318.46/291.56 WORST_CASE(Omega(n^1), ?) 318.46/291.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 318.46/291.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 318.46/291.57 318.46/291.57 318.46/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 318.46/291.57 318.46/291.57 (0) CpxTRS 318.46/291.57 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 318.46/291.57 (2) CpxTRS 318.46/291.57 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 318.46/291.57 (4) typed CpxTrs 318.46/291.57 (5) OrderProof [LOWER BOUND(ID), 0 ms] 318.46/291.57 (6) typed CpxTrs 318.46/291.57 (7) RewriteLemmaProof [LOWER BOUND(ID), 538 ms] 318.46/291.57 (8) BEST 318.46/291.57 (9) proven lower bound 318.46/291.57 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 318.46/291.57 (11) BOUNDS(n^1, INF) 318.46/291.57 (12) typed CpxTrs 318.46/291.57 (13) RewriteLemmaProof [LOWER BOUND(ID), 238 ms] 318.46/291.57 (14) typed CpxTrs 318.46/291.57 318.46/291.57 318.46/291.57 ---------------------------------------- 318.46/291.57 318.46/291.57 (0) 318.46/291.57 Obligation: 318.46/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 318.46/291.57 318.46/291.57 318.46/291.57 The TRS R consists of the following rules: 318.46/291.57 318.46/291.57 minus(x, 0) -> x 318.46/291.57 minus(s(x), s(y)) -> minus(x, y) 318.46/291.57 quot(0, s(y)) -> 0 318.46/291.57 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 318.46/291.57 minus(minus(x, y), z) -> minus(x, plus(y, z)) 318.46/291.57 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 318.46/291.57 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 318.46/291.57 plus(zero, y) -> y 318.46/291.57 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 318.46/291.57 id(x) -> x 318.46/291.57 if(true, x, y) -> x 318.46/291.57 if(false, x, y) -> y 318.46/291.57 not(x) -> if(x, false, true) 318.46/291.57 gt(s(x), zero) -> true 318.46/291.57 gt(zero, y) -> false 318.46/291.57 gt(s(x), s(y)) -> gt(x, y) 318.46/291.57 318.46/291.57 S is empty. 318.46/291.57 Rewrite Strategy: FULL 318.46/291.57 ---------------------------------------- 318.46/291.57 318.46/291.57 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 318.46/291.57 Renamed function symbols to avoid clashes with predefined symbol. 318.46/291.57 ---------------------------------------- 318.46/291.57 318.46/291.57 (2) 318.46/291.57 Obligation: 318.46/291.57 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 318.46/291.57 318.46/291.57 318.46/291.57 The TRS R consists of the following rules: 318.46/291.57 318.46/291.57 minus(x, 0') -> x 318.46/291.57 minus(s(x), s(y)) -> minus(x, y) 318.46/291.57 quot(0', s(y)) -> 0' 318.46/291.57 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 318.46/291.57 minus(minus(x, y), z) -> minus(x, plus(y, z)) 318.46/291.57 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 318.46/291.57 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 318.46/291.57 plus(zero, y) -> y 318.46/291.57 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 318.46/291.57 id(x) -> x 318.46/291.57 if(true, x, y) -> x 318.46/291.57 if(false, x, y) -> y 318.46/291.57 not(x) -> if(x, false, true) 318.46/291.57 gt(s(x), zero) -> true 318.46/291.57 gt(zero, y) -> false 318.46/291.57 gt(s(x), s(y)) -> gt(x, y) 318.46/291.57 318.46/291.57 S is empty. 318.46/291.57 Rewrite Strategy: FULL 318.46/291.57 ---------------------------------------- 318.46/291.57 318.46/291.57 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 318.46/291.57 Infered types. 318.46/291.57 ---------------------------------------- 318.46/291.57 318.46/291.57 (4) 318.46/291.57 Obligation: 318.46/291.57 TRS: 318.46/291.57 Rules: 318.46/291.57 minus(x, 0') -> x 318.46/291.57 minus(s(x), s(y)) -> minus(x, y) 318.46/291.57 quot(0', s(y)) -> 0' 318.46/291.57 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 318.46/291.57 minus(minus(x, y), z) -> minus(x, plus(y, z)) 318.46/291.57 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 318.46/291.57 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 318.46/291.57 plus(zero, y) -> y 318.46/291.57 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 318.46/291.57 id(x) -> x 318.46/291.57 