41.81/12.28 WORST_CASE(Omega(n^1), O(n^1)) 41.81/12.29 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 41.81/12.29 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 41.81/12.29 41.81/12.29 41.81/12.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.81/12.29 41.81/12.29 (0) CpxTRS 41.81/12.29 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 41.81/12.29 (2) CpxTRS 41.81/12.29 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 41.81/12.29 (4) CpxTRS 41.81/12.29 (5) CpxTrsMatchBoundsTAProof [FINISHED, 1016 ms] 41.81/12.29 (6) BOUNDS(1, n^1) 41.81/12.29 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 41.81/12.29 (8) TRS for Loop Detection 41.81/12.29 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 41.81/12.29 (10) BEST 41.81/12.29 (11) proven lower bound 41.81/12.29 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 41.81/12.29 (13) BOUNDS(n^1, INF) 41.81/12.29 (14) TRS for Loop Detection 41.81/12.29 41.81/12.29 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (0) 41.81/12.29 Obligation: 41.81/12.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.81/12.29 41.81/12.29 41.81/12.29 The TRS R consists of the following rules: 41.81/12.29 41.81/12.29 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 41.81/12.29 active(__(X, nil)) -> mark(X) 41.81/12.29 active(__(nil, X)) -> mark(X) 41.81/12.29 active(U11(tt, V)) -> mark(U12(isNeList(V))) 41.81/12.29 active(U12(tt)) -> mark(tt) 41.81/12.29 active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) 41.81/12.29 active(U22(tt, V2)) -> mark(U23(isList(V2))) 41.81/12.29 active(U23(tt)) -> mark(tt) 41.81/12.29 active(U31(tt, V)) -> mark(U32(isQid(V))) 41.81/12.29 active(U32(tt)) -> mark(tt) 41.81/12.29 active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) 41.81/12.29 active(U42(tt, V2)) -> mark(U43(isNeList(V2))) 41.81/12.29 active(U43(tt)) -> mark(tt) 41.81/12.29 active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) 41.81/12.29 active(U52(tt, V2)) -> mark(U53(isList(V2))) 41.81/12.29 active(U53(tt)) -> mark(tt) 41.81/12.29 active(U61(tt, V)) -> mark(U62(isQid(V))) 41.81/12.29 active(U62(tt)) -> mark(tt) 41.81/12.29 active(U71(tt, V)) -> mark(U72(isNePal(V))) 41.81/12.29 active(U72(tt)) -> mark(tt) 41.81/12.29 active(and(tt, X)) -> mark(X) 41.81/12.29 active(isList(V)) -> mark(U11(isPalListKind(V), V)) 41.81/12.29 active(isList(nil)) -> mark(tt) 41.81/12.29 active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) 41.81/12.29 active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) 41.81/12.29 active(isPal(V)) -> mark(U71(isPalListKind(V), V)) 41.81/12.29 active(isPal(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(a)) -> mark(tt) 41.81/12.29 active(isPalListKind(e)) -> mark(tt) 41.81/12.29 active(isPalListKind(i)) -> mark(tt) 41.81/12.29 active(isPalListKind(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(o)) -> mark(tt) 41.81/12.29 active(isPalListKind(u)) -> mark(tt) 41.81/12.29 active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) 41.81/12.29 active(isQid(a)) -> mark(tt) 41.81/12.29 active(isQid(e)) -> mark(tt) 41.81/12.29 active(isQid(i)) -> mark(tt) 41.81/12.29 active(isQid(o)) -> mark(tt) 41.81/12.29 active(isQid(u)) -> mark(tt) 41.81/12.29 active(__(X1, X2)) -> __(active(X1), X2) 41.81/12.29 active(__(X1, X2)) -> __(X1, active(X2)) 41.81/12.29 active(U11(X1, X2)) -> U11(active(X1), X2) 41.81/12.29 active(U12(X)) -> U12(active(X)) 41.81/12.29 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 41.81/12.29 active(U22(X1, X2)) -> U22(active(X1), X2) 41.81/12.29 active(U23(X)) -> U23(active(X)) 41.81/12.29 active(U31(X1, X2)) -> U31(active(X1), X2) 41.81/12.29 active(U32(X)) -> U32(active(X)) 41.81/12.29 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 41.81/12.29 active(U42(X1, X2)) -> U42(active(X1), X2) 41.81/12.29 active(U43(X)) -> U43(active(X)) 41.81/12.29 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 41.81/12.29 active(U52(X1, X2)) -> U52(active(X1), X2) 41.81/12.29 active(U53(X)) -> U53(active(X)) 41.81/12.29 active(U61(X1, X2)) -> U61(active(X1), X2) 41.81/12.29 active(U62(X)) -> U62(active(X)) 41.81/12.29 active(U71(X1, X2)) -> U71(active(X1), X2) 41.81/12.29 active(U72(X)) -> U72(active(X)) 41.81/12.29 active(and(X1, X2)) -> and(active(X1), X2) 41.81/12.29 __(mark(X1), X2) -> mark(__(X1, X2)) 41.81/12.29 __(X1, mark(X2)) -> mark(__(X1, X2)) 41.81/12.29 U11(mark(X1), X2) -> mark(U11(X1, X2)) 41.81/12.29 U12(mark(X)) -> mark(U12(X)) 41.81/12.29 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 41.81/12.29 U22(mark(X1), X2) -> mark(U22(X1, X2)) 41.81/12.29 U23(mark(X)) -> mark(U23(X)) 41.81/12.29 U31(mark(X1), X2) -> mark(U31(X1, X2)) 41.81/12.29 U32(mark(X)) -> mark(U32(X)) 41.81/12.29 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 41.81/12.29 U42(mark(X1), X2) -> mark(U42(X1, X2)) 41.81/12.29 U43(mark(X)) -> mark(U43(X)) 41.81/12.29 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 41.81/12.29 U52(mark(X1), X2) -> mark(U52(X1, X2)) 41.81/12.29 U53(mark(X)) -> mark(U53(X)) 41.81/12.29 U61(mark(X1), X2) -> mark(U61(X1, X2)) 41.81/12.29 U62(mark(X)) -> mark(U62(X)) 41.81/12.29 U71(mark(X1), X2) -> mark(U71(X1, X2)) 41.81/12.29 U72(mark(X)) -> mark(U72(X)) 41.81/12.29 and(mark(X1), X2) -> mark(and(X1, X2)) 41.81/12.29 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 41.81/12.29 proper(nil) -> ok(nil) 41.81/12.29 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 41.81/12.29 proper(tt) -> ok(tt) 41.81/12.29 proper(U12(X)) -> U12(proper(X)) 41.81/12.29 proper(isNeList(X)) -> isNeList(proper(X)) 41.81/12.29 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 41.81/12.29 proper(isList(X)) -> isList(proper(X)) 41.81/12.29 proper(U23(X)) -> U23(proper(X)) 41.81/12.29 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 41.81/12.29 proper(U32(X)) -> U32(proper(X)) 41.81/12.29 proper(isQid(X)) -> isQid(proper(X)) 41.81/12.29 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) 41.81/12.29 proper(U43(X)) -> U43(proper(X)) 41.81/12.29 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) 41.81/12.29 proper(U53(X)) -> U53(proper(X)) 41.81/12.29 