/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 266 ms] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (6) BEST (7) proven lower bound (8) LowerBoundPropagationProof [FINISHED, 0 ms] (9) BOUNDS(n^1, INF) (10) TRS for Loop Detection ---------------------------------------- (0) 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: rewrite(Op(Val(n), y)) -> Op(rewrite(y), Val(n)) rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x)) rewrite(Val(n)) -> Val(n) second(Op(x, y)) -> y isOp(Val(n)) -> False isOp(Op(x, y)) -> True first(Val(n)) -> Val(n) first(Op(x, y)) -> x assrewrite(exp) -> rewrite(exp) The (relative) TRS S consists of the following rules: rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y)) rewrite[Let][Let](Op(x, y), opab, a1, b1) -> rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y)) rewrite[Let][Let][Let](exp, a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1)))) Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (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: rewrite(Op(Val(n), y)) -> Op(rewrite(y), Val(n)) rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x)) rewrite(Val(n)) -> Val(n) second(Op(x, y)) -> y isOp(Val(n)) -> False isOp(Op(x, y)) -> True first(Val(n)) -> Val(n) first(Op(x, y)) -> x assrewrite(exp) -> rewrite(exp) The (relative) TRS S consists of the following rules: rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y)) rewrite[Let][Let](Op(x, y), opab, a1, b1) -> rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y)) rewrite[Let][Let][Let](exp, a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1)))) Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (4) 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: rewrite(Op(Val(n), y)) -> Op(rewrite(y), Val(n)) rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x)) rewrite(Val(n)) -> Val(n) second(Op(x, y)) -> y isOp(Val(n)) -> False isOp(Op(x, y)) -> True first(Val(n)) -> Val(n) first(Op(x, y)) -> x assrewrite(exp) -> rewrite(exp) The (relative) TRS S consists of the following rules: rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y)) rewrite[Let][Let](Op(x, y), opab, a1, b1) -> rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y)) rewrite[Let][Let][Let](exp, a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1)))) Rewrite Strategy: INNERMOST ---------------------------------------- (5) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence rewrite(Op(Val(n), y)) ->^+ Op(rewrite(y), Val(n)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [y / Op(Val(n), y)]. The result substitution is [ ]. ---------------------------------------- (6) Complex Obligation (BEST) ---------------------------------------- (7) 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: rewrite(Op(Val(n), y)) -> Op(rewrite(y), Val(n)) rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x)) rewrite(Val(n)) -> Val(n) second(Op(x, y)) -> y isOp(Val(n)) -> False isOp(Op(x, y)) -> True first(Val(n)) -> Val(n) first(Op(x, y)) -> x assrewrite(exp) -> rewrite(exp) The (relative) TRS S consists of the following rules: rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y)) rewrite[Let][Let](Op(x, y), opab, a1, b1) -> rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y)) rewrite[Let][Let][Let](exp, a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1)))) Rewrite Strategy: INNERMOST ---------------------------------------- (8) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (9) BOUNDS(n^1, INF) ---------------------------------------- (10) 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: rewrite(Op(Val(n), y)) -> Op(rewrite(y), Val(n)) rewrite(Op(Op(x, y), y')) -> rewrite[Let](Op(Op(x, y), y'), Op(x, y), rewrite(x)) rewrite(Val(n)) -> Val(n) second(Op(x, y)) -> y isOp(Val(n)) -> False isOp(Op(x, y)) -> True first(Val(n)) -> Val(n) first(Op(x, y)) -> x assrewrite(exp) -> rewrite(exp) The (relative) TRS S consists of the following rules: rewrite[Let](exp, Op(x, y), a1) -> rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y)) rewrite[Let][Let](Op(x, y), opab, a1, b1) -> rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y)) rewrite[Let][Let][Let](exp, a1, b1, c1) -> rewrite(Op(a1, Op(b1, rewrite(c1)))) Rewrite Strategy: INNERMOST