/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) a____(X, nil) -> mark(X) a____(nil, X) -> mark(X) a__U11(tt, V) -> a__U12(a__isNeList(V)) a__U12(tt) -> tt a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) a__U22(tt, V2) -> a__U23(a__isList(V2)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isQid(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) a__U42(tt, V2) -> a__U43(a__isNeList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) a__U52(tt, V2) -> a__U53(a__isList(V2)) a__U53(tt) -> tt a__U61(tt, V) -> a__U62(a__isQid(V)) a__U62(tt) -> tt a__U71(tt, V) -> a__U72(a__isNePal(V)) a__U72(tt) -> tt a__and(tt, X) -> mark(X) a__isList(V) -> a__U11(a__isPalListKind(V), V) a__isList(nil) -> tt a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNeList(V) -> a__U31(a__isPalListKind(V), V) a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNePal(V) -> a__U61(a__isPalListKind(V), V) a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) a__isPal(V) -> a__U71(a__isPalListKind(V), V) a__isPal(nil) -> tt a__isPalListKind(a) -> tt a__isPalListKind(e) -> tt a__isPalListKind(i) -> tt a__isPalListKind(nil) -> tt a__isPalListKind(o) -> tt a__isPalListKind(u) -> tt a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isList(X)) -> a__isList(X) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(isNePal(X)) -> a__isNePal(X) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isPalListKind(X)) -> a__isPalListKind(X) mark(isPal(X)) -> a__isPal(X) mark(nil) -> nil mark(tt) -> tt mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNeList(X) -> isNeList(X) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__U22(X1, X2) -> U22(X1, X2) a__isList(X) -> isList(X) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__isQid(X) -> isQid(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isNePal(X) -> isNePal(X) a__and(X1, X2) -> and(X1, X2) a__isPalListKind(X) -> isPalListKind(X) a__isPal(X) -> isPal(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) a____(X, nil) -> mark(X) a____(nil, X) -> mark(X) a__U11(tt, V) -> a__U12(a__isNeList(V)) a__U12(tt) -> tt a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) a__U22(tt, V2) -> a__U23(a__isList(V2)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isQid(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) a__U42(tt, V2) -> a__U43(a__isNeList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) a__U52(tt, V2) -> a__U53(a__isList(V2)) a__U53(tt) -> tt a__U61(tt, V) -> a__U62(a__isQid(V)) a__U62(tt) -> tt a__U71(tt, V) -> a__U72(a__isNePal(V)) a__U72(tt) -> tt a__and(tt, X) -> mark(X) a__isList(V) -> a__U11(a__isPalListKind(V), V) a__isList(nil) -> tt a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNeList(V) -> a__U31(a__isPalListKind(V), V) a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNePal(V) -> a__U61(a__isPalListKind(V), V) a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) a__isPal(V) -> a__U71(a__isPalListKind(V), V) a__isPal(nil) -> tt a__isPalListKind(a) -> tt a__isPalListKind(e) -> tt a__isPalListKind(i) -> tt a__isPalListKind(nil) -> tt a__isPalListKind(o) -> tt a__isPalListKind(u) -> tt a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isList(X)) -> a__isList(X) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(isNePal(X)) -> a__isNePal(X) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isPalListKind(X)) -> a__isPalListKind(X) mark(isPal(X)) -> a__isPal(X) mark(nil) -> nil mark(tt) -> tt mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNeList(X) -> isNeList(X) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__U22(X1, X2) -> U22(X1, X2) a__isList(X) -> isList(X) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__isQid(X) -> isQid(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isNePal(X) -> isNePal(X) a__and(X1, X2) -> and(X1, X2) a__isPalListKind(X) -> isPalListKind(X) a__isPal(X) -> isPal(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence mark(U12(X)) ->^+ a__U12(mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / U12(X)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) a____(X, nil) -> mark(X) a____(nil, X) -> mark(X) a__U11(tt, V) -> a__U12(a__isNeList(V)) a__U12(tt) -> tt a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) a__U22(tt, V2) -> a__U23(a__isList(V2)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isQid(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) a__U42(tt, V2) -> a__U43(a__isNeList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) a__U52(tt, V2) -> a__U53(a__isList(V2)) a__U53(tt) -> tt a__U61(tt, V) -> a__U62(a__isQid(V)) a__U62(tt) -> tt a__U71(tt, V) -> a__U72(a__isNePal(V)) a__U72(tt) -> tt a__and(tt, X) -> mark(X) a__isList(V) -> a__U11(a__isPalListKind(V), V) a__isList(nil) -> tt a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNeList(V) -> a__U31(a__isPalListKind(V), V) a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNePal(V) -> a__U61(a__isPalListKind(V), V) a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) a__isPal(V) -> a__U71(a__isPalListKind(V), V) a__isPal(nil) -> tt a__isPalListKind(a) -> tt a__isPalListKind(e) -> tt a__isPalListKind(i) -> tt a__isPalListKind(nil) -> tt a__isPalListKind(o) -> tt a__isPalListKind(u) -> tt a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isList(X)) -> a__isList(X) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(isNePal(X)) -> a__isNePal(X) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isPalListKind(X)) -> a__isPalListKind(X) mark(isPal(X)) -> a__isPal(X) mark(nil) -> nil mark(tt) -> tt mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNeList(X) -> isNeList(X) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__U22(X1, X2) -> U22(X1, X2) a__isList(X) -> isList(X) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__isQid(X) -> isQid(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isNePal(X) -> isNePal(X) a__and(X1, X2) -> and(X1, X2) a__isPalListKind(X) -> isPalListKind(X) a__isPal(X) -> isPal(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) a____(X, nil) -> mark(X) a____(nil, X) -> mark(X) a__U11(tt, V) -> a__U12(a__isNeList(V)) a__U12(tt) -> tt a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) a__U22(tt, V2) -> a__U23(a__isList(V2)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isQid(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) a__U42(tt, V2) -> a__U43(a__isNeList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) a__U52(tt, V2) -> a__U53(a__isList(V2)) a__U53(tt) -> tt a__U61(tt, V) -> a__U62(a__isQid(V)) a__U62(tt) -> tt a__U71(tt, V) -> a__U72(a__isNePal(V)) a__U72(tt) -> tt a__and(tt, X) -> mark(X) a__isList(V) -> a__U11(a__isPalListKind(V), V) a__isList(nil) -> tt a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNeList(V) -> a__U31(a__isPalListKind(V), V) a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) a__isNePal(V) -> a__U61(a__isPalListKind(V), V) a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) a__isPal(V) -> a__U71(a__isPalListKind(V), V) a__isPal(nil) -> tt a__isPalListKind(a) -> tt a__isPalListKind(e) -> tt a__isPalListKind(i) -> tt a__isPalListKind(nil) -> tt a__isPalListKind(o) -> tt a__isPalListKind(u) -> tt a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(isList(X)) -> a__isList(X) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(isNePal(X)) -> a__isNePal(X) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isPalListKind(X)) -> a__isPalListKind(X) mark(isPal(X)) -> a__isPal(X) mark(nil) -> nil mark(tt) -> tt mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNeList(X) -> isNeList(X) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__U22(X1, X2) -> U22(X1, X2) a__isList(X) -> isList(X) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__isQid(X) -> isQid(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isNePal(X) -> isNePal(X) a__and(X1, X2) -> and(X1, X2) a__isPalListKind(X) -> isPalListKind(X) a__isPal(X) -> isPal(X) S is empty. Rewrite Strategy: FULL