24.56/8.18 WORST_CASE(Omega(n^1), O(n^1)) 24.56/8.19 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.56/8.19 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.56/8.19 24.56/8.19 24.56/8.19 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.56/8.19 24.56/8.19 (0) CpxTRS 24.56/8.19 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 24.56/8.19 (2) CpxTRS 24.56/8.19 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 24.56/8.19 (4) CpxTRS 24.56/8.19 (5) CpxTrsMatchBoundsTAProof [FINISHED, 177 ms] 24.56/8.19 (6) BOUNDS(1, n^1) 24.56/8.19 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 24.56/8.19 (8) TRS for Loop Detection 24.56/8.19 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 24.56/8.19 (10) BEST 24.56/8.19 (11) proven lower bound 24.56/8.19 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 24.56/8.19 (13) BOUNDS(n^1, INF) 24.56/8.19 (14) TRS for Loop Detection 24.56/8.19 24.56/8.19 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (0) 24.56/8.19 Obligation: 24.56/8.19 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.56/8.19 24.56/8.19 24.56/8.19 The TRS R consists of the following rules: 24.56/8.19 24.56/8.19 active(f(a, b, X)) -> mark(f(X, X, X)) 24.56/8.19 active(c) -> mark(a) 24.56/8.19 active(c) -> mark(b) 24.56/8.19 active(f(X1, X2, X3)) -> f(active(X1), X2, X3) 24.56/8.19 active(f(X1, X2, X3)) -> f(X1, X2, active(X3)) 24.56/8.19 f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)) 24.56/8.19 f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)) 24.56/8.19 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 24.56/8.19 proper(a) -> ok(a) 24.56/8.19 proper(b) -> ok(b) 24.56/8.19 proper(c) -> ok(c) 24.56/8.19 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 24.56/8.19 top(mark(X)) -> top(proper(X)) 24.56/8.19 top(ok(X)) -> top(active(X)) 24.56/8.19 24.56/8.19 S is empty. 24.56/8.19 Rewrite Strategy: FULL 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 24.56/8.19 The following defined symbols can occur below the 0th argument of top: proper, active 24.56/8.19 The following defined symbols can occur below the 0th argument of proper: proper, active 24.56/8.19 The following defined symbols can occur below the 0th argument of active: proper, active 24.56/8.19 24.56/8.19 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 24.56/8.19 active(f(a, b, X)) -> mark(f(X, X, X)) 24.56/8.19 active(f(X1, X2, X3)) -> f(active(X1), X2, X3) 24.56/8.19 active(f(X1, X2, X3)) -> f(X1, X2, active(X3)) 24.56/8.19 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 24.56/8.19 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (2) 24.56/8.19 Obligation: 24.56/8.19 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.56/8.19 24.56/8.19 24.56/8.19 The TRS R consists of the following rules: 24.56/8.19 24.56/8.19 active(c) -> mark(a) 24.56/8.19 active(c) -> mark(b) 24.56/8.19 f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)) 24.56/8.19 f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)) 24.56/8.19 proper(a) -> ok(a) 24.56/8.19 proper(b) -> ok(b) 24.56/8.19 proper(c) -> ok(c) 24.56/8.19 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 24.56/8.19 top(mark(X)) -> top(proper(X)) 24.56/8.19 top(ok(X)) -> top(active(X)) 24.56/8.19 24.56/8.19 S is empty. 