/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 138 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 (11) DecreasingLoopProof [FINISHED, 65 ms] (12) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: disj(T) -> True disj(F) -> True conj(Or(o1, o2)) -> False conj(T) -> True conj(F) -> True disj(And(a1, a2)) -> conj(And(a1, a2)) disj(Or(t1, t2)) -> and(conj(t1), disj(t1)) conj(And(t1, t2)) -> and(disj(t1), conj(t1)) bool(T) -> True bool(F) -> True bool(And(a1, a2)) -> False bool(Or(o1, o2)) -> False isConsTerm(T, T) -> True isConsTerm(T, F) -> False isConsTerm(T, And(y1, y2)) -> False isConsTerm(T, Or(x1, x2)) -> False isConsTerm(F, T) -> False isConsTerm(F, F) -> True isConsTerm(F, And(y1, y2)) -> False isConsTerm(F, Or(x1, x2)) -> False isConsTerm(And(a1, a2), T) -> False isConsTerm(And(a1, a2), F) -> False isConsTerm(And(a1, a2), And(y1, y2)) -> True isConsTerm(And(a1, a2), Or(x1, x2)) -> False isConsTerm(Or(o1, o2), T) -> False isConsTerm(Or(o1, o2), F) -> False isConsTerm(Or(o1, o2), And(y1, y2)) -> False isConsTerm(Or(o1, o2), Or(x1, x2)) -> True isOp(T) -> False isOp(F) -> False isOp(And(t1, t2)) -> True isOp(Or(t1, t2)) -> True isAnd(T) -> False isAnd(F) -> False isAnd(And(t1, t2)) -> True isAnd(Or(t1, t2)) -> False getSecond(And(t1, t2)) -> t1 getSecond(Or(t1, t2)) -> t1 getFirst(And(t1, t2)) -> t1 getFirst(Or(t1, t2)) -> t1 disjconj(p) -> disj(p) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True 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(EXP, INF). The TRS R consists of the following rules: disj(T) -> True disj(F) -> True conj(Or(o1, o2)) -> False conj(T) -> True conj(F) -> True disj(And(a1, a2)) -> conj(And(a1, a2)) disj(Or(t1, t2)) -> and(conj(t1), disj(t1)) conj(And(t1, t2)) -> and(disj(t1), conj(t1)) bool(T) -> True bool(F) -> True bool(And(a1, a2)) -> False bool(Or(o1, o2)) -> False isConsTerm(T, T) -> True isConsTerm(T, F) -> False isConsTerm(T, And(y1, y2)) -> False isConsTerm(T, Or(x1, x2)) -> False isConsTerm(F, T) -> False isConsTerm(F, F) -> True isConsTerm(F, And(y1, y2)) -> False isConsTerm(F, Or(x1, x2)) -> False isConsTerm(And(a1, a2), T) -> False isConsTerm(And(a1, a2), F) -> False isConsTerm(And(a1, a2), And(y1, y2)) -> True isConsTerm(And(a1, a2), Or(x1, x2)) -> False isConsTerm(Or(o1, o2), T) -> False isConsTerm(Or(o1, o2), F) -> False isConsTerm(Or(o1, o2), And(y1, y2)) -> False isConsTerm(Or(o1, o2), Or(x1, x2)) -> True isOp(T) -> False isOp(F) -> False isOp(And(t1, t2)) -> True isOp(Or(t1, t2)) -> True isAnd(T) -> False isAnd(F) -> False isAnd(And(t1, t2)) -> True isAnd(Or(t1, t2)) -> False getSecond(And(t1, t2)) -> t1 getSecond(Or(t1, t2)) -> t1 getFirst(And(t1, t2)) -> t1 getFirst(Or(t1, t2)) -> t1 disjconj(p) -> disj(p) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True 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(EXP, INF). The TRS R consists of the following rules: disj(T) -> True disj(F) -> True conj(Or(o1, o2)) -> False conj(T) -> True conj(F) -> True disj(And(a1, a2)) -> conj(And(a1, a2)) disj(Or(t1, t2)) -> and(conj(t1), disj(t1)) conj(And(t1, t2)) -> and(disj(t1), conj(t1)) bool(T) -> True bool(F) -> True bool(And(a1, a2)) -> False bool(Or(o1, o2)) -> False isConsTerm(T, T) -> True isConsTerm(T, F) -> False isConsTerm(T, And(y1, y2)) -> False isConsTerm(T, Or(x1, x2)) -> False isConsTerm(F, T) -> False isConsTerm(F, F) -> True isConsTerm(F, And(y1, y2)) -> False isConsTerm(F, Or(x1, x2)) -> False isConsTerm(And(a1, a2), T) -> False isConsTerm(And(a1, a2), F) -> False isConsTerm(And(a1, a2), And(y1, y2)) -> True isConsTerm(And(a1, a2), Or(x1, x2)) -> False isConsTerm(Or(o1, o2), T) -> False isConsTerm(Or(o1, o2), F) -> False isConsTerm(Or(o1, o2), And(y1, y2)) -> False isConsTerm(Or(o1, o2), Or(x1, x2)) -> True isOp(T) -> False isOp(F) -> False isOp(And(t1, t2)) -> True isOp(Or(t1, t2)) -> True isAnd(T) -> False isAnd(F) -> False isAnd(And(t1, t2)) -> True isAnd(Or(t1, t2)) -> False getSecond(And(t1, t2)) -> t1 getSecond(Or(t1, t2)) -> t1 getFirst(And(t1, t2)) -> t1 getFirst(Or(t1, t2)) -> t1 disjconj(p) -> disj(p) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True 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 conj(And(t1, t2)) ->^+ and(disj(t1), conj(t1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [t1 / And(t1, t2)]. 