311.53/291.55 WORST_CASE(Omega(n^1), ?) 311.53/291.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 311.53/291.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 311.53/291.56 311.53/291.56 311.53/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.53/291.56 311.53/291.56 (0) CpxTRS 311.53/291.56 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 311.53/291.56 (2) CpxTRS 311.53/291.56 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 311.53/291.56 (4) typed CpxTrs 311.53/291.56 (5) OrderProof [LOWER BOUND(ID), 0 ms] 311.53/291.56 (6) typed CpxTrs 311.53/291.56 (7) RewriteLemmaProof [LOWER BOUND(ID), 301 ms] 311.53/291.56 (8) BEST 311.53/291.56 (9) proven lower bound 311.53/291.56 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 311.53/291.56 (11) BOUNDS(n^1, INF) 311.53/291.56 (12) typed CpxTrs 311.53/291.56 (13) RewriteLemmaProof [LOWER BOUND(ID), 28 ms] 311.53/291.56 (14) typed CpxTrs 311.53/291.56 (15) RewriteLemmaProof [LOWER BOUND(ID), 24 ms] 311.53/291.56 (16) typed CpxTrs 311.53/291.56 311.53/291.56 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (0) 311.53/291.56 Obligation: 311.53/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.53/291.56 311.53/291.56 311.53/291.56 The TRS R consists of the following rules: 311.53/291.56 311.53/291.56 le(0, y) -> true 311.53/291.56 le(s(x), 0) -> false 311.53/291.56 le(s(x), s(y)) -> le(x, y) 311.53/291.56 zero(0) -> true 311.53/291.56 zero(s(x)) -> false 311.53/291.56 id(0) -> 0 311.53/291.56 id(s(x)) -> s(id(x)) 311.53/291.56 minus(x, 0) -> x 311.53/291.56 minus(s(x), s(y)) -> minus(x, y) 311.53/291.56 mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) 311.53/291.56 if_mod(true, b1, b2, x, y) -> 0 311.53/291.56 if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) 311.53/291.56 if2(true, b2, x, y) -> 0 311.53/291.56 if2(false, b2, x, y) -> if3(b2, x, y) 311.53/291.56 if3(true, x, y) -> mod(minus(x, y), s(y)) 311.53/291.56 if3(false, x, y) -> x 311.53/291.56 311.53/291.56 S is empty. 311.53/291.56 Rewrite Strategy: FULL 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 311.53/291.56 Renamed function symbols to avoid clashes with predefined symbol. 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (2) 311.53/291.56 Obligation: 311.53/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.53/291.56 311.53/291.56 311.53/291.56 The TRS R consists of the following rules: 311.53/291.56 311.53/291.56 le(0', y) -> true 311.53/291.56 le(s(x), 0') -> false 311.53/291.56 le(s(x), s(y)) -> le(x, y) 311.53/291.56 zero(0') -> true 311.53/291.56 zero(s(x)) -> false 311.53/291.56 id(0') -> 0' 311.53/291.56 id(s(x)) -> s(id(x)) 311.53/291.56 minus(x, 0') -> x 311.53/291.56 minus(s(x), s(y)) -> minus(x, y) 311.53/291.56 mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) 311.53/291.56 if_mod(true, b1, b2, x, y) -> 0' 311.53/291.56 if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) 311.53/291.56 if2(true, b2, x, y) -> 0' 311.53/291.56 if2(false, b2, x, y) -> if3(b2, x, y) 311.53/291.56 if3(true, x, y) -> mod(minus(x, y), s(y)) 311.53/291.56 if3(false, x, y) -> x 311.53/291.56 311.53/291.56 S is empty. 311.53/291.56 Rewrite Strategy: FULL 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 311.53/291.56 Infered types. 