893.08/291.56 WORST_CASE(Omega(n^1), ?) 893.15/291.59 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 893.15/291.59 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 893.15/291.59 893.15/291.59 893.15/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 893.15/291.59 893.15/291.59 (0) CpxTRS 893.15/291.59 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 893.15/291.59 (2) TRS for Loop Detection 893.15/291.59 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 893.15/291.59 (4) BEST 893.15/291.59 (5) proven lower bound 893.15/291.59 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 893.15/291.59 (7) BOUNDS(n^1, INF) 893.15/291.59 (8) TRS for Loop Detection 893.15/291.59 893.15/291.59 893.15/291.59 ---------------------------------------- 893.15/291.59 893.15/291.59 (0) 893.15/291.59 Obligation: 893.15/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 893.15/291.59 893.15/291.59 893.15/291.59 The TRS R consists of the following rules: 893.15/291.59 893.15/291.59 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 893.15/291.59 a____(X, nil) -> mark(X) 893.15/291.59 a____(nil, X) -> mark(X) 893.15/291.59 a__U11(tt, V) -> a__U12(a__isNeList(V)) 893.15/291.59 a__U12(tt) -> tt 893.15/291.59 a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) 893.15/291.59 a__U22(tt, V2) -> a__U23(a__isList(V2)) 893.15/291.59 a__U23(tt) -> tt 893.15/291.59 a__U31(tt, V) -> a__U32(a__isQid(V)) 893.15/291.59 a__U32(tt) -> tt 893.15/291.59 a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) 893.15/291.59 a__U42(tt, V2) -> a__U43(a__isNeList(V2)) 893.15/291.59 a__U43(tt) -> tt 893.15/291.59 a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) 893.15/291.59 a__U52(tt, V2) -> a__U53(a__isList(V2)) 893.15/291.59 a__U53(tt) -> tt 893.15/291.59 a__U61(tt, V) -> a__U62(a__isQid(V)) 893.15/291.59 a__U62(tt) -> tt 893.15/291.59 a__U71(tt, V) -> a__U72(a__isNePal(V)) 893.15/291.59 a__U72(tt) -> tt 893.15/291.59 a__and(tt, X) -> mark(X) 893.15/291.59 a__isList(V) -> a__U11(a__isPalListKind(V), V) 893.15/291.59 a__isList(nil) -> tt 893.15/291.59 a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNeList(V) -> a__U31(a__isPalListKind(V), V) 893.15/291.59 a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNePal(V) -> a__U61(a__isPalListKind(V), V) 893.15/291.59 a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) 893.15/291.59 a__isPal(V) -> a__U71(a__isPalListKind(V), V) 893.15/291.59 a__isPal(nil) -> tt 893.15/291.59 a__isPalListKind(a) -> tt 893.15/291.59 a__isPalListKind(e) -> tt 893.15/291.59 a__isPalListKind(i) -> tt 893.15/291.59 a__isPalListKind(nil) -> tt 893.15/291.59 a__isPalListKind(o) -> tt 893.15/291.59 a__isPalListKind(u) -> tt 893.15/291.59 a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) 893.15/291.59 a__isQid(a) -> tt 893.15/291.59 a__isQid(e) -> tt 893.15/291.59 a__isQid(i) -> tt 893.15/291.59 a__isQid(o) -> tt 893.15/291.59 a__isQid(u) -> tt 893.15/291.59 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 893.15/291.59 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 893.15/291.59 mark(U12(X)) -> a__U12(mark(X)) 893.15/291.59 mark(isNeList(X)) -> a__isNeList(X) 893.15/291.59 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 893.15/291.59 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 893.15/291.59 mark(isList(X)) -> a__isList(X) 893.15/291.59 mark(U23(X)) -> a__U23(mark(X)) 893.15/291.59 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 893.15/291.59 mark(U32(X)) -> a__U32(mark(X)) 893.15/291.59 mark(isQid(X)) -> a__isQid(X) 893.15/291.59 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 893.15/291.59 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 893.15/291.59 