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 41.81/12.29 proper(U62(X)) -> U62(proper(X)) 41.81/12.29 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 41.81/12.29 proper(U72(X)) -> U72(proper(X)) 41.81/12.29 proper(isNePal(X)) -> isNePal(proper(X)) 41.81/12.29 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 41.81/12.29 proper(isPalListKind(X)) -> isPalListKind(proper(X)) 41.81/12.29 proper(isPal(X)) -> isPal(proper(X)) 41.81/12.29 proper(a) -> ok(a) 41.81/12.29 proper(e) -> ok(e) 41.81/12.29 proper(i) -> ok(i) 41.81/12.29 proper(o) -> ok(o) 41.81/12.29 proper(u) -> ok(u) 41.81/12.29 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 41.81/12.29 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 41.81/12.29 U12(ok(X)) -> ok(U12(X)) 41.81/12.29 isNeList(ok(X)) -> ok(isNeList(X)) 41.81/12.29 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 41.81/12.29 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 41.81/12.29 isList(ok(X)) -> ok(isList(X)) 41.81/12.29 U23(ok(X)) -> ok(U23(X)) 41.81/12.29 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 41.81/12.29 U32(ok(X)) -> ok(U32(X)) 41.81/12.29 isQid(ok(X)) -> ok(isQid(X)) 41.81/12.29 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 41.81/12.29 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 41.81/12.29 U43(ok(X)) -> ok(U43(X)) 41.81/12.29 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 41.81/12.29 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 41.81/12.29 U53(ok(X)) -> ok(U53(X)) 41.81/12.29 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 41.81/12.29 U62(ok(X)) -> ok(U62(X)) 41.81/12.29 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 41.81/12.29 U72(ok(X)) -> ok(U72(X)) 41.81/12.29 isNePal(ok(X)) -> ok(isNePal(X)) 41.81/12.29 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 41.81/12.29 isPalListKind(ok(X)) -> ok(isPalListKind(X)) 41.81/12.29 isPal(ok(X)) -> ok(isPal(X)) 41.81/12.29 top(mark(X)) -> top(proper(X)) 41.81/12.29 top(ok(X)) -> top(active(X)) 41.81/12.29 41.81/12.29 S is empty. 41.81/12.29 Rewrite Strategy: FULL 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 41.81/12.29 The following defined symbols can occur below the 0th argument of top: proper, active 41.81/12.29 The following defined symbols can occur below the 0th argument of proper: proper, active 41.81/12.29 The following defined symbols can occur below the 0th argument of active: proper, active 41.81/12.29 41.81/12.29 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 41.81/12.29 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 41.81/12.29 active(__(X, nil)) -> mark(X) 41.81/12.29 active(__(nil, X)) -> mark(X) 41.81/12.29 active(U11(tt, V)) -> mark(U12(isNeList(V))) 41.81/12.29 active(U12(tt)) -> mark(tt) 41.81/12.29 active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) 41.81/12.29 active(U22(tt, V2)) -> mark(U23(isList(V2))) 41.81/12.29 active(U23(tt)) -> mark(tt) 41.81/12.29 active(U31(tt, V)) -> mark(U32(isQid(V))) 41.81/12.29 active(U32(tt)) -> mark(tt) 41.81/12.29 active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) 41.81/12.29 active(U42(tt, V2)) -> mark(U43(isNeList(V2))) 41.81/12.29 active(U43(tt)) -> mark(tt) 41.81/12.29 active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) 41.81/12.29 active(U52(tt, V2)) -> mark(U53(isList(V2))) 41.81/12.29 active(U53(tt)) -> mark(tt) 41.81/12.29 active(U61(tt, V)) -> mark(U62(isQid(V))) 41.81/12.29 active(U62(tt)) -> mark(tt) 41.81/12.29 active(U71(tt, V)) -> mark(U72(isNePal(V))) 41.81/12.29 active(U72(tt)) -> mark(tt) 41.81/12.29 active(and(tt, X)) -> mark(X) 41.81/12.29 active(isList(V)) -> mark(U11(isPalListKind(V), V)) 41.81/12.29 active(isList(nil)) -> mark(tt) 41.81/12.29 active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) 41.81/12.29 active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) 41.81/12.29 active(isPal(V)) -> mark(U71(isPalListKind(V), V)) 41.81/12.29 active(isPal(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(a)) -> mark(tt) 41.81/12.29 active(isPalListKind(e)) -> mark(tt) 41.81/12.29 active(isPalListKind(i)) -> mark(tt) 41.81/12.29 active(isPalListKind(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(o)) -> mark(tt) 41.81/12.29 active(isPalListKind(u)) -> mark(tt) 41.81/12.29 active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) 41.81/12.29 active(isQid(a)) -> mark(tt) 41.81/12.29 active(isQid(e)) -> mark(tt) 41.81/12.29 active(isQid(i)) -> mark(tt) 41.81/12.29 active(isQid(o)) -> mark(tt) 41.81/12.29 active(isQid(u)) -> mark(tt) 41.81/12.29 active(__(X1, X2)) -> __(active(X1), X2) 41.81/12.29 active(__(X1, X2)) -> __(X1, active(X2)) 41.81/12.29 active(U11(X1, X2)) -> U11(active(X1), X2) 41.81/12.29 active(U12(X)) -> U12(active(X)) 41.81/12.29 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 41.81/12.29 active(U22(X1, X2)) -> U22(active(X1), X2) 41.81/12.29 active(U23(X)) -> U23(active(X)) 41.81/12.29 active(U31(X1, X2)) -> U31(active(X1), X2) 41.81/12.29 active(U32(X)) -> U32(active(X)) 41.81/12.29 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 41.81/12.29 active(U42(X1, X2)) -> U42(active(X1), X2) 41.81/12.29 active(U43(X)) -> U43(active(X)) 41.81/12.29 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 41.81/12.29 active(U52(X1, X2)) -> U52(active(X1), X2) 41.81/12.29 active(U53(X)) -> U53(active(X)) 41.81/12.29 active(U61(X1, X2)) -> U61(active(X1), X2) 41.81/12.29 active(U62(X)) -> U62(active(X)) 41.81/12.29 active(U71(X1, X2)) -> U71(active(X1), X2) 41.81/12.29 active(U72(X)) -> U72(active(X)) 41.81/12.29 active(and(X1, X2)) -> and(active(X1), X2) 41.81/12.29 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 41.81/12.29 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 41.81/12.29 proper(U12(X)) -> U12(proper(X)) 41.81/12.29 proper(isNeList(X)) -> isNeList(proper(X)) 41.81/12.29 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 41.81/12.29 proper(isList(X)) -> isList(proper(X)) 41.81/12.29 proper(U23(X)) -> U23(proper(X)) 41.81/12.29 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 41.81/12.29 proper(U32(X)) -> U32(proper(X)) 41.81/12.29 proper(isQid(X)) -> isQid(proper(X)) 41.81/12.29 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) 41.81/12.29 proper(U43(X)) -> U43(proper(X)) 41.81/12.29 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) 41.81/12.29 proper(U53(X)) -> U53(proper(X)) 41.81/12.29 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 41.81/12.29 proper(U62(X)) -> U62(proper(X)) 41.81/12.29 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 41.81/12.29 proper(U72(X)) -> U72(proper(X)) 41.81/12.29 proper(isNePal(X)) -> isNePal(proper(X)) 41.81/12.29 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 41.81/12.29 proper(isPalListKind(X)) -> isPalListKind(proper(X)) 41.81/12.29 proper(isPal(X)) -> isPal(proper(X)) 41.81/12.29 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (2) 41.81/12.29 Obligation: 41.81/12.