1127.47/291.57 WORST_CASE(Omega(n^2), ?) 1127.47/291.63 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1127.47/291.63 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1127.47/291.63 1127.47/291.63 1127.47/291.63 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1127.47/291.63 1127.47/291.63 (0) CpxTRS 1127.47/291.63 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1127.47/291.63 (2) CpxTRS 1127.47/291.63 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1127.47/291.63 (4) typed CpxTrs 1127.47/291.63 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1127.47/291.63 (6) typed CpxTrs 1127.47/291.63 (7) RewriteLemmaProof [LOWER BOUND(ID), 264 ms] 1127.47/291.63 (8) BEST 1127.47/291.63 (9) proven lower bound 1127.47/291.63 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1127.47/291.63 (11) BOUNDS(n^1, INF) 1127.47/291.63 (12) typed CpxTrs 1127.47/291.63 (13) RewriteLemmaProof [LOWER BOUND(ID), 82 ms] 1127.47/291.63 (14) BEST 1127.47/291.63 (15) proven lower bound 1127.47/291.63 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 1127.47/291.63 (17) BOUNDS(n^2, INF) 1127.47/291.63 (18) typed CpxTrs 1127.47/291.63 (19) RewriteLemmaProof [LOWER BOUND(ID), 40 ms] 1127.47/291.63 (20) typed CpxTrs 1127.47/291.63 (21) RewriteLemmaProof [LOWER BOUND(ID), 63 ms] 1127.47/291.63 (22) typed CpxTrs 1127.47/291.63 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (0) 1127.47/291.63 Obligation: 1127.47/291.63 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1127.47/291.63 1127.47/291.63 1127.47/291.63 The TRS R consists of the following rules: 1127.47/291.63 1127.47/291.63 +(x, 0) -> x 1127.47/291.63 +(x, s(y)) -> s(+(x, y)) 1127.47/291.63 *(x, 0) -> 0 1127.47/291.63 *(x, s(y)) -> +(*(x, y), x) 1127.47/291.63 ge(x, 0) -> true 1127.47/291.63 ge(0, s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0) -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0)))) 1127.47/291.63 iffact(x, true) -> *(x, fact(-(x, s(0)))) 1127.47/291.63 iffact(x, false) -> s(0) 1127.47/291.63 1127.47/291.63 S is empty. 1127.47/291.63 Rewrite Strategy: INNERMOST 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1127.47/291.63 Renamed function symbols to avoid clashes with predefined symbol. 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (2) 1127.47/291.63 Obligation: 1127.47/291.63 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1127.47/291.63 1127.47/291.63 1127.47/291.63 The TRS R consists of the following rules: 1127.47/291.63 1127.47/291.63 +'(x, 0') -> x 1127.47/291.63 +'(x, s(y)) -> s(+'(x, y)) 1127.47/291.63 *'(x, 0') -> 0' 1127.47/291.63 *'(x, s(y)) -> +'(*'(x, y), x) 1127.47/291.63 ge(x, 0') -> true 1127.47/291.63 ge(0', s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0') -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0')))) 1127.47/291.63 iffact(x, true) -> *'(x, fact(-(x, s(0')))) 1127.47/291.63 iffact(x, false) -> s(0') 1127.47/291.63 1127.47/291.63 S is empty. 1127.47/291.63 Rewrite Strategy: INNERMOST 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1127.47/291.63 Infered types. 