23.62/8.36 WORST_CASE(Omega(n^1), O(n^1)) 23.62/8.37 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 23.62/8.37 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 23.62/8.37 23.62/8.37 23.62/8.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.62/8.37 23.62/8.37 (0) CpxTRS 23.62/8.37 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 23.62/8.37 (2) CpxTRS 23.62/8.37 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 23.62/8.37 (4) CpxTRS 23.62/8.37 (5) CpxTrsMatchBoundsTAProof [FINISHED, 99 ms] 23.62/8.37 (6) BOUNDS(1, n^1) 23.62/8.37 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 23.62/8.37 (8) CpxTRS 23.62/8.37 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 23.62/8.37 (10) typed CpxTrs 23.62/8.37 (11) OrderProof [LOWER BOUND(ID), 0 ms] 23.62/8.37 (12) typed CpxTrs 23.62/8.37 (13) RewriteLemmaProof [LOWER BOUND(ID), 469 ms] 23.62/8.37 (14) BEST 23.62/8.37 (15) proven lower bound 23.62/8.37 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 23.62/8.37 (17) BOUNDS(n^1, INF) 23.62/8.37 (18) typed CpxTrs 23.62/8.37 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (0) 23.62/8.37 Obligation: 23.62/8.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.62/8.37 23.62/8.37 23.62/8.37 The TRS R consists of the following rules: 23.62/8.37 23.62/8.37 active(f(X, X)) -> mark(f(a, b)) 23.62/8.37 active(b) -> mark(a) 23.62/8.37 active(f(X1, X2)) -> f(active(X1), X2) 23.62/8.37 f(mark(X1), X2) -> mark(f(X1, X2)) 23.62/8.37 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 23.62/8.37 proper(a) -> ok(a) 23.62/8.37 proper(b) -> ok(b) 23.62/8.37 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 23.62/8.37 top(mark(X)) -> top(proper(X)) 23.62/8.37 top(ok(X)) -> top(active(X)) 23.62/8.37 23.62/8.37 S is empty. 23.62/8.37 Rewrite Strategy: FULL 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 23.62/8.37 The following defined symbols can occur below the 0th argument of top: proper, active 23.62/8.37 The following defined symbols can occur below the 0th argument of proper: proper, active 23.62/8.37 The following defined symbols can occur below the 0th argument of active: proper, active 23.62/8.37 23.62/8.37 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 23.62/8.37 active(f(X, X)) -> mark(f(a, b)) 23.62/8.37 active(f(X1, X2)) -> f(active(X1), X2) 23.62/8.37 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (2) 23.62/8.37 Obligation: 23.62/8.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.62/8.37 23.62/8.37 23.62/8.37 The TRS R consists of the following rules: 23.62/8.37 23.62/8.37 active(b) -> mark(a) 23.62/8.37 f(mark(X1), X2) -> mark(f(X1, X2)) 23.62/8.37 proper(a) -> ok(a) 23.62/8.37 proper(b) -> ok(b) 23.62/8.37 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 23.62/8.37 top(mark(X)) -> top(proper(X)) 23.62/8.37 top(ok(X)) -> top(active(X)) 23.62/8.37 23.62/8.37 S is empty. 23.62/8.37 Rewrite Strategy: FULL 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 23.62/8.37 transformed relative TRS to TRS 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (4) 23.62/8.37 Obligation: 23.62/8.