mark(U43(X)) -> a__U43(mark(X)) 893.15/291.59 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 893.15/291.59 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 893.15/291.59 mark(U53(X)) -> a__U53(mark(X)) 893.15/291.59 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 893.15/291.59 mark(U62(X)) -> a__U62(mark(X)) 893.15/291.59 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 893.15/291.59 mark(U72(X)) -> a__U72(mark(X)) 893.15/291.59 mark(isNePal(X)) -> a__isNePal(X) 893.15/291.59 mark(and(X1, X2)) -> a__and(mark(X1), X2) 893.15/291.59 mark(isPalListKind(X)) -> a__isPalListKind(X) 893.15/291.59 mark(isPal(X)) -> a__isPal(X) 893.15/291.59 mark(nil) -> nil 893.15/291.59 mark(tt) -> tt 893.15/291.59 mark(a) -> a 893.15/291.59 mark(e) -> e 893.15/291.59 mark(i) -> i 893.15/291.59 mark(o) -> o 893.15/291.59 mark(u) -> u 893.15/291.59 a____(X1, X2) -> __(X1, X2) 893.15/291.59 a__U11(X1, X2) -> U11(X1, X2) 893.15/291.59 a__U12(X) -> U12(X) 893.15/291.59 a__isNeList(X) -> isNeList(X) 893.15/291.59 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 893.15/291.59 a__U22(X1, X2) -> U22(X1, X2) 893.15/291.59 a__isList(X) -> isList(X) 893.15/291.59 a__U23(X) -> U23(X) 893.15/291.59 a__U31(X1, X2) -> U31(X1, X2) 893.15/291.59 a__U32(X) -> U32(X) 893.15/291.59 a__isQid(X) -> isQid(X) 893.15/291.59 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 893.15/291.59 a__U42(X1, X2) -> U42(X1, X2) 893.15/291.59 a__U43(X) -> U43(X) 893.15/291.59 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 893.15/291.59 a__U52(X1, X2) -> U52(X1, X2) 893.15/291.59 a__U53(X) -> U53(X) 893.15/291.59 a__U61(X1, X2) -> U61(X1, X2) 893.15/291.59 a__U62(X) -> U62(X) 893.15/291.59 a__U71(X1, X2) -> U71(X1, X2) 893.15/291.59 a__U72(X) -> U72(X) 893.15/291.59 a__isNePal(X) -> isNePal(X) 893.15/291.59 a__and(X1, X2) -> and(X1, X2) 893.15/291.59 a__isPalListKind(X) -> isPalListKind(X) 893.15/291.59 a__isPal(X) -> isPal(X) 893.15/291.59 893.15/291.59 S is empty. 893.15/291.59 Rewrite Strategy: FULL 893.15/291.59 ---------------------------------------- 893.15/291.59 893.15/291.59 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 893.15/291.59 Transformed a relative TRS into a decreasing-loop problem. 893.15/291.59 ---------------------------------------- 893.15/291.59 893.15/291.59 (2) 893.15/291.59 Obligation: 893.15/291.59 Analyzing the following TRS for decreasing loops: 893.15/291.59 893.15/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 893.15/291.59 893.15/291.59 893.15/291.59 The TRS R consists of the following rules: 893.15/291.59 893.15/291.59 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 893.15/291.59 a____(X, nil) -> mark(X) 893.15/291.59 a____(nil, X) -> mark(X) 893.15/291.59 a__U11(tt, V) -> a__U12(a__isNeList(V)) 893.15/291.59 a__U12(tt) -> tt 893.15/291.59 a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) 893.15/291.59 a__U22(tt, V2) -> a__U23(a__isList(V2)) 893.15/291.59 a__U23(tt) -> tt 893.15/291.59 a__U31(tt, V) -> a__U32(a__isQid(V)) 893.15/291.59 a__U32(tt) -> tt 893.15/291.59 a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) 893.15/291.59 a__U42(tt, V2) -> a__U43(a__isNeList(V2)) 893.15/291.59 a__U43(tt) -> tt 893.15/291.59 a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) 893.15/291.59 a__U52(tt, V2) -> a__U53(a__isList(V2)) 893.15/291.59 a__U53(tt) -> tt 893.15/291.59 a__U61(tt, V) -> a__U62(a__isQid(V)) 893.15/291.59 a__U62(tt) -> tt 893.15/291.59 a__U71(tt, V) -> a__U72(a__isNePal(V)) 893.15/291.59 a__U72(tt) -> tt 893.15/291.59 a__and(tt, X) -> mark(X) 893.15/291.59 a__isList(V) -> a__U11(a__isPalListKind(V), V) 893.15/291.59 a__isList(nil) -> tt 893.15/291.59 a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNeList(V) -> a__U31(a__isPalListKind(V), V) 893.15/291.59 a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNePal(V) -> a__U61(a__isPalListKind(V), V) 893.15/291.59 a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) 893.15/291.59 a__isPal(V) -> a__U71(a__isPalListKind(V), V) 893.15/291.59 a__isPal(nil) -> tt 893.15/291.59 a__isPalListKind(a) -> tt 893.15/291.59 a__isPalListKind(e) -> tt 893.15/291.59 a__isPalListKind(i) -> tt 893.15/291.59 a__isPalListKind(nil) -> tt 893.15/291.59 a__isPalListKind(o) -> tt 893.15/291.59 a__isPalListKind(u) -> tt 893.15/291.59 a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) 893.15/291.59 a__isQid(a) -> tt 893.15/291.59 a__isQid(e) -> tt 893.15/291.59 a__isQid(i) -> tt 893.15/291.59 a__isQid(o) -> tt 893.15/291.59 a__isQid(u) -> tt 893.15/291.59 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 893.15/291.59 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 893.15/291.59 mark(U12(X)) -> a__U12(mark(X)) 893.15/291.59 mark(isNeList(X)) -> a__isNeList(X) 893.15/291.59 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 893.15/291.59 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 893.15/291.59 mark(isList(X)) -> a__isList(X) 893.15/291.59 mark(U23(X)) -> a__U23(mark(X)) 893.15/291.59 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 893.15/291.59 mark(U32(X)) -> a__U32(mark(X)) 893.15/291.59 mark(isQid(X)) -> a__isQid(X) 893.15/291.59 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 893.15/291.59 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 893.15/291.59 mark(U43(X)) -> a__U43(mark(X)) 893.15/291.59 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 893.15/291.59 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 893.15/291.59 mark(U53(X)) -> a__U53(mark(X)) 893.15/291.59 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 893.15/291.59 mark(U62(X)) -> a__U62(mark(X)) 893.15/291.59 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 893.15/291.59 mark(U72(X)) -> a__U72(mark(X)) 893.15/291.59 mark(isNePal(X)) -> a__isNePal(X) 893.15/291.59 mark(and(X1, X2)) -> a__and(mark(X1), X2) 893.15/291.59 mark(isPalListKind(X)) -> a__isPalListKind(X) 893.15/291.59 mark(isPal(X)) -> a__isPal(X) 893.15/291.59 mark(nil) -> nil 893.15/291.59 mark(tt) -> tt 893.15/291.59 mark(a) -> a 893.15/291.59 mark(e) -> e 893.15/291.59 mark(i) -> i 893.15/291.59 mark(o) -> o 893.15/291.59 mark(u) -> u 893.15/291.59 a____(X1, X2) -> __(X1, X2) 893.15/291.59 a__U11(X1, X2) -> U11(X1, X2) 893.15/291.59 a__U12(X) -> U12(X) 893.15/291.59 a__isNeList(X) -> isNeList(X) 893.15/291.59 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 893.15/291.59 a__U22(X1, X2) -> U22(X1, X2) 893.15/291.59 a__isList(X) -> isList(X) 893.15/291.59 a__U23(X) -> U23(X) 893.15/291.59 a__U31(X1, X2) -> U31(X1, X2) 893.15/291.59 a__U32(X) -> U32(X) 893.15/291.59 a__isQid(X) -> isQid(X) 893.15/291.59 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 893.15/291.59 a__U42(X1, X2) -> U42(X1, X2) 893.15/291.59 a__U43(X) -> U43(X) 893.15/291.59 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 893.15/291.59 a__U52(X1, X2) -> U52(X1, X2) 893.15/291.59 a__U53(X) -> U53(X) 893.15/291.59 a__U61(X1, X2) -> U61(X1, X2) 893.15/291.59 a__U62(X) -> U62(X) 893.15/291.59 a__U71(X1, X2) -> U71(X1, X2) 893.15/291.59 a__U72(X) -> U72(X) 893.15/291.59 a__isNePal(X) -> isNePal(X) 893.15/291.59 a__and(X1, X2) -> and(X1, X2) 893.15/291.59 a__isPalListKind(X) -> isPalListKind(X) 893.15/291.59 a__isPal(X) -> isPal(X) 893.15/291.59 893.15/291.59 S is empty. 