/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 513 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection (13) InfiniteLowerBoundProof [FINISHED, 0 ms] (14) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) dbls(nil) -> nil dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) indx(nil, X) -> nil indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) from(X) -> cons(X, from(s(X))) dbl1(0) -> 01 dbl1(s(X)) -> s1(s1(dbl1(X))) sel1(0, cons(X, Y)) -> X sel1(s(X), cons(Y, Z)) -> sel1(X, Z) quote(0) -> 01 quote(s(X)) -> s1(quote(X)) quote(dbl(X)) -> dbl1(X) quote(sel(X, Y)) -> sel1(X, Y) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(01) -> 01 encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) dbls(nil) -> nil dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) indx(nil, X) -> nil indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) from(X) -> cons(X, from(s(X))) dbl1(0) -> 01 dbl1(s(X)) -> s1(s1(dbl1(X))) sel1(0, cons(X, Y)) -> X sel1(s(X), cons(Y, Z)) -> sel1(X, Z) quote(0) -> 01 quote(s(X)) -> s1(quote(X)) quote(dbl(X)) -> dbl1(X) quote(sel(X, Y)) -> sel1(X, Y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(01) -> 01 encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) dbls(nil) -> nil dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) indx(nil, X) -> nil indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) from(X) -> cons(X, from(s(X))) dbl1(0) -> 01 dbl1(s(X)) -> s1(s1(dbl1(X))) sel1(0, cons(X, Y)) -> X sel1(s(X), cons(Y, Z)) -> sel1(X, Z) quote(0) -> 01 quote(s(X)) -> s1(quote(X)) quote(dbl(X)) -> dbl1(X) quote(sel(X, Y)) -> sel1(X, Y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(01) -> 01 encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) dbls(nil) -> nil dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) indx(nil, X) -> nil indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) from(X) -> cons(X, from(s(X))) dbl1(0) -> 01 dbl1(s(X)) -> s1(s1(dbl1(X))) sel1(0, cons(X, Y)) -> X sel1(s(X), cons(Y, Z)) -> sel1(X, Z) quote(0) -> 01 quote(s(X)) -> s1(quote(X)) quote(dbl(X)) -> dbl1(X) quote(sel(X, Y)) -> sel1(X, Y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(01) -> 01 encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence dbls(cons(X, Y)) ->^+ cons(dbl(X), dbls(Y)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [Y / cons(X, Y)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) dbls(nil) -> nil dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) indx(nil, X) -> nil indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) from(X) -> cons(X, from(s(X))) dbl1(0) -> 01 dbl1(s(X)) -> s1(s1(dbl1(X))) sel1(0, cons(X, Y)) -> X sel1(s(X), cons(Y, Z)) -> sel1(X, Z) quote(0) -> 01 quote(s(X)) -> s1(quote(X)) quote(dbl(X)) -> dbl1(X) quote(sel(X, Y)) -> sel1(X, Y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(01) -> 01 encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) dbls(nil) -> nil dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) indx(nil, X) -> nil indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) from(X) -> cons(X, from(s(X))) dbl1(0) -> 01 dbl1(s(X)) -> s1(s1(dbl1(X))) sel1(0, cons(X, Y)) -> X sel1(s(X), cons(Y, Z)) -> sel1(X, Z) quote(0) -> 01 quote(s(X)) -> s1(quote(X)) quote(dbl(X)) -> dbl1(X) quote(sel(X, Y)) -> sel1(X, Y) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(01) -> 01 encArg(s1(x_1)) -> s1(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) InfiniteLowerBoundProof (FINISHED) The following loop proves infinite runtime complexity: The rewrite sequence from(X) ->^+ cons(X, from(s(X))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [ ]. The result substitution is [X / s(X)]. ---------------------------------------- (14) BOUNDS(INF, INF)