22.14/7.43 WORST_CASE(Omega(n^1), O(n^1)) 22.14/7.44 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 22.14/7.44 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 22.14/7.44 22.14/7.44 22.14/7.44 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.14/7.44 22.14/7.44 (0) CpxTRS 22.14/7.44 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 22.14/7.44 (2) CpxTRS 22.14/7.44 (3) RelTrsToTrsProof [UPPER BOUND(ID), 1 ms] 22.14/7.44 (4) CpxTRS 22.14/7.44 (5) CpxTrsMatchBoundsTAProof [FINISHED, 152 ms] 22.14/7.44 (6) BOUNDS(1, n^1) 22.14/7.44 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 22.14/7.44 (8) CpxTRS 22.14/7.44 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 22.14/7.44 (10) typed CpxTrs 22.14/7.44 (11) OrderProof [LOWER BOUND(ID), 0 ms] 22.14/7.44 (12) typed CpxTrs 22.14/7.44 (13) RewriteLemmaProof [LOWER BOUND(ID), 529 ms] 22.14/7.44 (14) BEST 22.14/7.44 (15) proven lower bound 22.14/7.44 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 22.14/7.44 (17) BOUNDS(n^1, INF) 22.14/7.44 (18) typed CpxTrs 22.14/7.44 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (0) 22.14/7.44 Obligation: 22.14/7.44 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 22.14/7.44 22.14/7.44 22.14/7.44 The TRS R consists of the following rules: 22.14/7.44 22.14/7.44 active(f(b, X, c)) -> mark(f(X, c, X)) 22.14/7.44 active(c) -> mark(b) 22.14/7.44 active(f(X1, X2, X3)) -> f(X1, active(X2), X3) 22.14/7.44 f(X1, mark(X2), X3) -> mark(f(X1, X2, X3)) 22.14/7.44 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 22.14/7.44 proper(b) -> ok(b) 22.14/7.44 proper(c) -> ok(c) 22.14/7.44 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 22.14/7.44 top(mark(X)) -> top(proper(X)) 22.14/7.44 top(ok(X)) -> top(active(X)) 22.14/7.44 22.14/7.44 S is empty. 22.14/7.44 Rewrite Strategy: FULL 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 22.14/7.44 The following defined symbols can occur below the 0th argument of top: proper, active 22.14/7.44 The following defined symbols can occur below the 0th argument of proper: proper, active 22.14/7.44 The following defined symbols can occur below the 0th argument of active: proper, active 22.14/7.44 22.14/7.44 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 22.14/7.44 active(f(b, X, c)) -> mark(f(X, c, X)) 22.14/7.44 active(f(X1, X2, X3)) -> f(X1, active(X2), X3) 22.14/7.44 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (2) 22.14/7.44 Obligation: 22.14/7.44 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 22.14/7.44 22.14/7.44 22.14/7.44 The TRS R consists of the following rules: 22.14/7.44 22.14/7.44 active(c) -> mark(b) 22.14/7.44 f(X1, mark(X2), X3) -> mark(f(X1, X2, X3)) 22.14/7.44 proper(b) -> ok(b) 22.14/7.44 proper(c) -> ok(c) 22.14/7.44 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 22.14/7.44 top(mark(X)) -> top(proper(X)) 22.14/7.44 top(ok(X)) -> top(active(X)) 22.14/7.44 22.14/7.44 S is empty. 22.14/7.44 Rewrite Strategy: FULL 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 22.14/7.44 transformed relative TRS to TRS 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (4) 22.14/7.44 Obligation: 22.14/7.44 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 22.14/7.44 22.14/7.44 22.14/7.44 The TRS R consists of the following rules: 22.14/7.44 22.14/7.44 active(c) -> mark(b) 22.14/7.44 f(X1, mark(X2), X3) -> mark(f(X1, X2, X3)) 22.14/7.44 proper(b) -> ok(b) 22.14/7.44 proper(c) -> ok(c) 22.14/7.44 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 22.14/7.44 top(mark(X)) -> top(proper(X)) 22.14/7.44 top(ok(X)) -> top(active(X)) 22.14/7.44 22.14/7.44 S is empty. 