893.15/291.59 Rewrite Strategy: FULL 893.15/291.59 ---------------------------------------- 893.15/291.59 893.15/291.59 (3) DecreasingLoopProof (LOWER BOUND(ID)) 893.15/291.59 The following loop(s) give(s) rise to the lower bound Omega(n^1): 893.15/291.59 893.15/291.59 The rewrite sequence 893.15/291.59 893.15/291.59 mark(U12(X)) ->^+ a__U12(mark(X)) 893.15/291.59 893.15/291.59 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 893.15/291.59 893.15/291.59 The pumping substitution is [X / U12(X)]. 893.15/291.59 893.15/291.59 The result substitution is [ ]. 893.15/291.59 893.15/291.59 893.15/291.59 893.15/291.59 893.15/291.59 ---------------------------------------- 893.15/291.59 893.15/291.59 (4) 893.15/291.59 Complex Obligation (BEST) 893.15/291.59 893.15/291.59 ---------------------------------------- 893.15/291.59 893.15/291.59 (5) 893.15/291.59 Obligation: 893.15/291.59 Proved the lower bound n^1 for the following obligation: 893.15/291.59 893.15/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 893.15/291.59 893.15/291.59 893.15/291.59 The TRS R consists of the following rules: 893.15/291.59 893.15/291.59 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 893.15/291.59 a____(X, nil) -> mark(X) 893.15/291.59 a____(nil, X) -> mark(X) 893.15/291.59 a__U11(tt, V) -> a__U12(a__isNeList(V)) 893.15/291.59 a__U12(tt) -> tt 893.15/291.59 a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) 893.15/291.59 a__U22(tt, V2) -> a__U23(a__isList(V2)) 893.15/291.59 a__U23(tt) -> tt 893.15/291.59 a__U31(tt, V) -> a__U32(a__isQid(V)) 893.15/291.59 a__U32(tt) -> tt 893.15/291.59 a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) 893.15/291.59 a__U42(tt, V2) -> a__U43(a__isNeList(V2)) 893.15/291.59 a__U43(tt) -> tt 893.15/291.59 a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) 893.15/291.59 a__U52(tt, V2) -> a__U53(a__isList(V2)) 893.15/291.59 a__U53(tt) -> tt 893.15/291.59 a__U61(tt, V) -> a__U62(a__isQid(V)) 893.15/291.59 a__U62(tt) -> tt 893.15/291.59 a__U71(tt, V) -> a__U72(a__isNePal(V)) 893.15/291.59 a__U72(tt) -> tt 893.15/291.59 a__and(tt, X) -> mark(X) 893.15/291.59 a__isList(V) -> a__U11(a__isPalListKind(V), V) 893.15/291.59 a__isList(nil) -> tt 893.15/291.59 a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNeList(V) -> a__U31(a__isPalListKind(V), V) 893.15/291.59 a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNePal(V) -> a__U61(a__isPalListKind(V), V) 893.15/291.59 a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) 893.15/291.59 a__isPal(V) -> a__U71(a__isPalListKind(V), V) 893.15/291.59 a__isPal(nil) -> tt 893.15/291.59 a__isPalListKind(a) -> tt 893.15/291.59 a__isPalListKind(e) -> tt 893.15/291.59 a__isPalListKind(i) -> tt 893.15/291.59 a__isPalListKind(nil) -> tt 893.15/291.59 a__isPalListKind(o) -> tt 893.15/291.59 a__isPalListKind(u) -> tt 893.15/291.59 a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) 893.15/291.59 a__isQid(a) -> tt 893.15/291.59 a__isQid(e) -> tt 893.15/291.59 a__isQid(i) -> tt 893.15/291.59 a__isQid(o) -> tt 893.15/291.59 a__isQid(u) -> tt 893.15/291.59 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 893.15/291.59 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 893.15/291.59 mark(U12(X)) -> a__U12(mark(X)) 893.15/291.59 mark(isNeList(X)) -> a__isNeList(X) 893.15/291.59 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 893.15/291.59 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 893.15/291.59 mark(isList(X)) -> a__isList(X) 893.15/291.59 mark(U23(X)) -> a__U23(mark(X)) 893.15/291.59 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 893.15/291.59 mark(U32(X)) -> a__U32(mark(X)) 893.15/291.59 mark(isQid(X)) -> a__isQid(X) 893.15/291.59 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 893.15/291.59 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 893.15/291.59 mark(U43(X)) -> a__U43(mark(X)) 