WORST_CASE(Omega(n^1),O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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, 0 ms] (4) BOUNDS(1, n^1) (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTRS (7) SlicingProof [LOWER BOUND(ID), 0 ms] (8) CpxTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 4 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 314 ms] (14) proven lower bound (15) LowerBoundPropagationProof [FINISHED, 0 ms] (16) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: anchored(Cons(x, xs), y) -> anchored(xs, Cons(Cons(Nil, Nil), y)) anchored(Nil, y) -> y goal(x, y) -> anchored(x, y) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: anchored(Cons(x, xs), y) -> anchored(xs, Cons(Cons(Nil, Nil), y)) anchored(Nil, y) -> y goal(x, y) -> anchored(x, y) S is empty. Rewrite Strategy: 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 anchored0(0, 0) -> 1 goal0(0, 0) -> 2 Nil1() -> 5 Nil1() -> 6 Cons1(5, 6) -> 4 Cons1(4, 0) -> 3 anchored1(0, 3) -> 1 anchored1(0, 0) -> 2 Cons1(4, 3) -> 3 anchored1(0, 3) -> 2 0 -> 1 0 -> 2 3 -> 1 3 -> 2 ---------------------------------------- (4) BOUNDS(1, n^1) ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: anchored(Cons(x, xs), y) -> anchored(xs, Cons(Cons(Nil, Nil), y)) anchored(Nil, y) -> y goal(x, y) -> anchored(x, y) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) SlicingProof (LOWER BOUND(ID)) Sliced the following arguments: Cons/0 ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: anchored(Cons(xs), y) -> anchored(xs, Cons(y)) anchored(Nil, y) -> y goal(x, y) -> anchored(x, y) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: anchored(Cons(xs), y) -> anchored(xs, Cons(y)) anchored(Nil, y) -> y goal(x, y) -> anchored(x, y) Types: anchored :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil hole_Cons:Nil1_0 :: Cons:Nil gen_Cons:Nil2_0 :: Nat -> Cons:Nil ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: anchored ---------------------------------------- (12) Obligation: Innermost TRS: Rules: anchored(Cons(xs), y) -> anchored(xs, Cons(y)) anchored(Nil, y) -> y goal(x, y) -> anchored(x, y) Types: anchored :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil hole_Cons:Nil1_0 :: Cons:Nil gen_Cons:Nil2_0 :: Nat -> Cons:Nil Generator Equations: gen_Cons:Nil2_0(0) <=> Nil gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x)) The following defined symbols remain to be analysed: anchored ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: anchored(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) -> gen_Cons:Nil2_0(+(n4_0, b)), rt in Omega(1 + n4_0) Induction Base: anchored(gen_Cons:Nil2_0(0), gen_Cons:Nil2_0(b)) ->_R^Omega(1) gen_Cons:Nil2_0(b) Induction Step: anchored(gen_Cons:Nil2_0(+(n4_0, 1)), gen_Cons:Nil2_0(b)) ->_R^Omega(1) anchored(gen_Cons:Nil2_0(n4_0), Cons(gen_Cons:Nil2_0(b))) ->_IH gen_Cons:Nil2_0(+(+(b, 1), c5_0)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: anchored(Cons(xs), y) -> anchored(xs, Cons(y)) anchored(Nil, y) -> y goal(x, y) -> anchored(x, y) Types: anchored :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil hole_Cons:Nil1_0 :: Cons:Nil gen_Cons:Nil2_0 :: Nat -> Cons:Nil Generator Equations: gen_Cons:Nil2_0(0) <=> Nil gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x)) The following defined symbols remain to be analysed: anchored ---------------------------------------- (15) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (16) BOUNDS(n^1, INF)