/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 850 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__from(X) -> cons(mark(X), from(s(X))) a__2ndspos(0, Z) -> rnil a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) a__2ndsneg(0, Z) -> rnil a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) a__pi(X) -> a__2ndspos(mark(X), a__from(0)) a__plus(0, Y) -> mark(Y) a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) a__times(0, Y) -> 0 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) a__square(X) -> a__times(mark(X), mark(X)) mark(from(X)) -> a__from(mark(X)) mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) mark(pi(X)) -> a__pi(mark(X)) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) mark(square(X)) -> a__square(mark(X)) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(posrecip(X)) -> posrecip(mark(X)) mark(negrecip(X)) -> negrecip(mark(X)) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) mark(rnil) -> rnil mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) a__from(X) -> from(X) a__2ndspos(X1, X2) -> 2ndspos(X1, X2) a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) a__pi(X) -> pi(X) a__plus(X1, X2) -> plus(X1, X2) a__times(X1, X2) -> times(X1, X2) a__square(X) -> square(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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(rnil) -> rnil encArg(cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2)) encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2)) encArg(posrecip(x_1)) -> posrecip(encArg(x_1)) encArg(negrecip(x_1)) -> negrecip(encArg(x_1)) encArg(2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2)) encArg(2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2)) encArg(pi(x_1)) -> pi(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(square(x_1)) -> square(encArg(x_1)) encArg(nil) -> nil encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__2ndspos(x_1, x_2)) -> a__2ndspos(encArg(x_1), encArg(x_2)) encArg(cons_a__2ndsneg(x_1, x_2)) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encArg(cons_a__pi(x_1)) -> a__pi(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__times(x_1, x_2)) -> a__times(encArg(x_1), encArg(x_2)) encArg(cons_a__square(x_1)) -> a__square(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(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_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__2ndspos(x_1, x_2) -> a__2ndspos(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_rnil -> rnil encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2)) encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2)) encode_posrecip(x_1) -> posrecip(encArg(x_1)) encode_a__2ndsneg(x_1, x_2) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encode_negrecip(x_1) -> negrecip(encArg(x_1)) encode_a__pi(x_1) -> a__pi(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__times(x_1, x_2) -> a__times(encArg(x_1), encArg(x_2)) encode_a__square(x_1) -> a__square(encArg(x_1)) encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2)) encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2)) encode_pi(x_1) -> pi(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_square(x_1) -> square(encArg(x_1)) encode_nil -> nil ---------------------------------------- (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__from(X) -> cons(mark(X), from(s(X))) a__2ndspos(0, Z) -> rnil a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) a__2ndsneg(0, Z) -> rnil a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) a__pi(X) -> a__2ndspos(mark(X), a__from(0)) a__plus(0, Y) -> mark(Y) a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) a__times(0, Y) -> 0 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) a__square(X) -> a__times(mark(X), mark(X)) mark(from(X)) -> a__from(mark(X)) mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) mark(pi(X)) -> a__pi(mark(X)) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) mark(square(X)) -> a__square(mark(X)) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(posrecip(X)) -> posrecip(mark(X)) mark(negrecip(X)) -> negrecip(mark(X)) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) mark(rnil) -> rnil mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) a__from(X) -> from(X) a__2ndspos(X1, X2) -> 2ndspos(X1, X2) a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) a__pi(X) -> pi(X) a__plus(X1, X2) -> plus(X1, X2) a__times(X1, X2) -> times(X1, X2) a__square(X) -> square(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(rnil) -> rnil encArg(cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2)) encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2)) encArg(posrecip(x_1)) -> posrecip(encArg(x_1)) encArg(negrecip(x_1)) -> negrecip(encArg(x_1)) encArg(2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2)) encArg(2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2)) encArg(pi(x_1)) -> pi(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(square(x_1)) -> square(encArg(x_1)) encArg(nil) -> nil encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__2ndspos(x_1, x_2)) -> a__2ndspos(encArg(x_1), encArg(x_2)) encArg(cons_a__2ndsneg(x_1, x_2)) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encArg(cons_a__pi(x_1)) -> a__pi(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__times(x_1, x_2)) -> a__times(encArg(x_1), encArg(x_2)) encArg(cons_a__square(x_1)) -> a__square(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(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_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__2ndspos(x_1, x_2) -> a__2ndspos(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_rnil -> rnil encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2)) encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2)) encode_posrecip(x_1) -> posrecip(encArg(x_1)) encode_a__2ndsneg(x_1, x_2) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encode_negrecip(x_1) -> negrecip(encArg(x_1)) encode_a__pi(x_1) -> a__pi(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__times(x_1, x_2) -> a__times(encArg(x_1), encArg(x_2)) encode_a__square(x_1) -> a__square(encArg(x_1)) encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2)) encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2)) encode_pi(x_1) -> pi(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_square(x_1) -> square(encArg(x_1)) encode_nil -> nil 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__from(X) -> cons(mark(X), from(s(X))) a__2ndspos(0, Z) -> rnil a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) a__2ndsneg(0, Z) -> rnil a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) a__pi(X) -> a__2ndspos(mark(X), a__from(0)) a__plus(0, Y) -> mark(Y) a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) a__times(0, Y) -> 0 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) a__square(X) -> a__times(mark(X), mark(X)) mark(from(X)) -> a__from(mark(X)) mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) mark(pi(X)) -> a__pi(mark(X)) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) mark(square(X)) -> a__square(mark(X)) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(posrecip(X)) -> posrecip(mark(X)) mark(negrecip(X)) -> negrecip(mark(X)) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) mark(rnil) -> rnil mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) a__from(X) -> from(X) a__2ndspos(X1, X2) -> 2ndspos(X1, X2) a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) a__pi(X) -> pi(X) a__plus(X1, X2) -> plus(X1, X2) a__times(X1, X2) -> times(X1, X2) a__square(X) -> square(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(rnil) -> rnil encArg(cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2)) encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2)) encArg(posrecip(x_1)) -> posrecip(encArg(x_1)) encArg(negrecip(x_1)) -> negrecip(encArg(x_1)) encArg(2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2)) encArg(2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2)) encArg(pi(x_1)) -> pi(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(square(x_1)) -> square(encArg(x_1)) encArg(nil) -> nil encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__2ndspos(x_1, x_2)) -> a__2ndspos(encArg(x_1), encArg(x_2)) encArg(cons_a__2ndsneg(x_1, x_2)) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encArg(cons_a__pi(x_1)) -> a__pi(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__times(x_1, x_2)) -> a__times(encArg(x_1), encArg(x_2)) encArg(cons_a__square(x_1)) -> a__square(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(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_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__2ndspos(x_1, x_2) -> a__2ndspos(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_rnil -> rnil encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2)) encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2)) encode_posrecip(x_1) -> posrecip(encArg(x_1)) encode_a__2ndsneg(x_1, x_2) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encode_negrecip(x_1) -> negrecip(encArg(x_1)) encode_a__pi(x_1) -> a__pi(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__times(x_1, x_2) -> a__times(encArg(x_1), encArg(x_2)) encode_a__square(x_1) -> a__square(encArg(x_1)) encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2)) encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2)) encode_pi(x_1) -> pi(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_square(x_1) -> square(encArg(x_1)) encode_nil -> nil 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__from(X) -> cons(mark(X), from(s(X))) a__2ndspos(0, Z) -> rnil a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) a__2ndsneg(0, Z) -> rnil a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) a__pi(X) -> a__2ndspos(mark(X), a__from(0)) a__plus(0, Y) -> mark(Y) a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) a__times(0, Y) -> 0 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) a__square(X) -> a__times(mark(X), mark(X)) mark(from(X)) -> a__from(mark(X)) mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) mark(pi(X)) -> a__pi(mark(X)) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) mark(square(X)) -> a__square(mark(X)) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(posrecip(X)) -> posrecip(mark(X)) mark(negrecip(X)) -> negrecip(mark(X)) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) mark(rnil) -> rnil mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) a__from(X) -> from(X) a__2ndspos(X1, X2) -> 2ndspos(X1, X2) a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) a__pi(X) -> pi(X) a__plus(X1, X2) -> plus(X1, X2) a__times(X1, X2) -> times(X1, X2) a__square(X) -> square(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(rnil) -> rnil encArg(cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2)) encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2)) encArg(posrecip(x_1)) -> posrecip(encArg(x_1)) encArg(negrecip(x_1)) -> negrecip(encArg(x_1)) encArg(2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2)) encArg(2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2)) encArg(pi(x_1)) -> pi(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(square(x_1)) -> square(encArg(x_1)) encArg(nil) -> nil encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__2ndspos(x_1, x_2)) -> a__2ndspos(encArg(x_1), encArg(x_2)) encArg(cons_a__2ndsneg(x_1, x_2)) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encArg(cons_a__pi(x_1)) -> a__pi(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__times(x_1, x_2)) -> a__times(encArg(x_1), encArg(x_2)) encArg(cons_a__square(x_1)) -> a__square(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(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_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__2ndspos(x_1, x_2) -> a__2ndspos(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_rnil -> rnil encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2)) encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2)) encode_posrecip(x_1) -> posrecip(encArg(x_1)) encode_a__2ndsneg(x_1, x_2) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encode_negrecip(x_1) -> negrecip(encArg(x_1)) encode_a__pi(x_1) -> a__pi(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__times(x_1, x_2) -> a__times(encArg(x_1), encArg(x_2)) encode_a__square(x_1) -> a__square(encArg(x_1)) encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2)) encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2)) encode_pi(x_1) -> pi(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_square(x_1) -> square(encArg(x_1)) encode_nil -> nil 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(2ndsneg(X1, X2)) ->^+ a__2ndsneg(mark(X1), mark(X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / 2ndsneg(X1, X2)]. 