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (4) 1127.47/291.63 Obligation: 1127.47/291.63 Innermost TRS: 1127.47/291.63 Rules: 1127.47/291.63 +'(x, 0') -> x 1127.47/291.63 +'(x, s(y)) -> s(+'(x, y)) 1127.47/291.63 *'(x, 0') -> 0' 1127.47/291.63 *'(x, s(y)) -> +'(*'(x, y), x) 1127.47/291.63 ge(x, 0') -> true 1127.47/291.63 ge(0', s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0') -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0')))) 1127.47/291.63 iffact(x, true) -> *'(x, fact(-(x, s(0')))) 1127.47/291.63 iffact(x, false) -> s(0') 1127.47/291.63 1127.47/291.63 Types: 1127.47/291.63 +' :: 0':s -> 0':s -> 0':s 1127.47/291.63 0' :: 0':s 1127.47/291.63 s :: 0':s -> 0':s 1127.47/291.63 *' :: 0':s -> 0':s -> 0':s 1127.47/291.63 ge :: 0':s -> 0':s -> true:false 1127.47/291.63 true :: true:false 1127.47/291.63 false :: true:false 1127.47/291.63 - :: 0':s -> 0':s -> 0':s 1127.47/291.63 fact :: 0':s -> 0':s 1127.47/291.63 iffact :: 0':s -> true:false -> 0':s 1127.47/291.63 hole_0':s1_0 :: 0':s 1127.47/291.63 hole_true:false2_0 :: true:false 1127.47/291.63 gen_0':s3_0 :: Nat -> 0':s 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (5) OrderProof (LOWER BOUND(ID)) 1127.47/291.63 Heuristically decided to analyse the following defined symbols: 1127.47/291.63 +', *', ge, -, fact 1127.47/291.63 1127.47/291.63 They will be analysed ascendingly in the following order: 1127.47/291.63 +' < *' 1127.47/291.63 *' < fact 1127.47/291.63 ge < fact 1127.47/291.63 - < fact 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (6) 1127.47/291.63 Obligation: 1127.47/291.63 Innermost TRS: 1127.47/291.63 Rules: 1127.47/291.63 +'(x, 0') -> x 1127.47/291.63 +'(x, s(y)) -> s(+'(x, y)) 1127.47/291.63 *'(x, 0') -> 0' 1127.47/291.63 *'(x, s(y)) -> +'(*'(x, y), x) 1127.47/291.63 ge(x, 0') -> true 1127.47/291.63 ge(0', s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0') -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0')))) 1127.47/291.63 iffact(x, true) -> *'(x, fact(-(x, s(0')))) 1127.47/291.63 iffact(x, false) -> s(0') 1127.47/291.63 1127.47/291.63 Types: 1127.47/291.63 +' :: 0':s -> 0':s -> 0':s 1127.47/291.63 0' :: 0':s 1127.47/291.63 s :: 0':s -> 0':s 1127.47/291.63 *' :: 0':s -> 0':s -> 0':s 1127.47/291.63 ge :: 0':s -> 0':s -> true:false 1127.47/291.63 true :: true:false 1127.47/291.63 false :: true:false 1127.47/291.63 - :: 0':s -> 0':s -> 0':s 1127.47/291.63 fact :: 0':s -> 0':s 1127.47/291.63 iffact :: 0':s -> true:false -> 0':s 1127.47/291.63 hole_0':s1_0 :: 0':s 1127.47/291.63 hole_true:false2_0 :: true:false 1127.47/291.63 gen_0':s3_0 :: Nat -> 0':s 1127.47/291.63 1127.47/291.63 1127.47/291.63 Generator Equations: 1127.47/291.63 gen_0':s3_0(0) <=> 0' 1127.47/291.63 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1127.47/291.63 1127.47/291.63 1127.47/291.63 The following defined symbols remain to be analysed: 1127.47/291.63 +', *', ge, -, fact 1127.47/291.63 1127.47/291.63 They will be analysed ascendingly in the following order: 1127.47/291.63 +' < *' 1127.47/291.63 *' < fact 1127.47/291.63 ge < fact 1127.47/291.63 - < fact 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1127.47/291.63 Proved the following rewrite lemma: 1127.47/291.63 +'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1127.47/291.63 1127.47/291.63 Induction Base: 1127.47/291.63 +'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1) 1127.47/291.63 gen_0':s3_0(a) 1127.47/291.63 1127.47/291.63 Induction Step: 1127.47/291.63 +'(gen_0':s3_0(a), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 1127.47/291.63 s(+'(gen_0':s3_0(a), gen_0':s3_0(n5_0))) ->_IH 1127.47/291.63 s(gen_0':s3_0(+(a, c6_0))) 1127.47/291.63 1127.47/291.63 