WORST_CASE(?,O(n^1)) proof of input_3gKBMSn6i7.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms] (4) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: f(X) -> g(n__h(n__f(X))) h(X) -> n__h(X) f(X) -> n__f(X) activate(n__h(X)) -> h(activate(X)) activate(n__f(X)) -> f(activate(X)) activate(X) -> X S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: f(X) -> g(n__h(n__f(X))) h(X) -> n__h(X) f(X) -> n__f(X) activate(n__h(X)) -> h(activate(X)) activate(n__f(X)) -> f(activate(X)) activate(X) -> X S is empty. Rewrite Strategy: PARALLEL_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 2. The certificate found is represented by the following graph. "[1, 2, 3, 4, 5, 6, 7, 8] {(1,2,[f_1|0, h_1|0, activate_1|0, n__f_1|1, n__h_1|1, g_1|1]), (1,3,[g_1|1]), (1,5,[h_1|1, n__h_1|2]), (1,6,[f_1|1, n__f_1|2]), (1,7,[g_1|2]), (2,2,[g_1|0, n__h_1|0, n__f_1|0]), (3,4,[n__h_1|1]), (4,2,[n__f_1|1]), (5,2,[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]), (5,5,[h_1|1, n__h_1|2]), (5,6,[f_1|1, n__f_1|2]), (5,7,[g_1|2]), (6,2,[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]), (6,5,[h_1|1, n__h_1|2]), (6,6,[f_1|1, n__f_1|2]), (6,7,[g_1|2]), (7,8,[n__h_1|2]), (8,6,[n__f_1|2])}" ---------------------------------------- (4) BOUNDS(1, n^1)