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (4) 311.53/291.56 Obligation: 311.53/291.56 TRS: 311.53/291.56 Rules: 311.53/291.56 le(0', y) -> true 311.53/291.56 le(s(x), 0') -> false 311.53/291.56 le(s(x), s(y)) -> le(x, y) 311.53/291.56 zero(0') -> true 311.53/291.56 zero(s(x)) -> false 311.53/291.56 id(0') -> 0' 311.53/291.56 id(s(x)) -> s(id(x)) 311.53/291.56 minus(x, 0') -> x 311.53/291.56 minus(s(x), s(y)) -> minus(x, y) 311.53/291.56 mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) 311.53/291.56 if_mod(true, b1, b2, x, y) -> 0' 311.53/291.56 if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) 311.53/291.56 if2(true, b2, x, y) -> 0' 311.53/291.56 if2(false, b2, x, y) -> if3(b2, x, y) 311.53/291.56 if3(true, x, y) -> mod(minus(x, y), s(y)) 311.53/291.56 if3(false, x, y) -> x 311.53/291.56 311.53/291.56 Types: 311.53/291.56 le :: 0':s -> 0':s -> true:false 311.53/291.56 0' :: 0':s 311.53/291.56 true :: true:false 311.53/291.56 s :: 0':s -> 0':s 311.53/291.56 false :: true:false 311.53/291.56 zero :: 0':s -> true:false 311.53/291.56 id :: 0':s -> 0':s 311.53/291.56 minus :: 0':s -> 0':s -> 0':s 311.53/291.56 mod :: 0':s -> 0':s -> 0':s 311.53/291.56 if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if3 :: true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 hole_true:false1_0 :: true:false 311.53/291.56 hole_0':s2_0 :: 0':s 311.53/291.56 gen_0':s3_0 :: Nat -> 0':s 311.53/291.56 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (5) OrderProof (LOWER BOUND(ID)) 311.53/291.56 Heuristically decided to analyse the following defined symbols: 311.53/291.56 le, id, minus, mod 311.53/291.56 311.53/291.56 They will be analysed ascendingly in the following order: 311.53/291.56 le < mod 311.53/291.56 id < mod 311.53/291.56 minus < mod 311.53/291.56 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (6) 311.53/291.56 Obligation: 311.53/291.56 TRS: 311.53/291.56 Rules: 311.53/291.56 le(0', y) -> true 311.53/291.56 le(s(x), 0') -> false 311.53/291.56 le(s(x), s(y)) -> le(x, y) 311.53/291.56 zero(0') -> true 311.53/291.56 zero(s(x)) -> false 311.53/291.56 id(0') -> 0' 311.53/291.56 id(s(x)) -> s(id(x)) 311.53/291.56 minus(x, 0') -> x 311.53/291.56 minus(s(x), s(y)) -> minus(x, y) 311.53/291.56 mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) 311.53/291.56 if_mod(true, b1, b2, x, y) -> 0' 311.53/291.56 if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) 311.53/291.56 if2(true, b2, x, y) -> 0' 311.53/291.56 if2(false, b2, x, y) -> if3(b2, x, y) 311.53/291.56 if3(true, x, y) -> mod(minus(x, y), s(y)) 311.53/291.56 if3(false, x, y) -> x 311.53/291.56 311.53/291.56 Types: 311.53/291.56 le :: 0':s -> 0':s -> true:false 311.53/291.56 0' :: 0':s 311.53/291.56 true :: true:false 311.53/291.56 s :: 0':s -> 0':s 311.53/291.56 false :: true:false 311.53/291.56 zero :: 0':s -> true:false 311.53/291.56 id :: 0':s -> 0':s 311.53/291.56 minus :: 0':s -> 0':s -> 0':s 311.53/291.56 mod :: 0':s -> 0':s -> 0':s 311.53/291.56 if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if3 :: true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 hole_true:false1_0 :: true:false 311.53/291.56 hole_0':s2_0 :: 0':s 311.53/291.56 gen_0':s3_0 :: Nat -> 0':s 311.53/291.56 311.53/291.56 311.53/291.56 Generator Equations: 311.53/291.56 gen_0':s3_0(0) <=> 0' 311.53/291.56 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.53/291.56 311.53/291.56 311.53/291.56 The following defined symbols remain to be analysed: 311.53/291.56 le, id, minus, mod 311.53/291.56 311.53/291.56 They will be analysed ascendingly in the following order: 