/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF).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), 831 ms] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 343 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, 20.2 s] (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: head(Cons(x, xs)) -> x e(Cons(F, Nil), b) -> False e(Cons(T, Nil), b) -> False e(Cons(B, Nil), b) -> False e(Cons(A, Nil), b) -> e[Match][Cons][Ite][True][Match](A, Nil, b) e(Cons(F, Cons(x, xs)), b) -> False e(Cons(T, Cons(x, xs)), b) -> False e(Cons(B, Cons(x, xs)), b) -> False e(Cons(A, Cons(x, xs)), b) -> False equal(F, F) -> True equal(F, T) -> False equal(F, B) -> False equal(F, A) -> False equal(T, F) -> False equal(T, T) -> True equal(T, B) -> False equal(T, A) -> False equal(B, F) -> False equal(B, T) -> False equal(B, B) -> True equal(B, A) -> False equal(A, F) -> False equal(A, T) -> False equal(A, B) -> False equal(A, A) -> True notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False e(Nil, b) -> False t(x, y) -> t[Ite](e(x, y), x, y) r(x, y) -> r[Ite](e(x, y), x, y) q(x, y) -> q[Ite](e(x, y), x, y) p(x, y) -> p[Ite](e(x, y), x, y) goal(x, y) -> q(x, y) 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 q[Ite](False, x', Cons(F, Cons(F, xs))) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(F, Cons(F, xs))), q(x', Cons(F, xs))), x', Cons(F, Cons(F, xs))) q[Ite](False, x, Cons(F, Cons(T, xs))) -> False q[Ite](False, x, Cons(F, Cons(B, xs))) -> False q[Ite](False, x, Cons(F, Cons(A, xs))) -> False q[Ite](False, x, Cons(T, Cons(F, xs))) -> False q[Ite](False, x, Cons(T, Cons(T, xs))) -> False q[Ite](False, x, Cons(T, Cons(B, xs))) -> False q[Ite](False, x, Cons(T, Cons(A, xs))) -> False q[Ite](False, x, Cons(B, Cons(F, xs))) -> False q[Ite](False, x, Cons(B, Cons(T, xs))) -> False q[Ite](False, x, Cons(B, Cons(B, xs))) -> False q[Ite](False, x, Cons(B, Cons(A, xs))) -> False q[Ite](False, x, Cons(A, Cons(F, xs))) -> False q[Ite](False, x, Cons(A, Cons(T, xs))) -> False q[Ite](False, x, Cons(A, Cons(B, xs))) -> False q[Ite](False, x, Cons(A, Cons(A, xs))) -> False q[Ite](False, x', Cons(F, Nil)) -> q[Ite][False][Ite](and(True, and(True, and(False, equal(head(Nil), F)))), x', Cons(F, Nil)) q[Ite](False, x', Cons(T, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(T, Nil)) q[Ite](False, x', Cons(B, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(B, Nil)) q[Ite](False, x', Cons(A, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(A, Nil)) r[Ite](False, x', Cons(F, xs)) -> r[Ite][False][Ite][True][Ite](and(q(x', xs), r(x', xs)), x', Cons(F, xs)) r[Ite](False, x, Cons(T, xs)) -> False r[Ite](False, x, Cons(B, xs)) -> False r[Ite](False, x, Cons(A, xs)) -> False p[Ite](False, x', Cons(F, xs)) -> and(r(x', Cons(F, xs)), p(x', xs)) p[Ite](False, x, Cons(T, xs)) -> False p[Ite](False, x, Cons(B, xs)) -> False p[Ite](False, x, Cons(A, xs)) -> False q[Ite][False][Ite](True, x', Cons(x, xs)) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(x, xs)), q(x', xs)), x', Cons(x, xs)) t[Ite](False, x, y) -> True t[Ite](True, x, y) -> True r[Ite](True, x, y) -> True q[Ite](True, x, y) -> True q[Ite][False][Ite](False, x, y) -> False p[Ite](True, x, y) -> 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: head(Cons(x, xs)) -> x e(Cons(F, Nil), b) -> False e(Cons(T, Nil), b) -> False e(Cons(B, Nil), b) -> False