WORST_CASE(Omega(n^1),O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms] (4) BOUNDS(1, n^1) (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: a__f(f(X)) -> a__c(f(g(f(X)))) a__c(X) -> d(X) a__h(X) -> a__c(d(X)) mark(f(X)) -> a__f(mark(X)) mark(c(X)) -> a__c(X) mark(h(X)) -> a__h(mark(X)) mark(g(X)) -> g(X) mark(d(X)) -> d(X) a__f(X) -> f(X) a__c(X) -> c(X) a__h(X) -> h(X) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a__f(f(X)) -> a__c(f(g(f(X)))) a__c(X) -> d(X) a__h(X) -> a__c(d(X)) mark(f(X)) -> a__f(mark(X)) mark(c(X)) -> a__c(X) mark(h(X)) -> a__h(mark(X)) mark(g(X)) -> g(X) mark(d(X)) -> d(X) a__f(X) -> f(X) a__c(X) -> c(X) a__h(X) -> h(X) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. The certificate found is represented by the following graph. "[5, 6, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] {(5,6,[a__f_1|0, a__c_1|0, a__h_1|0, mark_1|0, f_1|1, d_1|1, c_1|1, h_1|1, a__c_1|1, g_1|1, d_1|2, c_1|2]), (5,11,[a__c_1|1, d_1|2, c_1|2]), (5,14,[a__c_1|1, d_1|2, c_1|2]), (5,15,[a__f_1|1, f_1|2]), (5,16,[a__h_1|1, h_1|2]), (5,17,[a__c_1|2, d_1|3, c_1|3]), (5,18,[a__c_1|2, d_1|3, c_1|3]), (6,6,[f_1|0, g_1|0, d_1|0, c_1|0, h_1|0]), (11,12,[f_1|1]), (12,13,[g_1|1]), (13,6,[f_1|1]), (14,6,[d_1|1]), (15,6,[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]), (15,15,[a__f_1|1, f_1|2]), (15,16,[a__h_1|1, h_1|2]), (15,17,[a__c_1|2, d_1|3, c_1|3]), (15,18,[a__c_1|2, d_1|3, c_1|3]), (16,6,[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]), (16,15,[a__f_1|1, f_1|2]), (16,16,[a__h_1|1, h_1|2]), (16,17,[a__c_1|2, d_1|3, c_1|3]), (16,18,[a__c_1|2, d_1|3, c_1|3]), (17,16,[d_1|2]), (18,19,[f_1|2]), (19,20,[g_1|2]), (20,15,[f_1|2])}" ---------------------------------------- (4) BOUNDS(1, n^1) ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: a__f(f(X)) -> a__c(f(g(f(X)))) a__c(X) -> d(X) a__h(X) -> a__c(d(X)) mark(f(X)) -> a__f(mark(X)) mark(c(X)) -> a__c(X) mark(h(X)) -> a__h(mark(X)) mark(g(X)) -> g(X) mark(d(X)) -> d(X) a__f(X) -> f(X) a__c(X) -> c(X) a__h(X) -> h(X) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence mark(f(X)) ->^+ a__f(mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / f(X)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: a__f(f(X)) -> a__c(f(g(f(X)))) a__c(X) -> d(X) a__h(X) -> a__c(d(X)) mark(f(X)) -> a__f(mark(X)) mark(c(X)) -> a__c(X) mark(h(X)) -> a__h(mark(X)) mark(g(X)) -> g(X) mark(d(X)) -> d(X) a__f(X) -> f(X) a__c(X) -> c(X) a__h(X) -> h(X) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: a__f(f(X)) -> a__c(f(g(f(X)))) a__c(X) -> d(X) a__h(X) -> a__c(d(X)) mark(f(X)) -> a__f(mark(X)) mark(c(X)) -> a__c(X) mark(h(X)) -> a__h(mark(X)) mark(g(X)) -> g(X) mark(d(X)) -> d(X) a__f(X) -> f(X) a__c(X) -> c(X) a__h(X) -> h(X) S is empty. Rewrite Strategy: INNERMOST