1112.70/291.49 WORST_CASE(Omega(n^3), ?) 1112.91/291.53 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1112.91/291.53 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1112.91/291.53 1112.91/291.53 1112.91/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF). 1112.91/291.53 1112.91/291.53 (0) CpxTRS 1112.91/291.53 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1112.91/291.53 (2) CpxTRS 1112.91/291.53 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1112.91/291.53 (4) typed CpxTrs 1112.91/291.53 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1112.91/291.53 (6) typed CpxTrs 1112.91/291.53 (7) RewriteLemmaProof [LOWER BOUND(ID), 250 ms] 1112.91/291.53 (8) BEST 1112.91/291.53 (9) proven lower bound 1112.91/291.53 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1112.91/291.53 (11) BOUNDS(n^1, INF) 1112.91/291.53 (12) typed CpxTrs 1112.91/291.53 (13) RewriteLemmaProof [LOWER BOUND(ID), 46 ms] 1112.91/291.53 (14) BEST 1112.91/291.53 (15) proven lower bound 1112.91/291.53 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 1112.91/291.53 (17) BOUNDS(n^3, INF) 1112.91/291.53 (18) typed CpxTrs 1112.91/291.53 (19) RewriteLemmaProof [LOWER BOUND(ID), 74 ms] 1112.91/291.53 (20) typed CpxTrs 1112.91/291.53 (21) RewriteLemmaProof [LOWER BOUND(ID), 45 ms] 1112.91/291.53 (22) typed CpxTrs 1112.91/291.53 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (0) 1112.91/291.53 Obligation: 1112.91/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF). 1112.91/291.53 1112.91/291.53 1112.91/291.53 The TRS R consists of the following rules: 1112.91/291.53 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0) -> x 1112.91/291.53 plus(0, y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0, y) -> 0 1112.91/291.53 times(s(0), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0, y) -> 0 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0, s(y), z) -> 0 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0, s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0, 0) -> true 1112.91/291.53 eq(s(x), 0) -> false 1112.91/291.53 eq(0, s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0)) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 S is empty. 1112.91/291.53 Rewrite Strategy: FULL 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1112.91/291.53 Renamed function symbols to avoid clashes with predefined symbol. 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (2) 1112.91/291.53 Obligation: 1112.91/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF). 1112.91/291.53 1112.91/291.53 1112.91/291.53 The TRS R consists of the following rules: 1112.91/291.53 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0') -> x 1112.91/291.53 plus(0', y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0', y) -> 0' 1112.91/291.53 times(s(0'), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0', y) -> 0' 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0', s(y), z) -> 0' 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0', s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0', 0') -> true 1112.91/291.53 eq(s(x), 0') -> false 1112.91/291.53 eq(0', s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0')) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 S is empty. 1112.91/291.53 Rewrite Strategy: FULL 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1112.91/291.53 Infered types. 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (4) 1112.91/291.53 Obligation: 1112.91/291.53 TRS: 1112.91/291.53 Rules: 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0') -> x 1112.91/291.53 plus(0', y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0', y) -> 0' 1112.91/291.53 times(s(0'), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0', y) -> 0' 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0', s(y), z) -> 0' 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0', s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0', 0') -> true 1112.91/291.53 eq(s(x), 0') -> false 1112.91/291.53 eq(0', s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0')) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 Types: 1112.91/291.53 p :: s:0' -> s:0' 1112.91/291.53 s :: s:0' -> s:0' 1112.91/291.53 plus :: s:0' -> s:0' -> s:0' 1112.91/291.53 0' :: s:0' 1112.91/291.53 times :: s:0' -> s:0' -> s:0' 1112.91/291.53 div :: s:0' -> s:0' -> s:0' 1112.91/291.53 quot :: s:0' -> s:0' -> s:0' -> s:0' 1112.91/291.53 eq :: s:0' -> s:0' -> true:false 1112.91/291.53 true :: true:false 1112.91/291.53 false :: true:false 1112.91/291.53 divides :: s:0' -> s:0' -> true:false 1112.91/291.53 prime :: s:0' -> true:false 1112.91/291.53 pr :: s:0' -> s:0' -> true:false 1112.91/291.53 if :: true:false -> s:0' -> s:0' -> true:false 1112.91/291.53 hole_s:0'1_0 :: s:0' 1112.91/291.53 hole_true:false2_0 :: true:false 1112.91/291.53 gen_s:0'3_0 :: Nat -> s:0' 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (5) OrderProof (LOWER BOUND(ID)) 1112.91/291.53 Heuristically decided to analyse the following defined symbols: 1112.91/291.53 plus, times, div, quot, eq, pr 1112.91/291.53 1112.91/291.53 They will be analysed ascendingly in the following order: 1112.91/291.53 plus < times 1112.91/291.53 times < div 1112.91/291.53 div = quot 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (6) 1112.91/291.53 Obligation: 1112.91/291.53 TRS: 1112.91/291.53 Rules: 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0') -> x 1112.91/291.53 plus(0', y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0', y) -> 0' 1112.91/291.53 times(s(0'), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0', y) -> 0' 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0', s(y), z) -> 0' 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0', s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0', 0') -> true 1112.91/291.53 eq(s(x), 0') -> false 1112.91/291.53 eq(0', s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0')) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 Types: 1112.91/291.53 p :: s:0' -> s:0' 1112.91/291.53 s :: s:0' -> s:0' 1112.91/291.53 plus :: s:0' -> s:0' -> s:0' 1112.91/291.53 0' :: s:0' 1112.91/291.53 times :: s:0' -> s:0' -> s:0' 1112.91/291.53 div :: s:0' -> s:0' -> s:0' 1112.91/291.53 quot :: s:0' -> s:0' -> s:0' -> s:0' 1112.91/291.53 eq :: s:0' -> s:0' -> true:false 1112.91/291.53 true :: true:false 1112.91/291.53 false :: true:false 1112.91/291.53 divides :: s:0' -> s:0' -> true:false 1112.91/291.53 prime :: s:0' -> true:false 1112.91/291.53 pr :: s:0' -> s:0' -> true:false 1112.91/291.53 if :: true:false -> s:0' -> s:0' -> true:false 1112.91/291.53 hole_s:0'1_0 :: s:0' 1112.91/291.53 hole_true:false2_0 :: true:false 1112.91/291.53 gen_s:0'3_0 :: Nat -> s:0' 1112.91/291.53 1112.91/291.53 1112.91/291.53 Generator Equations: 1112.91/291.53 gen_s:0'3_0(0) <=> 0' 1112.91/291.53 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 1112.91/291.53 1112.91/291.53 1112.91/291.53 The following defined