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (8) 1127.47/291.63 Complex Obligation (BEST) 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (9) 1127.47/291.63 Obligation: 1127.47/291.63 Proved the lower bound n^1 for the following obligation: 1127.47/291.63 1127.47/291.63 Innermost TRS: 1127.47/291.63 Rules: 1127.47/291.63 +'(x, 0') -> x 1127.47/291.63 +'(x, s(y)) -> s(+'(x, y)) 1127.47/291.63 *'(x, 0') -> 0' 1127.47/291.63 *'(x, s(y)) -> +'(*'(x, y), x) 1127.47/291.63 ge(x, 0') -> true 1127.47/291.63 ge(0', s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0') -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0')))) 1127.47/291.63 iffact(x, true) -> *'(x, fact(-(x, s(0')))) 1127.47/291.63 iffact(x, false) -> s(0') 1127.47/291.63 1127.47/291.63 Types: 1127.47/291.63 +' :: 0':s -> 0':s -> 0':s 1127.47/291.63 0' :: 0':s 1127.47/291.63 s :: 0':s -> 0':s 1127.47/291.63 *' :: 0':s -> 0':s -> 0':s 1127.47/291.63 ge :: 0':s -> 0':s -> true:false 1127.47/291.63 true :: true:false 1127.47/291.63 false :: true:false 1127.47/291.63 - :: 0':s -> 0':s -> 0':s 1127.47/291.63 fact :: 0':s -> 0':s 1127.47/291.63 iffact :: 0':s -> true:false -> 0':s 1127.47/291.63 hole_0':s1_0 :: 0':s 1127.47/291.63 hole_true:false2_0 :: true:false 1127.47/291.63 gen_0':s3_0 :: Nat -> 0':s 1127.47/291.63 1127.47/291.63 1127.47/291.63 Generator Equations: 1127.47/291.63 gen_0':s3_0(0) <=> 0' 1127.47/291.63 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1127.47/291.63 1127.47/291.63 1127.47/291.63 The following defined symbols remain to be analysed: 1127.47/291.63 +', *', ge, -, fact 1127.47/291.63 1127.47/291.63 They will be analysed ascendingly in the following order: 1127.47/291.63 +' < *' 1127.47/291.63 *' < fact 1127.47/291.63 ge < fact 1127.47/291.63 - < fact 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (10) LowerBoundPropagationProof (FINISHED) 1127.47/291.63 Propagated lower bound. 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (11) 1127.47/291.63 BOUNDS(n^1, INF) 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (12) 1127.47/291.63 Obligation: 1127.47/291.63 Innermost TRS: 1127.47/291.63 Rules: 1127.47/291.63 +'(x, 0') -> x 1127.47/291.63 +'(x, s(y)) -> s(+'(x, y)) 1127.47/291.63 *'(x, 0') -> 0' 1127.47/291.63 *'(x, s(y)) -> +'(*'(x, y), x) 1127.47/291.63 ge(x, 0') -> true 1127.47/291.63 ge(0', s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0') -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0')))) 1127.47/291.63 iffact(x, true) -> *'(x, fact(-(x, s(0')))) 1127.47/291.63 iffact(x, false) -> s(0') 1127.47/291.63 1127.47/291.63 Types: 1127.47/291.63 +' :: 0':s -> 0':s -> 0':s 1127.47/291.63 0' :: 0':s 1127.47/291.63 s :: 0':s -> 0':s 1127.47/291.63 *' :: 0':s -> 0':s -> 0':s 1127.47/291.63 ge :: 0':s -> 0':s -> true:false 1127.47/291.63 true :: true:false 1127.47/291.63 false :: true:false 1127.47/291.63 - :: 0':s -> 0':s -> 0':s 1127.47/291.63 fact :: 0':s -> 0':s 1127.47/291.63 iffact :: 0':s -> true:false -> 0':s 1127.47/291.63 hole_0':s1_0 :: 0':s 1127.47/291.63 hole_true:false2_0 :: true:false 1127.47/291.63 gen_0':s3_0 :: Nat -> 0':s 1127.47/291.63 1127.47/291.63 1127.47/291.63 Lemmas: 1127.47/291.63 +'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1127.47/291.63 1127.47/291.63 1127.47/291.63 Generator Equations: 1127.47/291.63 gen_0':s3_0(0) <=> 0' 1127.47/291.63 