3.70/1.75 WORST_CASE(Omega(n^1), O(n^1)) 3.70/1.75 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.70/1.75 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.70/1.75 3.70/1.75 3.70/1.75 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.70/1.75 3.70/1.75 (0) CpxTRS 3.70/1.75 (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 3.70/1.75 (2) CpxTRS 3.70/1.75 (3) CpxTrsMatchBoundsTAProof [FINISHED, 28 ms] 3.70/1.75 (4) BOUNDS(1, n^1) 3.70/1.75 (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 3.70/1.75 (6) TRS for Loop Detection 3.70/1.75 (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 3.70/1.75 (8) BEST 3.70/1.75 (9) proven lower bound 3.70/1.75 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 3.70/1.75 (11) BOUNDS(n^1, INF) 3.70/1.75 (12) TRS for Loop Detection 3.70/1.75 3.70/1.75 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (0) 3.70/1.75 Obligation: 3.70/1.75 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.70/1.75 3.70/1.75 3.70/1.75 The TRS R consists of the following rules: 3.70/1.75 3.70/1.75 rev_l#2(x8, x10) -> Cons(x10, x8) 3.70/1.75 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)) 3.70/1.75 step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5) 3.70/1.75 foldr#3(Nil) -> fleft_op_e_xs_1 3.70/1.75 foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6)) 3.70/1.75 main(Nil) -> Nil 3.70/1.75 main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), Nil) 3.70/1.75 3.70/1.75 S is empty. 3.70/1.75 Rewrite Strategy: INNERMOST 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (1) RelTrsToTrsProof (UPPER BOUND(ID)) 3.70/1.75 transformed relative TRS to TRS 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (2) 3.70/1.75 Obligation: 3.70/1.75 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 3.70/1.75 3.70/1.75 3.70/1.75 The TRS R consists of the following rules: 3.70/1.75 3.70/1.75 rev_l#2(x8, x10) -> Cons(x10, x8) 3.70/1.75 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)) 3.70/1.75 step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5) 3.70/1.75 foldr#3(Nil) -> fleft_op_e_xs_1 3.70/1.75 foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6)) 3.70/1.75 main(Nil) -> Nil 3.70/1.75 main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), Nil) 3.70/1.75 3.70/1.75 S is empty. 3.70/1.75 Rewrite Strategy: INNERMOST 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (3) CpxTrsMatchBoundsTAProof (FINISHED) 3.70/1.75 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. 3.70/1.75 3.70/1.75 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 3.70/1.75 final states : [1, 2, 3, 4] 3.70/1.75 transitions: 3.70/1.75 Cons0(0, 0) -> 0 3.70/1.75 rev_l0() -> 0 3.70/1.75 step_x_f0(0, 0, 0) -> 0 3.70/1.75 fleft_op_e_xs_10() -> 0 3.70/1.75 Nil0() -> 0 3.70/1.75 rev_l#20(0, 0) -> 1 3.70/1.75 step_x_f#10(0, 0, 0, 0) -> 2 3.70/1.75 foldr#30(0) -> 3 3.70/1.75 main0(0) -> 4 3.70/1.75 Cons1(0, 0) -> 1 3.70/1.75 rev_l#21(0, 0) -> 5 3.70/1.75 step_x_f#11(0, 0, 0, 5) -> 2 3.70/1.75 rev_l#21(0, 0) -> 2 3.70/1.75 fleft_op_e_xs_11() -> 3 3.70/1.75 rev_l1() -> 6 3.70/1.75 foldr#31(0) -> 7 3.70/1.75 step_x_f1(6, 0, 7) -> 3 3.70/1.75 Nil1() -> 4 3.70/1.75 rev_l1() -> 8 3.70/1.75 foldr#31(0) -> 9 3.70/1.75 Nil1() -> 10 3.70/1.75 step_x_f#11(8, 0, 9, 10) -> 4 3.70/1.75 Cons2(0, 0) -> 2 3.70/1.75 Cons2(0, 0) -> 5 3.70/1.75 rev_l#21(5, 0) -> 5 3.70/1.75 rev_l#21(5, 0) -> 2 3.70/1.75 fleft_op_e_xs_11() -> 7 3.70/1.75 fleft_op_e_xs_11() -> 9 3.70/1.75 step_x_f1(6, 0, 7) -> 7 3.70/1.75 step_x_f1(6, 0, 7) -> 9 3.70/1.75 Cons2(0, 5) -> 2 3.70/1.75 Cons2(0, 5) -> 5 3.70/1.75 rev_l#22(10, 0) -> 11 3.70/1.75 step_x_f#12(6, 0, 7, 11) -> 4 3.70/1.75 rev_l#22(10, 0) -> 4 3.70/1.75 Cons3(0, 10) -> 4 3.70/1.75 Cons3(0, 10) -> 11 3.70/1.75 rev_l#22(11, 0) -> 11 3.70/1.75 rev_l#22(11, 0) -> 4 3.70/1.75 Cons3(0, 11) -> 4 3.70/1.75 Cons3(0, 11) -> 11 3.70/1.75 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (4) 3.70/1.75 BOUNDS(1, n^1) 3.70/1.75 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 3.70/1.75 Transformed a relative TRS into a decreasing-loop problem. 