893.15/291.59 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 893.15/291.59 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 893.15/291.59 mark(U53(X)) -> a__U53(mark(X)) 893.15/291.59 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 893.15/291.59 mark(U62(X)) -> a__U62(mark(X)) 893.15/291.59 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 893.15/291.59 mark(U72(X)) -> a__U72(mark(X)) 893.15/291.59 mark(isNePal(X)) -> a__isNePal(X) 893.15/291.59 mark(and(X1, X2)) -> a__and(mark(X1), X2) 893.15/291.59 mark(isPalListKind(X)) -> a__isPalListKind(X) 893.15/291.59 mark(isPal(X)) -> a__isPal(X) 893.15/291.59 mark(nil) -> nil 893.15/291.59 mark(tt) -> tt 893.15/291.59 mark(a) -> a 893.15/291.59 mark(e) -> e 893.15/291.59 mark(i) -> i 893.15/291.59 mark(o) -> o 893.15/291.59 mark(u) -> u 893.15/291.59 a____(X1, X2) -> __(X1, X2) 893.15/291.59 a__U11(X1, X2) -> U11(X1, X2) 893.15/291.59 a__U12(X) -> U12(X) 893.15/291.59 a__isNeList(X) -> isNeList(X) 893.15/291.59 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 893.15/291.59 a__U22(X1, X2) -> U22(X1, X2) 893.15/291.59 a__isList(X) -> isList(X) 893.15/291.59 a__U23(X) -> U23(X) 893.15/291.59 a__U31(X1, X2) -> U31(X1, X2) 893.15/291.59 a__U32(X) -> U32(X) 893.15/291.59 a__isQid(X) -> isQid(X) 893.15/291.59 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 893.15/291.59 a__U42(X1, X2) -> U42(X1, X2) 893.15/291.59 a__U43(X) -> U43(X) 893.15/291.59 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 893.15/291.59 a__U52(X1, X2) -> U52(X1, X2) 893.15/291.59 a__U53(X) -> U53(X) 893.15/291.59 a__U61(X1, X2) -> U61(X1, X2) 893.15/291.59 a__U62(X) -> U62(X) 893.15/291.59 a__U71(X1, X2) -> U71(X1, X2) 893.15/291.59 a__U72(X) -> U72(X) 893.15/291.59 a__isNePal(X) -> isNePal(X) 893.15/291.59 a__and(X1, X2) -> and(X1, X2) 893.15/291.59 a__isPalListKind(X) -> isPalListKind(X) 893.15/291.59 a__isPal(X) -> isPal(X) 893.15/291.59 893.15/291.59 S is empty. 893.15/291.59 Rewrite Strategy: FULL 893.15/291.59 ---------------------------------------- 893.15/291.59 893.15/291.59 (6) LowerBoundPropagationProof (FINISHED) 893.15/291.59 Propagated lower bound. 893.15/291.59 ---------------------------------------- 893.15/291.59 893.15/291.59 (7) 893.15/291.59 BOUNDS(n^1, INF) 893.15/291.59 893.15/291.59 ---------------------------------------- 893.15/291.59 893.15/291.59 (8) 893.15/291.59 Obligation: 893.15/291.59 Analyzing the following TRS for decreasing loops: 893.15/291.59 893.15/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 893.15/291.59 893.15/291.59 893.15/291.59 The TRS R consists of the following rules: 893.15/291.59 893.15/291.59 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 893.15/291.59 a____(X, nil) -> mark(X) 893.15/291.59 a____(nil, X) -> mark(X) 893.15/291.59 a__U11(tt, V) -> a__U12(a__isNeList(V)) 893.15/291.59 a__U12(tt) -> tt 893.15/291.59 a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) 893.15/291.59 a__U22(tt, V2) -> a__U23(a__isList(V2)) 893.15/291.59 a__U23(tt) -> tt 893.15/291.59 a__U31(tt, V) -> a__U32(a__isQid(V)) 893.15/291.59 a__U32(tt) -> tt 893.15/291.59 a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) 893.15/291.59 a__U42(tt, V2) -> a__U43(a__isNeList(V2)) 893.15/291.59 a__U43(tt) -> tt 893.15/291.59 a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) 893.15/291.59 a__U52(tt, V2) -> a__U53(a__isList(V2)) 893.15/291.59 a__U53(tt) -> tt 893.15/291.59 a__U61(tt, V) -> a__U62(a__isQid(V)) 893.15/291.59 a__U62(tt) -> tt 893.15/291.59 a__U71(tt, V) -> a__U72(a__isNePal(V)) 893.15/291.59 a__U72(tt) -> tt 893.15/291.59 a__and(tt, X) -> mark(X) 893.15/291.59 a__isList(V) -> a__U11(a__isPalListKind(V), V) 893.15/291.59 