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1127.47/291.63 1127.47/291.63 1127.47/291.63 The following defined symbols remain to be analysed: 1127.47/291.63 *', ge, -, fact 1127.47/291.63 1127.47/291.63 They will be analysed ascendingly in the following order: 1127.47/291.63 *' < fact 1127.47/291.63 ge < fact 1127.47/291.63 - < fact 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1127.47/291.63 Proved the following rewrite lemma: 1127.47/291.63 *'(gen_0':s3_0(a), gen_0':s3_0(n620_0)) -> gen_0':s3_0(*(n620_0, a)), rt in Omega(1 + a*n620_0 + n620_0) 1127.47/291.63 1127.47/291.63 Induction Base: 1127.47/291.63 *'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1) 1127.47/291.63 0' 1127.47/291.63 1127.47/291.63 Induction Step: 1127.47/291.63 *'(gen_0':s3_0(a), gen_0':s3_0(+(n620_0, 1))) ->_R^Omega(1) 1127.47/291.63 +'(*'(gen_0':s3_0(a), gen_0':s3_0(n620_0)), gen_0':s3_0(a)) ->_IH 1127.47/291.63 +'(gen_0':s3_0(*(c621_0, a)), gen_0':s3_0(a)) ->_L^Omega(1 + a) 1127.47/291.63 gen_0':s3_0(+(a, *(n620_0, a))) 1127.47/291.63 1127.47/291.63 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (14) 1127.47/291.63 Complex Obligation (BEST) 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (15) 1127.47/291.63 Obligation: 1127.47/291.63 Proved the lower bound n^2 for the following obligation: 1127.47/291.63 1127.47/291.63 Innermost TRS: 1127.47/291.63 Rules: 1127.47/291.63 +'(x, 0') -> x 1127.47/291.63 +'(x, s(y)) -> s(+'(x, y)) 1127.47/291.63 *'(x, 0') -> 0' 1127.47/291.63 *'(x, s(y)) -> +'(*'(x, y), x) 1127.47/291.63 ge(x, 0') -> true 1127.47/291.63 ge(0', s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0') -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0')))) 1127.47/291.63 iffact(x, true) -> *'(x, fact(-(x, s(0')))) 1127.47/291.63 iffact(x, false) -> s(0') 1127.47/291.63 1127.47/291.63 Types: 1127.47/291.63 +' :: 0':s -> 0':s -> 0':s 1127.47/291.63 0' :: 0':s 1127.47/291.63 s :: 0':s -> 0':s 1127.47/291.63 *' :: 0':s -> 0':s -> 0':s 1127.47/291.63 ge :: 0':s -> 0':s -> true:false 1127.47/291.63 true :: true:false 1127.47/291.63 false :: true:false 1127.47/291.63 - :: 0':s -> 0':s -> 0':s 1127.47/291.63 fact :: 0':s -> 0':s 1127.47/291.63 iffact :: 0':s -> true:false -> 0':s 1127.47/291.63 hole_0':s1_0 :: 0':s 1127.47/291.63 hole_true:false2_0 :: true:false 1127.47/291.63 gen_0':s3_0 :: Nat -> 0':s 1127.47/291.63 1127.47/291.63 1127.47/291.63 Lemmas: 1127.47/291.63 +'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1127.47/291.63 1127.47/291.63 1127.47/291.63 Generator Equations: 1127.47/291.63 gen_0':s3_0(0) <=> 0' 1127.47/291.63 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1127.47/291.63 1127.47/291.63 1127.47/291.63 The following defined symbols remain to be analysed: 1127.47/291.63 *', ge, -, fact 1127.47/291.63 1127.47/291.63 They will be analysed ascendingly in the following order: 1127.47/291.63 *' < fact 1127.47/291.63 ge < fact 1127.47/291.63 - < fact 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (16) LowerBoundPropagationProof (FINISHED) 1127.47/291.63 Propagated lower bound. 