1116.97/291.55 WORST_CASE(Omega(n^1), ?) 1116.97/291.61 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1116.97/291.61 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1116.97/291.61 1116.97/291.61 1116.97/291.61 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1116.97/291.61 1116.97/291.61 (0) CpxTRS 1116.97/291.61 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1116.97/291.61 (2) CpxTRS 1116.97/291.61 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1116.97/291.61 (4) typed CpxTrs 1116.97/291.61 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1116.97/291.61 (6) typed CpxTrs 1116.97/291.61 (7) RewriteLemmaProof [LOWER BOUND(ID), 317 ms] 1116.97/291.61 (8) BEST 1116.97/291.61 (9) proven lower bound 1116.97/291.61 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1116.97/291.61 (11) BOUNDS(n^1, INF) 1116.97/291.61 (12) typed CpxTrs 1116.97/291.61 (13) RewriteLemmaProof [LOWER BOUND(ID), 66 ms] 1116.97/291.61 (14) typed CpxTrs 1116.97/291.61 (15) RewriteLemmaProof [LOWER BOUND(ID), 70 ms] 1116.97/291.61 (16) BOUNDS(1, INF) 1116.97/291.61 1116.97/291.61 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (0) 1116.97/291.61 Obligation: 1116.97/291.61 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1116.97/291.61 1116.97/291.61 1116.97/291.61 The TRS R consists of the following rules: 1116.97/291.61 1116.97/291.61 minus_active(0, y) -> 0 1116.97/291.61 mark(0) -> 0 1116.97/291.61 minus_active(s(x), s(y)) -> minus_active(x, y) 1116.97/291.61 mark(s(x)) -> s(mark(x)) 1116.97/291.61 ge_active(x, 0) -> true 1116.97/291.61 mark(minus(x, y)) -> minus_active(x, y) 1116.97/291.61 ge_active(0, s(y)) -> false 1116.97/291.61 mark(ge(x, y)) -> ge_active(x, y) 1116.97/291.61 ge_active(s(x), s(y)) -> ge_active(x, y) 1116.97/291.61 mark(div(x, y)) -> div_active(mark(x), y) 1116.97/291.61 div_active(0, s(y)) -> 0 1116.97/291.61 mark(if(x, y, z)) -> if_active(mark(x), y, z) 1116.97/291.61 div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0) 1116.97/291.61 if_active(true, x, y) -> mark(x) 1116.97/291.61 minus_active(x, y) -> minus(x, y) 1116.97/291.61 if_active(false, x, y) -> mark(y) 1116.97/291.61 ge_active(x, y) -> ge(x, y) 1116.97/291.61 if_active(x, y, z) -> if(x, y, z) 1116.97/291.61 div_active(x, y) -> div(x, y) 1116.97/291.61 1116.97/291.61 S is empty. 1116.97/291.61 Rewrite Strategy: INNERMOST 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1116.97/291.61 Renamed function symbols to avoid clashes with predefined symbol. 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (2) 1116.97/291.61 Obligation: 1116.97/291.61 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1116.97/291.61 1116.97/291.61 1116.97/291.61 The TRS R consists of the following rules: 1116.97/291.61 1116.97/291.61 minus_active(0', y) -> 0' 1116.97/291.61 mark(0') -> 0' 1116.97/291.61 minus_active(s(x), s(y)) -> minus_active(x, y) 1116.97/291.61 mark(s(x)) -> s(mark(x)) 1116.97/291.61 ge_active(x, 0') -> true 1116.97/291.61 mark(minus(x, y)) -> minus_active(x, y) 1116.97/291.61 ge_active(0', s(y)) -> false 1116.97/291.61 mark(ge(x, y)) -> ge_active(x, y) 1116.97/291.61 ge_active(s(x), s(y)) -> ge_active(x, y) 1116.97/291.61 mark(div(x, y)) -> div_active(mark(x), y) 1116.97/291.61 div_active(0', s(y)) -> 0' 1116.97/291.61 mark(if(x, y, z)) -> if_active(mark(x), y, z) 1116.97/291.61 div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0') 1116.97/291.61 if_active(true, x, y) -> mark(x) 1116.97/291.61 minus_active(x, y) -> minus(x, y) 1116.97/291.61 if_active(false, x, y) -> mark(y) 1116.97/291.61 ge_active(x, y) -> ge(x, y) 1116.97/291.61 if_active(x, y, z) -> if(x, y, z) 1116.97/291.61 div_active(x, y) -> div(x, y) 1116.97/291.61 1116.97/291.61 S is empty. 