WORST_CASE(Omega(n^1),O(n^1)) proof of input_3pyry9DJWH.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) CpxTrsMatchBoundsTAProof [FINISHED, 100 ms] (4) BOUNDS(1, n^1) (5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxRelTRS (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 9 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 377 ms] (16) BEST (17) proven lower bound (18) LowerBoundPropagationProof [FINISHED, 0 ms] (19) BOUNDS(n^1, INF) (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 3 ms] (22) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: U11(tt, M, N) -> U12(tt, activate(M), activate(N)) U12(tt, M, N) -> s(plus(activate(N), activate(M))) plus(N, 0) -> N plus(N, s(M)) -> U11(tt, M, N) activate(X) -> 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: U11(tt, M, N) -> U12(tt, activate(M), activate(N)) U12(tt, M, N) -> s(plus(activate(N), activate(M))) plus(N, 0) -> N plus(N, s(M)) -> U11(tt, M, N) activate(X) -> X 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: tt0() -> 0 s0(0) -> 0 00() -> 0 U110(0, 0, 0) -> 1 U120(0, 0, 0) -> 2 plus0(0, 0) -> 3 activate0(0) -> 4 tt1() -> 5 activate1(0) -> 6 activate1(0) -> 7 U121(5, 6, 7) -> 1 activate1(0) -> 9 activate1(0) -> 10 plus1(9, 10) -> 8 s1(8) -> 2 tt1() -> 11 U111(11, 0, 0) -> 3 tt2() -> 12 activate2(0) -> 13 activate2(0) -> 14 U122(12, 13, 14) -> 3 activate2(7) -> 16 activate2(6) -> 17 plus2(16, 17) -> 15 s2(15) -> 1 U111(11, 0, 9) -> 8 activate3(14) -> 19 activate3(13) -> 20 plus3(19, 20) -> 18 s3(18) -> 3 activate2(9) -> 14 U122(12, 13, 14) -> 8 U111(11, 0, 16) -> 15 activate2(16) -> 14 U122(12, 13, 14) -> 15 s3(18) -> 8 U111(11, 0, 19) -> 18 activate2(19) -> 14 U122(12, 13, 14) -> 18 s3(18) -> 15 s3(18) -> 18 0 -> 3 0 -> 4 0 -> 6 0 -> 7 0 -> 9 0 -> 10 0 -> 13 0 -> 14 9 -> 8 9 -> 14 7 -> 16 6 -> 17 16 -> 15 16 -> 14 14 -> 19 13 -> 20 19 -> 18 19 -> 14 ---------------------------------------- (4) BOUNDS(1, n^1) ---------------------------------------- (5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> U11(tt, z1, z0) activate(z0) -> z0 Tuples: U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0)) U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0)) PLUS(z0, 0) -> c4 PLUS(z0, s(z1)) -> c5(U11'(tt, z1, z0)) ACTIVATE(z0) -> c6 S tuples: U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0)) U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0)) PLUS(z0, 0) -> c4 PLUS(z0, s(z1)) -> c5(U11'(tt, z1, z0)) ACTIVATE(z0) -> c6 K tuples:none Defined Rule Symbols: U11_3, U12_3, plus_2, activate_1 Defined Pair Symbols: U11'_3, U12'_3, PLUS_2, ACTIVATE_1 Compound Symbols: c_2, c1_2, c2_2, c3_2, c4, c5_1, c6 ---------------------------------------- (7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0)) U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0)) PLUS(z0, 0) -> c4 PLUS(z0, s(z1)) -> c5(U11'(tt, z1, z0)) ACTIVATE(z0) -> c6 The (relative) TRS S consists of the following rules: U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> U11(tt, z1, z0) activate(z0) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (9) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (10) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0)) U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0)) PLUS(z0, 0') -> c4 PLUS(z0, s(z1)) -> c5(U11'(tt, z1, z0)) ACTIVATE(z0) -> c6 The (relative) TRS S consists of