311.53/291.56 le < mod 311.53/291.56 id < mod 311.53/291.56 minus < mod 311.53/291.56 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (7) RewriteLemmaProof (LOWER BOUND(ID)) 311.53/291.56 Proved the following rewrite lemma: 311.53/291.56 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 311.53/291.56 311.53/291.56 Induction Base: 311.53/291.56 le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 311.53/291.56 true 311.53/291.56 311.53/291.56 Induction Step: 311.53/291.56 le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 311.53/291.56 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH 311.53/291.56 true 311.53/291.56 311.53/291.56 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (8) 311.53/291.56 Complex Obligation (BEST) 311.53/291.56 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (9) 311.53/291.56 Obligation: 311.53/291.56 Proved the lower bound n^1 for the following obligation: 311.53/291.56 311.53/291.56 TRS: 311.53/291.56 Rules: 311.53/291.56 le(0', y) -> true 311.53/291.56 le(s(x), 0') -> false 311.53/291.56 le(s(x), s(y)) -> le(x, y) 311.53/291.56 zero(0') -> true 311.53/291.56 zero(s(x)) -> false 311.53/291.56 id(0') -> 0' 311.53/291.56 id(s(x)) -> s(id(x)) 311.53/291.56 minus(x, 0') -> x 311.53/291.56 minus(s(x), s(y)) -> minus(x, y) 311.53/291.56 mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) 311.53/291.56 if_mod(true, b1, b2, x, y) -> 0' 311.53/291.56 if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) 311.53/291.56 if2(true, b2, x, y) -> 0' 311.53/291.56 if2(false, b2, x, y) -> if3(b2, x, y) 311.53/291.56 if3(true, x, y) -> mod(minus(x, y), s(y)) 311.53/291.56 if3(false, x, y) -> x 311.53/291.56 311.53/291.56 Types: 311.53/291.56 le :: 0':s -> 0':s -> true:false 311.53/291.56 0' :: 0':s 311.53/291.56 true :: true:false 311.53/291.56 s :: 0':s -> 0':s 311.53/291.56 false :: true:false 311.53/291.56 zero :: 0':s -> true:false 311.53/291.56 id :: 0':s -> 0':s 311.53/291.56 minus :: 0':s -> 0':s -> 0':s 311.53/291.56 mod :: 0':s -> 0':s -> 0':s 311.53/291.56 if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if3 :: true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 hole_true:false1_0 :: true:false 311.53/291.56 hole_0':s2_0 :: 0':s 311.53/291.56 gen_0':s3_0 :: Nat -> 0':s 311.53/291.56 311.53/291.56 311.53/291.56 Generator Equations: 311.53/291.56 gen_0':s3_0(0) <=> 0' 311.53/291.56 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.53/291.56 311.53/291.56 311.53/291.56 The following defined symbols remain to be analysed: 311.53/291.56 le, id, minus, mod 311.53/291.56 311.53/291.56 They will be analysed ascendingly in the following order: 311.53/291.56 le < mod 311.53/291.56 id < mod 311.53/291.56 minus < mod 311.53/291.56 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (10) LowerBoundPropagationProof (FINISHED) 311.53/291.56 Propagated lower bound. 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (11) 311.53/291.56 BOUNDS(n^1, INF) 311.53/291.56 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (12) 311.53/291.56 Obligation: 311.53/291.56 TRS: 311.53/291.56 Rules: 311.53/291.56 le(0', y) -> true 311.53/291.56 le(s(x), 0') -> false 311.53/291.56 le(s(x), s(y)) -> le(x, y) 311.53/291.56 zero(0') -> true 311.53/291.56 zero(s(x)) -> false 311.53/291.56 id(0') -> 0' 311.53/291.56 id(s(x)) -> s(id(x)) 311.53/291.56 minus(x, 0') -> x 311.53/291.56 minus(s(x), s(y)) -> minus(x, y) 311.53/291.56 mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) 311.53/291.56 if_mod(true, b1, b2, x, y) -> 0' 311.53/291.56 if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) 311.53/291.56 if2(true, b2, x, y) -> 0' 311.53/291.56 if2(false, b2, x, y) -> if3(b2, x, y) 311.53/291.56 if3(true, x, y) -> mod(minus(x, y), s(y)) 311.53/291.56 if3(false, x, y) -> x 311.53/291.56 311.53/291.56 Types: 311.53/291.56 le :: 0':s -> 0':s -> true:false 311.53/291.56 0' :: 0':s 311.53/291.56 true :: true:false 311.53/291.56 s :: 0':s -> 0':s 311.53/291.56 false :: true:false 311.53/291.56 zero :: 0':s -> true:false 311.53/291.56 id :: 0':s -> 0':s 311.53/291.56 minus :: 0':s -> 0':s -> 0':s 311.53/291.56 mod :: 0':s -> 0':s -> 0':s 311.53/291.56 if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if3 :: true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 hole_true:false1_0 :: true:false 311.53/291.56 hole_0':s2_0 :: 0':s 311.53/291.56 gen_0':s3_0 :: Nat -> 0':s 311.53/291.56 311.53/291.56 311.53/291.56 Lemmas: 311.53/291.56 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 311.53/291.56 311.53/291.56 311.53/291.56 Generator Equations: 311.53/291.56 gen_0':s3_0(0) <=> 0' 311.53/291.56 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.53/291.56 311.53/291.56 311.53/291.56 The following defined symbols remain to be analysed: 311.53/291.56 id, minus, mod 311.53/291.56 311.53/291.56 They will be analysed ascendingly in the following order: 311.53/291.56 id < mod 311.53/291.56 minus < mod 311.53/291.56 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (13) RewriteLemmaProof (LOWER BOUND(ID)) 311.53/291.56 Proved the following rewrite lemma: 311.53/291.56 id(gen_0':s3_0(n288_0)) -> gen_0':s3_0(n288_0), rt in Omega(1 + n288_0) 311.53/291.56 311.53/291.56 Induction Base: 311.53/291.56 id(gen_0':s3_0(0)) ->_R^Omega(1) 311.53/291.56 0' 311.53/291.56 311.53/291.56 Induction Step: 311.53/291.56 id(gen_0':s3_0(+(n288_0, 1))) ->_R^Omega(1) 311.53/291.56 s(id(gen_0':s3_0(n288_0))) ->_IH 311.53/291.56 s(gen_0':s3_0(c289_0)) 311.53/291.56 311.53/291.56 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (14) 311.53/291.56 Obligation: 311.53/291.56 TRS: 311.53/291.56 Rules: 311.53/291.56 le(0', y) -> true 311.53/291.56 le(s(x), 0') -> false 311.53/291.56 le(s(x), s(y)) -> le(x, y) 311.53/291.56 zero(0') -> true 311.53/291.56 zero(s(x)) -> false 311.53/291.56 id(0') -> 0' 311.53/291.56 id(s(x)) -> s(id(x)) 311.53/291.56 minus(x, 0') -> x 311.53/291.56 minus(s(x), s(y)) -> minus(x, y) 311.53/291.56 mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) 311.53/291.56 if_mod(true, b1, b2, x, y) -> 0' 311.53/291.56 if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) 311.53/291.56 if2(true, b2, x, y) -> 0' 311.53/291.56 if2(false, b2, x, y) -> if3(b2, x, y) 311.53/291.56 if3(true, x, y) -> mod(minus(x, y), s(y)) 311.53/291.56 if3(false, x, y) -> x 311.53/291.56 311.53/291.56 Types: 311.53/291.56 le :: 0':s -> 0':s -> true:false 311.53/291.56 0' :: 0':s 311.53/291.56 true :: true:false 311.53/291.56 s :: 0':s -> 0':s 311.53/291.56 false :: true:false 311.53/291.56 zero :: 0':s -> true:false 311.53/291.56 id :: 0':s -> 0':s 311.53/291.56 minus :: 0':s -> 0':s -> 0':s 311.53/291.56 mod :: 0':s -> 0':s -> 0':s 311.53/291.56 if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if3 :: true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 hole_true:false1_0 :: true:false 