WORST_CASE(?,O(n^1)) proof of input_a0kBz9bROh.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 198 ms] (2) CpxRelTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 44 ms] (6) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: f(C(x1, x2)) -> C(f(x1), f(x2)) f(Z) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 g(x) -> x The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: f(C(x1, x2)) -> C(f(x1), f(x2)) f(Z) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 g(x) -> x The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) 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(C(x1, x2)) -> C(f(x1), f(x2)) f(Z) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 g(x) -> x and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) 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, 3, 4, 5, 6] transitions: C0(0, 0) -> 0 Z0() -> 0 False0() -> 0 True0() -> 0 f0(0) -> 1 eqZList0(0, 0) -> 2 second0(0) -> 3 first0(0) -> 4 g0(0) -> 5 and0(0, 0) -> 6 f1(0) -> 7 f1(0) -> 8 C1(7, 8) -> 1 Z1() -> 1 eqZList1(0, 0) -> 9 eqZList1(0, 0) -> 10 and1(9, 10) -> 2 False1() -> 2 True1() -> 2 False1() -> 6 True1() -> 6 C1(7, 8) -> 7 C1(7, 8) -> 8 Z1() -> 7 Z1() -> 8 and1(9, 10) -> 9 and1(9, 10) -> 10 False1() -> 9 False1() -> 10 True1() -> 9 True1() -> 10 False2() -> 2 False2() -> 9 False2() -> 10 True2() -> 2 True2() -> 9 True2() -> 10 0 -> 3 0 -> 4 0 -> 5 ---------------------------------------- (6) BOUNDS(1, n^1)