e(Cons(A, Nil), b) -> e[Match][Cons][Ite][True][Match](A, Nil, b) e(Cons(F, Cons(x, xs)), b) -> False e(Cons(T, Cons(x, xs)), b) -> False e(Cons(B, Cons(x, xs)), b) -> False e(Cons(A, Cons(x, xs)), b) -> False equal(F, F) -> True equal(F, T) -> False equal(F, B) -> False equal(F, A) -> False equal(T, F) -> False equal(T, T) -> True equal(T, B) -> False equal(T, A) -> False equal(B, F) -> False equal(B, T) -> False equal(B, B) -> True equal(B, A) -> False equal(A, F) -> False equal(A, T) -> False equal(A, B) -> False equal(A, A) -> True notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False e(Nil, b) -> False t(x, y) -> t[Ite](e(x, y), x, y) r(x, y) -> r[Ite](e(x, y), x, y) q(x, y) -> q[Ite](e(x, y), x, y) p(x, y) -> p[Ite](e(x, y), x, y) goal(x, y) -> q(x, y) 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 q[Ite](False, x', Cons(F, Cons(F, xs))) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(F, Cons(F, xs))), q(x', Cons(F, xs))), x', Cons(F, Cons(F, xs))) q[Ite](False, x, Cons(F, Cons(T, xs))) -> False q[Ite](False, x, Cons(F, Cons(B, xs))) -> False q[Ite](False, x, Cons(F, Cons(A, xs))) -> False q[Ite](False, x, Cons(T, Cons(F, xs))) -> False q[Ite](False, x, Cons(T, Cons(T, xs))) -> False q[Ite](False, x, Cons(T, Cons(B, xs))) -> False q[Ite](False, x, Cons(T, Cons(A, xs))) -> False q[Ite](False, x, Cons(B, Cons(F, xs))) -> False q[Ite](False, x, Cons(B, Cons(T, xs))) -> False q[Ite](False, x, Cons(B, Cons(B, xs))) -> False q[Ite](False, x, Cons(B, Cons(A, xs))) -> False q[Ite](False, x, Cons(A, Cons(F, xs))) -> False q[Ite](False, x, Cons(A, Cons(T, xs))) -> False q[Ite](False, x, Cons(A, Cons(B, xs))) -> False q[Ite](False, x, Cons(A, Cons(A, xs))) -> False q[Ite](False, x', Cons(F, Nil)) -> q[Ite][False][Ite](and(True, and(True, and(False, equal(head(Nil), F)))), x', Cons(F, Nil)) q[Ite](False, x', Cons(T, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(T, Nil)) q[Ite](False, x', Cons(B, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(B, Nil)) q[Ite](False, x', Cons(A, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(A, Nil)) r[Ite](False, x', Cons(F, xs)) -> r[Ite][False][Ite][True][Ite](and(q(x', xs), r(x', xs)), x', Cons(F, xs)) r[Ite](False, x, Cons(T, xs)) -> False r[Ite](False, x, Cons(B, xs)) -> False r[Ite](False, x, Cons(A, xs)) -> False p[Ite](False, x', Cons(F, xs)) -> and(r(x', Cons(F, xs)), p(x', xs)) p[Ite](False, x, Cons(T, xs)) -> False p[Ite](False, x, Cons(B, xs)) -> False p[Ite](False, x, Cons(A, xs)) -> False q[Ite][False][Ite](True, x', Cons(x, xs)) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(x, xs)), q(x', xs)), x', Cons(x, xs)) t[Ite](False, x, y) -> True t[Ite](True, x, y) -> True r[Ite](True, x, y) -> True q[Ite](True, x, y) -> True q[Ite][False][Ite](False, x, y) -> False p[Ite](True, x, y) -> 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: head(Cons(x, xs)) -> x e(Cons(F, Nil), b) -> False e(Cons(T, Nil), b) -> False e(Cons(B, Nil), b) -> False e(Cons(A, Nil), b) -> e[Match][Cons][Ite][True][Match](A, Nil, b) e(Cons(F, Cons(x, xs)), b) -> False e(Cons(T, Cons(x, xs)), b) -> False e(Cons(B, Cons(x, xs)), b) -> False e(Cons(A, Cons(x, xs)), b) -> False equal(F, F) -> True equal(F, T) -> False equal(F, B) -> False equal(F, A) -> False equal(T, F) -> False equal(T, T) -> True equal(T, B) -> False equal(T, A) -> False equal(B, F) -> False equal(B, T) -> False equal(B, B) -> True equal(B, A) -> False