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (6) 3.70/1.75 Obligation: 3.70/1.75 Analyzing the following TRS for decreasing loops: 3.70/1.75 3.70/1.75 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.70/1.75 3.70/1.75 3.70/1.75 The TRS R consists of the following rules: 3.70/1.75 3.70/1.75 rev_l#2(x8, x10) -> Cons(x10, x8) 3.70/1.75 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)) 3.70/1.75 step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5) 3.70/1.75 foldr#3(Nil) -> fleft_op_e_xs_1 3.70/1.75 foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6)) 3.70/1.75 main(Nil) -> Nil 3.70/1.75 main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), Nil) 3.70/1.75 3.70/1.75 S is empty. 3.70/1.75 Rewrite Strategy: INNERMOST 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (7) DecreasingLoopProof (LOWER BOUND(ID)) 3.70/1.75 The following loop(s) give(s) rise to the lower bound Omega(n^1): 3.70/1.75 3.70/1.75 The rewrite sequence 3.70/1.75 3.70/1.75 foldr#3(Cons(x16, x6)) ->^+ step_x_f(rev_l, x16, foldr#3(x6)) 3.70/1.75 3.70/1.75 gives rise to a decreasing loop by considering the right hand sides subterm at position [2]. 3.70/1.75 3.70/1.75 The pumping substitution is [x6 / Cons(x16, x6)]. 3.70/1.75 3.70/1.75 The result substitution is [ ]. 3.70/1.75 3.70/1.75 3.70/1.75 3.70/1.75 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (8) 3.70/1.75 Complex Obligation (BEST) 3.70/1.75 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (9) 3.70/1.75 Obligation: 3.70/1.75 Proved the lower bound n^1 for the following obligation: 3.70/1.75 3.70/1.75 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.70/1.75 3.70/1.75 3.70/1.75 The TRS R consists of the following rules: 3.70/1.75 3.70/1.75 rev_l#2(x8, x10) -> Cons(x10, x8) 3.70/1.75 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)) 3.70/1.75 step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5) 3.70/1.75 foldr#3(Nil) -> fleft_op_e_xs_1 3.70/1.75 foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6)) 3.70/1.75 main(Nil) -> Nil 3.70/1.75 main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), Nil) 3.70/1.75 3.70/1.75 S is empty. 3.70/1.75 Rewrite Strategy: INNERMOST 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (10) LowerBoundPropagationProof (FINISHED) 3.70/1.75 Propagated lower bound. 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (11) 3.70/1.75 BOUNDS(n^1, INF) 3.70/1.75 3.70/1.75 ---------------------------------------- 3.70/1.75 3.70/1.75 (12) 3.70/1.75 Obligation: 3.70/1.75 Analyzing the following TRS for decreasing loops: 3.70/1.75 3.70/1.75 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.70/1.75 3.70/1.75 3.70/1.75 The TRS R consists of the following rules: 3.70/1.75 3.70/1.75 rev_l#2(x8, x10) -> Cons(x10, x8) 3.70/1.75 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)) 3.70/1.75 step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) -> rev_l#2(x3, x5) 3.70/1.75 foldr#3(Nil) -> fleft_op_e_xs_1 3.70/1.75 foldr#3(Cons(x16, x6)) -> step_x_f(rev_l, x16, foldr#3(x6)) 3.70/1.75 main(Nil) -> Nil 3.70/1.75 main(Cons(x8, x9)) -> step_x_f#1(rev_l, x8, foldr#3(x9), Nil) 3.70/1.75 3.70/1.75 S is empty. 3.70/1.75 Rewrite Strategy: INNERMOST 3.92/1.79 EOF