346.23/291.55 WORST_CASE(Omega(n^3), ?) 346.23/291.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 346.23/291.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 346.23/291.56 346.23/291.56 346.23/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF). 346.23/291.56 346.23/291.56 (0) CpxTRS 346.23/291.56 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 346.23/291.56 (2) CpxTRS 346.23/291.56 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 346.23/291.56 (4) typed CpxTrs 346.23/291.56 (5) OrderProof [LOWER BOUND(ID), 0 ms] 346.23/291.56 (6) typed CpxTrs 346.23/291.56 (7) RewriteLemmaProof [LOWER BOUND(ID), 222 ms] 346.23/291.56 (8) BEST 346.23/291.56 (9) proven lower bound 346.23/291.56 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 346.23/291.56 (11) BOUNDS(n^1, INF) 346.23/291.56 (12) typed CpxTrs 346.23/291.56 (13) RewriteLemmaProof [LOWER BOUND(ID), 101 ms] 346.23/291.56 (14) BEST 346.23/291.56 (15) proven lower bound 346.23/291.56 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 346.23/291.56 (17) BOUNDS(n^3, INF) 346.23/291.56 (18) typed CpxTrs 346.23/291.56 (19) RewriteLemmaProof [LOWER BOUND(ID), 77 ms] 346.23/291.56 (20) typed CpxTrs 346.23/291.56 (21) RewriteLemmaProof [LOWER BOUND(ID), 83 ms] 346.23/291.56 (22) typed CpxTrs 346.23/291.56 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (0) 346.23/291.56 Obligation: 346.23/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF). 346.23/291.56 346.23/291.56 346.23/291.56 The TRS R consists of the following rules: 346.23/291.56 346.23/291.56 p(0) -> 0 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0) -> x 346.23/291.56 plus(0, y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0, y) -> 0 346.23/291.56 times(s(0), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0, y) -> 0 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0, s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0, 0) -> true 346.23/291.56 eq(s(x), 0) -> false 346.23/291.56 eq(0, s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0)) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0)), plus(zero(0), 0), s(plus(0, zero(0)))) 346.23/291.56 346.23/291.56 S is empty. 346.23/291.56 Rewrite Strategy: FULL 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 346.23/291.56 Renamed function symbols to avoid clashes with predefined symbol. 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (2) 346.23/291.56 Obligation: 346.23/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF). 346.23/291.56 346.23/291.56 346.23/291.56 The TRS R consists of the following rules: 346.23/291.56 346.23/291.56 p(0') -> 0' 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0') -> x 346.23/291.56 plus(0', y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0', y) -> 0' 346.23/291.56 times(s(0'), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0', y) -> 0' 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0' 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0', s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0', 0') -> true 346.23/291.56 eq(s(x), 0') -> false 346.23/291.56 eq(0', s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0')) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0')))) 346.23/291.56 346.23/291.56 S is empty. 346.23/291.56 Rewrite Strategy: FULL 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 346.23/291.56 Infered types. 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (4) 346.23/291.56 Obligation: 346.23/291.56 TRS: 346.23/291.56 Rules: 346.23/291.56 p(0') -> 0' 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0') -> x 346.23/291.56 plus(0', y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0', y) -> 0' 346.23/291.56 times(s(0'), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0', y) -> 0' 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0' 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0', s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0', 0') -> true 346.23/291.56 eq(s(x), 0') -> false 346.23/291.56 eq(0', s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0')) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0')))) 