14.82/4.51 WORST_CASE(NON_POLY, ?) 14.82/4.52 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 14.82/4.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 14.82/4.52 14.82/4.52 14.82/4.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 14.82/4.52 14.82/4.52 (0) CpxTRS 14.82/4.52 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 14.82/4.52 (2) TRS for Loop Detection 14.82/4.52 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 14.82/4.52 (4) BEST 14.82/4.52 (5) proven lower bound 14.82/4.52 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 14.82/4.52 (7) BOUNDS(n^1, INF) 14.82/4.52 (8) TRS for Loop Detection 14.82/4.52 (9) DecreasingLoopProof [FINISHED, 2031 ms] 14.82/4.52 (10) BOUNDS(EXP, INF) 14.82/4.52 14.82/4.52 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (0) 14.82/4.52 Obligation: 14.82/4.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 14.82/4.52 14.82/4.52 14.82/4.52 The TRS R consists of the following rules: 14.82/4.52 14.82/4.52 a__zeros -> cons(0, zeros) 14.82/4.52 a__U11(tt, V1) -> a__U12(a__isNatList(V1)) 14.82/4.52 a__U12(tt) -> tt 14.82/4.52 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 14.82/4.52 a__U22(tt) -> tt 14.82/4.52 a__U31(tt, V) -> a__U32(a__isNatList(V)) 14.82/4.52 a__U32(tt) -> tt 14.82/4.52 a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) 14.82/4.52 a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) 14.82/4.52 a__U43(tt) -> tt 14.82/4.52 a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) 14.82/4.52 a__U52(tt, V2) -> a__U53(a__isNatList(V2)) 14.82/4.52 a__U53(tt) -> tt 14.82/4.52 a__U61(tt, L) -> s(a__length(mark(L))) 14.82/4.52 a__and(tt, X) -> mark(X) 14.82/4.52 a__isNat(0) -> tt 14.82/4.52 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 14.82/4.52 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 14.82/4.52 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 14.82/4.52 a__isNatIList(zeros) -> tt 14.82/4.52 a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 14.82/4.52 a__isNatIListKind(nil) -> tt 14.82/4.52 a__isNatIListKind(zeros) -> tt 14.82/4.52 a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 14.82/4.52 a__isNatKind(0) -> tt 14.82/4.52 a__isNatKind(length(V1)) -> a__isNatIListKind(V1) 14.82/4.52 a__isNatKind(s(V1)) -> a__isNatKind(V1) 14.82/4.52 a__isNatList(nil) -> tt 14.82/4.52 a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 14.82/4.52 a__length(nil) -> 0 14.82/4.52 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) 14.82/4.52 mark(zeros) -> a__zeros 14.82/4.52 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 14.82/4.52 mark(U12(X)) -> a__U12(mark(X)) 14.82/4.52 mark(isNatList(X)) -> a__isNatList(X) 14.82/4.52 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 14.82/4.52 mark(U22(X)) -> a__U22(mark(X)) 14.82/4.52 mark(isNat(X)) -> a__isNat(X) 14.82/4.52 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 14.82/4.52 mark(U32(X)) -> a__U32(mark(X)) 14.82/4.52 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 14.82/4.52 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 14.82/4.52 mark(U43(X)) -> a__U43(mark(X)) 14.82/4.52 mark(isNatIList(X)) -> a__isNatIList(X) 14.82/4.52 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 14.82/4.52 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 14.82/4.52 mark(U53(X)) -> a__U53(mark(X)) 14.82/4.52 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 14.82/4.52 mark(length(X)) -> a__length(mark(X)) 14.82/4.52 mark(and(X1, X2)) -> a__and(mark(X1), X2) 14.82/4.52 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 14.82/4.52 mark(isNatKind(X)) -> a__isNatKind(X) 14.82/4.52 mark(cons(X1, X2)) -> cons(mark(X1), X2) 14.82/4.52 mark(0) -> 0 14.82/4.52 mark(tt) -> tt 14.82/4.52 mark(s(X)) -> s(mark(X)) 14.82/4.52 mark(nil) -> nil 14.82/4.52 a__zeros -> zeros 14.82/4.52 a__U11(X1, X2) -> U11(X1, X2) 14.82/4.52 a__U12(X) -> U12(X) 14.82/4.52 a__isNatList(X) -> isNatList(X) 14.82/4.52 a__U21(X1, X2) -> U21(X1, X2) 14.82/4.52 a__U22(X) -> U22(X) 14.82/4.52 a__isNat(X) -> isNat(X) 14.82/4.52 a__U31(X1, X2) -> U31(X1, X2) 14.82/4.52 a__U32(X) -> U32(X) 14.82/4.52 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 14.82/4.52 a__U42(X1, X2) -> U42(X1, X2) 14.82/4.52 a__U43(X) -> U43(X) 14.82/4.52 a__isNatIList(X) -> isNatIList(X) 14.82/4.52 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 14.82/4.52 a__U52(X1, X2) -> U52(X1, X2) 14.82/4.52 a__U53(X) -> U53(X) 14.82/4.52 a__U61(X1, X2) -> U61(X1, X2) 14.82/4.52 a__length(X) -> length(X) 14.82/4.52 a__and(X1, X2) -> and(X1, X2) 14.82/4.52 a__isNatIListKind(X) -> isNatIListKind(X) 14.82/4.52 a__isNatKind(X) -> isNatKind(X) 14.82/4.52 14.82/4.52 S is empty. 14.82/4.52 Rewrite Strategy: FULL 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 14.82/4.52 Transformed a relative TRS into a decreasing-loop problem. 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (2) 14.82/4.52 Obligation: 14.82/4.52 Analyzing the following TRS for decreasing loops: 14.82/4.52 14.82/4.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 14.82/4.52 14.82/4.52 14.82/4.52 The TRS R consists of the following rules: 14.82/4.52 14.82/4.52 a__zeros -> cons(0, zeros) 14.82/4.52 a__U11(tt, V1) -> a__U12(a__isNatList(V1)) 14.82/4.52 a__U12(tt) -> tt 14.82/4.52 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 14.82/4.52 a__U22(tt) -> tt 14.82/4.52 a__U31(tt, V) -> a__U32(a__isNatList(V)) 14.82/4.52 a__U32(tt) -> tt 14.82/4.52 a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) 14.82/4.52 a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) 14.82/4.52 a__U43(tt) -> tt 14.82/4.52 a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) 14.82/4.52 a__U52(tt, V2) -> a__U53(a__isNatList(V2)) 14.82/4.52 a__U53(tt) -> tt 14.82/4.52 a__U61(tt, L) -> s(a__length(mark(L))) 14.82/4.52 a__and(tt, X) -> mark(X) 14.82/4.52 a__isNat(0) -> tt 14.82/4.52 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 14.82/4.52 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 14.82/4.52 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 14.82/4.52 a__isNatIList(zeros) -> tt 14.82/4.52 a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 14.82/4.52 a__isNatIListKind(nil) -> tt 14.82/4.52 a__isNatIListKind(zeros) -> tt 14.82/4.52 a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 14.82/4.52 a__isNatKind(0) -> tt 14.82/4.52 a__isNatKind(length(V1)) -> a__isNatIListKind(V1) 14.82/4.52 a__isNatKind(s(V1)) -> a__isNatKind(V1) 14.82/4.52 a__isNatList(nil) -> tt 14.82/4.52 a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 14.82/4.52 a__length(nil) -> 0 14.82/4.52 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) 14.82/4.52 mark(zeros) -> a__zeros 14.82/4.52 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 14.82/4.52 mark(U12(X)) -> a__U12(mark(X)) 14.82/4.52 mark(isNatList(X)) -> a__isNatList(X) 14.82/4.52 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 14.82/4.52 mark(U22(X)) -> a__U22(mark(X)) 14.82/4.52 mark(isNat(X)) -> a__isNat(X) 14.82/4.52 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 14.82/4.52 mark(U32(X)) -> a__U32(mark(X)) 14.82/4.52 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 14.82/4.52 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 14.82/4.52 mark(U43(X)) -> a__U43(mark(X)) 14.82/4.52 mark(isNatIList(X)) -> a__isNatIList(X) 14.82/4.52 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 14.82/4.52 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 14.82/4.52 mark(U53(X)) -> a__U53(mark(X)) 14.82/4.52 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 14.82/4.52 mark(length(X)) -> a__length(mark(X)) 