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 41.81/12.29 41.81/12.29 41.81/12.29 The TRS R consists of the following rules: 41.81/12.29 41.81/12.29 __(mark(X1), X2) -> mark(__(X1, X2)) 41.81/12.29 __(X1, mark(X2)) -> mark(__(X1, X2)) 41.81/12.29 U11(mark(X1), X2) -> mark(U11(X1, X2)) 41.81/12.29 U12(mark(X)) -> mark(U12(X)) 41.81/12.29 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 41.81/12.29 U22(mark(X1), X2) -> mark(U22(X1, X2)) 41.81/12.29 U23(mark(X)) -> mark(U23(X)) 41.81/12.29 U31(mark(X1), X2) -> mark(U31(X1, X2)) 41.81/12.29 U32(mark(X)) -> mark(U32(X)) 41.81/12.29 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 41.81/12.29 U42(mark(X1), X2) -> mark(U42(X1, X2)) 41.81/12.29 U43(mark(X)) -> mark(U43(X)) 41.81/12.29 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 41.81/12.29 U52(mark(X1), X2) -> mark(U52(X1, X2)) 41.81/12.29 U53(mark(X)) -> mark(U53(X)) 41.81/12.29 U61(mark(X1), X2) -> mark(U61(X1, X2)) 41.81/12.29 U62(mark(X)) -> mark(U62(X)) 41.81/12.29 U71(mark(X1), X2) -> mark(U71(X1, X2)) 41.81/12.29 U72(mark(X)) -> mark(U72(X)) 41.81/12.29 and(mark(X1), X2) -> mark(and(X1, X2)) 41.81/12.29 proper(nil) -> ok(nil) 41.81/12.29 proper(tt) -> ok(tt) 41.81/12.29 proper(a) -> ok(a) 41.81/12.29 proper(e) -> ok(e) 41.81/12.29 proper(i) -> ok(i) 41.81/12.29 proper(o) -> ok(o) 41.81/12.29 proper(u) -> ok(u) 41.81/12.29 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 41.81/12.29 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 41.81/12.29 U12(ok(X)) -> ok(U12(X)) 41.81/12.29 isNeList(ok(X)) -> ok(isNeList(X)) 41.81/12.29 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 41.81/12.29 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 41.81/12.29 isList(ok(X)) -> ok(isList(X)) 41.81/12.29 U23(ok(X)) -> ok(U23(X)) 41.81/12.29 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 41.81/12.29 U32(ok(X)) -> ok(U32(X)) 41.81/12.29 isQid(ok(X)) -> ok(isQid(X)) 41.81/12.29 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 41.81/12.29 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 41.81/12.29 U43(ok(X)) -> ok(U43(X)) 41.81/12.29 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 41.81/12.29 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 41.81/12.29 U53(ok(X)) -> ok(U53(X)) 41.81/12.29 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 41.81/12.29 U62(ok(X)) -> ok(U62(X)) 41.81/12.29 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 41.81/12.29 U72(ok(X)) -> ok(U72(X)) 41.81/12.29 isNePal(ok(X)) -> ok(isNePal(X)) 41.81/12.29 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 41.81/12.29 isPalListKind(ok(X)) -> ok(isPalListKind(X)) 41.81/12.29 isPal(ok(X)) -> ok(isPal(X)) 41.81/12.29 top(mark(X)) -> top(proper(X)) 41.81/12.29 top(ok(X)) -> top(active(X)) 41.81/12.29 41.81/12.29 S is empty. 41.81/12.29 Rewrite Strategy: FULL 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 41.81/12.29 transformed relative TRS to TRS 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (4) 41.81/12.29 Obligation: 41.81/12.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 41.81/12.29 41.81/12.29 41.81/12.29 The TRS R consists of the following rules: 41.81/12.29 41.81/12.29 __(mark(X1), X2) -> mark(__(X1, X2)) 41.81/12.29 __(X1, mark(X2)) -> mark(__(X1, X2)) 41.81/12.29 U11(mark(X1), X2) -> mark(U11(X1, X2)) 41.81/12.29 U12(mark(X)) -> mark(U12(X)) 41.81/12.29 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 41.81/12.29 U22(mark(X1), X2) -> mark(U22(X1, X2)) 41.81/12.29 U23(mark(X)) -> mark(U23(X)) 41.81/12.29 U31(mark(X1), X2) -> mark(U31(X1, X2)) 41.81/12.29 U32(mark(X)) -> mark(U32(X)) 41.81/12.29 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 41.81/12.29 U42(mark(X1), X2) -> mark(U42(X1, X2)) 41.81/12.29 U43(mark(X)) -> mark(U43(X)) 41.81/12.29 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 41.81/12.29 U52(mark(X1), X2) -> mark(U52(X1, X2)) 41.81/12.29 U53(mark(X)) -> mark(U53(X)) 41.81/12.29 U61(mark(X1), X2) -> mark(U61(X1, X2)) 41.81/12.29 U62(mark(X)) -> mark(U62(X)) 41.81/12.29 U71(mark(X1), X2) -> mark(U71(X1, X2)) 41.81/12.29 U72(mark(X)) -> mark(U72(X)) 41.81/12.29 and(mark(X1), X2) -> mark(and(X1, X2)) 41.81/12.29 proper(nil) -> ok(nil) 41.81/12.29 proper(tt) -> ok(tt) 41.81/12.29 proper(a) -> ok(a) 41.81/12.29 proper(e) -> ok(e) 41.81/12.29 proper(i) -> ok(i) 41.81/12.29 proper(o) -> ok(o) 41.81/12.29 proper(u) -> ok(u) 41.81/12.29 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 41.81/12.29 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 41.81/12.29 U12(ok(X)) -> ok(U12(X)) 41.81/12.29 isNeList(ok(X)) -> ok(isNeList(X)) 41.81/12.29 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 41.81/12.29 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 41.81/12.29 isList(ok(X)) -> ok(isList(X)) 41.81/12.29 U23(ok(X)) -> ok(U23(X)) 41.81/12.29 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 41.81/12.29 U32(ok(X)) -> ok(U32(X)) 41.81/12.29 isQid(ok(X)) -> ok(isQid(X)) 41.81/12.29 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 41.81/12.29 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 41.81/12.29 U43(ok(X)) -> ok(U43(X)) 41.81/12.29 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 41.81/12.29 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 41.81/12.29 U53(ok(X)) -> ok(U53(X)) 41.81/12.29 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 41.81/12.29 U62(ok(X)) -> ok(U62(X)) 41.81/12.29 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 41.81/12.29 U72(ok(X)) -> ok(U72(X)) 41.81/12.29 isNePal(ok(X)) -> ok(isNePal(X)) 41.81/12.29 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 41.81/12.29 isPalListKind(ok(X)) -> ok(isPalListKind(X)) 41.81/12.29 isPal(ok(X)) -> ok(isPal(X)) 41.81/12.29 top(mark(X)) -> top(proper(X)) 41.81/12.29 top(ok(X)) -> top(active(X)) 41.81/12.29 41.81/12.29 S is empty. 