if(true, x, y) -> x 318.46/291.58 if(false, x, y) -> y 318.46/291.58 not(x) -> if(x, false, true) 318.46/291.58 gt(s(x), zero) -> true 318.46/291.58 gt(zero, y) -> false 318.46/291.58 gt(s(x), s(y)) -> gt(x, y) 318.46/291.58 318.46/291.58 Types: 318.46/291.58 minus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 0' :: 0':s:zero:true:false 318.46/291.58 s :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 quot :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 plus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 if :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 gt :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 not :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 id :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 zero :: 0':s:zero:true:false 318.46/291.58 true :: 0':s:zero:true:false 318.46/291.58 false :: 0':s:zero:true:false 318.46/291.58 hole_0':s:zero:true:false1_0 :: 0':s:zero:true:false 318.46/291.58 gen_0':s:zero:true:false2_0 :: Nat -> 0':s:zero:true:false 318.46/291.58 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (5) OrderProof (LOWER BOUND(ID)) 318.46/291.58 Heuristically decided to analyse the following defined symbols: 318.46/291.58 minus, quot, plus, gt 318.46/291.58 318.46/291.58 They will be analysed ascendingly in the following order: 318.46/291.58 minus < quot 318.46/291.58 plus < minus 318.46/291.58 gt < plus 318.46/291.58 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (6) 318.46/291.58 Obligation: 318.46/291.58 TRS: 318.46/291.58 Rules: 318.46/291.58 minus(x, 0') -> x 318.46/291.58 minus(s(x), s(y)) -> minus(x, y) 318.46/291.58 quot(0', s(y)) -> 0' 318.46/291.58 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 318.46/291.58 minus(minus(x, y), z) -> minus(x, plus(y, z)) 318.46/291.58 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 318.46/291.58 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 318.46/291.58 plus(zero, y) -> y 318.46/291.58 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 318.46/291.58 id(x) -> x 318.46/291.58 if(true, x, y) -> x 318.46/291.58 if(false, x, y) -> y 318.46/291.58 not(x) -> if(x, false, true) 318.46/291.58 gt(s(x), zero) -> true 318.46/291.58 gt(zero, y) -> false 318.46/291.58 gt(s(x), s(y)) -> gt(x, y) 318.46/291.58 318.46/291.58 Types: 318.46/291.58 minus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 0' :: 0':s:zero:true:false 318.46/291.58 s :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 quot :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 plus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 if :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 gt :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 not :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 id :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 zero :: 0':s:zero:true:false 318.46/291.58 true :: 0':s:zero:true:false 318.46/291.58 false :: 0':s:zero:true:false 318.46/291.58 hole_0':s:zero:true:false1_0 :: 0':s:zero:true:false 318.46/291.58 gen_0':s:zero:true:false2_0 :: Nat -> 0':s:zero:true:false 318.46/291.58 318.46/291.58 318.46/291.58 Generator Equations: 318.46/291.58 gen_0':s:zero:true:false2_0(0) <=> 0' 318.46/291.58 gen_0':s:zero:true:false2_0(+(x, 1)) <=> s(gen_0':s:zero:true:false2_0(x)) 318.46/291.58 318.46/291.58 318.46/291.58 The following defined symbols remain to be analysed: 318.46/291.58 gt, minus, quot, plus 318.46/291.58 318.46/291.58 They will be analysed ascendingly in the following order: 318.46/291.58 minus < quot 318.46/291.58 plus < minus 318.46/291.58 gt < plus 318.46/291.58 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (7) RewriteLemmaProof (LOWER BOUND(ID)) 318.46/291.58 Proved the following rewrite lemma: 318.46/291.58 gt(gen_0':s:zero:true:false2_0(+(1, n4_0)), gen_0':s:zero:true:false2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) 318.46/291.58 318.46/291.58 Induction Base: 318.46/291.58 gt(gen_0':s:zero:true:false2_0(+(1, 0)), gen_0':s:zero:true:false2_0(+(1, 0))) 318.46/291.58 318.46/291.58 Induction Step: 318.46/291.58 gt(gen_0':s:zero:true:false2_0(+(1, +(n4_0, 1))), gen_0':s:zero:true:false2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) 