22.14/7.44 Rewrite Strategy: FULL 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (5) CpxTrsMatchBoundsTAProof (FINISHED) 22.14/7.44 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 4. 22.14/7.44 22.14/7.44 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 22.14/7.44 final states : [1, 2, 3, 4] 22.14/7.44 transitions: 22.14/7.44 c0() -> 0 22.14/7.44 mark0(0) -> 0 22.14/7.44 b0() -> 0 22.14/7.44 ok0(0) -> 0 22.14/7.44 active0(0) -> 1 22.14/7.44 f0(0, 0, 0) -> 2 22.14/7.44 proper0(0) -> 3 22.14/7.44 top0(0) -> 4 22.14/7.44 b1() -> 5 22.14/7.44 mark1(5) -> 1 22.14/7.44 f1(0, 0, 0) -> 6 22.14/7.44 mark1(6) -> 2 22.14/7.44 b1() -> 7 22.14/7.44 ok1(7) -> 3 22.14/7.44 c1() -> 8 22.14/7.44 ok1(8) -> 3 22.14/7.44 f1(0, 0, 0) -> 9 22.14/7.44 ok1(9) -> 2 22.14/7.44 proper1(0) -> 10 22.14/7.44 top1(10) -> 4 22.14/7.44 active1(0) -> 11 22.14/7.44 top1(11) -> 4 22.14/7.44 mark1(5) -> 11 22.14/7.44 mark1(6) -> 6 22.14/7.44 mark1(6) -> 9 22.14/7.44 ok1(7) -> 10 22.14/7.44 ok1(8) -> 10 22.14/7.44 ok1(9) -> 6 22.14/7.44 ok1(9) -> 9 22.14/7.44 proper2(5) -> 12 22.14/7.44 top2(12) -> 4 22.14/7.44 active2(7) -> 13 22.14/7.44 top2(13) -> 4 22.14/7.44 active2(8) -> 13 22.14/7.44 b2() -> 14 22.14/7.44 mark2(14) -> 13 22.14/7.44 b2() -> 15 22.14/7.44 ok2(15) -> 12 22.14/7.44 proper3(14) -> 16 22.14/7.44 top3(16) -> 4 22.14/7.44 active3(15) -> 17 22.14/7.44 top3(17) -> 4 22.14/7.44 b3() -> 18 22.14/7.44 ok3(18) -> 16 22.14/7.44 active4(18) -> 19 22.14/7.44 top4(19) -> 4 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (6) 22.14/7.44 BOUNDS(1, n^1) 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 22.14/7.44 Renamed function symbols to avoid clashes with predefined symbol. 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (8) 22.14/7.44 Obligation: 22.14/7.44 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 22.14/7.44 22.14/7.44 22.14/7.44 The TRS R consists of the following rules: 22.14/7.44 22.14/7.44 active(f(b, X, c)) -> mark(f(X, c, X)) 22.14/7.44 active(c) -> mark(b) 22.14/7.44 active(f(X1, X2, X3)) -> f(X1, active(X2), X3) 22.14/7.44 f(X1, mark(X2), X3) -> mark(f(X1, X2, X3)) 22.14/7.44 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 22.14/7.44 proper(b) -> ok(b) 22.14/7.44 proper(c) -> ok(c) 22.14/7.44 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 22.14/7.44 top(mark(X)) -> top(proper(X)) 22.14/7.44 top(ok(X)) -> top(active(X)) 22.14/7.44 22.14/7.44 S is empty. 22.14/7.44 Rewrite Strategy: FULL 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 22.14/7.44 Infered types. 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (10) 22.14/7.44 Obligation: 22.14/7.44 TRS: 22.14/7.44 Rules: 22.14/7.44 active(f(b, X, c)) -> mark(f(X, c, X)) 22.14/7.44 active(c) -> mark(b) 22.14/7.44 active(f(X1, X2, X3)) -> f(X1, active(X2), X3) 22.14/7.44 f(X1, mark(X2), X3) -> mark(f(X1, X2, X3)) 22.14/7.44 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 22.14/7.44 proper(b) -> ok(b) 22.14/7.44 proper(c) -> ok(c) 22.14/7.44 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 22.14/7.44 top(mark(X)) -> top(proper(X)) 22.14/7.44 top(ok(X)) -> top(active(X)) 22.14/7.44 22.14/7.44 Types: 22.14/7.44 active :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 f :: b:c:mark:ok -> b:c:mark:ok -> b:c:mark:ok -> b:c:mark:ok 22.14/7.44 b :: b:c:mark:ok 22.14/7.44 c :: b:c:mark:ok 22.14/7.44 mark :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 proper :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 ok :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 top :: b:c:mark:ok -> top 22.14/7.44 hole_b:c:mark:ok1_0 :: b:c:mark:ok 22.14/7.44 hole_top2_0 :: top 22.14/7.44 gen_b:c:mark:ok3_0 :: Nat -> b:c:mark:ok 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (11) OrderProof (LOWER BOUND(ID)) 22.14/7.44 Heuristically decided to analyse the following defined symbols: 22.14/7.44 active, f, proper, top 22.14/7.44 22.14/7.44 They will be analysed ascendingly in the following order: 22.14/7.44 f < active 22.14/7.44 active < top 22.14/7.44 f < proper 22.14/7.44 proper < top 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (12) 22.14/7.44 Obligation: 22.14/7.44 TRS: 22.14/7.44 Rules: 22.14/7.44 active(f(b, X, c)) -> mark(f(X, c, X)) 22.14/7.44 active(c) -> mark(b) 22.14/7.44 active(f(X1, X2, X3)) -> f(X1, active(X2), X3) 22.14/7.44 f(X1, mark(X2), X3) -> mark(f(X1, X2, X3)) 22.14/7.44 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 22.14/7.44 proper(b) -> ok(b) 22.14/7.44 proper(c) -> ok(c) 22.14/7.44 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 22.14/7.44 top(mark(X)) -> top(proper(X)) 22.14/7.44 top(ok(X)) -> top(active(X)) 22.14/7.44 22.14/7.44 Types: 22.14/7.44 active :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 f :: b:c:mark:ok -> b:c:mark:ok -> b:c:mark:ok -> b:c:mark:ok 22.14/7.44 b :: b:c:mark:ok 22.14/7.44 c :: b:c:mark:ok 22.14/7.44 mark :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 proper :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 ok :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 top :: b:c:mark:ok -> top 22.14/7.44 hole_b:c:mark:ok1_0 :: b:c:mark:ok 22.14/7.44 hole_top2_0 :: top 22.14/7.44 gen_b:c:mark:ok3_0 :: Nat -> b:c:mark:ok 22.14/7.44 22.14/7.44 22.14/7.44 Generator Equations: 22.14/7.44 gen_b:c:mark:ok3_0(0) <=> b 22.14/7.44 gen_b:c:mark:ok3_0(+(x, 1)) <=> mark(gen_b:c:mark:ok3_0(x)) 22.14/7.44 22.14/7.44 22.14/7.44 The following defined symbols remain to be analysed: 22.14/7.44 f, active, proper, top 22.14/7.44 22.14/7.44 They will be analysed ascendingly in the following order: 22.14/7.44 f < active 22.14/7.44 active < top 22.14/7.44 f < proper 22.14/7.44 proper < top 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (13) RewriteLemmaProof (LOWER BOUND(ID)) 22.14/7.44 Proved the following rewrite lemma: 22.14/7.44 f(gen_b:c:mark:ok3_0(a), gen_b:c:mark:ok3_0(+(1, n5_0)), gen_b:c:mark:ok3_0(c)) -> *4_0, rt in Omega(n5_0) 22.14/7.44 22.14/7.44 Induction Base: 22.14/7.44 f(gen_b:c:mark:ok3_0(a), gen_b:c:mark:ok3_0(+(1, 0)), gen_b:c:mark:ok3_0(c)) 22.14/7.44 22.14/7.44 Induction Step: 22.14/7.44 f(gen_b:c:mark:ok3_0(a), gen_b:c:mark:ok3_0(+(1, +(n5_0, 1))), gen_b:c:mark:ok3_0(c)) ->_R^Omega(1) 22.14/7.44 mark(f(gen_b:c:mark:ok3_0(a), gen_b:c:mark:ok3_0(+(1, n5_0)), gen_b:c:mark:ok3_0(c))) ->_IH 22.14/7.44 mark(*4_0) 22.14/7.44 22.14/7.44 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (14) 22.14/7.44 Complex Obligation (BEST) 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (15) 22.14/7.44 Obligation: 22.14/7.44 Proved the lower bound n^1 for the following obligation: 22.14/7.44 22.14/7.44 TRS: 22.14/7.44 Rules: 22.14/7.44 