41.81/12.29 Rewrite Strategy: FULL 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (5) CpxTrsMatchBoundsTAProof (FINISHED) 41.81/12.29 A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 41.81/12.29 41.81/12.29 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 41.81/12.29 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] 41.81/12.29 transitions: 41.81/12.29 mark0(0) -> 0 41.81/12.29 nil0() -> 0 41.81/12.29 ok0(0) -> 0 41.81/12.29 tt0() -> 0 41.81/12.29 a0() -> 0 41.81/12.29 e0() -> 0 41.81/12.29 i0() -> 0 41.81/12.29 o0() -> 0 41.81/12.29 u0() -> 0 41.81/12.29 active0(0) -> 0 41.81/12.29 __0(0, 0) -> 1 41.81/12.29 U110(0, 0) -> 2 41.81/12.29 U120(0) -> 3 41.81/12.29 U210(0, 0, 0) -> 4 41.81/12.29 U220(0, 0) -> 5 41.81/12.29 U230(0) -> 6 41.81/12.29 U310(0, 0) -> 7 41.81/12.29 U320(0) -> 8 41.81/12.29 U410(0, 0, 0) -> 9 41.81/12.29 U420(0, 0) -> 10 41.81/12.29 U430(0) -> 11 41.81/12.29 U510(0, 0, 0) -> 12 41.81/12.29 U520(0, 0) -> 13 41.81/12.29 U530(0) -> 14 41.81/12.29 U610(0, 0) -> 15 41.81/12.29 U620(0) -> 16 41.81/12.29 U710(0, 0) -> 17 41.81/12.29 U720(0) -> 18 41.81/12.29 and0(0, 0) -> 19 41.81/12.29 proper0(0) -> 20 41.81/12.29 isNeList0(0) -> 21 41.81/12.29 isList0(0) -> 22 41.81/12.29 isQid0(0) -> 23 41.81/12.29 isNePal0(0) -> 24 41.81/12.29 isPalListKind0(0) -> 25 41.81/12.29 isPal0(0) -> 26 41.81/12.29 top0(0) -> 27 41.81/12.29 __1(0, 0) -> 28 41.81/12.29 mark1(28) -> 1 41.81/12.29 U111(0, 0) -> 29 41.81/12.29 mark1(29) -> 2 41.81/12.29 U121(0) -> 30 41.81/12.29 mark1(30) -> 3 41.81/12.29 U211(0, 0, 0) -> 31 41.81/12.29 mark1(31) -> 4 41.81/12.29 U221(0, 0) -> 32 41.81/12.29 mark1(32) -> 5 41.81/12.29 U231(0) -> 33 41.81/12.29 mark1(33) -> 6 41.81/12.29 U311(0, 0) -> 34 41.81/12.29 mark1(34) -> 7 41.81/12.29 U321(0) -> 35 41.81/12.29 mark1(35) -> 8 41.81/12.29 U411(0, 0, 0) -> 36 41.81/12.29 mark1(36) -> 9 41.81/12.29 U421(0, 0) -> 37 41.81/12.29 mark1(37) -> 10 41.81/12.29 U431(0) -> 38 41.81/12.29 mark1(38) -> 11 41.81/12.29 U511(0, 0, 0) -> 39 41.81/12.29 mark1(39) -> 12 41.81/12.29 U521(0, 0) -> 40 41.81/12.29 mark1(40) -> 13 41.81/12.29 U531(0) -> 41 41.81/12.29 mark1(41) -> 14 41.81/12.29 U611(0, 0) -> 42 41.81/12.29 mark1(42) -> 15 41.81/12.29 U621(0) -> 43 41.81/12.29 mark1(43) -> 16 41.81/12.29 U711(0, 0) -> 44 41.81/12.29 mark1(44) -> 17 41.81/12.29 U721(0) -> 45 41.81/12.29 mark1(45) -> 18 41.81/12.29 and1(0, 0) -> 46 41.81/12.29 mark1(46) -> 19 41.81/12.29 nil1() -> 47 41.81/12.29 ok1(47) -> 20 41.81/12.29 tt1() -> 48 41.81/12.29 ok1(48) -> 20 41.81/12.29 a1() -> 49 41.81/12.29 ok1(49) -> 20 41.81/12.29 e1() -> 50 41.81/12.29 ok1(50) -> 20 41.81/12.29 i1() -> 51 41.81/12.29 ok1(51) -> 20 41.81/12.29 o1() -> 52 41.81/12.29 ok1(52) -> 20 41.81/12.29 u1() -> 53 41.81/12.29 ok1(53) -> 20 41.81/12.29 __1(0, 0) -> 54 41.81/12.29 ok1(54) -> 1 41.81/12.29 U111(0, 0) -> 55 41.81/12.29 ok1(55) -> 2 41.81/12.29 U121(0) -> 56 41.81/12.29 ok1(56) -> 3 41.81/12.29 isNeList1(0) -> 57 41.81/12.29 ok1(57) -> 21 41.81/12.29 U211(0, 0, 0) -> 58 41.81/12.29 ok1(58) -> 4 41.81/12.29 U221(0, 0) -> 59 41.81/12.29 ok1(59) -> 5 41.81/12.29 isList1(0) -> 60 41.81/12.29 ok1(60) -> 22 41.81/12.29 U231(0) -> 61 41.81/12.29 ok1(61) -> 6 41.81/12.29 U311(0, 0) -> 62 41.81/12.29 ok1(62) -> 7 41.81/12.29 U321(0) -> 63 41.81/12.29 ok1(63) -> 8 41.81/12.29 isQid1(0) -> 64 41.81/12.29 ok1(64) -> 23 41.81/12.29 U411(0, 0, 0) -> 65 41.81/12.29 ok1(65) -> 9 41.81/12.29 U421(0, 0) -> 66 41.81/12.29 ok1(66) -> 10 41.81/12.29 U431(0) -> 67 41.81/12.29 ok1(67) -> 11 41.81/12.29 U511(0, 0, 0) -> 68 41.81/12.29 ok1(68) -> 12 41.81/12.29 U521(0, 0) -> 69 41.81/12.29 ok1(69) -> 13 41.81/12.29 U531(0) -> 70 41.81/12.29 ok1(70) -> 14 41.81/12.29 U611(0, 0) -> 71 41.81/12.29 ok1(71) -> 15 41.81/12.29 U621(0) -> 72 41.81/12.29 ok1(72) -> 16 41.81/12.29 U711(0, 0) -> 73 41.81/12.29 ok1(73) -> 17 41.81/12.29 U721(0) -> 74 41.81/12.29 ok1(74) -> 18 41.81/12.29 isNePal1(0) -> 75 41.81/12.29 ok1(75) -> 24 41.81/12.29 and1(0, 0) -> 76 41.81/12.29 ok1(76) -> 19 41.81/12.29 isPalListKind1(0) -> 77 41.81/12.29 ok1(77) -> 25 41.81/12.29 isPal1(0) -> 78 41.81/12.29 ok1(78) -> 26 41.81/12.29 proper1(0) -> 79 41.81/12.29 top1(79) -> 27 41.81/12.29 active1(0) -> 80 41.81/12.29 top1(80) -> 27 41.81/12.29 mark1(28) -> 28 41.81/12.29 mark1(28) -> 54 41.81/12.29 mark1(29) -> 29 41.81/12.29 mark1(29) -> 55 41.81/12.29 mark1(30) -> 30 41.81/12.29 mark1(30) -> 56 41.81/12.29 mark1(31) -> 31 41.81/12.29 mark1(31) -> 58 41.81/12.29 mark1(32) -> 32 41.81/12.29 mark1(32) -> 59 41.81/12.29 mark1(33) -> 33 41.81/12.29 mark1(33) -> 61 41.81/12.29 mark1(34) -> 34 41.81/12.29 mark1(34) -> 62 41.81/12.29 mark1(35) -> 35 41.81/12.29 mark1(35) -> 63 41.81/12.29 mark1(36) -> 36 41.81/12.29 mark1(36) -> 65 41.81/12.29 mark1(37) -> 37 41.81/12.29 mark1(37) -> 66 41.81/12.29 mark1(38) -> 38 41.81/12.29 mark1(38) -> 67 41.81/12.29 mark1(39) -> 39 41.81/12.29 mark1(39) -> 68 41.81/12.29 mark1(40) -> 40 41.81/12.29 mark1(40) -> 69 41.81/12.29 mark1(41) -> 41 41.81/12.29 mark1(41) -> 70 41.81/12.29 mark1(42) -> 42 41.81/12.29 mark1(42) -> 71 41.81/12.29 mark1(43) -> 43 41.81/12.29 mark1(43) -> 72 41.81/12.29 mark1(44) -> 44 41.81/12.29 mark1(44) -> 73 41.81/12.29 mark1(45) -> 45 41.81/12.29 mark1(45) -> 74 41.81/12.29 mark1(46) -> 46 41.81/12.29 mark1(46) -> 76 41.81/12.29 ok1(47) -> 79 41.81/12.29 ok1(48) -> 79 41.81/12.29 ok1(49) -> 79 41.81/12.29 ok1(50) -> 79 41.81/12.29 ok1(51) -> 79 41.81/12.29 ok1(52) -> 79 41.81/12.29 ok1(53) -> 79 41.81/12.29 ok1(54) -> 28 41.81/12.29 ok1(54) -> 54 41.81/12.29 ok1(55) -> 29 41.81/12.29 ok1(55) -> 55 41.81/12.29 ok1(56) -> 30 