346.23/291.56 346.23/291.56 Types: 346.23/291.56 p :: 0':s:true:false -> 0':s:true:false 346.23/291.56 0' :: 0':s:true:false 346.23/291.56 s :: 0':s:true:false -> 0':s:true:false 346.23/291.56 plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 zero :: 0':s:true:false -> 0':s:true:false 346.23/291.56 eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 true :: 0':s:true:false 346.23/291.56 false :: 0':s:true:false 346.23/291.56 divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 prime :: 0':s:true:false -> 0':s:true:false 346.23/291.56 pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 hole_0':s:true:false1_0 :: 0':s:true:false 346.23/291.56 gen_0':s:true:false2_0 :: Nat -> 0':s:true:false 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (5) OrderProof (LOWER BOUND(ID)) 346.23/291.56 Heuristically decided to analyse the following defined symbols: 346.23/291.56 plus, times, div, quot, zero, eq, divides, pr, if 346.23/291.56 346.23/291.56 They will be analysed ascendingly in the following order: 346.23/291.56 plus < times 346.23/291.56 plus < zero 346.23/291.56 times < div 346.23/291.56 times < divides 346.23/291.56 div = quot 346.23/291.56 div = zero 346.23/291.56 div = divides 346.23/291.56 div = pr 346.23/291.56 div = if 346.23/291.56 quot = zero 346.23/291.56 quot = divides 346.23/291.56 quot = pr 346.23/291.56 quot = if 346.23/291.56 eq < zero 346.23/291.56 zero = divides 346.23/291.56 zero = pr 346.23/291.56 zero = if 346.23/291.56 eq < divides 346.23/291.56 divides = pr 346.23/291.56 divides = if 346.23/291.56 pr = if 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (6) 346.23/291.56 Obligation: 346.23/291.56 TRS: 346.23/291.56 Rules: 346.23/291.56 p(0') -> 0' 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0') -> x 346.23/291.56 plus(0', y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0', y) -> 0' 346.23/291.56 times(s(0'), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0', y) -> 0' 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0' 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0', s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0', 0') -> true 346.23/291.56 eq(s(x), 0') -> false 346.23/291.56 eq(0', s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0')) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0')))) 346.23/291.56 346.23/291.56 Types: 346.23/291.56 p :: 0':s:true:false -> 0':s:true:false 346.23/291.56 0' :: 0':s:true:false 346.23/291.56 s :: 0':s:true:false -> 0':s:true:false 346.23/291.56 plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 zero :: 0':s:true:false -> 0':s:true:false 346.23/291.56 eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 true :: 0':s:true:false 346.23/291.56 false :: 0':s:true:false 346.23/291.56 divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 prime :: 0':s:true:false -> 0':s:true:false 346.23/291.56 pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 hole_0':s:true:false1_0 :: 0':s:true:false 346.23/291.56 gen_0':s:true:false2_0 :: Nat -> 0':s:true:false 346.23/291.56 346.23/291.56 346.23/291.56 Generator Equations: 346.23/291.56 gen_0':s:true:false2_0(0) <=> 0' 346.23/291.56 gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x)) 346.23/291.56 346.23/291.56 346.23/291.56 The following defined symbols remain to be analysed: 346.23/291.56 plus, times, div, quot, zero, eq, divides, pr, if 346.23/291.56 346.23/291.56 They will be analysed ascendingly in the following order: 346.23/291.56 plus < times 346.23/291.56 plus < zero 346.23/291.56 times < div 346.23/291.56 times < divides 346.23/291.56 div = quot 346.23/291.56 div = zero 346.23/291.56 div = divides 346.23/291.56 div = pr 346.23/291.56 div = if 346.23/291.56 quot = zero 346.23/291.56 quot = divides 346.23/291.56 quot = pr 346.23/291.56 quot = if 346.23/291.56 eq < zero 346.23/291.56 zero = divides 346.23/291.56 zero = pr 346.23/291.56 zero = if 346.23/291.56 eq < divides 346.23/291.56 divides = pr 346.23/291.56 divides = if 346.23/291.56 pr = if 