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 26.57/8.04 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 26.57/8.04 a__U46(tt) -> tt 26.57/8.04 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 26.57/8.04 a__U52(tt) -> tt 26.57/8.04 a__U61(tt) -> tt 26.57/8.04 a__U71(tt) -> tt 26.57/8.04 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 26.57/8.04 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 26.57/8.04 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 26.57/8.05 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 26.57/8.05 a__U86(tt) -> tt 26.57/8.05 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 26.57/8.05 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 26.57/8.05 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 26.57/8.05 a__U94(tt, L) -> s(a__length(mark(L))) 26.57/8.05 a__isNat(0) -> tt 26.57/8.05 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 26.57/8.05 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 26.57/8.05 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 26.57/8.05 a__isNatIList(zeros) -> tt 26.57/8.05 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 26.57/8.05 a__isNatIListKind(nil) -> tt 26.57/8.05 a__isNatIListKind(zeros) -> tt 26.57/8.05 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 26.57/8.05 a__isNatKind(0) -> tt 26.57/8.05 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 26.57/8.05 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 26.57/8.05 a__isNatList(nil) -> tt 26.57/8.05 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 26.57/8.05 a__length(nil) -> 0 26.57/8.05 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 26.57/8.05 mark(zeros) -> a__zeros 26.57/8.05 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 26.57/8.05 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 26.57/8.05 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 26.57/8.05 mark(U13(X)) -> a__U13(mark(X)) 26.57/8.05 mark(isNatList(X)) -> a__isNatList(X) 26.57/8.05 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 26.57/8.05 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 26.57/8.05 mark(isNatKind(X)) -> a__isNatKind(X) 26.57/8.05 mark(U23(X)) -> a__U23(mark(X)) 26.57/8.05 mark(isNat(X)) -> a__isNat(X) 26.57/8.05 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 26.57/8.05 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 26.57/8.05 mark(U33(X)) -> a__U33(mark(X)) 26.57/8.05 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 26.57/8.05 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 26.57/8.05 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 26.57/8.05 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 26.57/8.05 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 26.57/8.05 mark(U46(X)) -> a__U46(mark(X)) 26.57/8.05 mark(isNatIList(X)) -> a__isNatIList(X) 26.57/8.05 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 26.57/8.05 mark(U52(X)) -> a__U52(mark(X)) 26.57/8.05 mark(U61(X)) -> a__U61(mark(X)) 26.57/8.05 mark(U71(X)) -> a__U71(mark(X)) 26.57/8.05 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 26.57/8.05 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 26.57/8.05 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 26.57/8.05 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 26.57/8.05 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 26.57/8.05 mark(U86(X)) -> a__U86(mark(X)) 26.57/8.05 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 26.57/8.05 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 26.57/8.05 