41.81/12.29 ok1(56) -> 56 41.81/12.29 ok1(57) -> 57 41.81/12.29 ok1(58) -> 31 41.81/12.29 ok1(58) -> 58 41.81/12.29 ok1(59) -> 32 41.81/12.29 ok1(59) -> 59 41.81/12.29 ok1(60) -> 60 41.81/12.29 ok1(61) -> 33 41.81/12.29 ok1(61) -> 61 41.81/12.29 ok1(62) -> 34 41.81/12.29 ok1(62) -> 62 41.81/12.29 ok1(63) -> 35 41.81/12.29 ok1(63) -> 63 41.81/12.29 ok1(64) -> 64 41.81/12.29 ok1(65) -> 36 41.81/12.29 ok1(65) -> 65 41.81/12.29 ok1(66) -> 37 41.81/12.29 ok1(66) -> 66 41.81/12.29 ok1(67) -> 38 41.81/12.29 ok1(67) -> 67 41.81/12.29 ok1(68) -> 39 41.81/12.29 ok1(68) -> 68 41.81/12.29 ok1(69) -> 40 41.81/12.29 ok1(69) -> 69 41.81/12.29 ok1(70) -> 41 41.81/12.29 ok1(70) -> 70 41.81/12.29 ok1(71) -> 42 41.81/12.29 ok1(71) -> 71 41.81/12.29 ok1(72) -> 43 41.81/12.29 ok1(72) -> 72 41.81/12.29 ok1(73) -> 44 41.81/12.29 ok1(73) -> 73 41.81/12.29 ok1(74) -> 45 41.81/12.29 ok1(74) -> 74 41.81/12.29 ok1(75) -> 75 41.81/12.29 ok1(76) -> 46 41.81/12.29 ok1(76) -> 76 41.81/12.29 ok1(77) -> 77 41.81/12.29 ok1(78) -> 78 41.81/12.29 active2(47) -> 81 41.81/12.29 top2(81) -> 27 41.81/12.29 active2(48) -> 81 41.81/12.29 active2(49) -> 81 41.81/12.29 active2(50) -> 81 41.81/12.29 active2(51) -> 81 41.81/12.29 active2(52) -> 81 41.81/12.29 active2(53) -> 81 41.81/12.29 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (6) 41.81/12.29 BOUNDS(1, n^1) 41.81/12.29 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 41.81/12.29 Transformed a relative TRS into a decreasing-loop problem. 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (8) 41.81/12.29 Obligation: 41.81/12.29 Analyzing the following TRS for decreasing loops: 41.81/12.29 41.81/12.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.81/12.29 41.81/12.29 41.81/12.29 The TRS R consists of the following rules: 41.81/12.29 41.81/12.29 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 41.81/12.29 active(__(X, nil)) -> mark(X) 41.81/12.29 active(__(nil, X)) -> mark(X) 41.81/12.29 active(U11(tt, V)) -> mark(U12(isNeList(V))) 41.81/12.29 active(U12(tt)) -> mark(tt) 41.81/12.29 active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) 41.81/12.29 active(U22(tt, V2)) -> mark(U23(isList(V2))) 41.81/12.29 active(U23(tt)) -> mark(tt) 41.81/12.29 active(U31(tt, V)) -> mark(U32(isQid(V))) 41.81/12.29 active(U32(tt)) -> mark(tt) 41.81/12.29 active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) 41.81/12.29 active(U42(tt, V2)) -> mark(U43(isNeList(V2))) 41.81/12.29 active(U43(tt)) -> mark(tt) 41.81/12.29 active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) 41.81/12.29 active(U52(tt, V2)) -> mark(U53(isList(V2))) 41.81/12.29 active(U53(tt)) -> mark(tt) 41.81/12.29 active(U61(tt, V)) -> mark(U62(isQid(V))) 41.81/12.29 active(U62(tt)) -> mark(tt) 41.81/12.29 active(U71(tt, V)) -> mark(U72(isNePal(V))) 41.81/12.29 active(U72(tt)) -> mark(tt) 41.81/12.29 active(and(tt, X)) -> mark(X) 41.81/12.29 active(isList(V)) -> mark(U11(isPalListKind(V), V)) 41.81/12.29 active(isList(nil)) -> mark(tt) 41.81/12.29 active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) 41.81/12.29 active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) 41.81/12.29 active(isPal(V)) -> mark(U71(isPalListKind(V), V)) 41.81/12.29 active(isPal(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(a)) -> mark(tt) 41.81/12.29 active(isPalListKind(e)) -> mark(tt) 41.81/12.29 active(isPalListKind(i)) -> mark(tt) 41.81/12.29 active(isPalListKind(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(o)) -> mark(tt) 41.81/12.29 active(isPalListKind(u)) -> mark(tt) 41.81/12.29 active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) 41.81/12.29 active(isQid(a)) -> mark(tt) 41.81/12.29 active(isQid(e)) -> mark(tt) 41.81/12.29 active(isQid(i)) -> mark(tt) 41.81/12.29 active(isQid(o)) -> mark(tt) 41.81/12.29 active(isQid(u)) -> mark(tt) 41.81/12.29 active(__(X1, X2)) -> __(active(X1), X2) 41.81/12.29 active(__(X1, X2)) -> __(X1, active(X2)) 41.81/12.29 active(U11(X1, X2)) -> U11(active(X1), X2) 41.81/12.29 active(U12(X)) -> U12(active(X)) 41.81/12.29 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 41.81/12.29 active(U22(X1, X2)) -> U22(active(X1), X2) 41.81/12.29 active(U23(X)) -> U23(active(X)) 41.81/12.29 active(U31(X1, X2)) -> U31(active(X1), X2) 41.81/12.29 active(U32(X)) -> U32(active(X)) 41.81/12.29 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 41.81/12.29 active(U42(X1, X2)) -> U42(active(X1), X2) 41.81/12.29 active(U43(X)) -> U43(active(X)) 41.81/12.29 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 41.81/12.29 active(U52(X1, X2)) -> U52(active(X1), X2) 41.81/12.29 active(U53(X)) -> U53(active(X)) 41.81/12.29 active(U61(X1, X2)) -> U61(active(X1), X2) 41.81/12.29 active(U62(X)) -> U62(active(X)) 41.81/12.29 active(U71(X1, X2)) -> U71(active(X1), X2) 41.81/12.29 active(U72(X)) -> U72(active(X)) 41.81/12.29 active(and(X1, X2)) -> and(active(X1), X2) 41.81/12.29 __(mark(X1), X2) -> mark(__(X1, X2)) 41.81/12.29 __(X1, mark(X2)) -> mark(__(X1, X2)) 41.81/12.29 U11(mark(X1), X2) -> mark(U11(X1, X2)) 41.81/12.29 U12(mark(X)) -> mark(U12(X)) 41.81/12.29 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 41.81/12.29 U22(mark(X1), X2) -> mark(U22(X1, X2)) 41.81/12.29 U23(mark(X)) -> mark(U23(X)) 41.81/12.29 U31(mark(X1), X2) -> mark(U31(X1, X2)) 41.81/12.29 U32(mark(X)) -> mark(U32(X)) 41.81/12.29 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 41.81/12.29 U42(mark(X1), X2) -> mark(U42(X1, X2)) 41.81/12.29 U43(mark(X)) -> mark(U43(X)) 41.81/12.29 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 41.81/12.29 U52(mark(X1), X2) -> mark(U52(X1, X2)) 41.81/12.29 U53(mark(X)) -> mark(U53(X)) 41.81/12.29 U61(mark(X1), X2) -> mark(U61(X1, X2)) 41.81/12.29 U62(mark(X)) -> mark(U62(X)) 41.81/12.29 U71(mark(X1), X2) -> mark(U71(X1, X2)) 41.81/12.29 U72(mark(X)) -> mark(U72(X)) 41.81/12.29 and(mark(X1), X2) -> mark(and(X1, X2)) 41.81/12.29 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 41.81/12.29 proper(nil) -> ok(nil) 41.81/12.29 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 41.81/12.29 proper(tt) -> ok(tt) 41.81/12.29 proper(U12(X)) -> U12(proper(X)) 41.81/12.29 