/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 218 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 ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) 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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) 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(n^1, INF). The TRS R consists of the following rules: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) 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(n^1, INF). The TRS R consists of the following rules: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) 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 circ(cons(lift, s), cons(lift, t)) ->^+ cons(lift, circ(s, t)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [s / cons(lift, s), t / cons(lift, t)]. 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(n^1, INF). The TRS R consists of the following rules: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) 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(n^1, INF). The TRS R consists of the following rules: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST