/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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, 102 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), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 256 ms] (14) BEST (15) proven lower bound (16) LowerBoundPropagationProof [FINISHED, 0 ms] (17) BOUNDS(n^1, INF) (18) typed CpxTrs ---------------------------------------- (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: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(x) 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: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(x) 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 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4, 5] transitions: Cons0(0, 0) -> 0 Nil0() -> 0 True0() -> 0 False0() -> 0 dec0(0) -> 1 isNilNil0(0) -> 2 nestdec0(0) -> 3 number170(0) -> 4 goal0(0) -> 5 Nil1() -> 1 Cons1(0, 0) -> 6 dec1(6) -> 1 Nil1() -> 7 dec1(7) -> 1 True1() -> 2 False1() -> 2 Nil1() -> 8 Nil1() -> 11 Cons1(8, 11) -> 10 Cons1(8, 10) -> 9 Cons1(8, 9) -> 9 Cons1(8, 9) -> 3 dec1(6) -> 12 nestdec1(12) -> 3 Cons1(8, 9) -> 4 nestdec1(0) -> 5 Nil1() -> 12 dec1(7) -> 12 Cons1(8, 9) -> 5 nestdec1(12) -> 5 Nil2() -> 13 Nil2() -> 16 Cons2(13, 16) -> 15 Cons2(13, 15) -> 14 Cons2(13, 14) -> 14 Cons2(13, 14) -> 3 Cons2(13, 14) -> 5 ---------------------------------------- (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: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(x) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) SlicingProof (LOWER BOUND(ID)) Sliced the following arguments: number17/0 ---------------------------------------- (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: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(x) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(x) Types: dec :: Nil:Cons -> Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons Nil :: Nil:Cons isNilNil :: Nil:Cons -> True:False True :: True:False False :: True:False nestdec :: Nil:Cons -> Nil:Cons number17 :: Nil:Cons goal :: Nil:Cons -> Nil:Cons hole_Nil:Cons1_0 :: Nil:Cons hole_True:False2_0 :: True:False gen_Nil:Cons3_0 :: Nat -> Nil:Cons ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: dec, nestdec They will be analysed ascendingly in the following order: dec < nestdec ---------------------------------------- (12) Obligation: Innermost TRS: Rules: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(x) Types: dec :: Nil:Cons -> Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons Nil :: Nil:Cons isNilNil :: Nil:Cons -> True:False True :: True:False False :: True:False nestdec :: Nil:Cons -> Nil:Cons number17 :: Nil:Cons goal :: Nil:Cons -> Nil:Cons hole_Nil:Cons1_0 :: Nil:Cons hole_True:False2_0 :: True:False gen_Nil:Cons3_0 :: Nat -> Nil:Cons Generator Equations: gen_Nil:Cons3_0(0) <=> Nil gen_Nil:Cons3_0(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons3_0(x)) The following defined symbols remain to be analysed: dec, nestdec They will be analysed ascendingly in the following order: dec < nestdec ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: dec(gen_Nil:Cons3_0(+(1, n5_0))) -> gen_Nil:Cons3_0(0), rt in Omega(1 + n5_0) Induction Base: dec(gen_Nil:Cons3_0(+(1, 0))) ->_R^Omega(1) Nil Induction Step: dec(gen_Nil:Cons3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1) dec(Cons(Nil, gen_Nil:Cons3_0(n5_0))) ->_IH gen_Nil:Cons3_0(0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Complex Obligation (BEST) ---------------------------------------- (15) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(x) Types: dec :: Nil:Cons -> Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons Nil :: Nil:Cons isNilNil :: Nil:Cons -> True:False True :: True:False False :: True:False nestdec :: Nil:Cons -> Nil:Cons number17 :: Nil:Cons goal :: Nil:Cons -> Nil:Cons hole_Nil:Cons1_0 :: Nil:Cons hole_True:False2_0 :: True:False gen_Nil:Cons3_0 :: Nat -> Nil:Cons Generator Equations: gen_Nil:Cons3_0(0) <=> Nil gen_Nil:Cons3_0(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons3_0(x)) The following defined symbols remain to be analysed: dec, nestdec They will be analysed ascendingly in the following order: dec < nestdec ---------------------------------------- (16) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (17) BOUNDS(n^1, INF) ---------------------------------------- (18) Obligation: Innermost TRS: Rules: dec(Cons(Nil, Nil)) -> Nil dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) isNilNil(Cons(Nil, Nil)) -> True isNilNil(Cons(Nil, Cons(x, xs))) -> False isNilNil(Cons(Cons(x, xs), Nil)) -> False isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nestdec(x) Types: dec :: Nil:Cons -> Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons Nil :: Nil:Cons isNilNil :: Nil:Cons -> True:False True :: True:False False :: True:False nestdec :: Nil:Cons -> Nil:Cons number17 :: Nil:Cons goal :: Nil:Cons -> Nil:Cons hole_Nil:Cons1_0 :: Nil:Cons hole_True:False2_0 :: True:False gen_Nil:Cons3_0 :: Nat -> Nil:Cons Lemmas: dec(gen_Nil:Cons3_0(+(1, n5_0))) -> gen_Nil:Cons3_0(0), rt in Omega(1 + n5_0) Generator Equations: gen_Nil:Cons3_0(0) <=> Nil gen_Nil:Cons3_0(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons3_0(x)) The following defined symbols remain to be analysed: nestdec