311.15/291.84 WORST_CASE(Omega(n^1), ?) 311.15/291.85 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 311.15/291.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 311.15/291.85 311.15/291.85 311.15/291.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.15/291.85 311.15/291.85 (0) CpxTRS 311.15/291.85 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 311.15/291.85 (2) CpxTRS 311.15/291.85 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 311.15/291.85 (4) typed CpxTrs 311.15/291.85 (5) OrderProof [LOWER BOUND(ID), 0 ms] 311.15/291.85 (6) typed CpxTrs 311.15/291.85 (7) RewriteLemmaProof [LOWER BOUND(ID), 259 ms] 311.15/291.85 (8) BEST 311.15/291.85 (9) proven lower bound 311.15/291.85 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 311.15/291.85 (11) BOUNDS(n^1, INF) 311.15/291.85 (12) typed CpxTrs 311.15/291.85 (13) RewriteLemmaProof [LOWER BOUND(ID), 84 ms] 311.15/291.85 (14) typed CpxTrs 311.15/291.85 (15) RewriteLemmaProof [LOWER BOUND(ID), 62 ms] 311.15/291.85 (16) typed CpxTrs 311.15/291.85 311.15/291.85 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (0) 311.15/291.85 Obligation: 311.15/291.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.15/291.85 311.15/291.85 311.15/291.85 The TRS R consists of the following rules: 311.15/291.85 311.15/291.85 f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0)))), plus(s(0), x), double(y)) 311.15/291.85 gt(0, v) -> false 311.15/291.85 gt(s(u), 0) -> true 311.15/291.85 gt(s(u), s(v)) -> gt(u, v) 311.15/291.85 and(x, true) -> x 311.15/291.85 and(x, false) -> false 311.15/291.85 plus(n, 0) -> n 311.15/291.85 plus(n, s(m)) -> s(plus(n, m)) 311.15/291.85 double(0) -> 0 311.15/291.85 double(s(x)) -> s(s(double(x))) 311.15/291.85 311.15/291.85 S is empty. 311.15/291.85 Rewrite Strategy: FULL 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 311.15/291.85 Renamed function symbols to avoid clashes with predefined symbol. 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (2) 311.15/291.85 Obligation: 311.15/291.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.15/291.85 311.15/291.85 311.15/291.85 The TRS R consists of the following rules: 311.15/291.85 311.15/291.85 f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y)) 311.15/291.85 gt(0', v) -> false 311.15/291.85 gt(s(u), 0') -> true 311.15/291.85 gt(s(u), s(v)) -> gt(u, v) 311.15/291.85 and(x, true) -> x 311.15/291.85 and(x, false) -> false 311.15/291.85 plus(n, 0') -> n 311.15/291.85 plus(n, s(m)) -> s(plus(n, m)) 311.15/291.85 double(0') -> 0' 311.15/291.85 double(s(x)) -> s(s(double(x))) 311.15/291.85 311.15/291.85 S is empty. 311.15/291.85 Rewrite Strategy: FULL 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 311.15/291.85 Infered types. 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (4) 311.15/291.85 Obligation: 311.15/291.85 TRS: 311.15/291.85 Rules: 311.15/291.85 f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y)) 311.15/291.85 gt(0', v) -> false 311.15/291.85 gt(s(u), 0') -> true 311.15/291.85 gt(s(u), s(v)) -> gt(u, v) 311.15/291.85 and(x, true) -> x 311.15/291.85 and(x, false) -> false 311.15/291.85 plus(n, 0') -> n 311.15/291.85 plus(n, s(m)) -> s(plus(n, m)) 311.15/291.85 double(0') -> 0' 311.15/291.85 double(s(x)) -> s(s(double(x))) 311.15/291.85 311.15/291.85 Types: 311.15/291.85 f :: true:false -> 0':s -> 0':s -> f 311.15/291.85 true :: true:false 311.15/291.85 and :: true:false -> true:false -> true:false 311.15/291.85 gt :: 0':s -> 0':s -> true:false 311.15/291.85 s :: 0':s -> 0':s 311.15/291.85 0' :: 0':s 311.15/291.85 plus :: 0':s -> 0':s -> 0':s 311.15/291.85 double :: 0':s -> 0':s 311.15/291.85 false :: true:false 