1116.97/291.61 Rewrite Strategy: INNERMOST 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1116.97/291.61 Infered types. 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (4) 1116.97/291.61 Obligation: 1116.97/291.61 Innermost TRS: 1116.97/291.61 Rules: 1116.97/291.61 minus_active(0', y) -> 0' 1116.97/291.61 mark(0') -> 0' 1116.97/291.61 minus_active(s(x), s(y)) -> minus_active(x, y) 1116.97/291.61 mark(s(x)) -> s(mark(x)) 1116.97/291.61 ge_active(x, 0') -> true 1116.97/291.61 mark(minus(x, y)) -> minus_active(x, y) 1116.97/291.61 ge_active(0', s(y)) -> false 1116.97/291.61 mark(ge(x, y)) -> ge_active(x, y) 1116.97/291.61 ge_active(s(x), s(y)) -> ge_active(x, y) 1116.97/291.61 mark(div(x, y)) -> div_active(mark(x), y) 1116.97/291.61 div_active(0', s(y)) -> 0' 1116.97/291.61 mark(if(x, y, z)) -> if_active(mark(x), y, z) 1116.97/291.61 div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0') 1116.97/291.61 if_active(true, x, y) -> mark(x) 1116.97/291.61 minus_active(x, y) -> minus(x, y) 1116.97/291.61 if_active(false, x, y) -> mark(y) 1116.97/291.61 ge_active(x, y) -> ge(x, y) 1116.97/291.61 if_active(x, y, z) -> if(x, y, z) 1116.97/291.61 div_active(x, y) -> div(x, y) 1116.97/291.61 1116.97/291.61 Types: 1116.97/291.61 minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 0' :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 true :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 false :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (5) OrderProof (LOWER BOUND(ID)) 1116.97/291.61 Heuristically decided to analyse the following defined symbols: 1116.97/291.61 minus_active, mark, ge_active 1116.97/291.61 1116.97/291.61 They will be analysed ascendingly in the following order: 1116.97/291.61 minus_active < mark 1116.97/291.61 ge_active < mark 1116.97/291.61 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (6) 1116.97/291.61 Obligation: 1116.97/291.61 Innermost TRS: 1116.97/291.61 Rules: 1116.97/291.61 minus_active(0', y) -> 0' 1116.97/291.61 mark(0') -> 0' 1116.97/291.61 minus_active(s(x), s(y)) -> minus_active(x, y) 1116.97/291.61 mark(s(x)) -> s(mark(x)) 1116.97/291.61 ge_active(x, 0') -> true 1116.97/291.61 mark(minus(x, y)) -> minus_active(x, y) 1116.97/291.61 ge_active(0', s(y)) -> false 1116.97/291.61 mark(ge(x, y)) -> ge_active(x, y) 1116.97/291.61 ge_active(s(x), s(y)) -> ge_active(x, y) 1116.97/291.61 mark(div(x, y)) -> div_active(mark(x), y) 1116.97/291.61 div_active(0', s(y)) -> 0' 1116.97/291.61 mark(if(x, y, z)) -> if_active(mark(x), y, z) 1116.97/291.61 div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0') 1116.97/291.61 if_active(true, x, y) -> mark(x) 1116.97/291.61 minus_active(x, y) -> minus(x, y) 1116.97/291.61 if_active(false, x, y) -> mark(y) 1116.97/291.61 ge_active(x, y) -> ge(x, y) 1116.97/291.61 if_active(x, y, z) -> if(x, y, z) 1116.97/291.61 div_active(x, y) -> div(x, y) 1116.97/291.61 1116.97/291.61 Types: 1116.97/291.61 minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 0' :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 true :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 false :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 1116.97/291.61 1116.97/291.61 Generator Equations: 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0(0) <=> 0' 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if2_0(x)) 1116.97/291.61 1116.97/291.61 1116.97/291.61 The following defined symbols remain to be analysed: 1116.97/291.61 minus_active, mark, ge_active 1116.97/291.61 1116.97/291.61 They will be analysed ascendingly in the