WORST_CASE(Omega(n^1),O(n^1)) proof of input_irqCoB3tKi.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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, 23 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), 217 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), 92 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: append(Cons(x, xs), ys) -> Cons(x, append(xs, ys)) append(Nil, ys) -> ys goal(x, y) -> append(x, y) 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: append(Cons(x, xs), ys) -> Cons(x, append(xs, ys)) append(Nil, ys) -> ys goal(x, y) -> append(x, y) 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 1. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2] transitions: Cons0(0, 0) -> 0 Nil0() -> 0 append0(0, 0) -> 1 goal0(0, 0) -> 2 append1(0, 0) -> 3 Cons1(0, 3) -> 1 append1(0, 0) -> 2 Cons1(0, 3) -> 2 Cons1(0, 3) -> 3 0 -> 1 0 -> 2 0 -> 3 ---------------------------------------- (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: append(Cons(z0, z1), z2) -> Cons(z0, append(z1, z2)) append(Nil, z0) -> z0 goal(z0, z1) -> append(z0, z1) Tuples: APPEND(Cons(z0, z1), z2) -> c(APPEND(z1, z2)) APPEND(Nil, z0) -> c1 GOAL(z0, z1) -> c2(APPEND(z0, z1)) S tuples: APPEND(Cons(z0, z1), z2) -> c(APPEND(z1, z2)) APPEND(Nil, z0) -> c1 GOAL(z0, z1) -> c2(APPEND(z0, z1)) K tuples:none Defined Rule Symbols: append_2, goal_2 Defined Pair Symbols: APPEND_2, GOAL_2 Compound Symbols: c_1, c1, c2_1 ---------------------------------------- (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: APPEND(Cons(z0, z1), z2) -> c(APPEND(z1, z2)) APPEND(Nil, z0) -> c1 GOAL(z0, z1) -> c2(APPEND(z0, z1)) The (relative) TRS S consists of the following rules: append(Cons(z0, z1), z2) -> Cons(z0, append(z1, z2)) append(Nil, z0) -> z0 goal(z0, z1) -> append(z0, z1) 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: APPEND(Cons(z0, z1), z2) -> c(APPEND(z1, z2)) APPEND(Nil, z0) -> c1 GOAL(z0, z1) -> c2(APPEND(z0, z1)) The (relative) TRS S consists of the following rules: append(Cons(z0, z1), z2) -> Cons(z0, append(z1, z2)) append(Nil, z0) -> z0 goal(z0, z1) -> append(z0, z1) Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: APPEND(Cons(z0, z1), z2) -> c(APPEND(z1, z2)) APPEND(Nil, z0) -> c1 GOAL(z0, z1) -> c2(APPEND(z0, z1)) append(Cons(z0, z1), z2) -> Cons(z0, append(z1, z2)) append(Nil, z0) -> z0 goal(z0, z1) -> append(z0, z1) Types: APPEND :: Cons:Nil -> b -> c:c1 Cons :: a -> Cons:Nil -> Cons:Nil c :: c:c1 -> c:c1 Nil :: Cons:Nil c1 :: c:c1 GOAL :: Cons:Nil -> b -> c2 c2 :: c:c1 -> c2 append :: Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil hole_c:c11_3 :: c:c1 hole_Cons:Nil2_3 :: Cons:Nil hole_b3_3 :: b hole_a4_3 :: a hole_c25_3 :: c2 gen_c:c16_3 :: Nat -> c:c1 gen_Cons:Nil7_3 :: Nat -> Cons:Nil ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: APPEND, append ---------------------------------------- (14) Obligation: Innermost TRS: Rules: APPEND(Cons(z0, z1), z2) -> c(APPEND(z1, z2)) APPEND(Nil, z0) -> c1 GOAL(z0, z1) -> c2(APPEND(z0, z1)) append(Cons(z0, z1), z2) -> Cons(z0, append(z1, z2)) append(Nil, z0) -> z0 goal(z0, z1) -> append(z0, z1) Types: APPEND :: Cons:Nil -> b -> c:c1 Cons :: a -> Cons:Nil -> Cons:Nil c :: c:c1 -> c:c1 Nil :: Cons:Nil c1 :: c:c1 GOAL :: Cons:Nil -> b -> c2 c2 :: c:c1 -> c2 append :: Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil hole_c:c11_3 :: c:c1 hole_Cons:Nil2_3 :: Cons:Nil hole_b3_3 :: b hole_a4_3 :: a hole_c25_3 :: c2 gen_c:c16_3 :: Nat -> c:c1 gen_Cons:Nil7_3 :: Nat -> Cons:Nil Generator Equations: gen_c:c16_3(0) <=> c1 gen_c:c16_3(+(x, 1)) <=> c(gen_c:c16_3(x)) gen_Cons:Nil7_3(0) <=> Nil gen_Cons:Nil7_3(+(x, 1)) <=> Cons(hole_a4_3, gen_Cons:Nil7_3(x)) The following defined symbols remain to be analysed: APPEND, append ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPEND(gen_Cons:Nil7_3(n9_3), hole_b3_3) -> gen_c:c16_3(n9_3), rt in Omega(1 + n9_3) Induction Base: APPEND(gen_Cons:Nil7_3(0), hole_b3_3) ->_R^Omega(1) c1 Induction Step: APPEND(gen_Cons:Nil7_3(+(n9_3, 1)), hole_b3_3) ->_R^Omega(1) c(APPEND(gen_Cons:Nil7_3(n9_3), hole_b3_3)) ->_IH c(gen_c:c16_3(c10_3)) 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: APPEND(Cons(z0, z1), z2) -> c(APPEND(z1, z2)) APPEND(Nil, z0) -> c1 GOAL(z0, z1) -> c2(APPEND(z0, z1)) append(Cons(z0, z1), z2) -> Cons(z0, append(z1, z2)) append(Nil, z0) -> z0 goal(z0, z1) -> append(z0, z1) Types: APPEND :: Cons:Nil -> b -> c:c1 Cons :: a -> Cons:Nil -> Cons:Nil c :: c:c1 -> c:c1 Nil :: Cons:Nil c1 :: c:c1 GOAL :: Cons:Nil -> b -> c2 c2 :: c:c1 -> c2 append :: Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil hole_c:c11_3 :: c:c1 hole_Cons:Nil2_3 :: Cons:Nil hole_b3_3 :: b hole_a4_3 :: a hole_c25_3 :: c2 gen_c:c16_3 :: Nat -> c:c1 gen_Cons:Nil7_3 :: Nat -> Cons:Nil Generator Equations: gen_c:c16_3(0) <=> c1 gen_c:c16_3(+(x, 1)) <=> c(gen_c:c16_3(x)) gen_Cons:Nil7_3(0) <=> Nil gen_Cons:Nil7_3(+(x, 1)) <=> Cons(hole_a4_3, gen_Cons:Nil7_3(x)) The following defined symbols remain to be analysed: APPEND, append ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Innermost TRS: Rules: APPEND(Cons(z0, z1), z2) -> c(APPEND(z1, z2)) APPEND(Nil, z0) -> c1 GOAL(z0, z1) -> c2(APPEND(z0, z1)) append(Cons(z0, z1), z2) -> Cons(z0, append(z1, z2)) append(Nil, z0) -> z0 goal(z0, z1) -> append(z0, z1) Types: APPEND :: Cons:Nil -> b -> c:c1 Cons :: a -> Cons:Nil -> Cons:Nil c :: c:c1 -> c:c1 Nil :: Cons:Nil c1 :: c:c1 GOAL :: Cons:Nil -> b -> c2 c2 :: c:c1 -> c2 append :: Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil hole_c:c11_3 :: c:c1 hole_Cons:Nil2_3 :: Cons:Nil hole_b3_3 :: b hole_a4_3 :: a hole_c25_3 :: c2 gen_c:c16_3 :: Nat -> c:c1 gen_Cons:Nil7_3 :: Nat -> Cons:Nil Lemmas: APPEND(gen_Cons:Nil7_3(n9_3), hole_b3_3) -> gen_c:c16_3(n9_3), rt in Omega(1 + n9_3) Generator Equations: gen_c:c16_3(0) <=> c1 gen_c:c16_3(+(x, 1)) <=> c(gen_c:c16_3(x)) gen_Cons:Nil7_3(0) <=> Nil gen_Cons:Nil7_3(+(x, 1)) <=> Cons(hole_a4_3, gen_Cons:Nil7_3(x)) The following defined symbols remain to be analysed: append ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: append(gen_Cons:Nil7_3(n225_3), gen_Cons:Nil7_3(b)) -> gen_Cons:Nil7_3(+(n225_3, b)), rt in Omega(0) Induction Base: append(gen_Cons:Nil7_3(0), gen_Cons:Nil7_3(b)) ->_R^Omega(0) gen_Cons:Nil7_3(b) Induction Step: append(gen_Cons:Nil7_3(+(n225_3, 1)), gen_Cons:Nil7_3(b)) ->_R^Omega(0) Cons(hole_a4_3, append(gen_Cons:Nil7_3(n225_3), gen_Cons:Nil7_3(b))) ->_IH Cons(hole_a4_3, gen_Cons:Nil7_3(+(b, c226_3))) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)