28.97/8.52 WORST_CASE(Omega(n^1), O(n^1)) 28.97/8.55 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 28.97/8.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.97/8.55 28.97/8.55 28.97/8.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.97/8.55 28.97/8.55 (0) CpxTRS 28.97/8.55 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 28.97/8.55 (2) CpxTRS 28.97/8.55 (3) CpxTrsToCdtProof [UPPER BOUND(ID), 3 ms] 28.97/8.55 (4) CdtProblem 28.97/8.55 (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 28.97/8.55 (6) CdtProblem 28.97/8.55 (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 28.97/8.55 (8) CdtProblem 28.97/8.55 (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 73 ms] 28.97/8.55 (10) CdtProblem 28.97/8.55 (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms] 28.97/8.55 (12) CdtProblem 28.97/8.55 (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] 28.97/8.55 (14) BOUNDS(1, 1) 28.97/8.55 (15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 28.97/8.55 (16) TRS for Loop Detection 28.97/8.55 (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 28.97/8.55 (18) BEST 28.97/8.55 (19) proven lower bound 28.97/8.55 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 28.97/8.55 (21) BOUNDS(n^1, INF) 28.97/8.55 (22) TRS for Loop Detection 28.97/8.55 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (0) 28.97/8.55 Obligation: 28.97/8.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.97/8.55 28.97/8.55 28.97/8.55 The TRS R consists of the following rules: 28.97/8.55 28.97/8.55 a__f(a, X, X) -> a__f(X, a__b, b) 28.97/8.55 a__b -> a 28.97/8.55 mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 a__f(X1, X2, X3) -> f(X1, X2, X3) 28.97/8.55 a__b -> b 28.97/8.55 28.97/8.55 S is empty. 28.97/8.55 Rewrite Strategy: FULL 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 28.97/8.55 Converted rc-obligation to irc-obligation. 28.97/8.55 28.97/8.55 As the TRS is a non-duplicating overlay system, we have rc = irc. 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (2) 28.97/8.55 Obligation: 28.97/8.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 28.97/8.55 28.97/8.55 28.97/8.55 The TRS R consists of the following rules: 28.97/8.55 28.97/8.55 a__f(a, X, X) -> a__f(X, a__b, b) 28.97/8.55 a__b -> a 28.97/8.55 mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 a__f(X1, X2, X3) -> f(X1, X2, X3) 28.97/8.55 a__b -> b 28.97/8.55 28.97/8.55 S is empty. 28.97/8.55 Rewrite Strategy: INNERMOST 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (3) CpxTrsToCdtProof (UPPER BOUND(ID)) 28.97/8.55 Converted Cpx (relative) TRS to CDT 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (4) 28.97/8.55 Obligation: 28.97/8.55 Complexity Dependency Tuples Problem 28.97/8.55 28.97/8.55 Rules: 28.97/8.55 a__f(a, z0, z0) -> a__f(z0, a__b, b) 28.97/8.55 a__f(z0, z1, z2) -> f(z0, z1, z2) 28.97/8.55 a__b -> a 28.97/8.55 a__b -> b 28.97/8.55 mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 Tuples: 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b), A__B) 28.97/8.55 A__F(z0, z1, z2) -> c1 28.97/8.55 A__B -> c2 28.97/8.55 A__B -> c3 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 MARK(b) -> c5(A__B) 28.97/8.55 MARK(a) -> c6 28.97/8.55 S tuples: 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b), A__B) 28.97/8.55 A__F(z0, z1, z2) -> c1 28.97/8.55 A__B -> c2 28.97/8.55 A__B -> c3 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 MARK(b) -> c5(A__B) 28.97/8.55 MARK(a) -> c6 28.97/8.55 K tuples:none 28.97/8.55 Defined Rule Symbols: a__f_3, a__b, mark_1 28.97/8.55 28.97/8.55 Defined Pair Symbols: A__F_3, A__B, MARK_1 28.97/8.55 28.97/8.55 Compound Symbols: c_2, c1, c2, c3, c4_2, c5_1, c6 28.97/8.55 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 28.97/8.55 Removed 5 trailing nodes: 28.97/8.55 MARK(b) -> c5(A__B) 28.97/8.55 