24.56/8.19 Rewrite Strategy: FULL 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 24.56/8.19 transformed relative TRS to TRS 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (4) 24.56/8.19 Obligation: 24.56/8.19 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.56/8.19 24.56/8.19 24.56/8.19 The TRS R consists of the following rules: 24.56/8.19 24.56/8.19 active(c) -> mark(a) 24.56/8.19 active(c) -> mark(b) 24.56/8.19 f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)) 24.56/8.19 f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)) 24.56/8.19 proper(a) -> ok(a) 24.56/8.19 proper(b) -> ok(b) 24.56/8.19 proper(c) -> ok(c) 24.56/8.19 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 24.56/8.19 top(mark(X)) -> top(proper(X)) 24.56/8.19 top(ok(X)) -> top(active(X)) 24.56/8.19 24.56/8.19 S is empty. 24.56/8.19 Rewrite Strategy: FULL 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (5) CpxTrsMatchBoundsTAProof (FINISHED) 24.56/8.19 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. 24.56/8.19 24.56/8.19 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 24.56/8.19 final states : [1, 2, 3, 4] 24.56/8.19 transitions: 24.56/8.19 c0() -> 0 24.56/8.19 mark0(0) -> 0 24.56/8.19 a0() -> 0 24.56/8.19 b0() -> 0 24.56/8.19 ok0(0) -> 0 24.56/8.19 active0(0) -> 1 24.56/8.19 f0(0, 0, 0) -> 2 24.56/8.19 proper0(0) -> 3 24.56/8.19 top0(0) -> 4 24.56/8.19 a1() -> 5 24.56/8.19 mark1(5) -> 1 24.56/8.19 b1() -> 6 24.56/8.19 mark1(6) -> 1 24.56/8.19 f1(0, 0, 0) -> 7 24.56/8.19 mark1(7) -> 2 24.56/8.19 a1() -> 8 24.56/8.19 ok1(8) -> 3 24.56/8.19 b1() -> 9 24.56/8.19 ok1(9) -> 3 24.56/8.19 c1() -> 10 24.56/8.19 ok1(10) -> 3 24.56/8.19 f1(0, 0, 0) -> 11 24.56/8.19 ok1(11) -> 2 24.56/8.19 proper1(0) -> 12 24.56/8.19 top1(12) -> 4 24.56/8.19 active1(0) -> 13 24.56/8.19 top1(13) -> 4 24.56/8.19 mark1(5) -> 13 24.56/8.19 mark1(6) -> 13 24.56/8.19 mark1(7) -> 7 24.56/8.19 mark1(7) -> 11 24.56/8.19 ok1(8) -> 12 24.56/8.19 ok1(9) -> 12 24.56/8.19 ok1(10) -> 12 24.56/8.19 ok1(11) -> 7 24.56/8.19 ok1(11) -> 11 24.56/8.19 proper2(5) -> 14 24.56/8.19 top2(14) -> 4 24.56/8.19 proper2(6) -> 14 24.56/8.19 active2(8) -> 15 24.56/8.19 top2(15) -> 4 24.56/8.19 active2(9) -> 15 24.56/8.19 active2(10) -> 15 24.56/8.19 a2() -> 16 24.56/8.19 mark2(16) -> 15 24.56/8.19 b2() -> 17 24.56/8.19 mark2(17) -> 15 24.56/8.19 a2() -> 18 24.56/8.19 ok2(18) -> 14 24.56/8.19 b2() -> 19 24.56/8.19 ok2(19) -> 14 24.56/8.19 proper3(16) -> 20 24.56/8.19 top3(20) -> 4 24.56/8.19 proper3(17) -> 20 24.56/8.19 active3(18) -> 21 24.56/8.19 top3(21) -> 4 24.56/8.19 active3(19) -> 21 24.56/8.19 a3() -> 22 24.56/8.19 ok3(22) -> 20 24.56/8.19 b3() -> 23 24.56/8.19 ok3(23) -> 20 24.56/8.19 active4(22) -> 24 24.56/8.19 top4(24) -> 4 24.56/8.19 active4(23) -> 24 24.56/8.19 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (6) 24.56/8.19 BOUNDS(1, n^1) 24.56/8.19 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 24.56/8.19 Transformed a relative TRS into a decreasing-loop problem. 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (8) 24.56/8.19 Obligation: 24.56/8.19 Analyzing the following TRS for decreasing loops: 24.56/8.19 24.56/8.19 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.56/8.19 24.56/8.19 24.56/8.19 The TRS R consists of the following rules: 24.56/8.19 24.56/8.19 active(f(a, b, X)) -> mark(f(X, X, X)) 24.56/8.19 active(c) -> mark(a) 24.56/8.19 active(c) -> mark(b) 24.56/8.19 active(f(X1, X2, X3)) -> f(active(X1), X2, X3) 24.56/8.19 active(f(X1, X2, X3)) -> f(X1, X2, active(X3)) 24.56/8.19 f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)) 24.56/8.19 f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)) 24.56/8.19 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 24.56/8.19 proper(a) -> ok(a) 24.56/8.19 proper(b) -> ok(b) 24.56/8.19 proper(c) -> ok(c) 24.56/8.19 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 24.56/8.19 top(mark(X)) -> top(proper(X)) 24.56/8.19 top(ok(X)) -> top(active(X)) 24.56/8.19 24.56/8.19 S is empty. 