/export/starexec/sandbox/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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 (9) DecreasingLoopProof [FINISHED, 388 ms] (10) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: and(true, y) -> y and(false, y) -> false eq(nil, nil) -> true eq(cons(t, l), nil) -> false eq(nil, cons(t, l)) -> false eq(cons(t, l), cons(t', l')) -> and(eq(t, t'), eq(l, l')) eq(var(l), var(l')) -> eq(l, l') eq(var(l), apply(t, s)) -> false eq(var(l), lambda(x, t)) -> false eq(apply(t, s), var(l)) -> false eq(apply(t, s), apply(t', s')) -> and(eq(t, t'), eq(s, s')) eq(apply(t, s), lambda(x, t)) -> false eq(lambda(x, t), var(l)) -> false eq(lambda(x, t), apply(t, s)) -> false eq(lambda(x, t), lambda(x', t')) -> and(eq(x, x'), eq(t, t')) if(true, var(k), var(l')) -> var(k) if(false, var(k), var(l')) -> var(l') ren(var(l), var(k), var(l')) -> if(eq(l, l'), var(k), var(l')) ren(x, y, apply(t, s)) -> apply(ren(x, y, t), ren(x, y, s)) ren(x, y, lambda(z, t)) -> lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))) 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(EXP, INF). The TRS R consists of the following rules: and(true, y) -> y and(false, y) -> false eq(nil, nil) -> true eq(cons(t, l), nil) -> false eq(nil, cons(t, l)) -> false eq(cons(t, l), cons(t', l')) -> and(eq(t, t'), eq(l, l')) eq(var(l), var(l')) -> eq(l, l') eq(var(l), apply(t, s)) -> false eq(var(l), lambda(x, t)) -> false eq(apply(t, s), var(l)) -> false eq(apply(t, s), apply(t', s')) -> and(eq(t, t'), eq(s, s')) eq(apply(t, s), lambda(x, t)) -> false eq(lambda(x, t), var(l)) -> false eq(lambda(x, t), apply(t, s)) -> false eq(lambda(x, t), lambda(x', t')) -> and(eq(x, x'), eq(t, t')) if(true, var(k), var(l')) -> var(k) if(false, var(k), var(l')) -> var(l') ren(var(l), var(k), var(l')) -> if(eq(l, l'), var(k), var(l')) ren(x, y, apply(t, s)) -> apply(ren(x, y, t), ren(x, y, s)) ren(x, y, lambda(z, t)) -> lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))) 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 eq(lambda(x, t), lambda(x', t')) ->^+ and(eq(x, x'), eq(t, t')) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x / lambda(x, t), x' / lambda(x', t')]. 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(EXP, INF). The TRS R consists of the following rules: and(true, y) -> y and(false, y) -> false eq(nil, nil) -> true eq(cons(t, l), nil) -> false eq(nil, cons(t, l)) -> false eq(cons(t, l), cons(t', l')) -> and(eq(t, t'), eq(l, l')) eq(var(l), var(l')) -> eq(l, l') eq(var(l), apply(t, s)) -> false eq(var(l), lambda(x, t)) -> false eq(apply(t, s), var(l)) -> false eq(apply(t, s), apply(t', s')) -> and(eq(t, t'), eq(s, s')) eq(apply(t, s), lambda(x, t)) -> false eq(lambda(x, t), var(l)) -> false eq(lambda(x, t), apply(t, s)) -> false eq(lambda(x, t), lambda(x', t')) -> and(eq(x, x'), eq(t, t')) if(true, var(k), var(l')) -> var(k) if(false, var(k), var(l')) -> var(l') ren(var(l), var(k), var(l')) -> if(eq(l, l'), var(k), var(l')) ren(x, y, apply(t, s)) -> apply(ren(x, y, t), ren(x, y, s)) ren(x, y, lambda(z, t)) -> lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))) 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(EXP, INF). The TRS R consists of the following rules: and(true, y) -> y and(false, y) -> false eq(nil, nil) -> true eq(cons(t, l), nil) -> false eq(nil, cons(t, l)) -> false eq(cons(t, l), cons(t', l')) -> and(eq(t, t'), eq(l, l')) eq(var(l), var(l')) -> eq(l, l') eq(var(l), apply(t, s)) -> false eq(var(l), lambda(x, t)) -> false eq(apply(t, s), var(l)) -> false eq(apply(t, s), apply(t', s')) -> and(eq(t, t'), eq(s, s')) eq(apply(t, s), lambda(x, t)) -> false eq(lambda(x, t), var(l)) -> false eq(lambda(x, t), apply(t, s)) -> false eq(lambda(x, t), lambda(x', t')) -> and(eq(x, x'), eq(t, t')) if(true, var(k), var(l')) -> var(k) if(false, var(k), var(l')) -> var(l') ren(var(l), var(k), var(l')) -> if(eq(l, l'), var(k), var(l')) ren(x, y, apply(t, s)) -> apply(ren(x, y, t), ren(x, y, s)) ren(x, y, lambda(z, t)) -> lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence ren(x, y, lambda(z, lambda(z3_0, t4_0))) ->^+ lambda(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), lambda(var(cons(x, cons(y, cons(lambda(var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), ren(z, var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), ren(z3_0, var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), t4_0))), nil)))), ren(x, y, ren(var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), var(cons(x, cons(y, cons(lambda(var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), ren(z, var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), ren(z3_0, var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), t4_0))), nil)))), ren(z, var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), ren(z3_0, var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), t4_0)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0,1,1,0,1,2]. The pumping substitution is [t4_0 / lambda(z, lambda(z3_0, t4_0))]. The result substitution is [x / z3_0, y / var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil))))]. The rewrite sequence ren(x, y, lambda(z, lambda(z3_0, t4_0))) ->^+ lambda(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), lambda(var(cons(x, cons(y, cons(lambda(var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), ren(z, var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), ren(z3_0, var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), t4_0))), nil)))), ren(x, y, ren(var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), var(cons(x, cons(y, cons(lambda(var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), ren(z, var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), ren(z3_0, var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), t4_0))), nil)))), ren(z, var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), ren(z3_0, var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil)))), t4_0)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,2,1,0,1,1,0,1,2]. The pumping substitution is [t4_0 / lambda(z, lambda(z3_0, t4_0))]. The result substitution is [x / z3_0, y / var(cons(z, cons(var(cons(x, cons(y, cons(lambda(z, lambda(z3_0, t4_0)), nil)))), cons(lambda(z3_0, t4_0), nil))))]. ---------------------------------------- (10) BOUNDS(EXP, INF)