symbols remain to be analysed: 1112.91/291.53 plus, times, div, quot, eq, pr 1112.91/291.53 1112.91/291.53 They will be analysed ascendingly in the following order: 1112.91/291.53 plus < times 1112.91/291.53 times < div 1112.91/291.53 div = quot 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1112.91/291.53 Proved the following rewrite lemma: 1112.91/291.53 plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1112.91/291.53 1112.91/291.53 Induction Base: 1112.91/291.53 plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1) 1112.91/291.53 gen_s:0'3_0(a) 1112.91/291.53 1112.91/291.53 Induction Step: 1112.91/291.53 plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1) 1112.91/291.53 s(plus(gen_s:0'3_0(a), p(s(gen_s:0'3_0(n5_0))))) ->_R^Omega(1) 1112.91/291.53 s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0))) ->_IH 1112.91/291.53 s(gen_s:0'3_0(+(a, c6_0))) 1112.91/291.53 1112.91/291.53 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (8) 1112.91/291.53 Complex Obligation (BEST) 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (9) 1112.91/291.53 Obligation: 1112.91/291.53 Proved the lower bound n^1 for the following obligation: 1112.91/291.53 1112.91/291.53 TRS: 1112.91/291.53 Rules: 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0') -> x 1112.91/291.53 plus(0', y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0', y) -> 0' 1112.91/291.53 times(s(0'), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0', y) -> 0' 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0', s(y), z) -> 0' 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0', s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0', 0') -> true 1112.91/291.53 eq(s(x), 0') -> false 1112.91/291.53 eq(0', s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0')) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 Types: 1112.91/291.53 p :: s:0' -> s:0' 1112.91/291.53 s :: s:0' -> s:0' 1112.91/291.53 plus :: s:0' -> s:0' -> s:0' 1112.91/291.53 0' :: s:0' 1112.91/291.53 times :: s:0' -> s:0' -> s:0' 1112.91/291.53 div :: s:0' -> s:0' -> s:0' 1112.91/291.53 quot :: s:0' -> s:0' -> s:0' -> s:0' 1112.91/291.53 eq :: s:0' -> s:0' -> true:false 1112.91/291.53 true :: true:false 1112.91/291.53 false :: true:false 1112.91/291.53 divides :: s:0' -> s:0' -> true:false 1112.91/291.53 prime :: s:0' -> true:false 1112.91/291.53 pr :: s:0' -> s:0' -> true:false 1112.91/291.53 if :: true:false -> s:0' -> s:0' -> true:false 1112.91/291.53 hole_s:0'1_0 :: s:0' 1112.91/291.53 hole_true:false2_0 :: true:false 1112.91/291.53 gen_s:0'3_0 :: Nat -> s:0' 1112.91/291.53 1112.91/291.53 1112.91/291.53 Generator Equations: 1112.91/291.53 gen_s:0'3_0(0) <=> 0' 1112.91/291.53 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 1112.91/291.53 1112.91/291.53 1112.91/291.53 The following defined symbols remain to be analysed: 1112.91/291.53 plus, times, div, quot, eq, pr 1112.91/291.53 1112.91/291.53 They will be analysed ascendingly in the following order: 1112.91/291.53 plus < times 1112.91/291.53 times < div 1112.91/291.53 div = quot 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (10) LowerBoundPropagationProof (FINISHED) 1112.91/291.53 Propagated lower bound. 