the following rules: U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) plus(z0, 0') -> z0 plus(z0, s(z1)) -> U11(tt, z1, z0) activate(z0) -> z0 Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0)) U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0)) PLUS(z0, 0') -> c4 PLUS(z0, s(z1)) -> c5(U11'(tt, z1, z0)) ACTIVATE(z0) -> c6 U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) plus(z0, 0') -> z0 plus(z0, s(z1)) -> U11(tt, z1, z0) activate(z0) -> z0 Types: U11' :: tt -> 0':s -> 0':s -> c:c1 tt :: tt c :: c2:c3 -> c6 -> c:c1 U12' :: tt -> 0':s -> 0':s -> c2:c3 activate :: 0':s -> 0':s ACTIVATE :: 0':s -> c6 c1 :: c2:c3 -> c6 -> c:c1 c2 :: c4:c5 -> c6 -> c2:c3 PLUS :: 0':s -> 0':s -> c4:c5 c3 :: c4:c5 -> c6 -> c2:c3 0' :: 0':s c4 :: c4:c5 s :: 0':s -> 0':s c5 :: c:c1 -> c4:c5 c6 :: c6 U11 :: tt -> 0':s -> 0':s -> 0':s U12 :: tt -> 0':s -> 0':s -> 0':s plus :: 0':s -> 0':s -> 0':s hole_c:c11_7 :: c:c1 hole_tt2_7 :: tt hole_0':s3_7 :: 0':s hole_c2:c34_7 :: c2:c3 hole_c65_7 :: c6 hole_c4:c56_7 :: c4:c5 gen_0':s7_7 :: Nat -> 0':s ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: PLUS, plus ---------------------------------------- (14) Obligation: Innermost TRS: Rules: U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0)) U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0)) PLUS(z0, 0') -> c4 PLUS(z0, s(z1)) -> c5(U11'(tt, z1, z0)) ACTIVATE(z0) -> c6 U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) plus(z0, 0') -> z0 plus(z0, s(z1)) -> U11(tt, z1, z0) activate(z0) -> z0 Types: U11' :: tt -> 0':s -> 0':s -> c:c1 tt :: tt c :: c2:c3 -> c6 -> c:c1 U12' :: tt -> 0':s -> 0':s -> c2:c3 activate :: 0':s -> 0':s ACTIVATE :: 0':s -> c6 c1 :: c2:c3 -> c6 -> c:c1 c2 :: c4:c5 -> c6 -> c2:c3 PLUS :: 0':s -> 0':s -> c4:c5 c3 :: c4:c5 -> c6 -> c2:c3 0' :: 0':s c4 :: c4:c5 s :: 0':s -> 0':s c5 :: c:c1 -> c4:c5 c6 :: c6 U11 :: tt -> 0':s -> 0':s -> 0':s U12 :: tt -> 0':s -> 0':s -> 0':s plus :: 0':s -> 0':s -> 0':s hole_c:c11_7 :: c:c1 hole_tt2_7 :: tt hole_0':s3_7 :: 0':s hole_c2:c34_7 :: c2:c3 hole_c65_7 :: c6 hole_c4:c56_7 :: c4:c5 gen_0':s7_7 :: Nat -> 0':s Generator Equations: gen_0':s7_7(0) <=> 0' gen_0':s7_7(+(x, 1)) <=> s(gen_0':s7_7(x)) The following defined symbols remain to be analysed: PLUS, plus ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: PLUS(gen_0':s7_7(a), gen_0':s7_7(n9_7)) -> *8_7, rt in Omega(n9_7) Induction Base: PLUS(gen_0':s7_7(a), gen_0':s7_7(0)) Induction Step: PLUS(gen_0':s7_7(a), gen_0':s7_7(+(n9_7, 1))) ->_R^Omega(1) c5(U11'(tt, gen_0':s7_7(n9_7), gen_0':s7_7(a))) ->_R^Omega(1) c5(c(U12'(tt, activate(gen_0':s7_7(n9_7)), activate(gen_0':s7_7(a))), ACTIVATE(gen_0':s7_7(n9_7)))) ->_R^Omega(0) c5(c(U12'(tt, gen_0':s7_7(n9_7), activate(gen_0':s7_7(a))), ACTIVATE(gen_0':s7_7(n9_7)))) ->_R^Omega(0) c5(c(U12'(tt, gen_0':s7_7(n9_7), gen_0':s7_7(a)), ACTIVATE(gen_0':s7_7(n9_7)))) ->_R^Omega(1) c5(c(c2(PLUS(activate(gen_0':s7_7(a)), activate(gen_0':s7_7(n9_7))), ACTIVATE(gen_0':s7_7(a))), ACTIVATE(gen_0':s7_7(n9_7)))) ->_R^Omega(0) c5(c(c2(PLUS(gen_0':s7_7(a), activate(gen_0':s7_7(n9_7))), ACTIVATE(gen_0':s7_7(a))), ACTIVATE(gen_0':s7_7(n9_7)))) ->_R^Omega(0) c5(c(c2(PLUS(gen_0':s7_7(a), gen_0':s7_7(n9_7)), ACTIVATE(gen_0':s7_7(a))), ACTIVATE(gen_0':s7_7(n9_7)))) ->_IH