equal(A, F) -> False equal(A, T) -> False equal(A, B) -> False equal(A, A) -> True notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False e(Nil, b) -> False t(x, y) -> t[Ite](e(x, y), x, y) r(x, y) -> r[Ite](e(x, y), x, y) q(x, y) -> q[Ite](e(x, y), x, y) p(x, y) -> p[Ite](e(x, y), x, y) goal(x, y) -> q(x, y) 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 q[Ite](False, x', Cons(F, Cons(F, xs))) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(F, Cons(F, xs))), q(x', Cons(F, xs))), x', Cons(F, Cons(F, xs))) q[Ite](False, x, Cons(F, Cons(T, xs))) -> False q[Ite](False, x, Cons(F, Cons(B, xs))) -> False q[Ite](False, x, Cons(F, Cons(A, xs))) -> False q[Ite](False, x, Cons(T, Cons(F, xs))) -> False q[Ite](False, x, Cons(T, Cons(T, xs))) -> False q[Ite](False, x, Cons(T, Cons(B, xs))) -> False q[Ite](False, x, Cons(T, Cons(A, xs))) -> False q[Ite](False, x, Cons(B, Cons(F, xs))) -> False q[Ite](False, x, Cons(B, Cons(T, xs))) -> False q[Ite](False, x, Cons(B, Cons(B, xs))) -> False q[Ite](False, x, Cons(B, Cons(A, xs))) -> False q[Ite](False, x, Cons(A, Cons(F, xs))) -> False q[Ite](False, x, Cons(A, Cons(T, xs))) -> False q[Ite](False, x, Cons(A, Cons(B, xs))) -> False q[Ite](False, x, Cons(A, Cons(A, xs))) -> False q[Ite](False, x', Cons(F, Nil)) -> q[Ite][False][Ite](and(True, and(True, and(False, equal(head(Nil), F)))), x', Cons(F, Nil)) q[Ite](False, x', Cons(T, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(T, Nil)) q[Ite](False, x', Cons(B, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(B, Nil)) q[Ite](False, x', Cons(A, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(A, Nil)) r[Ite](False, x', Cons(F, xs)) -> r[Ite][False][Ite][True][Ite](and(q(x', xs), r(x', xs)), x', Cons(F, xs)) r[Ite](False, x, Cons(T, xs)) -> False r[Ite](False, x, Cons(B, xs)) -> False r[Ite](False, x, Cons(A, xs)) -> False p[Ite](False, x', Cons(F, xs)) -> and(r(x', Cons(F, xs)), p(x', xs)) p[Ite](False, x, Cons(T, xs)) -> False p[Ite](False, x, Cons(B, xs)) -> False p[Ite](False, x, Cons(A, xs)) -> False q[Ite][False][Ite](True, x', Cons(x, xs)) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(x, xs)), q(x', xs)), x', Cons(x, xs)) t[Ite](False, x, y) -> True t[Ite](True, x, y) -> True r[Ite](True, x, y) -> True q[Ite](True, x, y) -> True q[Ite][False][Ite](False, x, y) -> False p[Ite](True, x, y) -> 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 r(Cons(B, Cons(x1_0, xs2_0)), Cons(F, xs2_1)) ->^+ r[Ite][False][Ite][True][Ite](and(q(Cons(B, Cons(x1_0, xs2_0)), xs2_1), r(Cons(B, Cons(x1_0, xs2_0)), xs2_1)), Cons(B, Cons(x1_0, xs2_0)), Cons(F, xs2_1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1]. The pumping substitution is [xs2_1 / Cons(F, xs2_1)]. 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: head(Cons(x, xs)) -> x e(Cons(F, Nil), b) -> False e(Cons(T, Nil), b) -> False e(Cons(B, Nil), b) -> False e(Cons(A, Nil), b) -> e[Match][Cons][Ite][True][Match](A, Nil, b) e(Cons(F, Cons(x, xs)), b) -> False e(Cons(T, Cons(x, xs)), b) -> False e(Cons(B, Cons(x, xs)), b) -> False e(Cons(A, Cons(x, xs)), b) -> False equal(F, F) -> True equal(F, T) -> False equal(F, B) -> False equal(F, A) -> False equal(T, F) -> False equal(T, T) -> True equal(T, B) -> False equal(T, A) -> False equal(B, F) -> False equal(B, T) -> False equal(B, B) -> True equal(B, A) -> False equal(A, F) -> False equal(A, T) -> False equal(A, B) -> False