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (11) 1112.91/291.53 BOUNDS(n^1, INF) 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (12) 1112.91/291.53 Obligation: 1112.91/291.53 TRS: 1112.91/291.53 Rules: 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0') -> x 1112.91/291.53 plus(0', y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0', y) -> 0' 1112.91/291.53 times(s(0'), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0', y) -> 0' 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0', s(y), z) -> 0' 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0', s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0', 0') -> true 1112.91/291.53 eq(s(x), 0') -> false 1112.91/291.53 eq(0', s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0')) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 Types: 1112.91/291.53 p :: s:0' -> s:0' 1112.91/291.53 s :: s:0' -> s:0' 1112.91/291.53 plus :: s:0' -> s:0' -> s:0' 1112.91/291.53 0' :: s:0' 1112.91/291.53 times :: s:0' -> s:0' -> s:0' 1112.91/291.53 div :: s:0' -> s:0' -> s:0' 1112.91/291.53 quot :: s:0' -> s:0' -> s:0' -> s:0' 1112.91/291.53 eq :: s:0' -> s:0' -> true:false 1112.91/291.53 true :: true:false 1112.91/291.53 false :: true:false 1112.91/291.53 divides :: s:0' -> s:0' -> true:false 1112.91/291.53 prime :: s:0' -> true:false 1112.91/291.53 pr :: s:0' -> s:0' -> true:false 1112.91/291.53 if :: true:false -> s:0' -> s:0' -> true:false 1112.91/291.53 hole_s:0'1_0 :: s:0' 1112.91/291.53 hole_true:false2_0 :: true:false 1112.91/291.53 gen_s:0'3_0 :: Nat -> s:0' 1112.91/291.53 1112.91/291.53 1112.91/291.53 Lemmas: 1112.91/291.53 plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1112.91/291.53 1112.91/291.53 1112.91/291.53 Generator Equations: 1112.91/291.53 gen_s:0'3_0(0) <=> 0' 1112.91/291.53 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 1112.91/291.53 1112.91/291.53 1112.91/291.53 The following defined symbols remain to be analysed: 1112.91/291.53 times, div, quot, eq, pr 1112.91/291.53 1112.91/291.53 They will be analysed ascendingly in the following order: 1112.91/291.53 times < div 1112.91/291.53 div = quot 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1112.91/291.53 Proved the following rewrite lemma: 1112.91/291.53 times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b)) -> gen_s:0'3_0(*(n854_0, b)), rt in Omega(1 + b*n854_0^2 + n854_0) 1112.91/291.53 1112.91/291.53 Induction Base: 1112.91/291.53 times(gen_s:0'3_0(0), gen_s:0'3_0(b)) ->_R^Omega(1) 1112.91/291.53 0' 1112.91/291.53 1112.91/291.53 Induction Step: 1112.91/291.53 times(gen_s:0'3_0(+(n854_0, 1)), gen_s:0'3_0(b)) ->_R^Omega(1) 1112.91/291.53 plus(gen_s:0'3_0(b), times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b))) ->_IH 1112.91/291.53 plus(gen_s:0'3_0(b), gen_s:0'3_0(*(c855_0, b))) ->_L^Omega(1 + b*n854_0) 1112.91/291.53 gen_s:0'3_0(+(*(n854_0, b), b)) 1112.91/291.53 1112.91/291.53 We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3). 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (14) 1112.91/291.53 Complex Obligation (BEST) 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (15) 1112.91/291.53 Obligation: 1112.91/291.53 Proved the lower bound n^3 for the following obligation: 1112.91/291.53 1112.91/291.53 TRS: 1112.91/291.53 Rules: 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0') -> x 1112.91/291.53 plus(0', y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0', y) -> 0' 1112.91/291.53 times(s(0'), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0', y) -> 0' 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0', s(y), z) -> 0' 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0', s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0', 0') -> true 1112.91/291.53 eq(s(x), 