311.15/291.85 hole_f1_0 :: f 311.15/291.85 hole_true:false2_0 :: true:false 311.15/291.85 hole_0':s3_0 :: 0':s 311.15/291.85 gen_0':s4_0 :: Nat -> 0':s 311.15/291.85 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (5) OrderProof (LOWER BOUND(ID)) 311.15/291.85 Heuristically decided to analyse the following defined symbols: 311.15/291.85 f, gt, plus, double 311.15/291.85 311.15/291.85 They will be analysed ascendingly in the following order: 311.15/291.85 gt < f 311.15/291.85 plus < f 311.15/291.85 double < f 311.15/291.85 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (6) 311.15/291.85 Obligation: 311.15/291.85 TRS: 311.15/291.85 Rules: 311.15/291.85 f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y)) 311.15/291.85 gt(0', v) -> false 311.15/291.85 gt(s(u), 0') -> true 311.15/291.85 gt(s(u), s(v)) -> gt(u, v) 311.15/291.85 and(x, true) -> x 311.15/291.85 and(x, false) -> false 311.15/291.85 plus(n, 0') -> n 311.15/291.85 plus(n, s(m)) -> s(plus(n, m)) 311.15/291.85 double(0') -> 0' 311.15/291.85 double(s(x)) -> s(s(double(x))) 311.15/291.85 311.15/291.85 Types: 311.15/291.85 f :: true:false -> 0':s -> 0':s -> f 311.15/291.85 true :: true:false 311.15/291.85 and :: true:false -> true:false -> true:false 311.15/291.85 gt :: 0':s -> 0':s -> true:false 311.15/291.85 s :: 0':s -> 0':s 311.15/291.85 0' :: 0':s 311.15/291.85 plus :: 0':s -> 0':s -> 0':s 311.15/291.85 double :: 0':s -> 0':s 311.15/291.85 false :: true:false 311.15/291.85 hole_f1_0 :: f 311.15/291.85 hole_true:false2_0 :: true:false 311.15/291.85 hole_0':s3_0 :: 0':s 311.15/291.85 gen_0':s4_0 :: Nat -> 0':s 311.15/291.85 311.15/291.85 311.15/291.85 Generator Equations: 311.15/291.85 gen_0':s4_0(0) <=> 0' 311.15/291.85 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 311.15/291.85 311.15/291.85 311.15/291.85 The following defined symbols remain to be analysed: 311.15/291.85 gt, f, plus, double 311.15/291.85 311.15/291.85 They will be analysed ascendingly in the following order: 311.15/291.85 gt < f 311.15/291.85 plus < f 311.15/291.85 double < f 311.15/291.85 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (7) RewriteLemmaProof (LOWER BOUND(ID)) 311.15/291.85 Proved the following rewrite lemma: 311.15/291.85 gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 311.15/291.85 311.15/291.85 Induction Base: 311.15/291.85 gt(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 311.15/291.85 false 311.15/291.85 311.15/291.85 Induction Step: 311.15/291.85 gt(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1) 311.15/291.85 gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH 311.15/291.85 false 311.15/291.85 311.15/291.85 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (8) 311.15/291.85 Complex Obligation (BEST) 311.15/291.85 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (9) 311.15/291.85 Obligation: 311.15/291.85 Proved the lower bound n^1 for the following obligation: 311.15/291.85 311.15/291.85 TRS: 311.15/291.85 Rules: 311.15/291.85 f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y)) 311.15/291.85 gt(0', v) -> false 311.15/291.85 gt(s(u), 0') -> true 311.15/291.85 gt(s(u), s(v)) -> gt(u, v) 311.15/291.85 and(x, true) -> x 311.15/291.85 and(x, false) -> false 311.15/291.85 plus(n, 0') -> n 311.15/291.85 plus(n, s(m)) -> s(plus(n, m)) 311.15/291.85 double(0') -> 0' 311.15/291.85 double(s(x)) -> s(s(double(x))) 311.15/291.85 311.15/291.85 Types: 311.15/291.85 f :: true:false -> 0':s -> 0':s -> f 311.15/291.85 true :: true:false 311.15/291.85 and :: true:false -> true:false -> true:false 311.15/291.85 gt :: 0':s -> 0':s -> true:false 311.15/291.85 s :: 0':s -> 0':s 311.15/291.85 0' :: 0':s 311.15/291.85 