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (7) RewriteLemmaProof (LOWER BOUND(ID)) 346.23/291.56 Proved the following rewrite lemma: 346.23/291.56 plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0) 346.23/291.56 346.23/291.56 Induction Base: 346.23/291.56 plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(0)) ->_R^Omega(1) 346.23/291.56 gen_0':s:true:false2_0(a) 346.23/291.56 346.23/291.56 Induction Step: 346.23/291.56 plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(+(n4_0, 1))) ->_R^Omega(1) 346.23/291.56 s(plus(gen_0':s:true:false2_0(a), p(s(gen_0':s:true:false2_0(n4_0))))) ->_R^Omega(1) 346.23/291.56 s(plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0))) ->_IH 346.23/291.56 s(gen_0':s:true:false2_0(+(a, c5_0))) 346.23/291.56 346.23/291.56 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (8) 346.23/291.56 Complex Obligation (BEST) 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (9) 346.23/291.56 Obligation: 346.23/291.56 Proved the lower bound n^1 for the following obligation: 346.23/291.56 346.23/291.56 TRS: 346.23/291.56 Rules: 346.23/291.56 p(0') -> 0' 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0') -> x 346.23/291.56 plus(0', y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0', y) -> 0' 346.23/291.56 times(s(0'), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0', y) -> 0' 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0' 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0', s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0', 0') -> true 346.23/291.56 eq(s(x), 0') -> false 346.23/291.56 eq(0', s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0')) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0')))) 346.23/291.56 346.23/291.56 Types: 346.23/291.56 p :: 0':s:true:false -> 0':s:true:false 346.23/291.56 0' :: 0':s:true:false 346.23/291.56 s :: 0':s:true:false -> 0':s:true:false 346.23/291.56 plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 zero :: 0':s:true:false -> 0':s:true:false 346.23/291.56 eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 true :: 0':s:true:false 346.23/291.56 false :: 0':s:true:false 346.23/291.56 divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 prime :: 0':s:true:false -> 0':s:true:false 346.23/291.56 pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 hole_0':s:true:false1_0 :: 0':s:true:false 346.23/291.56 gen_0':s:true:false2_0 :: Nat -> 0':s:true:false 346.23/291.56 346.23/291.56 346.23/291.56 Generator Equations: 346.23/291.56 gen_0':s:true:false2_0(0) <=> 0' 346.23/291.56 gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x)) 346.23/291.56 346.23/291.56 346.23/291.56 The following defined symbols remain to be analysed: 346.23/291.56 plus, times, div, quot, zero, eq, divides, pr, if 346.23/291.56 346.23/291.56 They will be analysed ascendingly in the following order: 346.23/291.56 plus < times 346.23/291.56 plus < zero 346.23/291.56 times < div 346.23/291.56 times < divides 346.23/291.56 div = quot 346.23/291.56 div = zero 346.23/291.56 div = divides 346.23/291.56 div = pr 346.23/291.56 div = if 346.23/291.56 quot = zero 346.23/291.56 quot = divides 346.23/291.56 quot = pr 346.23/291.56 quot = if 346.23/291.56 eq < zero 346.23/291.56 zero = divides 346.23/291.56 zero = pr 346.23/291.56 zero = if 346.23/291.56 eq < divides 346.23/291.56 divides = pr 346.23/291.56 divides = if 346.23/291.56 pr = if 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (10) LowerBoundPropagationProof (FINISHED) 346.23/291.56 Propagated lower bound. 