proper(isNeList(X)) -> isNeList(proper(X)) 41.81/12.29 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 41.81/12.29 proper(isList(X)) -> isList(proper(X)) 41.81/12.29 proper(U23(X)) -> U23(proper(X)) 41.81/12.29 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 41.81/12.29 proper(U32(X)) -> U32(proper(X)) 41.81/12.29 proper(isQid(X)) -> isQid(proper(X)) 41.81/12.29 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) 41.81/12.29 proper(U43(X)) -> U43(proper(X)) 41.81/12.29 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) 41.81/12.29 proper(U53(X)) -> U53(proper(X)) 41.81/12.29 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 41.81/12.29 proper(U62(X)) -> U62(proper(X)) 41.81/12.29 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 41.81/12.29 proper(U72(X)) -> U72(proper(X)) 41.81/12.29 proper(isNePal(X)) -> isNePal(proper(X)) 41.81/12.29 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 41.81/12.29 proper(isPalListKind(X)) -> isPalListKind(proper(X)) 41.81/12.29 proper(isPal(X)) -> isPal(proper(X)) 41.81/12.29 proper(a) -> ok(a) 41.81/12.29 proper(e) -> ok(e) 41.81/12.29 proper(i) -> ok(i) 41.81/12.29 proper(o) -> ok(o) 41.81/12.29 proper(u) -> ok(u) 41.81/12.29 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 41.81/12.29 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 41.81/12.29 U12(ok(X)) -> ok(U12(X)) 41.81/12.29 isNeList(ok(X)) -> ok(isNeList(X)) 41.81/12.29 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 41.81/12.29 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 41.81/12.29 isList(ok(X)) -> ok(isList(X)) 41.81/12.29 U23(ok(X)) -> ok(U23(X)) 41.81/12.29 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 41.81/12.29 U32(ok(X)) -> ok(U32(X)) 41.81/12.29 isQid(ok(X)) -> ok(isQid(X)) 41.81/12.29 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 41.81/12.29 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 41.81/12.29 U43(ok(X)) -> ok(U43(X)) 41.81/12.29 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 41.81/12.29 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 41.81/12.29 U53(ok(X)) -> ok(U53(X)) 41.81/12.29 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 41.81/12.29 U62(ok(X)) -> ok(U62(X)) 41.81/12.29 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 41.81/12.29 U72(ok(X)) -> ok(U72(X)) 41.81/12.29 isNePal(ok(X)) -> ok(isNePal(X)) 41.81/12.29 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 41.81/12.29 isPalListKind(ok(X)) -> ok(isPalListKind(X)) 41.81/12.29 isPal(ok(X)) -> ok(isPal(X)) 41.81/12.29 top(mark(X)) -> top(proper(X)) 41.81/12.29 top(ok(X)) -> top(active(X)) 41.81/12.29 41.81/12.29 S is empty. 41.81/12.29 Rewrite Strategy: FULL 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (9) DecreasingLoopProof (LOWER BOUND(ID)) 41.81/12.29 The following loop(s) give(s) rise to the lower bound Omega(n^1): 41.81/12.29 41.81/12.29 The rewrite sequence 41.81/12.29 41.81/12.29 U52(ok(X1), ok(X2)) ->^+ ok(U52(X1, X2)) 41.81/12.29 41.81/12.29 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 41.81/12.29 41.81/12.29 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 41.81/12.29 41.81/12.29 The result substitution is [ ]. 41.81/12.29 41.81/12.29 41.81/12.29 41.81/12.29 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (10) 41.81/12.29 Complex Obligation (BEST) 41.81/12.29 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (11) 41.81/12.29 Obligation: 41.81/12.29 Proved the lower bound n^1 for the following obligation: 41.81/12.29 41.81/12.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.81/12.29 41.81/12.29 41.81/12.29 The TRS R consists of the following rules: 41.81/12.29 41.81/12.29 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 41.81/12.29 active(__(X, nil)) -> mark(X) 41.81/12.29 active(__(nil, X)) -> mark(X) 41.81/12.29 active(U11(tt, V)) -> mark(U12(isNeList(V))) 41.81/12.29 active(U12(tt)) -> mark(tt) 41.81/12.29 active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) 41.81/12.29 active(U22(tt, V2)) -> mark(U23(isList(V2))) 41.81/12.29 active(U23(tt)) -> mark(tt) 41.81/12.29 active(U31(tt, V)) -> mark(U32(isQid(V))) 41.81/12.29 active(U32(tt)) -> mark(tt) 41.81/12.29 active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) 41.81/12.29 active(U42(tt, V2)) -> mark(U43(isNeList(V2))) 41.81/12.29 active(U43(tt)) -> mark(tt) 41.81/12.29 active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) 41.81/12.29 active(U52(tt, V2)) -> mark(U53(isList(V2))) 41.81/12.29 active(U53(tt)) -> mark(tt) 41.81/12.29 active(U61(tt, V)) -> mark(U62(isQid(V))) 41.81/12.29 active(U62(tt)) -> mark(tt) 41.81/12.29 active(U71(tt, V)) -> mark(U72(isNePal(V))) 41.81/12.29 active(U72(tt)) -> mark(tt) 41.81/12.29 active(and(tt, X)) -> mark(X) 41.81/12.29 active(isList(V)) -> mark(U11(isPalListKind(V), V)) 41.81/12.29 active(isList(nil)) -> mark(tt) 41.81/12.29 active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) 41.81/12.29 active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) 41.81/12.29 active(isPal(V)) -> mark(U71(isPalListKind(V), V)) 41.81/12.29 active(isPal(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(a)) -> mark(tt) 41.81/12.29 active(isPalListKind(e)) -> mark(tt) 41.81/12.29 active(isPalListKind(i)) -> mark(tt) 41.81/12.29 active(isPalListKind(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(o)) -> mark(tt) 41.81/12.29 active(isPalListKind(u)) -> mark(tt) 41.81/12.29 active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) 41.81/12.29 active(isQid(a)) -> mark(tt) 41.81/12.29 active(isQid(e)) -> mark(tt) 41.81/12.29 active(isQid(i)) -> mark(tt) 41.81/12.29 active(isQid(o)) -> mark(tt) 41.81/12.29 active(isQid(u)) -> mark(tt) 41.81/12.29 active(__(X1, X2)) -> __(active(X1), X2) 41.81/12.29 active(__(X1, X2)) -> __(X1, active(X2)) 41.81/12.29 active(U11(X1, X2)) -> U11(active(X1), X2) 41.81/12.29 active(U12(X)) -> U12(active(X)) 41.81/12.29 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 41.81/12.29 active(U22(X1, X2)) -> U22(active(X1), X2) 41.81/12.29 active(U23(X)) -> U23(active(X)) 41.81/12.29 active(U31(X1, X2)) -> U31(active(X1), X2) 41.81/12.29 active(U32(X)) -> U32(active(X)) 