plus :: 0':s -> 0':s -> 0':s 311.15/291.85 double :: 0':s -> 0':s 311.15/291.85 false :: true:false 311.15/291.85 hole_f1_0 :: f 311.15/291.85 hole_true:false2_0 :: true:false 311.15/291.85 hole_0':s3_0 :: 0':s 311.15/291.85 gen_0':s4_0 :: Nat -> 0':s 311.15/291.85 311.15/291.85 311.15/291.85 Generator Equations: 311.15/291.85 gen_0':s4_0(0) <=> 0' 311.15/291.85 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 311.15/291.85 311.15/291.85 311.15/291.85 The following defined symbols remain to be analysed: 311.15/291.85 gt, f, plus, double 311.15/291.85 311.15/291.85 They will be analysed ascendingly in the following order: 311.15/291.85 gt < f 311.15/291.85 plus < f 311.15/291.85 double < f 311.15/291.85 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (10) LowerBoundPropagationProof (FINISHED) 311.15/291.85 Propagated lower bound. 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (11) 311.15/291.85 BOUNDS(n^1, INF) 311.15/291.85 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (12) 311.15/291.85 Obligation: 311.15/291.85 TRS: 311.15/291.85 Rules: 311.15/291.85 f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y)) 311.15/291.85 gt(0', v) -> false 311.15/291.85 gt(s(u), 0') -> true 311.15/291.85 gt(s(u), s(v)) -> gt(u, v) 311.15/291.85 and(x, true) -> x 311.15/291.85 and(x, false) -> false 311.15/291.85 plus(n, 0') -> n 311.15/291.85 plus(n, s(m)) -> s(plus(n, m)) 311.15/291.85 double(0') -> 0' 311.15/291.85 double(s(x)) -> s(s(double(x))) 311.15/291.85 311.15/291.85 Types: 311.15/291.85 f :: true:false -> 0':s -> 0':s -> f 311.15/291.85 true :: true:false 311.15/291.85 and :: true:false -> true:false -> true:false 311.15/291.85 gt :: 0':s -> 0':s -> true:false 311.15/291.85 s :: 0':s -> 0':s 311.15/291.85 0' :: 0':s 311.15/291.85 plus :: 0':s -> 0':s -> 0':s 311.15/291.85 double :: 0':s -> 0':s 311.15/291.85 false :: true:false 311.15/291.85 hole_f1_0 :: f 311.15/291.85 hole_true:false2_0 :: true:false 311.15/291.85 hole_0':s3_0 :: 0':s 311.15/291.85 gen_0':s4_0 :: Nat -> 0':s 311.15/291.85 311.15/291.85 311.15/291.85 Lemmas: 311.15/291.85 gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 311.15/291.85 311.15/291.85 311.15/291.85 Generator Equations: 311.15/291.85 gen_0':s4_0(0) <=> 0' 311.15/291.85 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 311.15/291.85 311.15/291.85 311.15/291.85 The following defined symbols remain to be analysed: 311.15/291.85 plus, f, double 311.15/291.85 311.15/291.85 They will be analysed ascendingly in the following order: 311.15/291.85 plus < f 311.15/291.85 double < f 311.15/291.85 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (13) RewriteLemmaProof (LOWER BOUND(ID)) 311.15/291.85 Proved the following rewrite lemma: 311.15/291.85 plus(gen_0':s4_0(a), gen_0':s4_0(n253_0)) -> gen_0':s4_0(+(n253_0, a)), rt in Omega(1 + n253_0) 311.15/291.85 311.15/291.85 Induction Base: 311.15/291.85 plus(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1) 311.15/291.85 gen_0':s4_0(a) 311.15/291.85 311.15/291.85 Induction Step: 311.15/291.85 plus(gen_0':s4_0(a), gen_0':s4_0(+(n253_0, 1))) ->_R^Omega(1) 311.15/291.85 s(plus(gen_0':s4_0(a), gen_0':s4_0(n253_0))) ->_IH 311.15/291.85 s(gen_0':s4_0(+(a, c254_0))) 311.15/291.85 311.15/291.85 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (14) 311.15/291.85 Obligation: 311.15/291.85 TRS: 311.15/291.85 Rules: 311.15/291.85 f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y)) 311.15/291.85 gt(0', v) -> false 311.15/291.85 gt(s(u), 0') -> true 311.15/291.85 gt(s(u), s(v)) -> gt(u, v) 311.15/291.85 and(x, true) -> x 311.15/291.85 and(x, false) -> false 311.15/291.85 plus(n, 0') -> n 311.15/291.85 