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 26.57/8.05 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 26.57/8.05 mark(length(X)) -> a__length(mark(X)) 26.57/8.05 mark(cons(X1, X2)) -> cons(mark(X1), X2) 26.57/8.05 mark(0) -> 0 26.57/8.05 mark(tt) -> tt 26.57/8.05 mark(s(X)) -> s(mark(X)) 26.57/8.05 mark(nil) -> nil 26.57/8.05 a__zeros -> zeros 26.57/8.05 a__U11(X1, X2) -> U11(X1, X2) 26.57/8.05 a__U12(X1, X2) -> U12(X1, X2) 26.57/8.05 a__isNatIListKind(X) -> isNatIListKind(X) 26.57/8.05 a__U13(X) -> U13(X) 26.57/8.05 a__isNatList(X) -> isNatList(X) 26.57/8.05 a__U21(X1, X2) -> U21(X1, X2) 26.57/8.05 a__U22(X1, X2) -> U22(X1, X2) 26.57/8.05 a__isNatKind(X) -> isNatKind(X) 26.57/8.05 a__U23(X) -> U23(X) 26.57/8.05 a__isNat(X) -> isNat(X) 26.57/8.05 a__U31(X1, X2) -> U31(X1, X2) 26.57/8.05 a__U32(X1, X2) -> U32(X1, X2) 26.57/8.05 a__U33(X) -> U33(X) 26.57/8.05 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 26.57/8.05 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 26.57/8.05 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 26.57/8.05 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 26.57/8.05 a__U45(X1, X2) -> U45(X1, X2) 26.57/8.05 a__U46(X) -> U46(X) 26.57/8.05 a__isNatIList(X) -> isNatIList(X) 26.57/8.05 a__U51(X1, X2) -> U51(X1, X2) 26.57/8.05 a__U52(X) -> U52(X) 26.57/8.05 a__U61(X) -> U61(X) 26.57/8.05 a__U71(X) -> U71(X) 26.57/8.05 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 26.57/8.05 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 26.57/8.05 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 26.57/8.05 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 26.57/8.05 a__U85(X1, X2) -> U85(X1, X2) 26.57/8.05 a__U86(X) -> U86(X) 26.57/8.05 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 26.57/8.05 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 26.57/8.05 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 26.57/8.05 a__U94(X1, X2) -> U94(X1, X2) 26.57/8.05 a__length(X) -> length(X) 26.57/8.05 26.57/8.05 The set Q consists of the following terms: 26.57/8.05 26.57/8.05 a__zeros 26.57/8.05 a__isNatIList(x0) 26.57/8.05 mark(zeros) 26.57/8.05 mark(U11(x0, x1)) 26.57/8.05 mark(U12(x0, x1)) 26.57/8.05 mark(isNatIListKind(x0)) 26.57/8.05 mark(U13(x0)) 26.57/8.05 mark(isNatList(x0)) 26.57/8.05 mark(U21(x0, x1)) 26.57/8.05 mark(U22(x0, x1)) 26.57/8.05 mark(isNatKind(x0)) 26.57/8.05 mark(U23(x0)) 26.57/8.05 mark(isNat(x0)) 26.57/8.05 mark(U31(x0, x1)) 26.57/8.05 mark(U32(x0, x1)) 26.57/8.05 mark(U33(x0)) 26.57/8.05 mark(U41(x0, x1, x2)) 26.57/8.05 mark(U42(x0, x1, x2)) 26.57/8.05 mark(U43(x0, x1, x2)) 26.57/8.05 mark(U44(x0, x1, x2)) 26.57/8.05 mark(U45(x0, x1)) 26.57/8.05 mark(U46(x0)) 26.57/8.05 mark(isNatIList(x0)) 26.57/8.05 mark(U51(x0, x1)) 26.57/8.05 mark(U52(x0)) 26.57/8.05 mark(U61(x0)) 26.57/8.05 mark(U71(x0)) 26.57/8.05 mark(U81(x0, x1, x2)) 26.57/8.05 mark(U82(x0, x1, x2)) 26.57/8.05 mark(U83(x0, x1, x2)) 26.57/8.05 mark(U84(x0, x1, x2)) 26.57/8.05 mark(U85(x0, x1)) 26.57/8.05 mark(U86(x0)) 26.57/8.05 mark(U91(x0, x1, x2)) 26.57/8.05 mark(U92(x0, x1, x2)) 26.57/8.05 mark(U93(x0, x1, x2)) 26.57/8.05 mark(U94(x0, x1)) 26.57/8.05 mark(length(x0)) 26.57/8.05 mark(cons(x0, x1)) 26.57/8.05 mark(0) 26.57/8.05 mark(tt) 26.57/8.05 mark(s(x0)) 26.57/8.05 mark(nil) 26.57/8.05 a__U11(x0, x1) 26.57/8.05 a__U12(x0, x1) 26.57/8.05 a__isNatIListKind(x0) 26.57/8.05 a__U13(x0) 26.57/8.05 a__isNatList(x0) 26.57/8.05 a__U21(x0, x1) 26.57/8.05 a__U22(x0, x1) 26.57/8.05 a__isNatKind(x0) 26.57/8.05 a__U23(x0) 26.57/8.05 a__isNat(x0) 26.57/8.05 a__U31(x0, x1) 26.57/8.05 