A__B -> c3 28.97/8.55 A__F(z0, z1, z2) -> c1 28.97/8.55 MARK(a) -> c6 28.97/8.55 A__B -> c2 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (6) 28.97/8.55 Obligation: 28.97/8.55 Complexity Dependency Tuples Problem 28.97/8.55 28.97/8.55 Rules: 28.97/8.55 a__f(a, z0, z0) -> a__f(z0, a__b, b) 28.97/8.55 a__f(z0, z1, z2) -> f(z0, z1, z2) 28.97/8.55 a__b -> a 28.97/8.55 a__b -> b 28.97/8.55 mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 Tuples: 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b), A__B) 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 S tuples: 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b), A__B) 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 K tuples:none 28.97/8.55 Defined Rule Symbols: a__f_3, a__b, mark_1 28.97/8.55 28.97/8.55 Defined Pair Symbols: A__F_3, MARK_1 28.97/8.55 28.97/8.55 Compound Symbols: c_2, c4_2 28.97/8.55 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 28.97/8.55 Removed 1 trailing tuple parts 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (8) 28.97/8.55 Obligation: 28.97/8.55 Complexity Dependency Tuples Problem 28.97/8.55 28.97/8.55 Rules: 28.97/8.55 a__f(a, z0, z0) -> a__f(z0, a__b, b) 28.97/8.55 a__f(z0, z1, z2) -> f(z0, z1, z2) 28.97/8.55 a__b -> a 28.97/8.55 a__b -> b 28.97/8.55 mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 Tuples: 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b)) 28.97/8.55 S tuples: 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b)) 28.97/8.55 K tuples:none 28.97/8.55 Defined Rule Symbols: a__f_3, a__b, mark_1 28.97/8.55 28.97/8.55 Defined Pair Symbols: MARK_1, A__F_3 28.97/8.55 28.97/8.55 Compound Symbols: c4_2, c_1 28.97/8.55 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) 28.97/8.55 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b)) 28.97/8.55 We considered the (Usable) Rules:none 28.97/8.55 And the Tuples: 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b)) 28.97/8.55 The order we found is given by the following interpretation: 28.97/8.55 28.97/8.55 Polynomial interpretation : 28.97/8.55 28.97/8.55 POL(A__F(x_1, x_2, x_3)) = x_1 + x_3 28.97/8.55 POL(MARK(x_1)) = x_1 28.97/8.55 POL(a) = [1] 28.97/8.55 POL(a__b) = [1] 28.97/8.55 POL(a__f(x_1, x_2, x_3)) = x_1 + x_2 + x_3 28.97/8.55 POL(b) = 0 28.97/8.55 POL(c(x_1)) = x_1 28.97/8.55 POL(c4(x_1, x_2)) = x_1 + x_2 28.97/8.55 POL(f(x_1, x_2, x_3)) = x_1 + x_2 + x_3 28.97/8.55 POL(mark(x_1)) = [1] + x_1 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (10) 28.97/8.55 Obligation: 28.97/8.55 Complexity Dependency Tuples Problem 28.97/8.55 28.97/8.55 Rules: 28.97/8.55 a__f(a, z0, z0) -> a__f(z0, a__b, b) 28.97/8.55 a__f(z0, z1, z2) -> f(z0, z1, z2) 28.97/8.55 a__b -> a 28.97/8.55 a__b -> b 28.97/8.55 mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 Tuples: 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b)) 28.97/8.55 S tuples: 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 K tuples: 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b)) 28.97/8.55 Defined Rule Symbols: a__f_3, a__b, mark_1 28.97/8.55 28.97/8.55 Defined Pair Symbols: MARK_1, A__F_3 28.97/8.55 28.97/8.55 Compound Symbols: c4_2, c_1 28.97/8.55 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) 28.97/8.55 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 We considered the (Usable) Rules:none 28.97/8.55 And the Tuples: 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b)) 28.97/8.55 The order we found is given by the following interpretation: 28.97/8.55 28.97/8.55 Polynomial interpretation : 28.97/8.55 28.97/8.55 POL(A__F(x_1, x_2, x_3)) = x_1 + x_3 28.97/8.55 POL(MARK(x_1)) = x_1 28.97/8.55 POL(a) = [1] 28.97/8.55 POL(a__b) = [1] 28.97/8.55 POL(a__f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 28.97/8.55 POL(b) = 0 28.97/8.55 POL(c(x_1)) = x_1 28.97/8.55 