plus(n, s(m)) -> s(plus(n, m)) 311.15/291.85 double(0') -> 0' 311.15/291.85 double(s(x)) -> s(s(double(x))) 311.15/291.85 311.15/291.85 Types: 311.15/291.85 f :: true:false -> 0':s -> 0':s -> f 311.15/291.85 true :: true:false 311.15/291.85 and :: true:false -> true:false -> true:false 311.15/291.85 gt :: 0':s -> 0':s -> true:false 311.15/291.85 s :: 0':s -> 0':s 311.15/291.85 0' :: 0':s 311.15/291.85 plus :: 0':s -> 0':s -> 0':s 311.15/291.85 double :: 0':s -> 0':s 311.15/291.85 false :: true:false 311.15/291.85 hole_f1_0 :: f 311.15/291.85 hole_true:false2_0 :: true:false 311.15/291.85 hole_0':s3_0 :: 0':s 311.15/291.85 gen_0':s4_0 :: Nat -> 0':s 311.15/291.85 311.15/291.85 311.15/291.85 Lemmas: 311.15/291.85 gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 311.15/291.85 plus(gen_0':s4_0(a), gen_0':s4_0(n253_0)) -> gen_0':s4_0(+(n253_0, a)), rt in Omega(1 + n253_0) 311.15/291.85 311.15/291.85 311.15/291.85 Generator Equations: 311.15/291.85 gen_0':s4_0(0) <=> 0' 311.15/291.85 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 311.15/291.85 311.15/291.85 311.15/291.85 The following defined symbols remain to be analysed: 311.15/291.85 double, f 311.15/291.85 311.15/291.85 They will be analysed ascendingly in the following order: 311.15/291.85 double < f 311.15/291.85 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (15) RewriteLemmaProof (LOWER BOUND(ID)) 311.15/291.85 Proved the following rewrite lemma: 311.15/291.85 double(gen_0':s4_0(n778_0)) -> gen_0':s4_0(*(2, n778_0)), rt in Omega(1 + n778_0) 311.15/291.85 311.15/291.85 Induction Base: 311.15/291.85 double(gen_0':s4_0(0)) ->_R^Omega(1) 311.15/291.85 0' 311.15/291.85 311.15/291.85 Induction Step: 311.15/291.85 double(gen_0':s4_0(+(n778_0, 1))) ->_R^Omega(1) 311.15/291.85 s(s(double(gen_0':s4_0(n778_0)))) ->_IH 311.15/291.85 s(s(gen_0':s4_0(*(2, c779_0)))) 311.15/291.85 311.15/291.85 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.15/291.85 ---------------------------------------- 311.15/291.85 311.15/291.85 (16) 311.15/291.85 Obligation: 311.15/291.85 TRS: 311.15/291.85 Rules: 311.15/291.85 f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y)) 311.15/291.85 gt(0', v) -> false 311.15/291.85 gt(s(u), 0') -> true 311.15/291.85 gt(s(u), s(v)) -> gt(u, v) 311.15/291.85 and(x, true) -> x 311.15/291.85 and(x, false) -> false 311.15/291.85 plus(n, 0') -> n 311.15/291.85 plus(n, s(m)) -> s(plus(n, m)) 311.15/291.85 double(0') -> 0' 311.15/291.85 double(s(x)) -> s(s(double(x))) 311.15/291.85 311.15/291.85 Types: 311.15/291.85 f :: true:false -> 0':s -> 0':s -> f 311.15/291.85 true :: true:false 311.15/291.85 and :: true:false -> true:false -> true:false 311.15/291.85 gt :: 0':s -> 0':s -> true:false 311.15/291.85 s :: 0':s -> 0':s 311.15/291.85 0' :: 0':s 311.15/291.85 plus :: 0':s -> 0':s -> 0':s 311.15/291.85 double :: 0':s -> 0':s 311.15/291.85 false :: true:false 311.15/291.85 hole_f1_0 :: f 311.15/291.85 hole_true:false2_0 :: true:false 311.15/291.85 hole_0':s3_0 :: 0':s 311.15/291.85 gen_0':s4_0 :: Nat -> 0':s 311.15/291.85 311.15/291.85 311.15/291.85 Lemmas: 311.15/291.85 gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 311.15/291.85 plus(gen_0':s4_0(a), gen_0':s4_0(n253_0)) -> gen_0':s4_0(+(n253_0, a)), rt in Omega(1 + n253_0) 311.15/291.85 double(gen_0':s4_0(n778_0)) -> gen_0':s4_0(*(2, n778_0)), rt in Omega(1 + n778_0) 311.15/291.85 311.15/291.85 311.15/291.85 Generator Equations: 311.15/291.85 gen_0':s4_0(0) <=> 0' 311.15/291.85 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 311.15/291.85 311.15/291.85 311.15/291.85 The following defined symbols remain to be analysed: 311.15/291.85 f 311.21/291.89 EOF