active(f(b, X, c)) -> mark(f(X, c, X)) 22.14/7.44 active(c) -> mark(b) 22.14/7.44 active(f(X1, X2, X3)) -> f(X1, active(X2), X3) 22.14/7.44 f(X1, mark(X2), X3) -> mark(f(X1, X2, X3)) 22.14/7.44 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 22.14/7.44 proper(b) -> ok(b) 22.14/7.44 proper(c) -> ok(c) 22.14/7.44 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 22.14/7.44 top(mark(X)) -> top(proper(X)) 22.14/7.44 top(ok(X)) -> top(active(X)) 22.14/7.44 22.14/7.44 Types: 22.14/7.44 active :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 f :: b:c:mark:ok -> b:c:mark:ok -> b:c:mark:ok -> b:c:mark:ok 22.14/7.44 b :: b:c:mark:ok 22.14/7.44 c :: b:c:mark:ok 22.14/7.44 mark :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 proper :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 ok :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 top :: b:c:mark:ok -> top 22.14/7.44 hole_b:c:mark:ok1_0 :: b:c:mark:ok 22.14/7.44 hole_top2_0 :: top 22.14/7.44 gen_b:c:mark:ok3_0 :: Nat -> b:c:mark:ok 22.14/7.44 22.14/7.44 22.14/7.44 Generator Equations: 22.14/7.44 gen_b:c:mark:ok3_0(0) <=> b 22.14/7.44 gen_b:c:mark:ok3_0(+(x, 1)) <=> mark(gen_b:c:mark:ok3_0(x)) 22.14/7.44 22.14/7.44 22.14/7.44 The following defined symbols remain to be analysed: 22.14/7.44 f, active, proper, top 22.14/7.44 22.14/7.44 They will be analysed ascendingly in the following order: 22.14/7.44 f < active 22.14/7.44 active < top 22.14/7.44 f < proper 22.14/7.44 proper < top 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (16) LowerBoundPropagationProof (FINISHED) 22.14/7.44 Propagated lower bound. 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (17) 22.14/7.44 BOUNDS(n^1, INF) 22.14/7.44 22.14/7.44 ---------------------------------------- 22.14/7.44 22.14/7.44 (18) 22.14/7.44 Obligation: 22.14/7.44 TRS: 22.14/7.44 Rules: 22.14/7.44 active(f(b, X, c)) -> mark(f(X, c, X)) 22.14/7.44 active(c) -> mark(b) 22.14/7.44 active(f(X1, X2, X3)) -> f(X1, active(X2), X3) 22.14/7.44 f(X1, mark(X2), X3) -> mark(f(X1, X2, X3)) 22.14/7.44 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 22.14/7.44 proper(b) -> ok(b) 22.14/7.44 proper(c) -> ok(c) 22.14/7.44 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 22.14/7.44 top(mark(X)) -> top(proper(X)) 22.14/7.44 top(ok(X)) -> top(active(X)) 22.14/7.44 22.14/7.44 Types: 22.14/7.44 active :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 f :: b:c:mark:ok -> b:c:mark:ok -> b:c:mark:ok -> b:c:mark:ok 22.14/7.44 b :: b:c:mark:ok 22.14/7.44 c :: b:c:mark:ok 22.14/7.44 mark :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 proper :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 ok :: b:c:mark:ok -> b:c:mark:ok 22.14/7.44 top :: b:c:mark:ok -> top 22.14/7.44 hole_b:c:mark:ok1_0 :: b:c:mark:ok 22.14/7.44 hole_top2_0 :: top 22.14/7.44 gen_b:c:mark:ok3_0 :: Nat -> b:c:mark:ok 22.14/7.44 22.14/7.44 22.14/7.44 Lemmas: 22.14/7.44 f(gen_b:c:mark:ok3_0(a), gen_b:c:mark:ok3_0(+(1, n5_0)), gen_b:c:mark:ok3_0(c)) -> *4_0, rt in Omega(n5_0) 22.14/7.44 22.14/7.44 22.14/7.44 Generator Equations: 22.14/7.44 gen_b:c:mark:ok3_0(0) <=> b 22.14/7.44 gen_b:c:mark:ok3_0(+(x, 1)) <=> mark(gen_b:c:mark:ok3_0(x)) 22.14/7.44 22.14/7.44 22.14/7.44 The following defined symbols remain to be analysed: 22.14/7.44 active, proper, top 22.14/7.44 22.14/7.44 They will be analysed ascendingly in the following order: 22.14/7.44 active < top 22.14/7.44 proper < top 22.41/9.15 EOF