a__isList(nil) -> tt 893.15/291.59 a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNeList(V) -> a__U31(a__isPalListKind(V), V) 893.15/291.59 a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 893.15/291.59 a__isNePal(V) -> a__U61(a__isPalListKind(V), V) 893.15/291.59 a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) 893.15/291.59 a__isPal(V) -> a__U71(a__isPalListKind(V), V) 893.15/291.59 a__isPal(nil) -> tt 893.15/291.59 a__isPalListKind(a) -> tt 893.15/291.59 a__isPalListKind(e) -> tt 893.15/291.59 a__isPalListKind(i) -> tt 893.15/291.59 a__isPalListKind(nil) -> tt 893.15/291.59 a__isPalListKind(o) -> tt 893.15/291.59 a__isPalListKind(u) -> tt 893.15/291.59 a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) 893.15/291.59 a__isQid(a) -> tt 893.15/291.59 a__isQid(e) -> tt 893.15/291.59 a__isQid(i) -> tt 893.15/291.59 a__isQid(o) -> tt 893.15/291.59 a__isQid(u) -> tt 893.15/291.59 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 893.15/291.59 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 893.15/291.59 mark(U12(X)) -> a__U12(mark(X)) 893.15/291.59 mark(isNeList(X)) -> a__isNeList(X) 893.15/291.59 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 893.15/291.59 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 893.15/291.59 mark(isList(X)) -> a__isList(X) 893.15/291.59 mark(U23(X)) -> a__U23(mark(X)) 893.15/291.59 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 893.15/291.59 mark(U32(X)) -> a__U32(mark(X)) 893.15/291.59 mark(isQid(X)) -> a__isQid(X) 893.15/291.59 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 893.15/291.59 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 893.15/291.59 mark(U43(X)) -> a__U43(mark(X)) 893.15/291.59 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 893.15/291.59 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 893.15/291.59 mark(U53(X)) -> a__U53(mark(X)) 893.15/291.59 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 893.15/291.59 mark(U62(X)) -> a__U62(mark(X)) 893.15/291.59 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 893.15/291.59 mark(U72(X)) -> a__U72(mark(X)) 893.15/291.59 mark(isNePal(X)) -> a__isNePal(X) 893.15/291.59 mark(and(X1, X2)) -> a__and(mark(X1), X2) 893.15/291.59 mark(isPalListKind(X)) -> a__isPalListKind(X) 893.15/291.59 mark(isPal(X)) -> a__isPal(X) 893.15/291.59 mark(nil) -> nil 893.15/291.59 mark(tt) -> tt 893.15/291.59 mark(a) -> a 893.15/291.59 mark(e) -> e 893.15/291.59 mark(i) -> i 893.15/291.59 mark(o) -> o 893.15/291.59 mark(u) -> u 893.15/291.59 a____(X1, X2) -> __(X1, X2) 893.15/291.59 a__U11(X1, X2) -> U11(X1, X2) 893.15/291.59 a__U12(X) -> U12(X) 893.15/291.59 a__isNeList(X) -> isNeList(X) 893.15/291.59 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 893.15/291.59 a__U22(X1, X2) -> U22(X1, X2) 893.15/291.59 a__isList(X) -> isList(X) 893.15/291.59 a__U23(X) -> U23(X) 893.15/291.59 a__U31(X1, X2) -> U31(X1, X2) 893.15/291.59 a__U32(X) -> U32(X) 893.15/291.59 a__isQid(X) -> isQid(X) 893.15/291.59 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 893.15/291.59 a__U42(X1, X2) -> U42(X1, X2) 893.15/291.59 a__U43(X) -> U43(X) 893.15/291.59 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 893.15/291.59 a__U52(X1, X2) -> U52(X1, X2) 893.15/291.59 a__U53(X) -> U53(X) 893.15/291.59 a__U61(X1, X2) -> U61(X1, X2) 893.15/291.59 a__U62(X) -> U62(X) 893.15/291.59 a__U71(X1, X2) -> U71(X1, X2) 893.15/291.59 a__U72(X) -> U72(X) 893.15/291.59 a__isNePal(X) -> isNePal(X) 893.15/291.59 a__and(X1, X2) -> and(X1, X2) 893.15/291.59 a__isPalListKind(X) -> isPalListKind(X) 893.15/291.59 a__isPal(X) -> isPal(X) 893.15/291.59 893.15/291.59 S is empty. 893.15/291.59 Rewrite Strategy: FULL 893.27/291.63 EOF