WORST_CASE(?,O(n^1)) proof of input_E7L0N8IFey.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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, 52 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: rev_l#2(x8, x10) -> Cons(x10, x8) step_x_f#1(rev_l, x5, step_x_f(x2, x3, x4), x1) -> step_x_f#1(x2, x3, x4, rev_l#2(x1, x5)) step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5) foldr#3(Nil) -> fleft_op_e_xs_1 foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6)) main(Nil) -> Nil main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), Nil) 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: rev_l#2(x8, x10) -> Cons(x10, x8) step_x_f#1(rev_l, x5, step_x_f(x2, x3, x4), x1) -> step_x_f#1(x2, x3, x4, rev_l#2(x1, x5)) step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5) foldr#3(Nil) -> fleft_op_e_xs_1 foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6)) main(Nil) -> Nil main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), Nil) 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 3. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4] transitions: Cons0(0, 0) -> 0 rev_l0() -> 0 step_x_f0(0, 0, 0) -> 0 fleft_op_e_xs_10() -> 0 Nil0() -> 0 rev_l#20(0, 0) -> 1 step_x_f#10(0, 0, 0, 0) -> 2 foldr#30(0) -> 3 main0(0) -> 4 Cons1(0, 0) -> 1 rev_l#21(0, 0) -> 5 step_x_f#11(0, 0, 0, 5) -> 2 rev_l#21(0, 0) -> 2 fleft_op_e_xs_11() -> 3 rev_l1() -> 6 foldr#31(0) -> 7 step_x_f1(6, 0, 7) -> 3 Nil1() -> 4 rev_l1() -> 8 foldr#31(0) -> 9 Nil1() -> 10 step_x_f#11(8, 0, 9, 10) -> 4 Cons2(0, 0) -> 2 Cons2(0, 0) -> 5 rev_l#21(5, 0) -> 5 rev_l#21(5, 0) -> 2 fleft_op_e_xs_11() -> 7 fleft_op_e_xs_11() -> 9 step_x_f1(6, 0, 7) -> 7 step_x_f1(6, 0, 7) -> 9 Cons2(0, 5) -> 2 Cons2(0, 5) -> 5 rev_l#22(10, 0) -> 11 step_x_f#12(6, 0, 7, 11) -> 4 rev_l#22(10, 0) -> 4 Cons3(0, 10) -> 4 Cons3(0, 10) -> 11 rev_l#22(11, 0) -> 11 rev_l#22(11, 0) -> 4 Cons3(0, 11) -> 4 Cons3(0, 11) -> 11 ---------------------------------------- (4) BOUNDS(1, n^1)