318.46/291.58 gt(gen_0':s:zero:true:false2_0(+(1, n4_0)), gen_0':s:zero:true:false2_0(+(1, n4_0))) ->_IH 318.46/291.58 *3_0 318.46/291.58 318.46/291.58 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (8) 318.46/291.58 Complex Obligation (BEST) 318.46/291.58 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (9) 318.46/291.58 Obligation: 318.46/291.58 Proved the lower bound n^1 for the following obligation: 318.46/291.58 318.46/291.58 TRS: 318.46/291.58 Rules: 318.46/291.58 minus(x, 0') -> x 318.46/291.58 minus(s(x), s(y)) -> minus(x, y) 318.46/291.58 quot(0', s(y)) -> 0' 318.46/291.58 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 318.46/291.58 minus(minus(x, y), z) -> minus(x, plus(y, z)) 318.46/291.58 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 318.46/291.58 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 318.46/291.58 plus(zero, y) -> y 318.46/291.58 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 318.46/291.58 id(x) -> x 318.46/291.58 if(true, x, y) -> x 318.46/291.58 if(false, x, y) -> y 318.46/291.58 not(x) -> if(x, false, true) 318.46/291.58 gt(s(x), zero) -> true 318.46/291.58 gt(zero, y) -> false 318.46/291.58 gt(s(x), s(y)) -> gt(x, y) 318.46/291.58 318.46/291.58 Types: 318.46/291.58 minus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 0' :: 0':s:zero:true:false 318.46/291.58 s :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 quot :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 plus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 if :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 gt :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 not :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 id :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 zero :: 0':s:zero:true:false 318.46/291.58 true :: 0':s:zero:true:false 318.46/291.58 false :: 0':s:zero:true:false 318.46/291.58 hole_0':s:zero:true:false1_0 :: 0':s:zero:true:false 318.46/291.58 gen_0':s:zero:true:false2_0 :: Nat -> 0':s:zero:true:false 318.46/291.58 318.46/291.58 318.46/291.58 Generator Equations: 318.46/291.58 gen_0':s:zero:true:false2_0(0) <=> 0' 318.46/291.58 gen_0':s:zero:true:false2_0(+(x, 1)) <=> s(gen_0':s:zero:true:false2_0(x)) 318.46/291.58 318.46/291.58 318.46/291.58 The following defined symbols remain to be analysed: 318.46/291.58 gt, minus, quot, plus 318.46/291.58 318.46/291.58 They will be analysed ascendingly in the following order: 318.46/291.58 minus < quot 318.46/291.58 plus < minus 318.46/291.58 gt < plus 318.46/291.58 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (10) LowerBoundPropagationProof (FINISHED) 318.46/291.58 Propagated lower bound. 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (11) 318.46/291.58 BOUNDS(n^1, INF) 318.46/291.58 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (12) 318.46/291.58 Obligation: 318.46/291.58 TRS: 318.46/291.58 Rules: 318.46/291.58 minus(x, 0') -> x 318.46/291.58 minus(s(x), s(y)) -> minus(x, y) 318.46/291.58 quot(0', s(y)) -> 0' 318.46/291.58 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 318.46/291.58 minus(minus(x, y), z) -> minus(x, plus(y, z)) 318.46/291.58 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 318.46/291.58 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 318.46/291.58 plus(zero, y) -> y 318.46/291.58 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 318.46/291.58 id(x) -> x 318.46/291.58 if(true, x, y) -> x 318.46/291.58 if(false, x, y) -> y 318.46/291.58 not(x) -> if(x, false, true) 318.46/291.58 gt(s(x), zero) -> true 318.46/291.58 gt(zero, y) -> false 318.46/291.58 gt(s(x), s(y)) -> gt(x, y) 318.46/291.58 318.46/291.58 Types: 318.46/291.58 minus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 0' :: 0':s:zero:true:false 318.46/291.58 s :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 quot :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 plus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 if :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 gt :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 not :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 id :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 zero :: 0':s:zero:true:false 318.46/291.58 true :: 0':s:zero:true:false 318.46/291.58 false :: 0':s:zero:true:false 318.46/291.58 hole_0':s:zero:true:false1_0 :: 0':s:zero:true:false 318.46/291.58 gen_0':s:zero:true:false2_0 :: Nat -> 0':s:zero:true:false 318.46/291.58 318.46/291.58 318.46/291.58 Lemmas: 318.46/291.58 gt(gen_0':s:zero:true:false2_0(+(1, n4_0)), gen_0':s:zero:true:false2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) 318.46/291.58 318.46/291.58 318.46/291.58 Generator Equations: 318.46/291.58 gen_0':s:zero:true:false2_0(0) <=> 0' 318.46/291.58 gen_0':s:zero:true:false2_0(+(x, 1)) <=> s(gen_0':s:zero:true:false2_0(x)) 318.46/291.58 318.46/291.58 318.46/291.58 The following defined symbols remain to be analysed: 318.46/291.58 plus, minus, quot 318.46/291.58 318.46/291.58 They will be analysed ascendingly in the following order: 318.46/291.58 minus < quot 318.46/291.58 plus < minus 318.46/291.58 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (13) RewriteLemmaProof (LOWER BOUND(ID)) 318.46/291.58 Proved the following rewrite lemma: 318.46/291.58 minus(gen_0':s:zero:true:false2_0(+(1, n2196_0)), gen_0':s:zero:true:false2_0(+(1, n2196_0))) -> *3_0, rt in Omega(n2196_0) 318.46/291.58 318.46/291.58 Induction Base: 318.46/291.58 minus(gen_0':s:zero:true:false2_0(+(1, 0)), gen_0':s:zero:true:false2_0(+(1, 0))) 318.46/291.58 318.46/291.58 Induction Step: 318.46/291.58 minus(gen_0':s:zero:true:false2_0(+(1, +(n2196_0, 1))), gen_0':s:zero:true:false2_0(+(1, +(n2196_0, 1)))) ->_R^Omega(1) 318.46/291.58 minus(gen_0':s:zero:true:false2_0(+(1, n2196_0)), gen_0':s:zero:true:false2_0(+(1, n2196_0))) ->_IH 318.46/291.58 *3_0 318.46/291.58 318.46/291.58 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 318.46/291.58 ---------------------------------------- 318.46/291.58 318.46/291.58 (14) 318.46/291.58 Obligation: 318.46/291.58 TRS: 318.46/291.58 Rules: 318.46/291.58 minus(x, 0') -> x 318.46/291.58 minus(s(x), s(y)) -> minus(x, y) 318.46/291.58 quot(0', s(y)) -> 0' 318.46/291.58 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 318.46/291.58 minus(minus(x, y), z) -> minus(x, plus(y, z)) 318.46/291.58 plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 318.46/291.58 plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) 318.46/291.58 plus(zero, y) -> y 318.46/291.58 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) 318.46/291.58 id(x) -> x 318.46/291.58 if(true, x, y) -> x 318.46/291.58 if(false, x, y) -> y 318.46/291.58 not(x) -> if(x, false, true) 318.46/291.58 gt(s(x), zero) -> true 318.46/291.58 gt(zero, y) -> false 318.46/291.58 gt(s(x), s(y)) -> gt(x, y) 318.46/291.58 318.46/291.58 Types: 318.46/291.58 minus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 0' :: 0':s:zero:true:false 318.46/291.58 s :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 quot :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 plus :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 if :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 gt :: 0':s:zero:true:false -> 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 not :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 id :: 0':s:zero:true:false -> 0':s:zero:true:false 318.46/291.58 zero :: 0':s:zero:true:false 318.46/291.58 true :: 0':s:zero:true:false 318.46/291.58 false :: 0':s:zero:true:false 318.46/291.58 hole_0':s:zero:true:false1_0 :: 0':s:zero:true:false 318.46/291.58 gen_0':s:zero:true:false2_0 :: Nat -> 0':s:zero:true:false 318.46/291.58 318.46/291.58 318.46/291.58 Lemmas: 318.46/291.58 gt(gen_0':s:zero:true:false2_0(+(1, n4_0)), gen_0':s:zero:true:false2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) 318.46/291.58 minus(gen_0':s:zero:true:false2_0(+(1, n2196_0)), gen_0':s:zero:true:false2_0(+(1, n2196_0))) -> *3_0, rt in Omega(n2196_0) 318.46/291.58 318.46/291.58 318.46/291.58 Generator Equations: 318.46/291.58 gen_0':s:zero:true:false2_0(0) <=> 0' 318.46/291.58 gen_0':s:zero:true:false2_0(+(x, 1)) <=> s(gen_0':s:zero:true:false2_0(x)) 318.46/291.58 318.46/291.58 318.46/291.58 The following defined symbols remain to be analysed: 318.46/291.58 quot 318.46/291.61 EOF