14.82/4.52 mark(and(X1, X2)) -> a__and(mark(X1), X2) 14.82/4.52 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 14.82/4.52 mark(isNatKind(X)) -> a__isNatKind(X) 14.82/4.52 mark(cons(X1, X2)) -> cons(mark(X1), X2) 14.82/4.52 mark(0) -> 0 14.82/4.52 mark(tt) -> tt 14.82/4.52 mark(s(X)) -> s(mark(X)) 14.82/4.52 mark(nil) -> nil 14.82/4.52 a__zeros -> zeros 14.82/4.52 a__U11(X1, X2) -> U11(X1, X2) 14.82/4.52 a__U12(X) -> U12(X) 14.82/4.52 a__isNatList(X) -> isNatList(X) 14.82/4.52 a__U21(X1, X2) -> U21(X1, X2) 14.82/4.52 a__U22(X) -> U22(X) 14.82/4.52 a__isNat(X) -> isNat(X) 14.82/4.52 a__U31(X1, X2) -> U31(X1, X2) 14.82/4.52 a__U32(X) -> U32(X) 14.82/4.52 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 14.82/4.52 a__U42(X1, X2) -> U42(X1, X2) 14.82/4.52 a__U43(X) -> U43(X) 14.82/4.52 a__isNatIList(X) -> isNatIList(X) 14.82/4.52 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 14.82/4.52 a__U52(X1, X2) -> U52(X1, X2) 14.82/4.52 a__U53(X) -> U53(X) 14.82/4.52 a__U61(X1, X2) -> U61(X1, X2) 14.82/4.52 a__length(X) -> length(X) 14.82/4.52 a__and(X1, X2) -> and(X1, X2) 14.82/4.52 a__isNatIListKind(X) -> isNatIListKind(X) 14.82/4.52 a__isNatKind(X) -> isNatKind(X) 14.82/4.52 14.82/4.52 S is empty. 14.82/4.52 Rewrite Strategy: FULL 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (3) DecreasingLoopProof (LOWER BOUND(ID)) 14.82/4.52 The following loop(s) give(s) rise to the lower bound Omega(n^1): 14.82/4.52 14.82/4.52 The rewrite sequence 14.82/4.52 14.82/4.52 mark(U12(X)) ->^+ a__U12(mark(X)) 14.82/4.52 14.82/4.52 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 14.82/4.52 14.82/4.52 The pumping substitution is [X / U12(X)]. 14.82/4.52 14.82/4.52 The result substitution is [ ]. 14.82/4.52 14.82/4.52 14.82/4.52 14.82/4.52 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (4) 14.82/4.52 Complex Obligation (BEST) 14.82/4.52 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (5) 14.82/4.52 Obligation: 14.82/4.52 Proved the lower bound n^1 for the following obligation: 14.82/4.52 14.82/4.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 14.82/4.52 14.82/4.52 14.82/4.52 The TRS R consists of the following rules: 14.82/4.52 14.82/4.52 a__zeros -> cons(0, zeros) 14.82/4.52 a__U11(tt, V1) -> a__U12(a__isNatList(V1)) 14.82/4.52 a__U12(tt) -> tt 14.82/4.52 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 14.82/4.52 a__U22(tt) -> tt 14.82/4.52 a__U31(tt, V) -> a__U32(a__isNatList(V)) 14.82/4.52 a__U32(tt) -> tt 14.82/4.52 a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) 14.82/4.52 a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) 14.82/4.52 a__U43(tt) -> tt 14.82/4.52 a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) 14.82/4.52 a__U52(tt, V2) -> a__U53(a__isNatList(V2)) 14.82/4.52 a__U53(tt) -> tt 14.82/4.52 a__U61(tt, L) -> s(a__length(mark(L))) 14.82/4.52 a__and(tt, X) -> mark(X) 14.82/4.52 a__isNat(0) -> tt 14.82/4.52 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 14.82/4.52 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 14.82/4.52 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 14.82/4.52 a__isNatIList(zeros) -> tt 14.82/4.52 a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 14.82/4.52 a__isNatIListKind(nil) -> tt 14.82/4.52 a__isNatIListKind(zeros) -> tt 14.82/4.52 a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 14.82/4.52 a__isNatKind(0) -> tt 14.82/4.52 a__isNatKind(length(V1)) -> a__isNatIListKind(V1) 14.82/4.52 a__isNatKind(s(V1)) -> a__isNatKind(V1) 14.82/4.52 a__isNatList(nil) -> tt 14.82/4.52 a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 14.82/4.52 a__length(nil) -> 0 14.82/4.52 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) 14.82/4.52 mark(zeros) -> a__zeros 14.82/4.52 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 14.82/4.52 