0') -> false 1112.91/291.53 eq(0', s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0')) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 Types: 1112.91/291.53 p :: s:0' -> s:0' 1112.91/291.53 s :: s:0' -> s:0' 1112.91/291.53 plus :: s:0' -> s:0' -> s:0' 1112.91/291.53 0' :: s:0' 1112.91/291.53 times :: s:0' -> s:0' -> s:0' 1112.91/291.53 div :: s:0' -> s:0' -> s:0' 1112.91/291.53 quot :: s:0' -> s:0' -> s:0' -> s:0' 1112.91/291.53 eq :: s:0' -> s:0' -> true:false 1112.91/291.53 true :: true:false 1112.91/291.53 false :: true:false 1112.91/291.53 divides :: s:0' -> s:0' -> true:false 1112.91/291.53 prime :: s:0' -> true:false 1112.91/291.53 pr :: s:0' -> s:0' -> true:false 1112.91/291.53 if :: true:false -> s:0' -> s:0' -> true:false 1112.91/291.53 hole_s:0'1_0 :: s:0' 1112.91/291.53 hole_true:false2_0 :: true:false 1112.91/291.53 gen_s:0'3_0 :: Nat -> s:0' 1112.91/291.53 1112.91/291.53 1112.91/291.53 Lemmas: 1112.91/291.53 plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1112.91/291.53 1112.91/291.53 1112.91/291.53 Generator Equations: 1112.91/291.53 gen_s:0'3_0(0) <=> 0' 1112.91/291.53 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 1112.91/291.53 1112.91/291.53 1112.91/291.53 The following defined symbols remain to be analysed: 1112.91/291.53 times, div, quot, eq, pr 1112.91/291.53 1112.91/291.53 They will be analysed ascendingly in the following order: 1112.91/291.53 times < div 1112.91/291.53 div = quot 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (16) LowerBoundPropagationProof (FINISHED) 1112.91/291.53 Propagated lower bound. 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (17) 1112.91/291.53 BOUNDS(n^3, INF) 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (18) 1112.91/291.53 Obligation: 1112.91/291.53 TRS: 1112.91/291.53 Rules: 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0') -> x 1112.91/291.53 plus(0', y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0', y) -> 0' 1112.91/291.53 times(s(0'), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0', y) -> 0' 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0', s(y), z) -> 0' 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0', s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0', 0') -> true 1112.91/291.53 eq(s(x), 0') -> false 1112.91/291.53 eq(0', s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0')) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 Types: 1112.91/291.53 p :: s:0' -> s:0' 1112.91/291.53 s :: s:0' -> s:0' 1112.91/291.53 plus :: s:0' -> s:0' -> s:0' 1112.91/291.53 0' :: s:0' 1112.91/291.53 times :: s:0' -> s:0' -> s:0' 1112.91/291.53 div :: s:0' -> s:0' -> s:0' 1112.91/291.53 quot :: s:0' -> s:0' -> s:0' -> s:0' 1112.91/291.53 eq :: s:0' -> s:0' -> true:false 1112.91/291.53 true :: true:false 1112.91/291.53 false :: true:false 1112.91/291.53 divides :: s:0' -> s:0' -> true:false 1112.91/291.53 prime :: s:0' -> true:false 1112.91/291.53 pr :: s:0' -> s:0' -> true:false 1112.91/291.53 if :: true:false -> s:0' -> s:0' -> true:false 1112.91/291.53 hole_s:0'1_0 :: s:0' 1112.91/291.53 hole_true:false2_0 :: true:false 1112.91/291.53 gen_s:0'3_0 :: Nat -> s:0' 1112.91/291.53 1112.91/291.53 1112.91/291.53 Lemmas: 1112.91/291.53 plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1112.91/291.53 times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b)) -> gen_s:0'3_0(*(n854_0, b)), rt in Omega(1 + b*n854_0^2 + n854_0) 1112.91/291.53 1112.91/291.53 1112.91/291.53 Generator