WORST_CASE(?,O(n^1)) proof of input_HkVn7xOqY5.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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) CpxTrsMatchBoundsTAProof [FINISHED, 62 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: concat(leaf, Y) -> Y concat(cons(U, V), Y) -> cons(U, concat(V, Y)) lessleaves(X, leaf) -> false lessleaves(leaf, cons(W, Z)) -> true lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z)) 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: concat(leaf, Y) -> Y concat(cons(U, V), Y) -> cons(U, concat(V, Y)) lessleaves(X, leaf) -> false lessleaves(leaf, cons(W, Z)) -> true lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) CpxTrsMatchBoundsTAProof (FINISHED) 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 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2] transitions: leaf0() -> 0 cons0(0, 0) -> 0 false0() -> 0 true0() -> 0 concat0(0, 0) -> 1 lessleaves0(0, 0) -> 2 concat1(0, 0) -> 3 cons1(0, 3) -> 1 false1() -> 2 true1() -> 2 concat1(0, 0) -> 4 concat1(0, 0) -> 5 lessleaves1(4, 5) -> 2 cons1(0, 3) -> 3 cons1(0, 3) -> 4 cons1(0, 3) -> 5 concat1(0, 3) -> 5 concat1(0, 3) -> 4 concat2(0, 3) -> 6 concat2(0, 3) -> 7 lessleaves2(6, 7) -> 2 concat1(0, 3) -> 3 0 -> 1 0 -> 3 0 -> 4 0 -> 5 3 -> 4 3 -> 5 3 -> 6 3 -> 7 ---------------------------------------- (4) BOUNDS(1, n^1)