POL(c4(x_1, x_2)) = x_1 + x_2 28.97/8.55 POL(f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 28.97/8.55 POL(mark(x_1)) = [1] + x_1 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (12) 28.97/8.55 Obligation: 28.97/8.55 Complexity Dependency Tuples Problem 28.97/8.55 28.97/8.55 Rules: 28.97/8.55 a__f(a, z0, z0) -> a__f(z0, a__b, b) 28.97/8.55 a__f(z0, z1, z2) -> f(z0, z1, z2) 28.97/8.55 a__b -> a 28.97/8.55 a__b -> b 28.97/8.55 mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 Tuples: 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b)) 28.97/8.55 S tuples:none 28.97/8.55 K tuples: 28.97/8.55 A__F(a, z0, z0) -> c(A__F(z0, a__b, b)) 28.97/8.55 MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1)) 28.97/8.55 Defined Rule Symbols: a__f_3, a__b, mark_1 28.97/8.55 28.97/8.55 Defined Pair Symbols: MARK_1, A__F_3 28.97/8.55 28.97/8.55 Compound Symbols: c4_2, c_1 28.97/8.55 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (13) SIsEmptyProof (BOTH BOUNDS(ID, ID)) 28.97/8.55 The set S is empty 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (14) 28.97/8.55 BOUNDS(1, 1) 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 28.97/8.55 Transformed a relative TRS into a decreasing-loop problem. 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (16) 28.97/8.55 Obligation: 28.97/8.55 Analyzing the following TRS for decreasing loops: 28.97/8.55 28.97/8.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.97/8.55 28.97/8.55 28.97/8.55 The TRS R consists of the following rules: 28.97/8.55 28.97/8.55 a__f(a, X, X) -> a__f(X, a__b, b) 28.97/8.55 a__b -> a 28.97/8.55 mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 a__f(X1, X2, X3) -> f(X1, X2, X3) 28.97/8.55 a__b -> b 28.97/8.55 28.97/8.55 S is empty. 28.97/8.55 Rewrite Strategy: FULL 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (17) DecreasingLoopProof (LOWER BOUND(ID)) 28.97/8.55 The following loop(s) give(s) rise to the lower bound Omega(n^1): 28.97/8.55 28.97/8.55 The rewrite sequence 28.97/8.55 28.97/8.55 mark(f(X1, X2, X3)) ->^+ a__f(X1, mark(X2), X3) 28.97/8.55 28.97/8.55 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 28.97/8.55 28.97/8.55 The pumping substitution is [X2 / f(X1, X2, X3)]. 28.97/8.55 28.97/8.55 The result substitution is [ ]. 28.97/8.55 28.97/8.55 28.97/8.55 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (18) 28.97/8.55 Complex Obligation (BEST) 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (19) 28.97/8.55 Obligation: 28.97/8.55 Proved the lower bound n^1 for the following obligation: 28.97/8.55 28.97/8.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.97/8.55 28.97/8.55 28.97/8.55 The TRS R consists of the following rules: 28.97/8.55 28.97/8.55 a__f(a, X, X) -> a__f(X, a__b, b) 28.97/8.55 a__b -> a 28.97/8.55 mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 a__f(X1, X2, X3) -> f(X1, X2, X3) 28.97/8.55 a__b -> b 28.97/8.55 28.97/8.55 S is empty. 28.97/8.55 Rewrite Strategy: FULL 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (20) LowerBoundPropagationProof (FINISHED) 28.97/8.55 Propagated lower bound. 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (21) 28.97/8.55 BOUNDS(n^1, INF) 28.97/8.55 28.97/8.55 ---------------------------------------- 28.97/8.55 28.97/8.55 (22) 28.97/8.55 Obligation: 28.97/8.55 Analyzing the following TRS for decreasing loops: 28.97/8.55 28.97/8.55 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.97/8.55 28.97/8.55 28.97/8.55 The TRS R consists of the following rules: 28.97/8.55 28.97/8.55 a__f(a, X, X) -> a__f(X, a__b, b) 28.97/8.55 a__b -> a 28.97/8.55 mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) 28.97/8.55 mark(b) -> a__b 28.97/8.55 mark(a) -> a 28.97/8.55 a__f(X1, X2, X3) -> f(X1, X2, X3) 28.97/8.55 a__b -> b 28.97/8.55 28.97/8.55 S is empty. 28.97/8.55 Rewrite Strategy: FULL 29.19/8.60 EOF