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.62/8.37 23.62/8.37 23.62/8.37 The TRS R consists of the following rules: 23.62/8.37 23.62/8.37 active(b) -> mark(a) 23.62/8.37 f(mark(X1), X2) -> mark(f(X1, X2)) 23.62/8.37 proper(a) -> ok(a) 23.62/8.37 proper(b) -> ok(b) 23.62/8.37 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 23.62/8.37 top(mark(X)) -> top(proper(X)) 23.62/8.37 top(ok(X)) -> top(active(X)) 23.62/8.37 23.62/8.37 S is empty. 23.62/8.37 Rewrite Strategy: FULL 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (5) CpxTrsMatchBoundsTAProof (FINISHED) 23.62/8.37 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. 23.62/8.37 23.62/8.37 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 23.62/8.37 final states : [1, 2, 3, 4] 23.62/8.37 transitions: 23.62/8.37 b0() -> 0 23.62/8.37 mark0(0) -> 0 23.62/8.37 a0() -> 0 23.62/8.37 ok0(0) -> 0 23.62/8.37 active0(0) -> 1 23.62/8.37 f0(0, 0) -> 2 23.62/8.37 proper0(0) -> 3 23.62/8.37 top0(0) -> 4 23.62/8.37 a1() -> 5 23.62/8.37 mark1(5) -> 1 23.62/8.37 f1(0, 0) -> 6 23.62/8.37 mark1(6) -> 2 23.62/8.37 a1() -> 7 23.62/8.37 ok1(7) -> 3 23.62/8.37 b1() -> 8 23.62/8.37 ok1(8) -> 3 23.62/8.37 f1(0, 0) -> 9 23.62/8.37 ok1(9) -> 2 23.62/8.37 proper1(0) -> 10 23.62/8.37 top1(10) -> 4 23.62/8.37 active1(0) -> 11 23.62/8.37 top1(11) -> 4 23.62/8.37 mark1(5) -> 11 23.62/8.37 mark1(6) -> 6 23.62/8.37 mark1(6) -> 9 23.62/8.37 ok1(7) -> 10 23.62/8.37 ok1(8) -> 10 23.62/8.37 ok1(9) -> 6 23.62/8.37 ok1(9) -> 9 23.62/8.37 proper2(5) -> 12 23.62/8.37 top2(12) -> 4 23.62/8.37 active2(7) -> 13 23.62/8.37 top2(13) -> 4 23.62/8.37 active2(8) -> 13 23.62/8.37 a2() -> 14 23.62/8.37 mark2(14) -> 13 23.62/8.37 a2() -> 15 23.62/8.37 ok2(15) -> 12 23.62/8.37 proper3(14) -> 16 23.62/8.37 top3(16) -> 4 23.62/8.37 active3(15) -> 17 23.62/8.37 top3(17) -> 4 23.62/8.37 a3() -> 18 23.62/8.37 ok3(18) -> 16 23.62/8.37 active4(18) -> 19 23.62/8.37 top4(19) -> 4 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (6) 23.62/8.37 BOUNDS(1, n^1) 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 23.62/8.37 Renamed function symbols to avoid clashes with predefined symbol. 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (8) 23.62/8.37 Obligation: 23.62/8.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 23.62/8.37 23.62/8.37 23.62/8.37 The TRS R consists of the following rules: 23.62/8.37 23.62/8.37 active(f(X, X)) -> mark(f(a, b)) 23.62/8.37 active(b) -> mark(a) 23.62/8.37 active(f(X1, X2)) -> f(active(X1), X2) 23.62/8.37 f(mark(X1), X2) -> mark(f(X1, X2)) 23.62/8.37 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 23.62/8.37 proper(a) -> ok(a) 23.62/8.37 proper(b) -> ok(b) 23.62/8.37 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 23.62/8.37 top(mark(X)) -> top(proper(X)) 23.62/8.37 top(ok(X)) -> top(active(X)) 23.62/8.37 23.62/8.37 S is empty. 23.62/8.37 Rewrite Strategy: FULL 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 23.62/8.37 Infered types. 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (10) 23.62/8.37 Obligation: 23.62/8.37 TRS: 23.62/8.37 Rules: 23.62/8.37 active(f(X, X)) -> mark(f(a, b)) 23.62/8.37 active(b) -> mark(a) 23.62/8.37 active(f(X1, X2)) -> f(active(X1), X2) 23.62/8.37 f(mark(X1), X2) -> mark(f(X1, X2)) 23.62/8.37 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 23.62/8.37 proper(a) -> ok(a) 23.62/8.37 proper(b) -> ok(b) 23.62/8.37 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 23.62/8.37 top(mark(X)) -> top(proper(X)) 23.62/8.37 top(ok(X)) -> top(active(X)) 23.62/8.37 23.62/8.37 