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (11) 346.23/291.56 BOUNDS(n^1, INF) 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (12) 346.23/291.56 Obligation: 346.23/291.56 TRS: 346.23/291.56 Rules: 346.23/291.56 p(0') -> 0' 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0') -> x 346.23/291.56 plus(0', y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0', y) -> 0' 346.23/291.56 times(s(0'), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0', y) -> 0' 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0' 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0', s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0', 0') -> true 346.23/291.56 eq(s(x), 0') -> false 346.23/291.56 eq(0', s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0')) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0')))) 346.23/291.56 346.23/291.56 Types: 346.23/291.56 p :: 0':s:true:false -> 0':s:true:false 346.23/291.56 0' :: 0':s:true:false 346.23/291.56 s :: 0':s:true:false -> 0':s:true:false 346.23/291.56 plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 zero :: 0':s:true:false -> 0':s:true:false 346.23/291.56 eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 true :: 0':s:true:false 346.23/291.56 false :: 0':s:true:false 346.23/291.56 divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 prime :: 0':s:true:false -> 0':s:true:false 346.23/291.56 pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 hole_0':s:true:false1_0 :: 0':s:true:false 346.23/291.56 gen_0':s:true:false2_0 :: Nat -> 0':s:true:false 346.23/291.56 346.23/291.56 346.23/291.56 Lemmas: 346.23/291.56 plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0) 346.23/291.56 346.23/291.56 346.23/291.56 Generator Equations: 346.23/291.56 gen_0':s:true:false2_0(0) <=> 0' 346.23/291.56 gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x)) 346.23/291.56 346.23/291.56 346.23/291.56 The following defined symbols remain to be analysed: 346.23/291.56 times, div, quot, zero, eq, divides, pr, if 346.23/291.56 346.23/291.56 They will be analysed ascendingly in the following order: 346.23/291.56 times < div 346.23/291.56 times < divides 346.23/291.56 div = quot 346.23/291.56 div = zero 346.23/291.56 div = divides 346.23/291.56 div = pr 346.23/291.56 div = if 346.23/291.56 quot = zero 346.23/291.56 quot = divides 346.23/291.56 quot = pr 346.23/291.56 quot = if 346.23/291.56 eq < zero 346.23/291.56 zero = divides 346.23/291.56 zero = pr 346.23/291.56 zero = if 346.23/291.56 eq < divides 346.23/291.56 divides = pr 346.23/291.56 divides = if 346.23/291.56 pr = if 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (13) RewriteLemmaProof (LOWER BOUND(ID)) 346.23/291.56 Proved the following rewrite lemma: 346.23/291.56 times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b)) -> gen_0':s:true:false2_0(*(n983_0, b)), rt in Omega(1 + b*n983_0^2 + n983_0) 346.23/291.56 346.23/291.56 Induction Base: 346.23/291.56 times(gen_0':s:true:false2_0(0), gen_0':s:true:false2_0(b)) ->_R^Omega(1) 346.23/291.56 0' 346.23/291.56 346.23/291.56 Induction Step: 346.23/291.56 times(gen_0':s:true:false2_0(+(n983_0, 1)), gen_0':s:true:false2_0(b)) ->_R^Omega(1) 346.23/291.56 plus(gen_0':s:true:false2_0(b), times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b))) ->_IH 346.23/291.56 plus(gen_0':s:true:false2_0(b), gen_0':s:true:false2_0(*(c984_0, b))) ->_L^Omega(1 + b*n983_0) 346.23/291.56 gen_0':s:true:false2_0(+(*(n983_0, b), b)) 346.23/291.56 346.23/291.56 We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3). 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (14) 346.23/291.56 Complex Obligation (BEST) 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (15) 346.23/291.56 Obligation: 346.23/291.56 Proved the lower bound n^3 for the following obligation: 346.23/291.56 346.23/291.56 TRS: 346.23/291.56 Rules: 346.23/291.56 p(0') -> 0' 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0') -> x 346.23/291.56 plus(0', y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0', y) -> 0' 346.23/291.56 times(s(0'), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0', y) -> 0' 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0' 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0', s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0', 0') -> true 346.23/291.56 eq(s(x), 0') -> false 346.23/291.56 eq(0', s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0')) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0')))) 346.23/291.56 346.23/291.56 Types: 346.23/291.56 p :: 0':s:true:false -> 0':s:true:false 346.23/291.56 0' :: 0':s:true:false 346.23/291.56 s :: 0':s:true:false -> 0':s:true:false 346.23/291.56 plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 zero :: 0':s:true:false -> 0':s:true:false 346.23/291.56 eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 true :: 0':s:true:false 346.23/291.56 false :: 0':s:true:false 346.23/291.56 divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 prime :: 0':s:true:false -> 0':s:true:false 346.23/291.56 pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 hole_0':s:true:false1_0 :: 0':s:true:false 346.23/291.56 gen_0':s:true:false2_0 :: Nat -> 0':s:true:false 346.23/291.56 346.23/291.56 346.23/291.56 Lemmas: 346.23/291.56 plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0) 346.23/291.56 346.23/291.56 346.23/291.56 Generator Equations: 346.23/291.56 gen_0':s:true:false2_0(0) <=> 0' 346.23/291.56 gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x)) 346.23/291.56 346.23/291.56 346.23/291.56 The following defined symbols remain to be analysed: 346.23/291.56 times, div, quot, zero, eq, divides, pr, if 346.23/291.56 346.23/291.56 They will be analysed ascendingly in the following order: 346.23/291.56 times < div 346.23/291.56 times < divides 346.23/291.56 div = quot 346.23/291.56 div = zero 346.23/291.56 div = divides 346.23/291.56 div = pr 346.23/291.56 div = if 346.23/291.56 quot = zero 346.23/291.56 quot = divides 346.23/291.56 quot = pr 346.23/291.56 quot = if 346.23/291.56 eq < zero 346.23/291.56 zero = divides 346.23/291.56 zero = pr 346.23/291.56 zero = if 346.23/291.56 eq < divides 346.23/291.56 divides = pr 346.23/291.56 divides = if 346.23/291.56 pr = if 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (16) LowerBoundPropagationProof (FINISHED) 346.23/291.56 Propagated lower bound. 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (17) 346.23/291.56 BOUNDS(n^3, INF) 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (18) 346.23/291.56 Obligation: 346.23/291.56 TRS: 346.23/291.56 Rules: 346.23/291.56 p(0') -> 0' 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0') -> x 346.23/291.56 plus(0', y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0', y) -> 0' 346.23/291.56 times(s(0'), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0', y) -> 0' 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0' 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0', s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0', 0') -> true 346.23/291.56 eq(s(x), 0') -> false 346.23/291.56 eq(0', s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0')) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0')))) 346.23/291.56 346.23/291.56 Types: 346.23/291.56 p :: 0':s:true:false -> 0':s:true:false 346.23/291.56 0' :: 0':s:true:false 346.23/291.56 s :: 0':s:true:false -> 0':s:true:false 346.23/291.56 plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 zero :: 0':s:true:false -> 0':s:true:false 346.23/291.56 eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 true :: 0':s:true:false 346.23/291.56 false :: 0':s:true:false 346.23/291.56 divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 prime :: 0':s:true:false -> 0':s:true:false 346.23/291.56 pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 hole_0':s:true:false1_0 :: 0':s:true:false 346.23/291.56 gen_0':s:true:false2_0 :: Nat -> 0':s:true:false 346.23/291.56 346.23/291.56 346.23/291.56 Lemmas: 346.23/291.56 plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0) 346.23/291.56 times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b)) -> gen_0':s:true:false2_0(*(n983_0, b)), rt in Omega(1 + b*n983_0^2 + n983_0) 346.23/291.56 346.23/291.56 346.23/291.56 Generator Equations: 346.23/291.56 gen_0':s:true:false2_0(0) <=> 0' 346.23/291.56 gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x)) 346.23/291.56 346.23/291.56 346.23/291.56 The following defined symbols remain to be analysed: 346.23/291.56 eq, div, quot, zero, divides, pr, if 346.23/291.56 346.23/291.56 They will be analysed ascendingly in the following order: 346.23/291.56 div = quot 346.23/291.56 div = zero 346.23/291.56 div = divides 346.23/291.56 div = pr 346.23/291.56 