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__from(X) -> cons(mark(X), from(s(X))) a__2ndspos(0, Z) -> rnil a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) a__2ndsneg(0, Z) -> rnil a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) a__pi(X) -> a__2ndspos(mark(X), a__from(0)) a__plus(0, Y) -> mark(Y) a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) a__times(0, Y) -> 0 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) a__square(X) -> a__times(mark(X), mark(X)) mark(from(X)) -> a__from(mark(X)) mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) mark(pi(X)) -> a__pi(mark(X)) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) mark(square(X)) -> a__square(mark(X)) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(posrecip(X)) -> posrecip(mark(X)) mark(negrecip(X)) -> negrecip(mark(X)) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) mark(rnil) -> rnil mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) a__from(X) -> from(X) a__2ndspos(X1, X2) -> 2ndspos(X1, X2) a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) a__pi(X) -> pi(X) a__plus(X1, X2) -> plus(X1, X2) a__times(X1, X2) -> times(X1, X2) a__square(X) -> square(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(rnil) -> rnil encArg(cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2)) encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2)) encArg(posrecip(x_1)) -> posrecip(encArg(x_1)) encArg(negrecip(x_1)) -> negrecip(encArg(x_1)) encArg(2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2)) encArg(2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2)) encArg(pi(x_1)) -> pi(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(square(x_1)) -> square(encArg(x_1)) encArg(nil) -> nil encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__2ndspos(x_1, x_2)) -> a__2ndspos(encArg(x_1), encArg(x_2)) encArg(cons_a__2ndsneg(x_1, x_2)) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encArg(cons_a__pi(x_1)) -> a__pi(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__times(x_1, x_2)) -> a__times(encArg(x_1), encArg(x_2)) encArg(cons_a__square(x_1)) -> a__square(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(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_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__2ndspos(x_1, x_2) -> a__2ndspos(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_rnil -> rnil encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2)) encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2)) encode_posrecip(x_1) -> posrecip(encArg(x_1)) encode_a__2ndsneg(x_1, x_2) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encode_negrecip(x_1) -> negrecip(encArg(x_1)) encode_a__pi(x_1) -> a__pi(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__times(x_1, x_2) -> a__times(encArg(x_1), encArg(x_2)) encode_a__square(x_1) -> a__square(encArg(x_1)) encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2)) encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2)) encode_pi(x_1) -> pi(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_square(x_1) -> square(encArg(x_1)) encode_nil -> nil 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__from(X) -> cons(mark(X), from(s(X))) a__2ndspos(0, Z) -> rnil a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) a__2ndsneg(0, Z) -> rnil a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) a__pi(X) -> a__2ndspos(mark(X), a__from(0)) a__plus(0, Y) -> mark(Y) a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) a__times(0, Y) -> 0 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) a__square(X) -> a__times(mark(X), mark(X)) mark(from(X)) -> a__from(mark(X)) mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) mark(pi(X)) -> a__pi(mark(X)) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) mark(square(X)) -> a__square(mark(X)) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(posrecip(X)) -> posrecip(mark(X)) mark(negrecip(X)) -> negrecip(mark(X)) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) mark(rnil) -> rnil mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) a__from(X) -> from(X) a__2ndspos(X1, X2) -> 2ndspos(X1, X2) a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) a__pi(X) -> pi(X) a__plus(X1, X2) -> plus(X1, X2) a__times(X1, X2) -> times(X1, X2) a__square(X) -> square(X) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(rnil) -> rnil encArg(cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2)) encArg(rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2)) encArg(posrecip(x_1)) -> posrecip(encArg(x_1)) encArg(negrecip(x_1)) -> negrecip(encArg(x_1)) encArg(2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2)) encArg(2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2)) encArg(pi(x_1)) -> pi(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(square(x_1)) -> square(encArg(x_1)) encArg(nil) -> nil encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__2ndspos(x_1, x_2)) -> a__2ndspos(encArg(x_1), encArg(x_2)) encArg(cons_a__2ndsneg(x_1, x_2)) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encArg(cons_a__pi(x_1)) -> a__pi(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__times(x_1, x_2)) -> a__times(encArg(x_1), encArg(x_2)) encArg(cons_a__square(x_1)) -> a__square(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(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_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__2ndspos(x_1, x_2) -> a__2ndspos(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_rnil -> rnil encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2)) encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2)) encode_posrecip(x_1) -> posrecip(encArg(x_1)) encode_a__2ndsneg(x_1, x_2) -> a__2ndsneg(encArg(x_1), encArg(x_2)) encode_negrecip(x_1) -> negrecip(encArg(x_1)) encode_a__pi(x_1) -> a__pi(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__times(x_1, x_2) -> a__times(encArg(x_1), encArg(x_2)) encode_a__square(x_1) -> a__square(encArg(x_1)) encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2)) encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2)) encode_pi(x_1) -> pi(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_square(x_1) -> square(encArg(x_1)) encode_nil -> nil Rewrite Strategy: INNERMOST