a__U32(x0, x1) 26.57/8.05 a__U33(x0) 26.57/8.05 a__U41(x0, x1, x2) 26.57/8.05 a__U42(x0, x1, x2) 26.57/8.05 a__U43(x0, x1, x2) 26.57/8.05 a__U44(x0, x1, x2) 26.57/8.05 a__U45(x0, x1) 26.57/8.05 a__U46(x0) 26.57/8.05 a__U51(x0, x1) 26.57/8.05 a__U52(x0) 26.57/8.05 a__U61(x0) 26.57/8.05 a__U71(x0) 26.57/8.05 a__U81(x0, x1, x2) 26.57/8.05 a__U82(x0, x1, x2) 26.57/8.05 a__U83(x0, x1, x2) 26.57/8.05 a__U84(x0, x1, x2) 26.57/8.05 a__U85(x0, x1) 26.57/8.05 a__U86(x0) 26.57/8.05 a__U91(x0, x1, x2) 26.57/8.05 a__U92(x0, x1, x2) 26.57/8.05 a__U93(x0, x1, x2) 26.57/8.05 a__U94(x0, x1) 26.57/8.05 a__length(x0) 26.57/8.05 26.57/8.05 We have to consider all minimal (P,Q,R)-chains. 26.57/8.05 ---------------------------------------- 26.57/8.05 26.57/8.05 (39) UsableRulesProof (EQUIVALENT) 26.57/8.05 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 26.57/8.05 ---------------------------------------- 26.57/8.05 26.57/8.05 (40) 26.57/8.05 Obligation: 26.57/8.05 Q DP problem: 26.57/8.05 The TRS P consists of the following rules: 26.57/8.05 26.57/8.05 MARK(U12(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U11(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U13(X)) -> MARK(X) 26.57/8.05 MARK(U21(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U22(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U23(X)) -> MARK(X) 26.57/8.05 MARK(U31(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U32(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U33(X)) -> MARK(X) 26.57/8.05 MARK(U41(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U42(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U43(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U44(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U45(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U46(X)) -> MARK(X) 26.57/8.05 MARK(U51(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U52(X)) -> MARK(X) 26.57/8.05 MARK(U61(X)) -> MARK(X) 26.57/8.05 MARK(U71(X)) -> MARK(X) 26.57/8.05 MARK(U81(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U82(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U83(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U84(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U85(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U86(X)) -> MARK(X) 26.57/8.05 MARK(s(X)) -> MARK(X) 26.57/8.05 26.57/8.05 R is empty. 26.57/8.05 The set Q consists of the following terms: 26.57/8.05 26.57/8.05 a__zeros 26.57/8.05 a__isNatIList(x0) 26.57/8.05 mark(zeros) 26.57/8.05 mark(U11(x0, x1)) 26.57/8.05 mark(U12(x0, x1)) 26.57/8.05 mark(isNatIListKind(x0)) 26.57/8.05 mark(U13(x0)) 26.57/8.05 mark(isNatList(x0)) 26.57/8.05 mark(U21(x0, x1)) 26.57/8.05 mark(U22(x0, x1)) 26.57/8.05 mark(isNatKind(x0)) 26.57/8.05 mark(U23(x0)) 26.57/8.05 mark(isNat(x0)) 26.57/8.05 mark(U31(x0, x1)) 26.57/8.05 mark(U32(x0, x1)) 26.57/8.05 mark(U33(x0)) 26.57/8.05 mark(U41(x0, x1, x2)) 26.57/8.05 mark(U42(x0, x1, x2)) 26.57/8.05 mark(U43(x0, x1, x2)) 26.57/8.05 mark(U44(x0, x1, x2)) 26.57/8.05 mark(U45(x0, x1)) 26.57/8.05 mark(U46(x0)) 26.57/8.05 mark(isNatIList(x0)) 26.57/8.05 mark(U51(x0, x1)) 26.57/8.05 mark(U52(x0)) 26.57/8.05 mark(U61(x0)) 26.57/8.05 mark(U71(x0)) 26.57/8.05 mark(U81(x0, x1, x2)) 26.57/8.05 mark(U82(x0, x1, x2)) 26.57/8.05 mark(U83(x0, x1, x2)) 26.57/8.05 mark(U84(x0, x1, x2)) 26.57/8.05 mark(U85(x0, x1)) 26.57/8.05 mark(U86(x0)) 26.57/8.05 mark(U91(x0, x1, x2)) 26.57/8.05 mark(U92(x0, x1, x2)) 26.57/8.05 mark(U93(x0, x1, x2)) 26.57/8.05 mark(U94(x0, x1)) 26.57/8.05 mark(length(x0)) 26.57/8.05 mark(cons(x0, x1)) 26.57/8.05 