div = if 346.23/291.56 quot = zero 346.23/291.56 quot = divides 346.23/291.56 quot = pr 346.23/291.56 quot = if 346.23/291.56 eq < zero 346.23/291.56 zero = divides 346.23/291.56 zero = pr 346.23/291.56 zero = if 346.23/291.56 eq < divides 346.23/291.56 divides = pr 346.23/291.56 divides = if 346.23/291.56 pr = if 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (19) RewriteLemmaProof (LOWER BOUND(ID)) 346.23/291.56 Proved the following rewrite lemma: 346.23/291.56 eq(gen_0':s:true:false2_0(n2234_0), gen_0':s:true:false2_0(n2234_0)) -> true, rt in Omega(1 + n2234_0) 346.23/291.56 346.23/291.56 Induction Base: 346.23/291.56 eq(gen_0':s:true:false2_0(0), gen_0':s:true:false2_0(0)) ->_R^Omega(1) 346.23/291.56 true 346.23/291.56 346.23/291.56 Induction Step: 346.23/291.56 eq(gen_0':s:true:false2_0(+(n2234_0, 1)), gen_0':s:true:false2_0(+(n2234_0, 1))) ->_R^Omega(1) 346.23/291.56 eq(gen_0':s:true:false2_0(n2234_0), gen_0':s:true:false2_0(n2234_0)) ->_IH 346.23/291.56 true 346.23/291.56 346.23/291.56 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (20) 346.23/291.56 Obligation: 346.23/291.56 TRS: 346.23/291.56 Rules: 346.23/291.56 p(0') -> 0' 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0') -> x 346.23/291.56 plus(0', y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0', y) -> 0' 346.23/291.56 times(s(0'), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0', y) -> 0' 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0' 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0', s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0', 0') -> true 346.23/291.56 eq(s(x), 0') -> false 346.23/291.56 eq(0', s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0')) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0')))) 346.23/291.56 346.23/291.56 Types: 346.23/291.56 p :: 0':s:true:false -> 0':s:true:false 346.23/291.56 0' :: 0':s:true:false 346.23/291.56 s :: 0':s:true:false -> 0':s:true:false 346.23/291.56 plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 zero :: 0':s:true:false -> 0':s:true:false 346.23/291.56 eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 true :: 0':s:true:false 346.23/291.56 false :: 0':s:true:false 346.23/291.56 divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 prime :: 0':s:true:false -> 0':s:true:false 346.23/291.56 pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 hole_0':s:true:false1_0 :: 0':s:true:false 346.23/291.56 gen_0':s:true:false2_0 :: Nat -> 0':s:true:false 346.23/291.56 346.23/291.56 346.23/291.56 Lemmas: 346.23/291.56 plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0) 346.23/291.56 times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b)) -> gen_0':s:true:false2_0(*(n983_0, b)), rt in Omega(1 + b*n983_0^2 + n983_0) 346.23/291.56 eq(gen_0':s:true:false2_0(n2234_0), gen_0':s:true:false2_0(n2234_0)) -> true, rt in Omega(1 + n2234_0) 346.23/291.56 346.23/291.56 346.23/291.56 Generator Equations: 346.23/291.56 gen_0':s:true:false2_0(0) <=> 0' 346.23/291.56 gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x)) 346.23/291.56 346.23/291.56 346.23/291.56 The following defined symbols remain to be analysed: 346.23/291.56 quot, div, zero, divides, pr, if 346.23/291.56 346.23/291.56 They will be analysed ascendingly in the following order: 346.23/291.56 div = quot 346.23/291.56 div = zero 346.23/291.56 div = divides 346.23/291.56 div = pr 346.23/291.56 div = if 346.23/291.56 quot = zero 346.23/291.56 quot = divides 346.23/291.56 quot = pr 346.23/291.56 quot = if 346.23/291.56 zero = divides 346.23/291.56 zero = pr 346.23/291.56 zero = if 346.23/291.56 divides = pr 346.23/291.56 divides = if 346.23/291.56 pr = if 346.23/291.56 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (21) RewriteLemmaProof (LOWER BOUND(ID)) 346.23/291.56 Proved the following rewrite lemma: 346.23/291.56 quot(gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(1)) -> gen_0':s:true:false2_0(1), rt in Omega(1 + n2831_0) 346.23/291.56 346.23/291.56 Induction Base: 346.23/291.56 quot(gen_0':s:true:false2_0(0), gen_0':s:true:false2_0(0), gen_0':s:true:false2_0(1)) ->_R^Omega(1) 