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(EXP, INF). The TRS R consists of the following rules: disj(T) -> True disj(F) -> True conj(Or(o1, o2)) -> False conj(T) -> True conj(F) -> True disj(And(a1, a2)) -> conj(And(a1, a2)) disj(Or(t1, t2)) -> and(conj(t1), disj(t1)) conj(And(t1, t2)) -> and(disj(t1), conj(t1)) bool(T) -> True bool(F) -> True bool(And(a1, a2)) -> False bool(Or(o1, o2)) -> False isConsTerm(T, T) -> True isConsTerm(T, F) -> False isConsTerm(T, And(y1, y2)) -> False isConsTerm(T, Or(x1, x2)) -> False isConsTerm(F, T) -> False isConsTerm(F, F) -> True isConsTerm(F, And(y1, y2)) -> False isConsTerm(F, Or(x1, x2)) -> False isConsTerm(And(a1, a2), T) -> False isConsTerm(And(a1, a2), F) -> False isConsTerm(And(a1, a2), And(y1, y2)) -> True isConsTerm(And(a1, a2), Or(x1, x2)) -> False isConsTerm(Or(o1, o2), T) -> False isConsTerm(Or(o1, o2), F) -> False isConsTerm(Or(o1, o2), And(y1, y2)) -> False isConsTerm(Or(o1, o2), Or(x1, x2)) -> True isOp(T) -> False isOp(F) -> False isOp(And(t1, t2)) -> True isOp(Or(t1, t2)) -> True isAnd(T) -> False isAnd(F) -> False isAnd(And(t1, t2)) -> True isAnd(Or(t1, t2)) -> False getSecond(And(t1, t2)) -> t1 getSecond(Or(t1, t2)) -> t1 getFirst(And(t1, t2)) -> t1 getFirst(Or(t1, t2)) -> t1 disjconj(p) -> disj(p) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True 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(EXP, INF). The TRS R consists of the following rules: disj(T) -> True disj(F) -> True conj(Or(o1, o2)) -> False conj(T) -> True conj(F) -> True disj(And(a1, a2)) -> conj(And(a1, a2)) disj(Or(t1, t2)) -> and(conj(t1), disj(t1)) conj(And(t1, t2)) -> and(disj(t1), conj(t1)) bool(T) -> True bool(F) -> True bool(And(a1, a2)) -> False bool(Or(o1, o2)) -> False isConsTerm(T, T) -> True isConsTerm(T, F) -> False isConsTerm(T, And(y1, y2)) -> False isConsTerm(T, Or(x1, x2)) -> False isConsTerm(F, T) -> False isConsTerm(F, F) -> True isConsTerm(F, And(y1, y2)) -> False isConsTerm(F, Or(x1, x2)) -> False isConsTerm(And(a1, a2), T) -> False isConsTerm(And(a1, a2), F) -> False isConsTerm(And(a1, a2), And(y1, y2)) -> True isConsTerm(And(a1, a2), Or(x1, x2)) -> False isConsTerm(Or(o1, o2), T) -> False isConsTerm(Or(o1, o2), F) -> False isConsTerm(Or(o1, o2), And(y1, y2)) -> False isConsTerm(Or(o1, o2), Or(x1, x2)) -> True isOp(T) -> False isOp(F) -> False isOp(And(t1, t2)) -> True isOp(Or(t1, t2)) -> True isAnd(T) -> False isAnd(F) -> False isAnd(And(t1, t2)) -> True isAnd(Or(t1, t2)) -> False getSecond(And(t1, t2)) -> t1 getSecond(Or(t1, t2)) -> t1 getFirst(And(t1, t2)) -> t1 getFirst(Or(t1, t2)) -> t1 disjconj(p) -> disj(p) The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: INNERMOST ---------------------------------------- (11) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence conj(And(And(a11_0, a22_0), t2)) ->^+ and(conj(And(a11_0, a22_0)), conj(And(a11_0, a22_0))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [a11_0 / And(a11_0, a22_0)]. The result substitution is [t2 / a22_0]. The rewrite sequence conj(And(And(a11_0, a22_0), t2)) ->^+ and(conj(And(a11_0, a22_0)), conj(And(a11_0, a22_0))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [a11_0 / And(a11_0, a22_0)]. The result substitution is [t2 / a22_0]. ---------------------------------------- (12) BOUNDS(EXP, INF)