41.81/12.29 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 41.81/12.29 active(U42(X1, X2)) -> U42(active(X1), X2) 41.81/12.29 active(U43(X)) -> U43(active(X)) 41.81/12.29 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 41.81/12.29 active(U52(X1, X2)) -> U52(active(X1), X2) 41.81/12.29 active(U53(X)) -> U53(active(X)) 41.81/12.29 active(U61(X1, X2)) -> U61(active(X1), X2) 41.81/12.29 active(U62(X)) -> U62(active(X)) 41.81/12.29 active(U71(X1, X2)) -> U71(active(X1), X2) 41.81/12.29 active(U72(X)) -> U72(active(X)) 41.81/12.29 active(and(X1, X2)) -> and(active(X1), X2) 41.81/12.29 __(mark(X1), X2) -> mark(__(X1, X2)) 41.81/12.29 __(X1, mark(X2)) -> mark(__(X1, X2)) 41.81/12.29 U11(mark(X1), X2) -> mark(U11(X1, X2)) 41.81/12.29 U12(mark(X)) -> mark(U12(X)) 41.81/12.29 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 41.81/12.29 U22(mark(X1), X2) -> mark(U22(X1, X2)) 41.81/12.29 U23(mark(X)) -> mark(U23(X)) 41.81/12.29 U31(mark(X1), X2) -> mark(U31(X1, X2)) 41.81/12.29 U32(mark(X)) -> mark(U32(X)) 41.81/12.29 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 41.81/12.29 U42(mark(X1), X2) -> mark(U42(X1, X2)) 41.81/12.29 U43(mark(X)) -> mark(U43(X)) 41.81/12.29 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 41.81/12.29 U52(mark(X1), X2) -> mark(U52(X1, X2)) 41.81/12.29 U53(mark(X)) -> mark(U53(X)) 41.81/12.29 U61(mark(X1), X2) -> mark(U61(X1, X2)) 41.81/12.29 U62(mark(X)) -> mark(U62(X)) 41.81/12.29 U71(mark(X1), X2) -> mark(U71(X1, X2)) 41.81/12.29 U72(mark(X)) -> mark(U72(X)) 41.81/12.29 and(mark(X1), X2) -> mark(and(X1, X2)) 41.81/12.29 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 41.81/12.29 proper(nil) -> ok(nil) 41.81/12.29 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 41.81/12.29 proper(tt) -> ok(tt) 41.81/12.29 proper(U12(X)) -> U12(proper(X)) 41.81/12.29 proper(isNeList(X)) -> isNeList(proper(X)) 41.81/12.29 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 41.81/12.29 proper(isList(X)) -> isList(proper(X)) 41.81/12.29 proper(U23(X)) -> U23(proper(X)) 41.81/12.29 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 41.81/12.29 proper(U32(X)) -> U32(proper(X)) 41.81/12.29 proper(isQid(X)) -> isQid(proper(X)) 41.81/12.29 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) 41.81/12.29 proper(U43(X)) -> U43(proper(X)) 41.81/12.29 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) 41.81/12.29 proper(U53(X)) -> U53(proper(X)) 41.81/12.29 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 41.81/12.29 proper(U62(X)) -> U62(proper(X)) 41.81/12.29 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 41.81/12.29 proper(U72(X)) -> U72(proper(X)) 41.81/12.29 proper(isNePal(X)) -> isNePal(proper(X)) 41.81/12.29 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 41.81/12.29 proper(isPalListKind(X)) -> isPalListKind(proper(X)) 41.81/12.29 proper(isPal(X)) -> isPal(proper(X)) 41.81/12.29 proper(a) -> ok(a) 41.81/12.29 proper(e) -> ok(e) 41.81/12.29 proper(i) -> ok(i) 41.81/12.29 proper(o) -> ok(o) 41.81/12.29 proper(u) -> ok(u) 41.81/12.29 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 41.81/12.29 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 41.81/12.29 U12(ok(X)) -> ok(U12(X)) 41.81/12.29 isNeList(ok(X)) -> ok(isNeList(X)) 41.81/12.29 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 41.81/12.29 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 41.81/12.29 isList(ok(X)) -> ok(isList(X)) 41.81/12.29 U23(ok(X)) -> ok(U23(X)) 41.81/12.29 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 41.81/12.29 U32(ok(X)) -> ok(U32(X)) 41.81/12.29 isQid(ok(X)) -> ok(isQid(X)) 41.81/12.29 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 41.81/12.29 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 41.81/12.29 U43(ok(X)) -> ok(U43(X)) 41.81/12.29 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 41.81/12.29 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 41.81/12.29 U53(ok(X)) -> ok(U53(X)) 41.81/12.29 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 41.81/12.29 U62(ok(X)) -> ok(U62(X)) 41.81/12.29 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 41.81/12.29 U72(ok(X)) -> ok(U72(X)) 41.81/12.29 isNePal(ok(X)) -> ok(isNePal(X)) 41.81/12.29 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 41.81/12.29 isPalListKind(ok(X)) -> ok(isPalListKind(X)) 41.81/12.29 isPal(ok(X)) -> ok(isPal(X)) 41.81/12.29 top(mark(X)) -> top(proper(X)) 41.81/12.29 top(ok(X)) -> top(active(X)) 41.81/12.29 41.81/12.29 S is empty. 41.81/12.29 Rewrite Strategy: FULL 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (12) LowerBoundPropagationProof (FINISHED) 41.81/12.29 Propagated lower bound. 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (13) 41.81/12.29 BOUNDS(n^1, INF) 41.81/12.29 41.81/12.29 ---------------------------------------- 41.81/12.29 41.81/12.29 (14) 41.81/12.29 Obligation: 41.81/12.29 Analyzing the following TRS for decreasing loops: 41.81/12.29 41.81/12.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.81/12.29 41.81/12.29 41.81/12.29 The TRS R consists of the following rules: 41.81/12.29 41.81/12.29 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 41.81/12.29 active(__(X, nil)) -> mark(X) 41.81/12.29 active(__(nil, X)) -> mark(X) 41.81/12.29 active(U11(tt, V)) -> mark(U12(isNeList(V))) 41.81/12.29 active(U12(tt)) -> mark(tt) 41.81/12.29 active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) 41.81/12.29 active(U22(tt, V2)) -> mark(U23(isList(V2))) 41.81/12.29 active(U23(tt)) -> mark(tt) 41.81/12.29 active(U31(tt, V)) -> mark(U32(isQid(V))) 41.81/12.29 active(U32(tt)) -> mark(tt) 41.81/12.29 active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) 41.81/12.29 active(U42(tt, V2)) -> mark(U43(isNeList(V2))) 41.81/12.29 active(U43(tt)) -> mark(tt) 41.81/12.29 active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) 41.81/12.29 active(U52(tt, V2)) -> mark(U53(isList(V2))) 41.81/12.29 active(U53(tt)) -> mark(tt) 41.81/12.29 active(U61(tt, V)) -> mark(U62(isQid(V))) 41.81/12.29 active(U62(tt)) -> mark(tt) 41.81/12.29 active(U71(tt, V)) -> mark(U72(isNePal(V))) 41.81/12.29 active(U72(tt)) -> mark(tt) 