mark(U12(X)) -> a__U12(mark(X)) 14.82/4.52 mark(isNatList(X)) -> a__isNatList(X) 14.82/4.52 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 14.82/4.52 mark(U22(X)) -> a__U22(mark(X)) 14.82/4.52 mark(isNat(X)) -> a__isNat(X) 14.82/4.52 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 14.82/4.52 mark(U32(X)) -> a__U32(mark(X)) 14.82/4.52 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 14.82/4.52 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 14.82/4.52 mark(U43(X)) -> a__U43(mark(X)) 14.82/4.52 mark(isNatIList(X)) -> a__isNatIList(X) 14.82/4.52 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 14.82/4.52 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 14.82/4.52 mark(U53(X)) -> a__U53(mark(X)) 14.82/4.52 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 14.82/4.52 mark(length(X)) -> a__length(mark(X)) 14.82/4.52 mark(and(X1, X2)) -> a__and(mark(X1), X2) 14.82/4.52 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 14.82/4.52 mark(isNatKind(X)) -> a__isNatKind(X) 14.82/4.52 mark(cons(X1, X2)) -> cons(mark(X1), X2) 14.82/4.52 mark(0) -> 0 14.82/4.52 mark(tt) -> tt 14.82/4.52 mark(s(X)) -> s(mark(X)) 14.82/4.52 mark(nil) -> nil 14.82/4.52 a__zeros -> zeros 14.82/4.52 a__U11(X1, X2) -> U11(X1, X2) 14.82/4.52 a__U12(X) -> U12(X) 14.82/4.52 a__isNatList(X) -> isNatList(X) 14.82/4.52 a__U21(X1, X2) -> U21(X1, X2) 14.82/4.52 a__U22(X) -> U22(X) 14.82/4.52 a__isNat(X) -> isNat(X) 14.82/4.52 a__U31(X1, X2) -> U31(X1, X2) 14.82/4.52 a__U32(X) -> U32(X) 14.82/4.52 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 14.82/4.52 a__U42(X1, X2) -> U42(X1, X2) 14.82/4.52 a__U43(X) -> U43(X) 14.82/4.52 a__isNatIList(X) -> isNatIList(X) 14.82/4.52 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 14.82/4.52 a__U52(X1, X2) -> U52(X1, X2) 14.82/4.52 a__U53(X) -> U53(X) 14.82/4.52 a__U61(X1, X2) -> U61(X1, X2) 14.82/4.52 a__length(X) -> length(X) 14.82/4.52 a__and(X1, X2) -> and(X1, X2) 14.82/4.52 a__isNatIListKind(X) -> isNatIListKind(X) 14.82/4.52 a__isNatKind(X) -> isNatKind(X) 14.82/4.52 14.82/4.52 S is empty. 14.82/4.52 Rewrite Strategy: FULL 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (6) LowerBoundPropagationProof (FINISHED) 14.82/4.52 Propagated lower bound. 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (7) 14.82/4.52 BOUNDS(n^1, INF) 14.82/4.52 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (8) 14.82/4.52 Obligation: 14.82/4.52 Analyzing the following TRS for decreasing loops: 14.82/4.52 14.82/4.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 14.82/4.52 14.82/4.52 14.82/4.52 The TRS R consists of the following rules: 14.82/4.52 14.82/4.52 a__zeros -> cons(0, zeros) 14.82/4.52 a__U11(tt, V1) -> a__U12(a__isNatList(V1)) 14.82/4.52 a__U12(tt) -> tt 14.82/4.52 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 14.82/4.52 a__U22(tt) -> tt 14.82/4.52 a__U31(tt, V) -> a__U32(a__isNatList(V)) 14.82/4.52 a__U32(tt) -> tt 14.82/4.52 a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) 14.82/4.52 a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) 14.82/4.52 a__U43(tt) -> tt 14.82/4.52 a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) 14.82/4.52 a__U52(tt, V2) -> a__U53(a__isNatList(V2)) 14.82/4.52 a__U53(tt) -> tt 14.82/4.52 a__U61(tt, L) -> s(a__length(mark(L))) 14.82/4.52 a__and(tt, X) -> mark(X) 14.82/4.52 a__isNat(0) -> tt 14.82/4.52 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 14.82/4.52 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 14.82/4.52 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 14.82/4.52 a__isNatIList(zeros) -> tt 14.82/4.52 a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 14.82/4.52 a__isNatIListKind(nil) -> tt 14.82/4.52 a__isNatIListKind(zeros) -> tt 14.82/4.52 a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 