Types: 23.62/8.37 active :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 f :: a:b:mark:ok -> a:b:mark:ok -> a:b:mark:ok 23.62/8.37 mark :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 a :: a:b:mark:ok 23.62/8.37 b :: a:b:mark:ok 23.62/8.37 proper :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 ok :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 top :: a:b:mark:ok -> top 23.62/8.37 hole_a:b:mark:ok1_0 :: a:b:mark:ok 23.62/8.37 hole_top2_0 :: top 23.62/8.37 gen_a:b:mark:ok3_0 :: Nat -> a:b:mark:ok 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (11) OrderProof (LOWER BOUND(ID)) 23.62/8.37 Heuristically decided to analyse the following defined symbols: 23.62/8.37 active, f, proper, top 23.62/8.37 23.62/8.37 They will be analysed ascendingly in the following order: 23.62/8.37 f < active 23.62/8.37 active < top 23.62/8.37 f < proper 23.62/8.37 proper < top 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (12) 23.62/8.37 Obligation: 23.62/8.37 TRS: 23.62/8.37 Rules: 23.62/8.37 active(f(X, X)) -> mark(f(a, b)) 23.62/8.37 active(b) -> mark(a) 23.62/8.37 active(f(X1, X2)) -> f(active(X1), X2) 23.62/8.37 f(mark(X1), X2) -> mark(f(X1, X2)) 23.62/8.37 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 23.62/8.37 proper(a) -> ok(a) 23.62/8.37 proper(b) -> ok(b) 23.62/8.37 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 23.62/8.37 top(mark(X)) -> top(proper(X)) 23.62/8.37 top(ok(X)) -> top(active(X)) 23.62/8.37 23.62/8.37 Types: 23.62/8.37 active :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 f :: a:b:mark:ok -> a:b:mark:ok -> a:b:mark:ok 23.62/8.37 mark :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 a :: a:b:mark:ok 23.62/8.37 b :: a:b:mark:ok 23.62/8.37 proper :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 ok :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 top :: a:b:mark:ok -> top 23.62/8.37 hole_a:b:mark:ok1_0 :: a:b:mark:ok 23.62/8.37 hole_top2_0 :: top 23.62/8.37 gen_a:b:mark:ok3_0 :: Nat -> a:b:mark:ok 23.62/8.37 23.62/8.37 23.62/8.37 Generator Equations: 23.62/8.37 gen_a:b:mark:ok3_0(0) <=> a 23.62/8.37 gen_a:b:mark:ok3_0(+(x, 1)) <=> mark(gen_a:b:mark:ok3_0(x)) 23.62/8.37 23.62/8.37 23.62/8.37 The following defined symbols remain to be analysed: 23.62/8.37 f, active, proper, top 23.62/8.37 23.62/8.37 They will be analysed ascendingly in the following order: 23.62/8.37 f < active 23.62/8.37 active < top 23.62/8.37 f < proper 23.62/8.37 proper < top 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (13) RewriteLemmaProof (LOWER BOUND(ID)) 23.62/8.37 Proved the following rewrite lemma: 23.62/8.37 f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 23.62/8.37 23.62/8.37 Induction Base: 23.62/8.37 f(gen_a:b:mark:ok3_0(+(1, 0)), gen_a:b:mark:ok3_0(b)) 23.62/8.37 23.62/8.37 Induction Step: 23.62/8.37 f(gen_a:b:mark:ok3_0(+(1, +(n5_0, 1))), gen_a:b:mark:ok3_0(b)) ->_R^Omega(1) 23.62/8.37 mark(f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b))) ->_IH 23.62/8.37 mark(*4_0) 23.62/8.37 23.62/8.37 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (14) 23.62/8.37 Complex Obligation (BEST) 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (15) 23.62/8.37 Obligation: 23.62/8.37 Proved the lower bound n^1 for the following obligation: 23.62/8.37 23.62/8.37 TRS: 23.62/8.37 Rules: 