equal(A, A) -> True notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False e(Nil, b) -> False t(x, y) -> t[Ite](e(x, y), x, y) r(x, y) -> r[Ite](e(x, y), x, y) q(x, y) -> q[Ite](e(x, y), x, y) p(x, y) -> p[Ite](e(x, y), x, y) goal(x, y) -> q(x, y) 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 q[Ite](False, x', Cons(F, Cons(F, xs))) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(F, Cons(F, xs))), q(x', Cons(F, xs))), x', Cons(F, Cons(F, xs))) q[Ite](False, x, Cons(F, Cons(T, xs))) -> False q[Ite](False, x, Cons(F, Cons(B, xs))) -> False q[Ite](False, x, Cons(F, Cons(A, xs))) -> False q[Ite](False, x, Cons(T, Cons(F, xs))) -> False q[Ite](False, x, Cons(T, Cons(T, xs))) -> False q[Ite](False, x, Cons(T, Cons(B, xs))) -> False q[Ite](False, x, Cons(T, Cons(A, xs))) -> False q[Ite](False, x, Cons(B, Cons(F, xs))) -> False q[Ite](False, x, Cons(B, Cons(T, xs))) -> False q[Ite](False, x, Cons(B, Cons(B, xs))) -> False q[Ite](False, x, Cons(B, Cons(A, xs))) -> False q[Ite](False, x, Cons(A, Cons(F, xs))) -> False q[Ite](False, x, Cons(A, Cons(T, xs))) -> False q[Ite](False, x, Cons(A, Cons(B, xs))) -> False q[Ite](False, x, Cons(A, Cons(A, xs))) -> False q[Ite](False, x', Cons(F, Nil)) -> q[Ite][False][Ite](and(True, and(True, and(False, equal(head(Nil), F)))), x', Cons(F, Nil)) q[Ite](False, x', Cons(T, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(T, Nil)) q[Ite](False, x', Cons(B, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(B, Nil)) q[Ite](False, x', Cons(A, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(A, Nil)) r[Ite](False, x', Cons(F, xs)) -> r[Ite][False][Ite][True][Ite](and(q(x', xs), r(x', xs)), x', Cons(F, xs)) r[Ite](False, x, Cons(T, xs)) -> False r[Ite](False, x, Cons(B, xs)) -> False r[Ite](False, x, Cons(A, xs)) -> False p[Ite](False, x', Cons(F, xs)) -> and(r(x', Cons(F, xs)), p(x', xs)) p[Ite](False, x, Cons(T, xs)) -> False p[Ite](False, x, Cons(B, xs)) -> False p[Ite](False, x, Cons(A, xs)) -> False q[Ite][False][Ite](True, x', Cons(x, xs)) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(x, xs)), q(x', xs)), x', Cons(x, xs)) t[Ite](False, x, y) -> True t[Ite](True, x, y) -> True r[Ite](True, x, y) -> True q[Ite](True, x, y) -> True q[Ite][False][Ite](False, x, y) -> False p[Ite](True, x, y) -> 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: head(Cons(x, xs)) -> x e(Cons(F, Nil), b) -> False e(Cons(T, Nil), b) -> False e(Cons(B, Nil), b) -> False e(Cons(A, Nil), b) -> e[Match][Cons][Ite][True][Match](A, Nil, b) e(Cons(F, Cons(x, xs)), b) -> False e(Cons(T, Cons(x, xs)), b) -> False e(Cons(B, Cons(x, xs)), b) -> False e(Cons(A, Cons(x, xs)), b) -> False equal(F, F) -> True equal(F, T) -> False equal(F, B) -> False equal(F, A) -> False equal(T, F) -> False equal(T, T) -> True equal(T, B) -> False equal(T, A) -> False equal(B, F) -> False equal(B, T) -> False equal(B, B) -> True equal(B, A) -> False equal(A, F) -> False equal(A, T) -> False equal(A, B) -> False equal(A, A) -> True notEmpty(Cons(x, xs)) -> True notEmpty(Nil) -> False e(Nil, b) -> False t(x, y) -> t[Ite](e(x, y), x, y) r(x, y) -> r[Ite](e(x, y), x, y) q(x, y) -> q[Ite](e(x, y), x, y) p(x, y) -> p[Ite](e(x, y), x, y) goal(x, y) -> q(x, y) 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 q[Ite](False, x', Cons(F, Cons(F, xs))) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(F, Cons(F, xs))), q(x', Cons(F, xs))), x', Cons(F, Cons(F, xs))) q[Ite](False, x, Cons(F, Cons(T, xs))) -> False