14.82/4.52 a__isNatKind(0) -> tt 14.82/4.52 a__isNatKind(length(V1)) -> a__isNatIListKind(V1) 14.82/4.52 a__isNatKind(s(V1)) -> a__isNatKind(V1) 14.82/4.52 a__isNatList(nil) -> tt 14.82/4.52 a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 14.82/4.52 a__length(nil) -> 0 14.82/4.52 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) 14.82/4.52 mark(zeros) -> a__zeros 14.82/4.52 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 14.82/4.52 mark(U12(X)) -> a__U12(mark(X)) 14.82/4.52 mark(isNatList(X)) -> a__isNatList(X) 14.82/4.52 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 14.82/4.52 mark(U22(X)) -> a__U22(mark(X)) 14.82/4.52 mark(isNat(X)) -> a__isNat(X) 14.82/4.52 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 14.82/4.52 mark(U32(X)) -> a__U32(mark(X)) 14.82/4.52 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 14.82/4.52 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 14.82/4.52 mark(U43(X)) -> a__U43(mark(X)) 14.82/4.52 mark(isNatIList(X)) -> a__isNatIList(X) 14.82/4.52 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 14.82/4.52 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 14.82/4.52 mark(U53(X)) -> a__U53(mark(X)) 14.82/4.52 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 14.82/4.52 mark(length(X)) -> a__length(mark(X)) 14.82/4.52 mark(and(X1, X2)) -> a__and(mark(X1), X2) 14.82/4.52 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 14.82/4.52 mark(isNatKind(X)) -> a__isNatKind(X) 14.82/4.52 mark(cons(X1, X2)) -> cons(mark(X1), X2) 14.82/4.52 mark(0) -> 0 14.82/4.52 mark(tt) -> tt 14.82/4.52 mark(s(X)) -> s(mark(X)) 14.82/4.52 mark(nil) -> nil 14.82/4.52 a__zeros -> zeros 14.82/4.52 a__U11(X1, X2) -> U11(X1, X2) 14.82/4.52 a__U12(X) -> U12(X) 14.82/4.52 a__isNatList(X) -> isNatList(X) 14.82/4.52 a__U21(X1, X2) -> U21(X1, X2) 14.82/4.52 a__U22(X) -> U22(X) 14.82/4.52 a__isNat(X) -> isNat(X) 14.82/4.52 a__U31(X1, X2) -> U31(X1, X2) 14.82/4.52 a__U32(X) -> U32(X) 14.82/4.52 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 14.82/4.52 a__U42(X1, X2) -> U42(X1, X2) 14.82/4.52 a__U43(X) -> U43(X) 14.82/4.52 a__isNatIList(X) -> isNatIList(X) 14.82/4.52 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 14.82/4.52 a__U52(X1, X2) -> U52(X1, X2) 14.82/4.52 a__U53(X) -> U53(X) 14.82/4.52 a__U61(X1, X2) -> U61(X1, X2) 14.82/4.52 a__length(X) -> length(X) 14.82/4.52 a__and(X1, X2) -> and(X1, X2) 14.82/4.52 a__isNatIListKind(X) -> isNatIListKind(X) 14.82/4.52 a__isNatKind(X) -> isNatKind(X) 14.82/4.52 14.82/4.52 S is empty. 14.82/4.52 Rewrite Strategy: FULL 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (9) DecreasingLoopProof (FINISHED) 14.82/4.52 The following loop(s) give(s) rise to the lower bound EXP: 14.82/4.52 14.82/4.52 The rewrite sequence 14.82/4.52 14.82/4.52 mark(length(cons(X11_0, X22_0))) ->^+ a__U61(a__and(a__and(a__isNatList(X22_0), isNatIListKind(X22_0)), and(isNat(mark(X11_0)), isNatKind(mark(X11_0)))), X22_0) 14.82/4.52 14.82/4.52 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0]. 14.82/4.52 14.82/4.52 The pumping substitution is [X11_0 / length(cons(X11_0, X22_0))]. 14.82/4.52 14.82/4.52 The result substitution is [ ]. 14.82/4.52 14.82/4.52 14.82/4.52 14.82/4.52 The rewrite sequence 14.82/4.52 14.82/4.52 mark(length(cons(X11_0, X22_0))) ->^+ a__U61(a__and(a__and(a__isNatList(X22_0), isNatIListKind(X22_0)), and(isNat(mark(X11_0)), isNatKind(mark(X11_0)))), X22_0) 14.82/4.52 14.82/4.52 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,1,0]. 14.82/4.52 14.82/4.52 The pumping substitution is [X11_0 / length(cons(X11_0, X22_0))]. 14.82/4.52 14.82/4.52 The result substitution is [ ]. 14.82/4.52 14.82/4.52 14.82/4.52 14.82/4.52 14.82/4.52 ---------------------------------------- 14.82/4.52 14.82/4.52 (10) 14.82/4.52 BOUNDS(EXP, INF) 15.39/4.97 EOF