Equations: 1112.91/291.53 gen_s:0'3_0(0) <=> 0' 1112.91/291.53 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 1112.91/291.53 1112.91/291.53 1112.91/291.53 The following defined symbols remain to be analysed: 1112.91/291.53 eq, div, quot, pr 1112.91/291.53 1112.91/291.53 They will be analysed ascendingly in the following order: 1112.91/291.53 div = quot 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (19) RewriteLemmaProof (LOWER BOUND(ID)) 1112.91/291.53 Proved the following rewrite lemma: 1112.91/291.53 eq(gen_s:0'3_0(n1961_0), gen_s:0'3_0(n1961_0)) -> true, rt in Omega(1 + n1961_0) 1112.91/291.53 1112.91/291.53 Induction Base: 1112.91/291.53 eq(gen_s:0'3_0(0), gen_s:0'3_0(0)) ->_R^Omega(1) 1112.91/291.53 true 1112.91/291.53 1112.91/291.53 Induction Step: 1112.91/291.53 eq(gen_s:0'3_0(+(n1961_0, 1)), gen_s:0'3_0(+(n1961_0, 1))) ->_R^Omega(1) 1112.91/291.53 eq(gen_s:0'3_0(n1961_0), gen_s:0'3_0(n1961_0)) ->_IH 1112.91/291.53 true 1112.91/291.53 1112.91/291.53 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (20) 1112.91/291.53 Obligation: 1112.91/291.53 TRS: 1112.91/291.53 Rules: 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0') -> x 1112.91/291.53 plus(0', y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0', y) -> 0' 1112.91/291.53 times(s(0'), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0', y) -> 0' 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0', s(y), z) -> 0' 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0', s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0', 0') -> true 1112.91/291.53 eq(s(x), 0') -> false 1112.91/291.53 eq(0', s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0')) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 Types: 1112.91/291.53 p :: s:0' -> s:0' 1112.91/291.53 s :: s:0' -> s:0' 1112.91/291.53 plus :: s:0' -> s:0' -> s:0' 1112.91/291.53 0' :: s:0' 1112.91/291.53 times :: s:0' -> s:0' -> s:0' 1112.91/291.53 div :: s:0' -> s:0' -> s:0' 1112.91/291.53 quot :: s:0' -> s:0' -> s:0' -> s:0' 1112.91/291.53 eq :: s:0' -> s:0' -> true:false 1112.91/291.53 true :: true:false 1112.91/291.53 false :: true:false 1112.91/291.53 divides :: s:0' -> s:0' -> true:false 1112.91/291.53 prime :: s:0' -> true:false 1112.91/291.53 pr :: s:0' -> s:0' -> true:false 1112.91/291.53 if :: true:false -> s:0' -> s:0' -> true:false 1112.91/291.53 hole_s:0'1_0 :: s:0' 1112.91/291.53 hole_true:false2_0 :: true:false 1112.91/291.53 gen_s:0'3_0 :: Nat -> s:0' 1112.91/291.53 1112.91/291.53 1112.91/291.53 Lemmas: 1112.91/291.53 plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1112.91/291.53 times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b)) -> gen_s:0'3_0(*(n854_0, b)), rt in Omega(1 + b*n854_0^2 + n854_0) 1112.91/291.53 eq(gen_s:0'3_0(n1961_0), gen_s:0'3_0(n1961_0)) -> true, rt in Omega(1 + n1961_0) 1112.91/291.53 1112.91/291.53 1112.91/291.53 Generator Equations: 1112.91/291.53 gen_s:0'3_0(0) <=> 0' 1112.91/291.53 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 1112.91/291.53 1112.91/291.53 1112.91/291.53 The following defined symbols remain to be analysed: 1112.91/291.53 pr, div, quot 1112.91/291.53 1112.91/291.53 They will be analysed ascendingly in the following order: 1112.91/291.53 div = quot 1112.91/291.53 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (21) RewriteLemmaProof (LOWER BOUND(ID)) 1112.91/291.53 Proved the following rewrite lemma: 1112.91/291.53 quot(gen_s:0'3_0(n2634_0), gen_s:0'3_0(+(1, n2634_0)), gen_s:0'3_0(c)) -> gen_s:0'3_0(0), rt