c5(c(c2(*8_7, ACTIVATE(gen_0':s7_7(a))), ACTIVATE(gen_0':s7_7(n9_7)))) ->_R^Omega(1) c5(c(c2(*8_7, c6), ACTIVATE(gen_0':s7_7(n9_7)))) ->_R^Omega(1) c5(c(c2(*8_7, c6), c6)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Complex Obligation (BEST) ---------------------------------------- (17) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0)) U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0)) PLUS(z0, 0') -> c4 PLUS(z0, s(z1)) -> c5(U11'(tt, z1, z0)) ACTIVATE(z0) -> c6 U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) plus(z0, 0') -> z0 plus(z0, s(z1)) -> U11(tt, z1, z0) activate(z0) -> z0 Types: U11' :: tt -> 0':s -> 0':s -> c:c1 tt :: tt c :: c2:c3 -> c6 -> c:c1 U12' :: tt -> 0':s -> 0':s -> c2:c3 activate :: 0':s -> 0':s ACTIVATE :: 0':s -> c6 c1 :: c2:c3 -> c6 -> c:c1 c2 :: c4:c5 -> c6 -> c2:c3 PLUS :: 0':s -> 0':s -> c4:c5 c3 :: c4:c5 -> c6 -> c2:c3 0' :: 0':s c4 :: c4:c5 s :: 0':s -> 0':s c5 :: c:c1 -> c4:c5 c6 :: c6 U11 :: tt -> 0':s -> 0':s -> 0':s U12 :: tt -> 0':s -> 0':s -> 0':s plus :: 0':s -> 0':s -> 0':s hole_c:c11_7 :: c:c1 hole_tt2_7 :: tt hole_0':s3_7 :: 0':s hole_c2:c34_7 :: c2:c3 hole_c65_7 :: c6 hole_c4:c56_7 :: c4:c5 gen_0':s7_7 :: Nat -> 0':s Generator Equations: gen_0':s7_7(0) <=> 0' gen_0':s7_7(+(x, 1)) <=> s(gen_0':s7_7(x)) The following defined symbols remain to be analysed: PLUS, plus ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Innermost TRS: Rules: U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0)) U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1)) U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0)) PLUS(z0, 0') -> c4 PLUS(z0, s(z1)) -> c5(U11'(tt, z1, z0)) ACTIVATE(z0) -> c6 U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) plus(z0, 0') -> z0 plus(z0, s(z1)) -> U11(tt, z1, z0) activate(z0) -> z0 Types: U11' :: tt -> 0':s -> 0':s -> c:c1 tt :: tt c :: c2:c3 -> c6 -> c:c1 U12' :: tt -> 0':s -> 0':s -> c2:c3 activate :: 0':s -> 0':s ACTIVATE :: 0':s -> c6 c1 :: c2:c3 -> c6 -> c:c1 c2 :: c4:c5 -> c6 -> c2:c3 PLUS :: 0':s -> 0':s -> c4:c5 c3 :: c4:c5 -> c6 -> c2:c3 0' :: 0':s c4 :: c4:c5 s :: 0':s -> 0':s c5 :: c:c1 -> c4:c5 c6 :: c6 U11 :: tt -> 0':s -> 0':s -> 0':s U12 :: tt -> 0':s -> 0':s -> 0':s plus :: 0':s -> 0':s -> 0':s hole_c:c11_7 :: c:c1 hole_tt2_7 :: tt hole_0':s3_7 :: 0':s hole_c2:c34_7 :: c2:c3 hole_c65_7 :: c6 hole_c4:c56_7 :: c4:c5 gen_0':s7_7 :: Nat -> 0':s Lemmas: PLUS(gen_0':s7_7(a), gen_0':s7_7(n9_7)) -> *8_7, rt in Omega(n9_7) Generator Equations: gen_0':s7_7(0) <=> 0' gen_0':s7_7(+(x, 1)) <=> s(gen_0':s7_7(x)) The following defined symbols remain to be analysed: plus ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_0':s7_7(a), gen_0':s7_7(n1552_7)) -> gen_0':s7_7(+(n1552_7, a)), rt in Omega(0) Induction Base: plus(gen_0':s7_7(a), gen_0':s7_7(0)) ->_R^Omega(0) gen_0':s7_7(a) Induction Step: plus(gen_0':s7_7(a), gen_0':s7_7(+(n1552_7, 1))) ->_R^Omega(0) U11(tt, gen_0':s7_7(n1552_7), gen_0':s7_7(a)) ->_R^Omega(0) U12(tt, activate(gen_0':s7_7(n1552_7)), activate(gen_0':s7_7(a))) ->_R^Omega(0) U12(tt, gen_0':s7_7(n1552_7), activate(gen_0':s7_7(a))) ->_R^Omega(0) U12(tt, gen_0':s7_7(n1552_7), gen_0':s7_7(a)) ->_R^Omega(0) s(plus(activate(gen_0':s7_7(a)), activate(gen_0':s7_7(n1552_7)))) ->_R^Omega(0) s(plus(gen_0':s7_7(a), activate(gen_0':s7_7(n1552_7)))) ->_R^Omega(0) s(plus(gen_0':s7_7(a), gen_0':s7_7(n1552_7))) ->_IH s(gen_0':s7_7(+(a, c1553_7))) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)