23.62/8.37 active(f(X, X)) -> mark(f(a, b)) 23.62/8.37 active(b) -> mark(a) 23.62/8.37 active(f(X1, X2)) -> f(active(X1), X2) 23.62/8.37 f(mark(X1), X2) -> mark(f(X1, X2)) 23.62/8.37 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 23.62/8.37 proper(a) -> ok(a) 23.62/8.37 proper(b) -> ok(b) 23.62/8.37 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 23.62/8.37 top(mark(X)) -> top(proper(X)) 23.62/8.37 top(ok(X)) -> top(active(X)) 23.62/8.37 23.62/8.37 Types: 23.62/8.37 active :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 f :: a:b:mark:ok -> a:b:mark:ok -> a:b:mark:ok 23.62/8.37 mark :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 a :: a:b:mark:ok 23.62/8.37 b :: a:b:mark:ok 23.62/8.37 proper :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 ok :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 top :: a:b:mark:ok -> top 23.62/8.37 hole_a:b:mark:ok1_0 :: a:b:mark:ok 23.62/8.37 hole_top2_0 :: top 23.62/8.37 gen_a:b:mark:ok3_0 :: Nat -> a:b:mark:ok 23.62/8.37 23.62/8.37 23.62/8.37 Generator Equations: 23.62/8.37 gen_a:b:mark:ok3_0(0) <=> a 23.62/8.37 gen_a:b:mark:ok3_0(+(x, 1)) <=> mark(gen_a:b:mark:ok3_0(x)) 23.62/8.37 23.62/8.37 23.62/8.37 The following defined symbols remain to be analysed: 23.62/8.37 f, active, proper, top 23.62/8.37 23.62/8.37 They will be analysed ascendingly in the following order: 23.62/8.37 f < active 23.62/8.37 active < top 23.62/8.37 f < proper 23.62/8.37 proper < top 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (16) LowerBoundPropagationProof (FINISHED) 23.62/8.37 Propagated lower bound. 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (17) 23.62/8.37 BOUNDS(n^1, INF) 23.62/8.37 23.62/8.37 ---------------------------------------- 23.62/8.37 23.62/8.37 (18) 23.62/8.37 Obligation: 23.62/8.37 TRS: 23.62/8.37 Rules: 23.62/8.37 active(f(X, X)) -> mark(f(a, b)) 23.62/8.37 active(b) -> mark(a) 23.62/8.37 active(f(X1, X2)) -> f(active(X1), X2) 23.62/8.37 f(mark(X1), X2) -> mark(f(X1, X2)) 23.62/8.37 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 23.62/8.37 proper(a) -> ok(a) 23.62/8.37 proper(b) -> ok(b) 23.62/8.37 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 23.62/8.37 top(mark(X)) -> top(proper(X)) 23.62/8.37 top(ok(X)) -> top(active(X)) 23.62/8.37 23.62/8.37 Types: 23.62/8.37 active :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 f :: a:b:mark:ok -> a:b:mark:ok -> a:b:mark:ok 23.62/8.37 mark :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 a :: a:b:mark:ok 23.62/8.37 b :: a:b:mark:ok 23.62/8.37 proper :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 ok :: a:b:mark:ok -> a:b:mark:ok 23.62/8.37 top :: a:b:mark:ok -> top 23.62/8.37 hole_a:b:mark:ok1_0 :: a:b:mark:ok 23.62/8.37 hole_top2_0 :: top 23.62/8.37 gen_a:b:mark:ok3_0 :: Nat -> a:b:mark:ok 23.62/8.37 23.62/8.37 23.62/8.37 Lemmas: 23.62/8.37 f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 23.62/8.37 23.62/8.37 23.62/8.37 Generator Equations: 23.62/8.37 gen_a:b:mark:ok3_0(0) <=> a 23.62/8.37 gen_a:b:mark:ok3_0(+(x, 1)) <=> mark(gen_a:b:mark:ok3_0(x)) 23.62/8.37 23.62/8.37 23.62/8.37 The following defined symbols remain to be analysed: 23.62/8.37 active, proper, top 23.62/8.37 23.62/8.37 They will be analysed ascendingly in the following order: 23.62/8.37 active < top 23.62/8.37 proper < top 23.86/8.41 EOF