/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 313 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: a__fst(0, Z) -> nil a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) a__from(X) -> cons(mark(X), from(s(X))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(add(X, Y)) a__len(nil) -> 0 a__len(cons(X, Z)) -> s(len(Z)) mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) a__fst(X1, X2) -> fst(X1, X2) a__from(X) -> from(X) a__add(X1, X2) -> add(X1, X2) a__len(X) -> len(X) 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(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(fst(x_1, x_2)) -> fst(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(len(x_1)) -> len(encArg(x_1)) encArg(cons_a__fst(x_1, x_2)) -> a__fst(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__len(x_1)) -> a__len(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fst(x_1, x_2) -> a__fst(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1, x_2) -> fst(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_a__len(x_1) -> a__len(encArg(x_1)) encode_len(x_1) -> len(encArg(x_1)) ---------------------------------------- (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: a__fst(0, Z) -> nil a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) a__from(X) -> cons(mark(X), from(s(X))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(add(X, Y)) a__len(nil) -> 0 a__len(cons(X, Z)) -> s(len(Z)) mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) a__fst(X1, X2) -> fst(X1, X2) a__from(X) -> from(X) a__add(X1, X2) -> add(X1, X2) a__len(X) -> len(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(fst(x_1, x_2)) -> fst(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(len(x_1)) -> len(encArg(x_1)) encArg(cons_a__fst(x_1, x_2)) -> a__fst(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__len(x_1)) -> a__len(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fst(x_1, x_2) -> a__fst(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1, x_2) -> fst(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_a__len(x_1) -> a__len(encArg(x_1)) encode_len(x_1) -> len(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(n^1, INF). The TRS R consists of the following rules: a__fst(0, Z) -> nil a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) a__from(X) -> cons(mark(X), from(s(X))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(add(X, Y)) a__len(nil) -> 0 a__len(cons(X, Z)) -> s(len(Z)) mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) a__fst(X1, X2) -> fst(X1, X2) a__from(X) -> from(X) a__add(X1, X2) -> add(X1, X2) a__len(X) -> len(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(fst(x_1, x_2)) -> fst(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(len(x_1)) -> len(encArg(x_1)) encArg(cons_a__fst(x_1, x_2)) -> a__fst(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__len(x_1)) -> a__len(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fst(x_1, x_2) -> a__fst(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1, x_2) -> fst(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_a__len(x_1) -> a__len(encArg(x_1)) encode_len(x_1) -> len(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(n^1, INF). The TRS R consists of the following rules: a__fst(0, Z) -> nil a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) a__from(X) -> cons(mark(X), from(s(X))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(add(X, Y)) a__len(nil) -> 0 a__len(cons(X, Z)) -> s(len(Z)) mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) a__fst(X1, X2) -> fst(X1, X2) a__from(X) -> from(X) a__add(X1, X2) -> add(X1, X2) a__len(X) -> len(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(fst(x_1, x_2)) -> fst(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(len(x_1)) -> len(encArg(x_1)) encArg(cons_a__fst(x_1, x_2)) -> a__fst(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__len(x_1)) -> a__len(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fst(x_1, x_2) -> a__fst(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1, x_2) -> fst(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_a__len(x_1) -> a__len(encArg(x_1)) encode_len(x_1) -> len(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 mark(len(X)) ->^+ a__len(mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / len(X)]. 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: a__fst(0, Z) -> nil a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) a__from(X) -> cons(mark(X), from(s(X))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(add(X, Y)) a__len(nil) -> 0 a__len(cons(X, Z)) -> s(len(Z)) mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) a__fst(X1, X2) -> fst(X1, X2) a__from(X) -> from(X) a__add(X1, X2) -> add(X1, X2) a__len(X) -> len(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(fst(x_1, x_2)) -> fst(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(len(x_1)) -> len(encArg(x_1)) encArg(cons_a__fst(x_1, x_2)) -> a__fst(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__len(x_1)) -> a__len(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fst(x_1, x_2) -> a__fst(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1, x_2) -> fst(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_a__len(x_1) -> a__len(encArg(x_1)) encode_len(x_1) -> len(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(n^1, INF). The TRS R consists of the following rules: a__fst(0, Z) -> nil a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) a__from(X) -> cons(mark(X), from(s(X))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(add(X, Y)) a__len(nil) -> 0 a__len(cons(X, Z)) -> s(len(Z)) mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) a__fst(X1, X2) -> fst(X1, X2) a__from(X) -> from(X) a__add(X1, X2) -> add(X1, X2) a__len(X) -> len(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(fst(x_1, x_2)) -> fst(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(len(x_1)) -> len(encArg(x_1)) encArg(cons_a__fst(x_1, x_2)) -> a__fst(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__len(x_1)) -> a__len(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fst(x_1, x_2) -> a__fst(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_fst(x_1, x_2) -> fst(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_a__len(x_1) -> a__len(encArg(x_1)) encode_len(x_1) -> len(encArg(x_1)) Rewrite Strategy: INNERMOST