WORST_CASE(?,O(n^1)) proof of input_MXeGf2171R.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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: a__f(f(a)) -> c(f(g(f(a)))) mark(f(X)) -> a__f(mark(X)) mark(a) -> a mark(c(X)) -> c(X) mark(g(X)) -> g(mark(X)) a__f(X) -> f(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: a__f(f(a)) -> c(f(g(f(a)))) mark(f(X)) -> a__f(mark(X)) mark(a) -> a mark(c(X)) -> c(X) mark(g(X)) -> g(mark(X)) a__f(X) -> f(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, 9, 10, 11, 12] {(1,2,[a__f_1|0, mark_1|0, f_1|1, a|1, c_1|1]), (1,3,[c_1|1]), (1,7,[a__f_1|1, f_1|2]), (1,8,[g_1|1]), (1,9,[c_1|2]), (2,2,[f_1|0, a|0, c_1|0, g_1|0]), (3,4,[f_1|1]), (4,5,[g_1|1]), (5,6,[f_1|1]), (6,2,[a|1]), (7,2,[mark_1|1, a|1, c_1|1]), (7,7,[a__f_1|1, f_1|2]), (7,8,[g_1|1]), (7,9,[c_1|2]), (8,2,[mark_1|1, a|1, c_1|1]), (8,7,[a__f_1|1, f_1|2]), (8,8,[g_1|1]), (8,9,[c_1|2]), (9,10,[f_1|2]), (10,11,[g_1|2]), (11,12,[f_1|2]), (12,2,[a|2])}" ---------------------------------------- (4) BOUNDS(1, n^1)