q[Ite](False, x, Cons(F, Cons(B, xs))) -> False q[Ite](False, x, Cons(F, Cons(A, xs))) -> False q[Ite](False, x, Cons(T, Cons(F, xs))) -> False q[Ite](False, x, Cons(T, Cons(T, xs))) -> False q[Ite](False, x, Cons(T, Cons(B, xs))) -> False q[Ite](False, x, Cons(T, Cons(A, xs))) -> False q[Ite](False, x, Cons(B, Cons(F, xs))) -> False q[Ite](False, x, Cons(B, Cons(T, xs))) -> False q[Ite](False, x, Cons(B, Cons(B, xs))) -> False q[Ite](False, x, Cons(B, Cons(A, xs))) -> False q[Ite](False, x, Cons(A, Cons(F, xs))) -> False q[Ite](False, x, Cons(A, Cons(T, xs))) -> False q[Ite](False, x, Cons(A, Cons(B, xs))) -> False q[Ite](False, x, Cons(A, Cons(A, xs))) -> False q[Ite](False, x', Cons(F, Nil)) -> q[Ite][False][Ite](and(True, and(True, and(False, equal(head(Nil), F)))), x', Cons(F, Nil)) q[Ite](False, x', Cons(T, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(T, Nil)) q[Ite](False, x', Cons(B, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(B, Nil)) q[Ite](False, x', Cons(A, Nil)) -> q[Ite][False][Ite](and(True, and(False, and(False, equal(head(Nil), F)))), x', Cons(A, Nil)) r[Ite](False, x', Cons(F, xs)) -> r[Ite][False][Ite][True][Ite](and(q(x', xs), r(x', xs)), x', Cons(F, xs)) r[Ite](False, x, Cons(T, xs)) -> False r[Ite](False, x, Cons(B, xs)) -> False r[Ite](False, x, Cons(A, xs)) -> False p[Ite](False, x', Cons(F, xs)) -> and(r(x', Cons(F, xs)), p(x', xs)) p[Ite](False, x, Cons(T, xs)) -> False p[Ite](False, x, Cons(B, xs)) -> False p[Ite](False, x, Cons(A, xs)) -> False q[Ite][False][Ite](True, x', Cons(x, xs)) -> q[Ite][False][Ite][True][Ite](and(p(x', Cons(x, xs)), q(x', xs)), x', Cons(x, xs)) t[Ite](False, x, y) -> True t[Ite](True, x, y) -> True r[Ite](True, x, y) -> True q[Ite](True, x, y) -> True q[Ite][False][Ite](False, x, y) -> False p[Ite](True, x, y) -> True Rewrite Strategy: INNERMOST ---------------------------------------- (11) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence r(Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, Cons(F, xs2_2)))) ->^+ r[Ite][False][Ite][True][Ite](and(q[Ite][False][Ite][True][Ite](and(and(r(Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, xs2_2))), p(Cons(B, Cons(x1_0, xs2_0)), Cons(F, xs2_2))), q(Cons(B, Cons(x1_0, xs2_0)), Cons(F, xs2_2))), Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, xs2_2))), r(Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, xs2_2)))), Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, Cons(F, xs2_2)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0]. The pumping substitution is [xs2_2 / Cons(F, xs2_2)]. The result substitution is [ ]. The rewrite sequence r(Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, Cons(F, xs2_2)))) ->^+ r[Ite][False][Ite][True][Ite](and(q[Ite][False][Ite][True][Ite](and(and(r(Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, xs2_2))), p(Cons(B, Cons(x1_0, xs2_0)), Cons(F, xs2_2))), q(Cons(B, Cons(x1_0, xs2_0)), Cons(F, xs2_2))), Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, xs2_2))), r(Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, xs2_2)))), Cons(B, Cons(x1_0, xs2_0)), Cons(F, Cons(F, Cons(F, xs2_2)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1]. The pumping substitution is [xs2_2 / Cons(F, xs2_2)]. The result substitution is [ ]. ---------------------------------------- (12) BOUNDS(EXP, INF)