/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 94 ms] (6) BOUNDS(1, n^1) (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 438 ms] (14) BEST (15) proven lower bound (16) LowerBoundPropagationProof [FINISHED, 0 ms] (17) BOUNDS(n^1, INF) (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 143 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 67 ms] (22) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(eq(0, 0)) -> mark(true) active(eq(s(X), s(Y))) -> mark(eq(X, Y)) active(eq(X, Y)) -> mark(false) active(inf(X)) -> mark(cons(X, inf(s(X)))) active(take(0, X)) -> mark(nil) active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) active(length(nil)) -> mark(0) active(length(cons(X, L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) proper(0) -> ok(0) proper(true) -> ok(true) proper(s(X)) -> s(proper(X)) proper(false) -> ok(false) proper(inf(X)) -> inf(proper(X)) proper(cons(any(X1), X2)) -> cons(any(any(proper(X1))), any(proper(X2))) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(length(X)) -> length(proper(X)) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: active(eq(0, 0)) -> mark(true) active(eq(s(X), s(Y))) -> mark(eq(X, Y)) active(eq(X, Y)) -> mark(false) active(inf(X)) -> mark(cons(X, inf(s(X)))) active(take(0, X)) -> mark(nil) active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) active(length(nil)) -> mark(0) active(length(cons(X, L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(length(X)) -> length(active(X)) proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(inf(X)) -> inf(proper(X)) proper(cons(any(X1), X2)) -> cons(any(any(proper(X1))), any(proper(X2))) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(length(X)) -> length(proper(X)) any(proper(X)) -> any(any(any(X))) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(0) -> ok(0) proper(true) -> ok(true) proper(false) -> ok(false) proper(nil) -> ok(nil) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(0) -> ok(0) proper(true) -> ok(true) proper(false) -> ok(false) proper(nil) -> ok(nil) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (5) CpxTrsMatchBoundsTAProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4, 5, 6, 7, 8, 9] transitions: mark0(0) -> 0 00() -> 0 ok0(0) -> 0 true0() -> 0 false0() -> 0 nil0() -> 0 active0(0) -> 0 inf0(0) -> 1 take0(0, 0) -> 2 length0(0) -> 3 proper0(0) -> 4 eq0(0, 0) -> 5 s0(0) -> 6 cons0(0, 0) -> 7 top0(0) -> 8 any0(0) -> 9 inf1(0) -> 10 mark1(10) -> 1 take1(0, 0) -> 11 mark1(11) -> 2 length1(0) -> 12 mark1(12) -> 3 01() -> 13 ok1(13) -> 4 true1() -> 14 ok1(14) -> 4 false1() -> 15 ok1(15) -> 4 nil1() -> 16 ok1(16) -> 4 eq1(0, 0) -> 17 ok1(17) -> 5 s1(0) -> 18 ok1(18) -> 6 inf1(0) -> 19 ok1(19) -> 1 cons1(0, 0) -> 20 ok1(20) -> 7 take1(0, 0) -> 21 ok1(21) -> 2 length1(0) -> 22 ok1(22) -> 3 proper1(0) -> 23 top1(23) -> 8 active1(0) -> 24 top1(24) -> 8 s1(0) -> 9 mark1(10) -> 10 mark1(10) -> 19 mark1(11) -> 11 mark1(11) -> 21 mark1(12) -> 12 mark1(12) -> 22 ok1(13) -> 23 ok1(14) -> 23 ok1(15) -> 23 ok1(16) -> 23 ok1(17) -> 17 ok1(18) -> 9 ok1(18) -> 18 ok1(19) -> 10 ok1(19) -> 19 ok1(20) -> 20 ok1(21) -> 11 ok1(21) -> 21 ok1(22) -> 12 ok1(22) -> 22 active2(13) -> 25 top2(25) -> 8 active2(14) -> 25 active2(15) -> 25 active2(16) -> 25 ---------------------------------------- (6) BOUNDS(1, n^1) ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(eq(0', 0')) -> mark(true) active(eq(s(X), s(Y))) -> mark(eq(X, Y)) active(eq(X, Y)) -> mark(false) active(inf(X)) -> mark(cons(X, inf(s(X)))) active(take(0', X)) -> mark(nil) active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) active(length(nil)) -> mark(0') active(length(cons(X, L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) proper(0') -> ok(0') proper(true) -> ok(true) proper(s(X)) -> s(proper(X)) proper(false) -> ok(false) proper(inf(X)) -> inf(proper(X)) proper(cons(any(X1), X2)) -> cons(any(any(proper(X1))), any(proper(X2))) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(length(X)) -> length(proper(X)) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: TRS: Rules: active(eq(0', 0')) -> mark(true) active(eq(s(X), s(Y))) -> mark(eq(X, Y)) active(eq(X, Y)) -> mark(false) active(inf(X)) -> mark(cons(X, inf(s(X)))) active(take(0', X)) -> mark(nil) active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) active(length(nil)) -> mark(0') active(length(cons(X, L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) proper(0') -> ok(0') proper(true) -> ok(true) proper(s(X)) -> s(proper(X)) proper(false) -> ok(false) proper(inf(X)) -> inf(proper(X)) proper(cons(any(X1), X2)) -> cons(any(any(proper(X1))), any(proper(X2))) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(length(X)) -> length(proper(X)) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) Types: active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 0' :: 0':true:mark:false:nil:ok mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok true :: 0':true:mark:false:nil:ok s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok false :: 0':true:mark:false:nil:ok inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok nil :: 0':true:mark:false:nil:ok length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok any :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok top :: 0':true:mark:false:nil:ok -> top hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok hole_top2_0 :: top gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, eq, cons, inf, s, take, length, proper, any, top They will be analysed ascendingly in the following order: eq < active cons < active inf < active s < active take < active length < active active < top eq < proper cons < proper inf < proper s < proper s < any take < proper