mark(0) 26.57/8.05 mark(tt) 26.57/8.05 mark(s(x0)) 26.57/8.05 mark(nil) 26.57/8.05 a__U11(x0, x1) 26.57/8.05 a__U12(x0, x1) 26.57/8.05 a__isNatIListKind(x0) 26.57/8.05 a__U13(x0) 26.57/8.05 a__isNatList(x0) 26.57/8.05 a__U21(x0, x1) 26.57/8.05 a__U22(x0, x1) 26.57/8.05 a__isNatKind(x0) 26.57/8.05 a__U23(x0) 26.57/8.05 a__isNat(x0) 26.57/8.05 a__U31(x0, x1) 26.57/8.05 a__U32(x0, x1) 26.57/8.05 a__U33(x0) 26.57/8.05 a__U41(x0, x1, x2) 26.57/8.05 a__U42(x0, x1, x2) 26.57/8.05 a__U43(x0, x1, x2) 26.57/8.05 a__U44(x0, x1, x2) 26.57/8.05 a__U45(x0, x1) 26.57/8.05 a__U46(x0) 26.57/8.05 a__U51(x0, x1) 26.57/8.05 a__U52(x0) 26.57/8.05 a__U61(x0) 26.57/8.05 a__U71(x0) 26.57/8.05 a__U81(x0, x1, x2) 26.57/8.05 a__U82(x0, x1, x2) 26.57/8.05 a__U83(x0, x1, x2) 26.57/8.05 a__U84(x0, x1, x2) 26.57/8.05 a__U85(x0, x1) 26.57/8.05 a__U86(x0) 26.57/8.05 a__U91(x0, x1, x2) 26.57/8.05 a__U92(x0, x1, x2) 26.57/8.05 a__U93(x0, x1, x2) 26.57/8.05 a__U94(x0, x1) 26.57/8.05 a__length(x0) 26.57/8.05 26.57/8.05 We have to consider all minimal (P,Q,R)-chains. 26.57/8.05 ---------------------------------------- 26.57/8.05 26.57/8.05 (41) QReductionProof (EQUIVALENT) 26.57/8.05 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 26.57/8.05 26.57/8.05 a__zeros 26.57/8.05 a__isNatIList(x0) 26.57/8.05 mark(zeros) 26.57/8.05 mark(U11(x0, x1)) 26.57/8.05 mark(U12(x0, x1)) 26.57/8.05 mark(isNatIListKind(x0)) 26.57/8.05 mark(U13(x0)) 26.57/8.05 mark(isNatList(x0)) 26.57/8.05 mark(U21(x0, x1)) 26.57/8.05 mark(U22(x0, x1)) 26.57/8.05 mark(isNatKind(x0)) 26.57/8.05 mark(U23(x0)) 26.57/8.05 mark(isNat(x0)) 26.57/8.05 mark(U31(x0, x1)) 26.57/8.05 mark(U32(x0, x1)) 26.57/8.05 mark(U33(x0)) 26.57/8.05 mark(U41(x0, x1, x2)) 26.57/8.05 mark(U42(x0, x1, x2)) 26.57/8.05 mark(U43(x0, x1, x2)) 26.57/8.05 mark(U44(x0, x1, x2)) 26.57/8.05 mark(U45(x0, x1)) 26.57/8.05 mark(U46(x0)) 26.57/8.05 mark(isNatIList(x0)) 26.57/8.05 mark(U51(x0, x1)) 26.57/8.05 mark(U52(x0)) 26.57/8.05 mark(U61(x0)) 26.57/8.05 mark(U71(x0)) 26.57/8.05 mark(U81(x0, x1, x2)) 26.57/8.05 mark(U82(x0, x1, x2)) 26.57/8.05 mark(U83(x0, x1, x2)) 26.57/8.05 mark(U84(x0, x1, x2)) 26.57/8.05 mark(U85(x0, x1)) 26.57/8.05 mark(U86(x0)) 26.57/8.05 mark(U91(x0, x1, x2)) 26.57/8.05 mark(U92(x0, x1, x2)) 26.57/8.05 mark(U93(x0, x1, x2)) 26.57/8.05 mark(U94(x0, x1)) 26.57/8.05 mark(length(x0)) 26.57/8.05 mark(cons(x0, x1)) 26.57/8.05 mark(0) 26.57/8.05 mark(tt) 26.57/8.05 mark(s(x0)) 26.57/8.05 mark(nil) 26.57/8.05 a__U11(x0, x1) 26.57/8.05 a__U12(x0, x1) 26.57/8.05 a__isNatIListKind(x0) 26.57/8.05 a__U13(x0) 26.57/8.05 a__isNatList(x0) 26.57/8.05 a__U21(x0, x1) 26.57/8.05 a__U22(x0, x1) 26.57/8.05 a__isNatKind(x0) 26.57/8.05 a__U23(x0) 26.57/8.05 a__isNat(x0) 26.57/8.05 a__U31(x0, x1) 26.57/8.05 a__U32(x0, x1) 26.57/8.05 a__U33(x0) 26.57/8.05 a__U41(x0, x1, x2) 26.57/8.05 a__U42(x0, x1, x2) 26.57/8.05 a__U43(x0, x1, x2) 26.57/8.05 a__U44(x0, x1, x2) 26.57/8.05 a__U45(x0, x1) 26.57/8.05 a__U46(x0) 26.57/8.05 a__U51(x0, x1) 26.57/8.05 a__U52(x0) 26.57/8.05 a__U61(x0) 26.57/8.05 a__U71(x0) 26.57/8.05 a__U81(x0, x1, x2) 26.57/8.05 a__U82(x0, x1, x2) 26.57/8.05 a__U83(x0, x1, x2) 26.57/8.05 a__U84(x0, x1, x2) 26.57/8.05 a__U85(x0, x1) 26.57/8.05 a__U86(x0) 26.57/8.05 a__U91(x0, x1, x2) 26.57/8.05 a__U92(x0, x1, x2) 26.57/8.05 a__U93(x0, x1, x2) 26.57/8.05 a__U94(x0, x1) 26.57/8.05 a__length(x0) 26.57/8.05 26.57/8.05 26.57/8.05 ---------------------------------------- 26.57/8.05 26.57/8.05 (42) 26.57/8.05 Obligation: 26.57/8.05 Q DP problem: 26.57/8.05 The TRS P consists of the following rules: 26.57/8.05 