24.56/8.19 Rewrite Strategy: FULL 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (9) DecreasingLoopProof (LOWER BOUND(ID)) 24.56/8.19 The following loop(s) give(s) rise to the lower bound Omega(n^1): 24.56/8.19 24.56/8.19 The rewrite sequence 24.56/8.19 24.56/8.19 f(X1, X2, mark(X3)) ->^+ mark(f(X1, X2, X3)) 24.56/8.19 24.56/8.19 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 24.56/8.19 24.56/8.19 The pumping substitution is [X3 / mark(X3)]. 24.56/8.19 24.56/8.19 The result substitution is [ ]. 24.56/8.19 24.56/8.19 24.56/8.19 24.56/8.19 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (10) 24.56/8.19 Complex Obligation (BEST) 24.56/8.19 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (11) 24.56/8.19 Obligation: 24.56/8.19 Proved the lower bound n^1 for the following obligation: 24.56/8.19 24.56/8.19 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.56/8.19 24.56/8.19 24.56/8.19 The TRS R consists of the following rules: 24.56/8.19 24.56/8.19 active(f(a, b, X)) -> mark(f(X, X, X)) 24.56/8.19 active(c) -> mark(a) 24.56/8.19 active(c) -> mark(b) 24.56/8.19 active(f(X1, X2, X3)) -> f(active(X1), X2, X3) 24.56/8.19 active(f(X1, X2, X3)) -> f(X1, X2, active(X3)) 24.56/8.19 f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)) 24.56/8.19 f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)) 24.56/8.19 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 24.56/8.19 proper(a) -> ok(a) 24.56/8.19 proper(b) -> ok(b) 24.56/8.19 proper(c) -> ok(c) 24.56/8.19 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 24.56/8.19 top(mark(X)) -> top(proper(X)) 24.56/8.19 top(ok(X)) -> top(active(X)) 24.56/8.19 24.56/8.19 S is empty. 24.56/8.19 Rewrite Strategy: FULL 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (12) LowerBoundPropagationProof (FINISHED) 24.56/8.19 Propagated lower bound. 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (13) 24.56/8.19 BOUNDS(n^1, INF) 24.56/8.19 24.56/8.19 ---------------------------------------- 24.56/8.19 24.56/8.19 (14) 24.56/8.19 Obligation: 24.56/8.19 Analyzing the following TRS for decreasing loops: 24.56/8.19 24.56/8.19 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.56/8.19 24.56/8.19 24.56/8.19 The TRS R consists of the following rules: 24.56/8.19 24.56/8.19 active(f(a, b, X)) -> mark(f(X, X, X)) 24.56/8.19 active(c) -> mark(a) 24.56/8.19 active(c) -> mark(b) 24.56/8.19 active(f(X1, X2, X3)) -> f(active(X1), X2, X3) 24.56/8.19 active(f(X1, X2, X3)) -> f(X1, X2, active(X3)) 24.56/8.19 f(mark(X1), X2, X3) -> mark(f(X1, X2, X3)) 24.56/8.19 f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3)) 24.56/8.19 proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3)) 24.56/8.19 proper(a) -> ok(a) 24.56/8.19 proper(b) -> ok(b) 24.56/8.19 proper(c) -> ok(c) 24.56/8.19 f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3)) 24.56/8.19 top(mark(X)) -> top(proper(X)) 24.56/8.19 top(ok(X)) -> top(active(X)) 24.56/8.19 24.56/8.19 S is empty. 24.56/8.19 Rewrite Strategy: FULL 24.56/8.22 EOF