346.23/291.56 s(div(gen_0':s:true:false2_0(0), s(gen_0':s:true:false2_0(0)))) ->_R^Omega(1) 346.23/291.56 s(0') 346.23/291.56 346.23/291.56 Induction Step: 346.23/291.56 quot(gen_0':s:true:false2_0(+(n2831_0, 1)), gen_0':s:true:false2_0(+(n2831_0, 1)), gen_0':s:true:false2_0(1)) ->_R^Omega(1) 346.23/291.56 quot(gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(1)) ->_IH 346.23/291.56 gen_0':s:true:false2_0(1) 346.23/291.56 346.23/291.56 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 346.23/291.56 ---------------------------------------- 346.23/291.56 346.23/291.56 (22) 346.23/291.56 Obligation: 346.23/291.56 TRS: 346.23/291.56 Rules: 346.23/291.56 p(0') -> 0' 346.23/291.56 p(s(x)) -> x 346.23/291.56 plus(x, 0') -> x 346.23/291.56 plus(0', y) -> y 346.23/291.56 plus(s(x), y) -> s(plus(x, y)) 346.23/291.56 plus(s(x), y) -> s(plus(p(s(x)), y)) 346.23/291.56 plus(x, s(y)) -> s(plus(x, p(s(y)))) 346.23/291.56 times(0', y) -> 0' 346.23/291.56 times(s(0'), y) -> y 346.23/291.56 times(s(x), y) -> plus(y, times(x, y)) 346.23/291.56 div(0', y) -> 0' 346.23/291.56 div(x, y) -> quot(x, y, y) 346.23/291.56 quot(zero(y), s(y), z) -> 0' 346.23/291.56 quot(s(x), s(y), z) -> quot(x, y, z) 346.23/291.56 quot(x, 0', s(z)) -> s(div(x, s(z))) 346.23/291.56 div(div(x, y), z) -> div(x, times(zero(y), z)) 346.23/291.56 eq(0', 0') -> true 346.23/291.56 eq(s(x), 0') -> false 346.23/291.56 eq(0', s(y)) -> false 346.23/291.56 eq(s(x), s(y)) -> eq(x, y) 346.23/291.56 divides(y, x) -> eq(x, times(div(x, y), y)) 346.23/291.56 prime(s(s(x))) -> pr(s(s(x)), s(x)) 346.23/291.56 pr(x, s(0')) -> true 346.23/291.56 pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) 346.23/291.56 if(true, x, y) -> false 346.23/291.56 if(false, x, y) -> pr(x, y) 346.23/291.56 zero(div(x, x)) -> x 346.23/291.56 zero(divides(x, x)) -> x 346.23/291.56 zero(times(x, x)) -> x 346.23/291.56 zero(quot(x, x, x)) -> x 346.23/291.56 zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0')))) 346.23/291.56 346.23/291.56 Types: 346.23/291.56 p :: 0':s:true:false -> 0':s:true:false 346.23/291.56 0' :: 0':s:true:false 346.23/291.56 s :: 0':s:true:false -> 0':s:true:false 346.23/291.56 plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 zero :: 0':s:true:false -> 0':s:true:false 346.23/291.56 eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 true :: 0':s:true:false 346.23/291.56 false :: 0':s:true:false 346.23/291.56 divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 prime :: 0':s:true:false -> 0':s:true:false 346.23/291.56 pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false 346.23/291.56 hole_0':s:true:false1_0 :: 0':s:true:false 346.23/291.56 gen_0':s:true:false2_0 :: Nat -> 0':s:true:false 346.23/291.56 346.23/291.56 346.23/291.56 Lemmas: 346.23/291.56 plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0) 346.23/291.56 times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b)) -> gen_0':s:true:false2_0(*(n983_0, b)), rt in Omega(1 + b*n983_0^2 + n983_0) 346.23/291.56 eq(gen_0':s:true:false2_0(n2234_0), gen_0':s:true:false2_0(n2234_0)) -> true, rt in Omega(1 + n2234_0) 346.23/291.56 quot(gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(1)) -> gen_0':s:true:false2_0(1), rt in Omega(1 + n2831_0) 346.23/291.56 346.23/291.56 346.23/291.56 Generator Equations: 346.23/291.56 gen_0':s:true:false2_0(0) <=> 0' 346.23/291.56 gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x)) 346.23/291.56 346.23/291.56 346.23/291.56 The following defined symbols remain to be analysed: 346.23/291.56 div, zero, divides, pr, if 346.23/291.56 346.23/291.56 They will be analysed ascendingly in the following order: 346.23/291.56 div = quot 346.23/291.56 div = zero 346.23/291.56 div = divides 346.23/291.56 div = pr 346.23/291.56 div = if 346.23/291.56 quot = zero 346.23/291.56 quot = divides 346.23/291.56 quot = pr 346.23/291.56 quot = if 346.23/291.56 zero = divides 346.23/291.56 zero = pr 346.23/291.56 zero = if 346.23/291.56 divides = pr 346.23/291.56 divides = if 346.23/291.56 pr = if 346.31/291.60 EOF