in Omega(1 + n2634_0) 1112.91/291.53 1112.91/291.53 Induction Base: 1112.91/291.53 quot(gen_s:0'3_0(0), gen_s:0'3_0(+(1, 0)), gen_s:0'3_0(c)) ->_R^Omega(1) 1112.91/291.53 0' 1112.91/291.53 1112.91/291.53 Induction Step: 1112.91/291.53 quot(gen_s:0'3_0(+(n2634_0, 1)), gen_s:0'3_0(+(1, +(n2634_0, 1))), gen_s:0'3_0(c)) ->_R^Omega(1) 1112.91/291.53 quot(gen_s:0'3_0(n2634_0), gen_s:0'3_0(+(1, n2634_0)), gen_s:0'3_0(c)) ->_IH 1112.91/291.53 gen_s:0'3_0(0) 1112.91/291.53 1112.91/291.53 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1112.91/291.53 ---------------------------------------- 1112.91/291.53 1112.91/291.53 (22) 1112.91/291.53 Obligation: 1112.91/291.53 TRS: 1112.91/291.53 Rules: 1112.91/291.53 p(s(x)) -> x 1112.91/291.53 plus(x, 0') -> x 1112.91/291.53 plus(0', y) -> y 1112.91/291.53 plus(s(x), y) -> s(plus(x, y)) 1112.91/291.53 plus(s(x), y) -> s(plus(p(s(x)), y)) 1112.91/291.53 plus(x, s(y)) -> s(plus(x, p(s(y)))) 1112.91/291.53 times(0', y) -> 0' 1112.91/291.53 times(s(0'), y) -> y 1112.91/291.53 times(s(x), y) -> plus(y, times(x, y)) 1112.91/291.53 div(0', y) -> 0' 1112.91/291.53 div(x, y) -> quot(x, y, y) 1112.91/291.53 quot(0', s(y), z) -> 0' 1112.91/291.53 quot(s(x), s(y), z) -> quot(x, y, z) 1112.91/291.53 quot(x, 0', s(z)) -> s(div(x, s(z))) 1112.91/291.53 div(div(x, y), z) -> div(x, times(y, z)) 1112.91/291.53 eq(0', 0') -> true 1112.91/291.53 eq(s(x), 0') -> false 1112.91/291.53 eq(0', s(y)) -> false 1112.91/291.53 eq(s(x), s(y)) -> eq(x, y) 1112.91/291.53 divides(y, x) -> eq(x, times(div(x, y), y)) 1112.91/291.53 prime(s(s(x))) -> pr(s(s(x)), s(x)) 1112.91/291.53 pr(x, s(0')) -> true 1112.91/291.53 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 1112.91/291.53 if(true, x, y) -> false 1112.91/291.53 if(false, x, y) -> pr(x, y) 1112.91/291.53 1112.91/291.53 Types: 1112.91/291.53 p :: s:0' -> s:0' 1112.91/291.53 s :: s:0' -> s:0' 1112.91/291.53 plus :: s:0' -> s:0' -> s:0' 1112.91/291.53 0' :: s:0' 1112.91/291.53 times :: s:0' -> s:0' -> s:0' 1112.91/291.53 div :: s:0' -> s:0' -> s:0' 1112.91/291.53 quot :: s:0' -> s:0' -> s:0' -> s:0' 1112.91/291.53 eq :: s:0' -> s:0' -> true:false 1112.91/291.53 true :: true:false 1112.91/291.53 false :: true:false 1112.91/291.53 divides :: s:0' -> s:0' -> true:false 1112.91/291.53 prime :: s:0' -> true:false 1112.91/291.53 pr :: s:0' -> s:0' -> true:false 1112.91/291.53 if :: true:false -> s:0' -> s:0' -> true:false 1112.91/291.53 hole_s:0'1_0 :: s:0' 1112.91/291.53 hole_true:false2_0 :: true:false 1112.91/291.53 gen_s:0'3_0 :: Nat -> s:0' 1112.91/291.53 1112.91/291.53 1112.91/291.53 Lemmas: 1112.91/291.53 plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0) 1112.91/291.53 times(gen_s:0'3_0(n854_0), gen_s:0'3_0(b)) -> gen_s:0'3_0(*(n854_0, b)), rt in Omega(1 + b*n854_0^2 + n854_0) 1112.91/291.53 eq(gen_s:0'3_0(n1961_0), gen_s:0'3_0(n1961_0)) -> true, rt in Omega(1 + n1961_0) 1112.91/291.53 quot(gen_s:0'3_0(n2634_0), gen_s:0'3_0(+(1, n2634_0)), gen_s:0'3_0(c)) -> gen_s:0'3_0(0), rt in Omega(1 + n2634_0) 1112.91/291.53 1112.91/291.53 1112.91/291.53 Generator Equations: 1112.91/291.53 gen_s:0'3_0(0) <=> 0' 1112.91/291.53 gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x)) 1112.91/291.53 1112.91/291.53 1112.91/291.53 The following defined symbols remain to be analysed: 1112.91/291.53 div 1112.91/291.53 1112.91/291.53 They will be analysed ascendingly in the following order: 1112.91/291.53 div = quot 1113.04/291.60 EOF