41.81/12.29 active(and(tt, X)) -> mark(X) 41.81/12.29 active(isList(V)) -> mark(U11(isPalListKind(V), V)) 41.81/12.29 active(isList(nil)) -> mark(tt) 41.81/12.29 active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) 41.81/12.29 active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) 41.81/12.29 active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) 41.81/12.29 active(isPal(V)) -> mark(U71(isPalListKind(V), V)) 41.81/12.29 active(isPal(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(a)) -> mark(tt) 41.81/12.29 active(isPalListKind(e)) -> mark(tt) 41.81/12.29 active(isPalListKind(i)) -> mark(tt) 41.81/12.29 active(isPalListKind(nil)) -> mark(tt) 41.81/12.29 active(isPalListKind(o)) -> mark(tt) 41.81/12.29 active(isPalListKind(u)) -> mark(tt) 41.81/12.29 active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) 41.81/12.29 active(isQid(a)) -> mark(tt) 41.81/12.29 active(isQid(e)) -> mark(tt) 41.81/12.29 active(isQid(i)) -> mark(tt) 41.81/12.29 active(isQid(o)) -> mark(tt) 41.81/12.29 active(isQid(u)) -> mark(tt) 41.81/12.29 active(__(X1, X2)) -> __(active(X1), X2) 41.81/12.29 active(__(X1, X2)) -> __(X1, active(X2)) 41.81/12.29 active(U11(X1, X2)) -> U11(active(X1), X2) 41.81/12.29 active(U12(X)) -> U12(active(X)) 41.81/12.29 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 41.81/12.29 active(U22(X1, X2)) -> U22(active(X1), X2) 41.81/12.29 active(U23(X)) -> U23(active(X)) 41.81/12.29 active(U31(X1, X2)) -> U31(active(X1), X2) 41.81/12.29 active(U32(X)) -> U32(active(X)) 41.81/12.29 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 41.81/12.29 active(U42(X1, X2)) -> U42(active(X1), X2) 41.81/12.29 active(U43(X)) -> U43(active(X)) 41.81/12.29 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 41.81/12.29 active(U52(X1, X2)) -> U52(active(X1), X2) 41.81/12.29 active(U53(X)) -> U53(active(X)) 41.81/12.29 active(U61(X1, X2)) -> U61(active(X1), X2) 41.81/12.29 active(U62(X)) -> U62(active(X)) 41.81/12.29 active(U71(X1, X2)) -> U71(active(X1), X2) 41.81/12.29 active(U72(X)) -> U72(active(X)) 41.81/12.29 active(and(X1, X2)) -> and(active(X1), X2) 41.81/12.29 __(mark(X1), X2) -> mark(__(X1, X2)) 41.81/12.29 __(X1, mark(X2)) -> mark(__(X1, X2)) 41.81/12.29 U11(mark(X1), X2) -> mark(U11(X1, X2)) 41.81/12.29 U12(mark(X)) -> mark(U12(X)) 41.81/12.29 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 41.81/12.29 U22(mark(X1), X2) -> mark(U22(X1, X2)) 41.81/12.29 U23(mark(X)) -> mark(U23(X)) 41.81/12.29 U31(mark(X1), X2) -> mark(U31(X1, X2)) 41.81/12.29 U32(mark(X)) -> mark(U32(X)) 41.81/12.29 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 41.81/12.29 U42(mark(X1), X2) -> mark(U42(X1, X2)) 41.81/12.29 U43(mark(X)) -> mark(U43(X)) 41.81/12.29 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 41.81/12.29 U52(mark(X1), X2) -> mark(U52(X1, X2)) 41.81/12.29 U53(mark(X)) -> mark(U53(X)) 41.81/12.29 U61(mark(X1), X2) -> mark(U61(X1, X2)) 41.81/12.29 U62(mark(X)) -> mark(U62(X)) 41.81/12.29 U71(mark(X1), X2) -> mark(U71(X1, X2)) 41.81/12.29 U72(mark(X)) -> mark(U72(X)) 41.81/12.29 and(mark(X1), X2) -> mark(and(X1, X2)) 41.81/12.29 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 41.81/12.29 proper(nil) -> ok(nil) 41.81/12.29 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 41.81/12.29 proper(tt) -> ok(tt) 41.81/12.29 proper(U12(X)) -> U12(proper(X)) 41.81/12.29 proper(isNeList(X)) -> isNeList(proper(X)) 41.81/12.29 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 41.81/12.29 proper(isList(X)) -> isList(proper(X)) 41.81/12.29 proper(U23(X)) -> U23(proper(X)) 41.81/12.29 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 41.81/12.29 proper(U32(X)) -> U32(proper(X)) 41.81/12.29 proper(isQid(X)) -> isQid(proper(X)) 41.81/12.29 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) 41.81/12.29 proper(U43(X)) -> U43(proper(X)) 41.81/12.29 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 41.81/12.29 proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) 41.81/12.29 proper(U53(X)) -> U53(proper(X)) 41.81/12.29 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 41.81/12.29 proper(U62(X)) -> U62(proper(X)) 41.81/12.29 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 41.81/12.29 proper(U72(X)) -> U72(proper(X)) 41.81/12.29 proper(isNePal(X)) -> isNePal(proper(X)) 41.81/12.29 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 41.81/12.29 proper(isPalListKind(X)) -> isPalListKind(proper(X)) 41.81/12.29 proper(isPal(X)) -> isPal(proper(X)) 41.81/12.29 proper(a) -> ok(a) 41.81/12.29 proper(e) -> ok(e) 41.81/12.29 proper(i) -> ok(i) 41.81/12.29 proper(o) -> ok(o) 41.81/12.29 proper(u) -> ok(u) 41.81/12.29 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 41.81/12.29 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 41.81/12.29 U12(ok(X)) -> ok(U12(X)) 41.81/12.29 isNeList(ok(X)) -> ok(isNeList(X)) 41.81/12.29 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 41.81/12.29 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 41.81/12.29 isList(ok(X)) -> ok(isList(X)) 41.81/12.29 U23(ok(X)) -> ok(U23(X)) 41.81/12.29 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 41.81/12.29 U32(ok(X)) -> ok(U32(X)) 41.81/12.29 isQid(ok(X)) -> ok(isQid(X)) 41.81/12.29 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 41.81/12.29 U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) 41.81/12.29 U43(ok(X)) -> ok(U43(X)) 41.81/12.29 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 41.81/12.29 U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) 41.81/12.29 U53(ok(X)) -> ok(U53(X)) 41.81/12.29 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 41.81/12.29 U62(ok(X)) -> ok(U62(X)) 41.81/12.29 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 41.81/12.29 U72(ok(X)) -> ok(U72(X)) 41.81/12.29 isNePal(ok(X)) -> ok(isNePal(X)) 41.81/12.29 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 41.81/12.29 isPalListKind(ok(X)) -> ok(isPalListKind(X)) 41.81/12.29 isPal(ok(X)) -> ok(isPal(X)) 41.81/12.29 top(mark(X)) -> top(proper(X)) 41.81/12.29 top(ok(X)) -> top(active(X)) 41.81/12.29 41.81/12.29 S is empty. 41.81/12.29 Rewrite Strategy: FULL 42.78/12.36 EOF