26.57/8.05 MARK(U12(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U11(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U13(X)) -> MARK(X) 26.57/8.05 MARK(U21(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U22(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U23(X)) -> MARK(X) 26.57/8.05 MARK(U31(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U32(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U33(X)) -> MARK(X) 26.57/8.05 MARK(U41(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U42(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U43(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U44(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U45(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U46(X)) -> MARK(X) 26.57/8.05 MARK(U51(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U52(X)) -> MARK(X) 26.57/8.05 MARK(U61(X)) -> MARK(X) 26.57/8.05 MARK(U71(X)) -> MARK(X) 26.57/8.05 MARK(U81(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U82(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U83(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U84(X1, X2, X3)) -> MARK(X1) 26.57/8.05 MARK(U85(X1, X2)) -> MARK(X1) 26.57/8.05 MARK(U86(X)) -> MARK(X) 26.57/8.05 MARK(s(X)) -> MARK(X) 26.57/8.05 26.57/8.05 R is empty. 26.57/8.05 Q is empty. 26.57/8.05 We have to consider all minimal (P,Q,R)-chains. 26.57/8.05 ---------------------------------------- 26.57/8.05 26.57/8.05 (43) QDPSizeChangeProof (EQUIVALENT) 26.57/8.05 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 26.57/8.05 26.57/8.05 From the DPs we obtained the following set of size-change graphs: 26.57/8.05 *MARK(U12(X1, X2)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U11(X1, X2)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U13(X)) -> MARK(X) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U21(X1, X2)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U22(X1, X2)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U23(X)) -> MARK(X) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U31(X1, X2)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U32(X1, X2)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U33(X)) -> MARK(X) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U41(X1, X2, X3)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U42(X1, X2, X3)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U43(X1, X2, X3)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U44(X1, X2, X3)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U45(X1, X2)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U46(X)) -> MARK(X) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U51(X1, X2)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U52(X)) -> MARK(X) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U61(X)) -> MARK(X) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U71(X)) -> MARK(X) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U81(X1, X2, X3)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U82(X1, X2, X3)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U83(X1, X2, X3)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U84(X1, X2, X3)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U85(X1, X2)) -> MARK(X1) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(U86(X)) -> MARK(X) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 *MARK(s(X)) -> MARK(X) 26.57/8.05 The graph contains the following edges 1 > 1 26.57/8.05 26.57/8.05 26.57/8.05 ---------------------------------------- 26.57/8.05 26.57/8.05 (44) 26.57/8.05 YES 26.83/8.11 EOF