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (17) 1127.47/291.63 BOUNDS(n^2, INF) 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (18) 1127.47/291.63 Obligation: 1127.47/291.63 Innermost TRS: 1127.47/291.63 Rules: 1127.47/291.63 +'(x, 0') -> x 1127.47/291.63 +'(x, s(y)) -> s(+'(x, y)) 1127.47/291.63 *'(x, 0') -> 0' 1127.47/291.63 *'(x, s(y)) -> +'(*'(x, y), x) 1127.47/291.63 ge(x, 0') -> true 1127.47/291.63 ge(0', s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0') -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0')))) 1127.47/291.63 iffact(x, true) -> *'(x, fact(-(x, s(0')))) 1127.47/291.63 iffact(x, false) -> s(0') 1127.47/291.63 1127.47/291.63 Types: 1127.47/291.63 +' :: 0':s -> 0':s -> 0':s 1127.47/291.63 0' :: 0':s 1127.47/291.63 s :: 0':s -> 0':s 1127.47/291.63 *' :: 0':s -> 0':s -> 0':s 1127.47/291.63 ge :: 0':s -> 0':s -> true:false 1127.47/291.63 true :: true:false 1127.47/291.63 false :: true:false 1127.47/291.63 - :: 0':s -> 0':s -> 0':s 1127.47/291.63 fact :: 0':s -> 0':s 1127.47/291.63 iffact :: 0':s -> true:false -> 0':s 1127.47/291.63 hole_0':s1_0 :: 0':s 1127.47/291.63 hole_true:false2_0 :: true:false 1127.47/291.63 gen_0':s3_0 :: Nat -> 0':s 1127.47/291.63 1127.47/291.63 1127.47/291.63 Lemmas: 1127.47/291.63 +'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1127.47/291.63 *'(gen_0':s3_0(a), gen_0':s3_0(n620_0)) -> gen_0':s3_0(*(n620_0, a)), rt in Omega(1 + a*n620_0 + n620_0) 1127.47/291.63 1127.47/291.63 1127.47/291.63 Generator Equations: 1127.47/291.63 gen_0':s3_0(0) <=> 0' 1127.47/291.63 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1127.47/291.63 1127.47/291.63 1127.47/291.63 The following defined symbols remain to be analysed: 1127.47/291.63 ge, -, fact 1127.47/291.63 1127.47/291.63 They will be analysed ascendingly in the following order: 1127.47/291.63 ge < fact 1127.47/291.63 - < fact 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (19) RewriteLemmaProof (LOWER BOUND(ID)) 1127.47/291.63 Proved the following rewrite lemma: 1127.47/291.63 ge(gen_0':s3_0(n1361_0), gen_0':s3_0(n1361_0)) -> true, rt in Omega(1 + n1361_0) 1127.47/291.63 1127.47/291.63 Induction Base: 1127.47/291.63 ge(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1127.47/291.63 true 1127.47/291.63 1127.47/291.63 Induction Step: 1127.47/291.63 ge(gen_0':s3_0(+(n1361_0, 1)), gen_0':s3_0(+(n1361_0, 1))) ->_R^Omega(1) 1127.47/291.63 ge(gen_0':s3_0(n1361_0), gen_0':s3_0(n1361_0)) ->_IH 1127.47/291.63 true 1127.47/291.63 1127.47/291.63 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (20) 1127.47/291.63 Obligation: 1127.47/291.63 Innermost TRS: 1127.47/291.63 Rules: 1127.47/291.63 +'(x, 0') -> x 1127.47/291.63 +'(x, s(y)) -> s(+'(x, y)) 1127.47/291.63 *'(x, 0') -> 0' 1127.47/291.63 *'(x, s(y)) -> +'(*'(x, y), x) 1127.47/291.63 ge(x, 0') -> true 1127.47/291.63 ge(0', s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0') -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0')))) 1127.47/291.63 iffact(x, true) -> *'(x, fact(-(x, s(0')))) 1127.47/291.63 iffact(x, false) -> s(0') 1127.47/291.63 1127.47/291.63 Types: 1127.47/291.63 +' :: 0':s -> 0':s -> 0':s 1127.47/291.63 0' :: 0':s 1127.47/291.63 s :: 0':s -> 0':s 1127.47/291.63 *' :: 0':s -> 0':s -> 0':s 1127.47/291.63 ge :: 0':s -> 0':s -> true:false 1127.47/291.63 true :: true:false 1127.47/291.63 false :: true:false 1127.47/291.63 - :: 0':s -> 0':s -> 0':s 1127.47/291.63 fact :: 0':s -> 0':s 1127.47/291.63 iffact :: 0':s -> true:false -> 0':s 1127.47/291.63 hole_0':s1_0 :: 0':s 1127.47/291.63 hole_true:false2_0 :: true:false 1127.47/291.63 gen_0':s3_0 :: Nat -> 0':s 1127.47/291.63 1127.47/291.63 1127.47/291.63 Lemmas: 1127.47/291.63 +'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1127.47/291.63 *'(gen_0':s3_0(a), gen_0':s3_0(n620_0)) -> gen_0':s3_0(*(n620_0, a)), rt in Omega(1 + a*n620_0 + n620_0) 1127.47/291.63 ge(gen_0':s3_0(n1361_0), gen_0':s3_0(n1361_0)) -> true, rt in Omega(1 + n1361_0) 1127.47/291.63 1127.47/291.63 1127.47/291.63 Generator Equations: 1127.47/291.63 gen_0':s3_0(0) <=> 0' 1127.47/291.63 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1127.47/291.63 1127.47/291.63 1127.47/291.63 The following defined symbols remain to be analysed: 1127.47/291.63 -, fact 1127.47/291.63 1127.47/291.63 They will be analysed ascendingly in the following order: 1127.47/291.63 - < fact 1127.47/291.63 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (21) RewriteLemmaProof (LOWER BOUND(ID)) 1127.47/291.63 Proved the following rewrite lemma: 1127.47/291.63 -(gen_0':s3_0(n1685_0), gen_0':s3_0(n1685_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1685_0) 1127.47/291.63 1127.47/291.63 Induction Base: 1127.47/291.63 -(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1127.47/291.63 gen_0':s3_0(0) 1127.47/291.63 1127.47/291.63 Induction Step: 1127.47/291.63 -(gen_0':s3_0(+(n1685_0, 1)), gen_0':s3_0(+(n1685_0, 1))) ->_R^Omega(1) 1127.47/291.63 -(gen_0':s3_0(n1685_0), gen_0':s3_0(n1685_0)) ->_IH 1127.47/291.63 gen_0':s3_0(0) 1127.47/291.63 1127.47/291.63 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1127.47/291.63 ---------------------------------------- 1127.47/291.63 1127.47/291.63 (22) 1127.47/291.63 Obligation: 1127.47/291.63 Innermost TRS: 1127.47/291.63 Rules: 1127.47/291.63 +'(x, 0') -> x 1127.47/291.63 +'(x, s(y)) -> s(+'(x, y)) 1127.47/291.63 *'(x, 0') -> 0' 1127.47/291.63 *'(x, s(y)) -> +'(*'(x, y), x) 1127.47/291.63 ge(x, 0') -> true 1127.47/291.63 ge(0', s(y)) -> false 1127.47/291.63 ge(s(x), s(y)) -> ge(x, y) 1127.47/291.63 -(x, 0') -> x 1127.47/291.63 -(s(x), s(y)) -> -(x, y) 1127.47/291.63 fact(x) -> iffact(x, ge(x, s(s(0')))) 1127.47/291.63 iffact(x, true) -> *'(x, fact(-(x, s(0')))) 1127.47/291.63 iffact(x, false) -> s(0') 1127.47/291.63 1127.47/291.63 Types: 1127.47/291.63 +' :: 0':s -> 0':s -> 0':s 1127.47/291.63 0' :: 0':s 1127.47/291.63 s :: 0':s -> 0':s 1127.47/291.63 *' :: 0':s -> 0':s -> 0':s 1127.47/291.63 ge :: 0':s -> 0':s -> true:false 1127.47/291.63 true :: true:false 1127.47/291.63 false :: true:false 1127.47/291.63 - :: 0':s -> 0':s -> 0':s 1127.47/291.63 fact :: 0':s -> 0':s 1127.47/291.63 iffact :: 0':s -> true:false -> 0':s 1127.47/291.63 hole_0':s1_0 :: 0':s 1127.47/291.63 hole_true:false2_0 :: true:false 1127.47/291.63 gen_0':s3_0 :: Nat -> 0':s 1127.47/291.63 1127.47/291.63 1127.47/291.63 Lemmas: 1127.47/291.63 +'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1127.47/291.63 *'(gen_0':s3_0(a), gen_0':s3_0(n620_0)) -> gen_0':s3_0(*(n620_0, a)), rt in Omega(1 + a*n620_0 + n620_0) 1127.47/291.63 ge(gen_0':s3_0(n1361_0), gen_0':s3_0(n1361_0)) -> true, rt in Omega(1 + n1361_0) 1127.47/291.63 -(gen_0':s3_0(n1685_0), gen_0':s3_0(n1685_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1685_0) 1127.47/291.63 1127.47/291.63 1127.47/291.63 Generator Equations: 1127.47/291.63 gen_0':s3_0(0) <=> 0' 1127.47/291.63 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1127.47/291.63 1127.47/291.63 1127.47/291.63 The following defined symbols remain to be analysed: 1127.47/291.63 fact 1127.88/291.72 EOF