length < proper any < proper proper < top ---------------------------------------- (12) Obligation: TRS: Rules: active(eq(0', 0')) -> mark(true) active(eq(s(X), s(Y))) -> mark(eq(X, Y)) active(eq(X, Y)) -> mark(false) active(inf(X)) -> mark(cons(X, inf(s(X)))) active(take(0', X)) -> mark(nil) active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) active(length(nil)) -> mark(0') active(length(cons(X, L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) proper(0') -> ok(0') proper(true) -> ok(true) proper(s(X)) -> s(proper(X)) proper(false) -> ok(false) proper(inf(X)) -> inf(proper(X)) proper(cons(any(X1), X2)) -> cons(any(any(proper(X1))), any(proper(X2))) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(length(X)) -> length(proper(X)) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) Types: active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 0' :: 0':true:mark:false:nil:ok mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok true :: 0':true:mark:false:nil:ok s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok false :: 0':true:mark:false:nil:ok inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok nil :: 0':true:mark:false:nil:ok length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok any :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok top :: 0':true:mark:false:nil:ok -> top hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok hole_top2_0 :: top gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok Generator Equations: gen_0':true:mark:false:nil:ok3_0(0) <=> 0' gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) The following defined symbols remain to be analysed: eq, active, cons, inf, s, take, length, proper, any, top They will be analysed ascendingly in the following order: eq < active cons < active inf < active s < active take < active length < active active < top eq < proper cons < proper inf < proper s < proper s < any take < proper length < proper any < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0))) -> *4_0, rt in Omega(n15_0) Induction Base: inf(gen_0':true:mark:false:nil:ok3_0(+(1, 0))) Induction Step: inf(gen_0':true:mark:false:nil:ok3_0(+(1, +(n15_0, 1)))) ->_R^Omega(1) mark(inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Complex Obligation (BEST) ---------------------------------------- (15) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(eq(0', 0')) -> mark(true) active(eq(s(X), s(Y))) -> mark(eq(X, Y)) active(eq(X, Y)) -> mark(false) active(inf(X)) -> mark(cons(X, inf(s(X)))) active(take(0', X)) -> mark(nil) active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) active(length(nil)) -> mark(0') active(length(cons(X, L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) proper(0') -> ok(0') proper(true) -> ok(true) proper(s(X)) -> s(proper(X)) proper(false) -> ok(false) proper(inf(X)) -> inf(proper(X)) proper(cons(any(X1), X2)) -> cons(any(any(proper(X1))), any(proper(X2))) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(length(X)) -> length(proper(X)) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) Types: active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 0' :: 0':true:mark:false:nil:ok mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok true :: 0':true:mark:false:nil:ok s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok false :: 0':true:mark:false:nil:ok inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok nil :: 0':true:mark:false:nil:ok length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok any :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok top :: 0':true:mark:false:nil:ok -> top hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok hole_top2_0 :: top gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok Generator Equations: gen_0':true:mark:false:nil:ok3_0(0) <=> 0' gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) The following defined symbols remain to be analysed: inf, active, s, take, length, proper, any, top They will be analysed ascendingly in the following order: inf < active s < active take < active length < active active < top inf < proper s < proper s < any take < proper length < proper any < proper proper < top ---------------------------------------- (16) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (17) BOUNDS(n^1, INF) ---------------------------------------- (18) Obligation: TRS: Rules: active(eq(0', 0')) -> mark(true) active(eq(s(X), s(Y))) -> mark(eq(X, Y)) active(eq(X, Y)) -> mark(false) active(inf(X)) -> mark(cons(X, inf(s(X)))) active(take(0', X)) -> mark(nil) active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) active(length(nil)) -> mark(0') active(length(cons(X, L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) proper(0') -> ok(0') proper(true) -> ok(true) proper(s(X)) -> s(proper(X)) proper(false) -> ok(false) proper(inf(X)) -> inf(proper(X)) proper(cons(any(X1), X2)) -> cons(any(any(proper(X1))), any(proper(X2))) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(length(X)) -> length(proper(X)) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) Types: active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 0' :: 0':true:mark:false:nil:ok mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok true :: 0':true:mark:false:nil:ok s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok false :: 0':true:mark:false:nil:ok inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok nil :: 0':true:mark:false:nil:ok length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok any :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok top :: 0':true:mark:false:nil:ok -> top hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok hole_top2_0 :: top gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok Lemmas: inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0))) -> *4_0, rt in Omega(n15_0) Generator Equations: gen_0':true:mark:false:nil:ok3_0(0) <=> 0' gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) The following defined symbols remain to be analysed: s, active, take, length, proper, any, top They will be analysed ascendingly in the following order: s < active take < active length < active active < top s < proper s < any take < proper length < proper any < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: take(gen_0':true:mark:false:nil:ok3_0(+(1, n397_0)), gen_0':true:mark:false:nil:ok3_0(b)) -> *4_0, rt in Omega(n397_0) Induction Base: take(gen_0':true:mark:false:nil:ok3_0(+(1, 0)), gen_0':true:mark:false:nil:ok3_0(b)) Induction Step: take(gen_0':true:mark:false:nil:ok3_0(+(1, +(n397_0, 1))), gen_0':true:mark:false:nil:ok3_0(b)) ->_R^Omega(1) mark(take(gen_0':true:mark:false:nil:ok3_0(+(1, n397_0)), gen_0':true:mark:false:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: active(eq(0', 0')) -> mark(true) active(eq(s(X), s(Y))) -> mark(eq(X, Y)) active(eq(X, Y)) -> mark(false) active(inf(X)) -> mark(cons(X, inf(s(X)))) active(take(0', X)) -> mark(nil) active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) active(length(nil)) -> mark(0') active(length(cons(X, L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) proper(0') -> ok(0') proper(true) -> ok(true) proper(s(X)) -> s(proper(X)) proper(false) -> ok(false) proper(inf(X)) -> inf(proper(X)) proper(cons(any(X1), X2)) -> cons(any(any(proper(X1))), any(proper(X2))) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(length(X)) -> length(proper(X)) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) Types: active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 0' :: 0':true:mark:false:nil:ok mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok true :: 0':true:mark:false:nil:ok s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok false :: 0':true:mark:false:nil:ok inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok nil :: 0':true:mark:false:nil:ok length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok any :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok top :: 0':true:mark:false:nil:ok -> top hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok hole_top2_0 :: top gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok Lemmas: inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0))) -> *4_0, rt in Omega(n15_0) take(gen_0':true:mark:false:nil:ok3_0(+(1, n397_0)), gen_0':true:mark:false:nil:ok3_0(b)) -> *4_0, rt in Omega(n397_0) Generator Equations: gen_0':true:mark:false:nil:ok3_0(0) <=> 0' gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) The following defined symbols remain to be analysed: length, active, proper, any, top They will be analysed ascendingly in the following order: length < active active < top length < proper any < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: length(gen_0':true:mark:false:nil:ok3_0(+(1, n1743_0))) -> *4_0, rt in Omega(n1743_0) Induction Base: length(gen_0':true:mark:false:nil:ok3_0(+(1, 0))) Induction Step: length(gen_0':true:mark:false:nil:ok3_0(+(1, +(n1743_0, 1)))) ->_R^Omega(1) mark(length(gen_0':true:mark:false:nil:ok3_0(+(1, n1743_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: TRS: Rules: active(eq(0', 0')) -> mark(true) active(eq(s(X), s(Y))) -> mark(eq(X, Y)) active(eq(X, Y)) -> mark(false) active(inf(X)) -> mark(cons(X, inf(s(X)))) active(take(0', X)) -> mark(nil) active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) active(length(nil)) -> mark(0') active(length(cons(X, L))) -> mark(s(length(L))) active(inf(X)) -> inf(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(length(X)) -> length(active(X)) inf(mark(X)) -> mark(inf(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) length(mark(X)) -> mark(length(X)) proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) proper(0') -> ok(0') proper(true) -> ok(true) proper(s(X)) -> s(proper(X)) proper(false) -> ok(false) proper(inf(X)) -> inf(proper(X)) proper(cons(any(X1), X2)) -> cons(any(any(proper(X1))), any(proper(X2))) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(length(X)) -> length(proper(X)) eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) s(ok(X)) -> ok(s(X)) inf(ok(X)) -> ok(inf(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) any(X) -> s(X) any(proper(X)) -> any(any(any(X))) Types: active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 0' :: 0':true:mark:false:nil:ok mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok true :: 0':true:mark:false:nil:ok s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok false :: 0':true:mark:false:nil:ok inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok nil :: 0':true:mark:false:nil:ok length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok any :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok top :: 0':true:mark:false:nil:ok -> top hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok hole_top2_0 :: top gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok Lemmas: inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0))) -> *4_0, rt in Omega(n15_0) take(gen_0':true:mark:false:nil:ok3_0(+(1, n397_0)), gen_0':true:mark:false:nil:ok3_0(b)) -> *4_0, rt in Omega(n397_0) length(gen_0':true:mark:false:nil:ok3_0(+(1, n1743_0))) -> *4_0, rt in Omega(n1743_0) Generator Equations: gen_0':true:mark:false:nil:ok3_0(0) <=> 0' gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) The following defined symbols remain to be analysed: active, proper, any, top They will be analysed ascendingly in the following order: active < top any < proper proper < top