311.53/291.56 hole_0':s2_0 :: 0':s 311.53/291.56 gen_0':s3_0 :: Nat -> 0':s 311.53/291.56 311.53/291.56 311.53/291.56 Lemmas: 311.53/291.56 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 311.53/291.56 id(gen_0':s3_0(n288_0)) -> gen_0':s3_0(n288_0), rt in Omega(1 + n288_0) 311.53/291.56 311.53/291.56 311.53/291.56 Generator Equations: 311.53/291.56 gen_0':s3_0(0) <=> 0' 311.53/291.56 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.53/291.56 311.53/291.56 311.53/291.56 The following defined symbols remain to be analysed: 311.53/291.56 minus, mod 311.53/291.56 311.53/291.56 They will be analysed ascendingly in the following order: 311.53/291.56 minus < mod 311.53/291.56 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (15) RewriteLemmaProof (LOWER BOUND(ID)) 311.53/291.56 Proved the following rewrite lemma: 311.53/291.56 minus(gen_0':s3_0(n502_0), gen_0':s3_0(n502_0)) -> gen_0':s3_0(0), rt in Omega(1 + n502_0) 311.53/291.56 311.53/291.56 Induction Base: 311.53/291.56 minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 311.53/291.56 gen_0':s3_0(0) 311.53/291.56 311.53/291.56 Induction Step: 311.53/291.56 minus(gen_0':s3_0(+(n502_0, 1)), gen_0':s3_0(+(n502_0, 1))) ->_R^Omega(1) 311.53/291.56 minus(gen_0':s3_0(n502_0), gen_0':s3_0(n502_0)) ->_IH 311.53/291.56 gen_0':s3_0(0) 311.53/291.56 311.53/291.56 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.53/291.56 ---------------------------------------- 311.53/291.56 311.53/291.56 (16) 311.53/291.56 Obligation: 311.53/291.56 TRS: 311.53/291.56 Rules: 311.53/291.56 le(0', y) -> true 311.53/291.56 le(s(x), 0') -> false 311.53/291.56 le(s(x), s(y)) -> le(x, y) 311.53/291.56 zero(0') -> true 311.53/291.56 zero(s(x)) -> false 311.53/291.56 id(0') -> 0' 311.53/291.56 id(s(x)) -> s(id(x)) 311.53/291.56 minus(x, 0') -> x 311.53/291.56 minus(s(x), s(y)) -> minus(x, y) 311.53/291.56 mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) 311.53/291.56 if_mod(true, b1, b2, x, y) -> 0' 311.53/291.56 if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y) 311.53/291.56 if2(true, b2, x, y) -> 0' 311.53/291.56 if2(false, b2, x, y) -> if3(b2, x, y) 311.53/291.56 if3(true, x, y) -> mod(minus(x, y), s(y)) 311.53/291.56 if3(false, x, y) -> x 311.53/291.56 311.53/291.56 Types: 311.53/291.56 le :: 0':s -> 0':s -> true:false 311.53/291.56 0' :: 0':s 311.53/291.56 true :: true:false 311.53/291.56 s :: 0':s -> 0':s 311.53/291.56 false :: true:false 311.53/291.56 zero :: 0':s -> true:false 311.53/291.56 id :: 0':s -> 0':s 311.53/291.56 minus :: 0':s -> 0':s -> 0':s 311.53/291.56 mod :: 0':s -> 0':s -> 0':s 311.53/291.56 if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 if3 :: true:false -> 0':s -> 0':s -> 0':s 311.53/291.56 hole_true:false1_0 :: true:false 311.53/291.56 hole_0':s2_0 :: 0':s 311.53/291.56 gen_0':s3_0 :: Nat -> 0':s 311.53/291.56 311.53/291.56 311.53/291.56 Lemmas: 311.53/291.56 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 311.53/291.56 id(gen_0':s3_0(n288_0)) -> gen_0':s3_0(n288_0), rt in Omega(1 + n288_0) 311.53/291.56 minus(gen_0':s3_0(n502_0), gen_0':s3_0(n502_0)) -> gen_0':s3_0(0), rt in Omega(1 + n502_0) 311.53/291.56 311.53/291.56 311.53/291.56 Generator Equations: 311.53/291.56 gen_0':s3_0(0) <=> 0' 311.53/291.56 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.53/291.56 311.53/291.56 311.53/291.56 The following defined symbols remain to be analysed: 311.53/291.56 mod 311.61/291.60 EOF