following order: 1116.97/291.61 minus_active < mark 1116.97/291.61 ge_active < mark 1116.97/291.61 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1116.97/291.61 Proved the following rewrite lemma: 1116.97/291.61 minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) -> gen_0':s:true:minus:false:ge:div:if2_0(0), rt in Omega(1 + n4_0) 1116.97/291.61 1116.97/291.61 Induction Base: 1116.97/291.61 minus_active(gen_0':s:true:minus:false:ge:div:if2_0(0), gen_0':s:true:minus:false:ge:div:if2_0(0)) ->_R^Omega(1) 1116.97/291.61 0' 1116.97/291.61 1116.97/291.61 Induction Step: 1116.97/291.61 minus_active(gen_0':s:true:minus:false:ge:div:if2_0(+(n4_0, 1)), gen_0':s:true:minus:false:ge:div:if2_0(+(n4_0, 1))) ->_R^Omega(1) 1116.97/291.61 minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) ->_IH 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0(0) 1116.97/291.61 1116.97/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (8) 1116.97/291.61 Complex Obligation (BEST) 1116.97/291.61 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (9) 1116.97/291.61 Obligation: 1116.97/291.61 Proved the lower bound n^1 for the following obligation: 1116.97/291.61 1116.97/291.61 Innermost TRS: 1116.97/291.61 Rules: 1116.97/291.61 minus_active(0', y) -> 0' 1116.97/291.61 mark(0') -> 0' 1116.97/291.61 minus_active(s(x), s(y)) -> minus_active(x, y) 1116.97/291.61 mark(s(x)) -> s(mark(x)) 1116.97/291.61 ge_active(x, 0') -> true 1116.97/291.61 mark(minus(x, y)) -> minus_active(x, y) 1116.97/291.61 ge_active(0', s(y)) -> false 1116.97/291.61 mark(ge(x, y)) -> ge_active(x, y) 1116.97/291.61 ge_active(s(x), s(y)) -> ge_active(x, y) 1116.97/291.61 mark(div(x, y)) -> div_active(mark(x), y) 1116.97/291.61 div_active(0', s(y)) -> 0' 1116.97/291.61 mark(if(x, y, z)) -> if_active(mark(x), y, z) 1116.97/291.61 div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0') 1116.97/291.61 if_active(true, x, y) -> mark(x) 1116.97/291.61 minus_active(x, y) -> minus(x, y) 1116.97/291.61 if_active(false, x, y) -> mark(y) 1116.97/291.61 ge_active(x, y) -> ge(x, y) 1116.97/291.61 if_active(x, y, z) -> if(x, y, z) 1116.97/291.61 div_active(x, y) -> div(x, y) 1116.97/291.61 1116.97/291.61 Types: 1116.97/291.61 minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 0' :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 true :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 false :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 1116.97/291.61 1116.97/291.61 Generator Equations: 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0(0) <=> 0' 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if2_0(x)) 1116.97/291.61 1116.97/291.61 1116.97/291.61 The following defined symbols remain to be analysed: 1116.97/291.61 minus_active, mark, ge_active 1116.97/291.61 1116.97/291.61 They will be analysed ascendingly in the following order: 1116.97/291.61 minus_active < mark 1116.97/291.61 ge_active < mark 1116.97/291.61 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (10) LowerBoundPropagationProof (FINISHED) 1116.97/291.61 Propagated lower bound. 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (11) 1116.97/291.61 BOUNDS(n^1, INF) 1116.97/291.61 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (12) 1116.97/291.61 Obligation: 1116.97/291.61 Innermost TRS: 1116.97/291.61 Rules: 1116.97/291.61 minus_active(0', y) -> 0' 1116.97/291.61 mark(0') -> 0' 1116.97/291.61 minus_active(s(x), s(y)) -> minus_active(x, y) 1116.97/291.61 mark(s(x)) -> s(mark(x)) 1116.97/291.61 ge_active(x, 0') -> true 1116.97/291.61 mark(minus(x, y)) -> minus_active(x, y) 1116.97/291.61 ge_active(0', s(y)) -> false 1116.97/291.61 mark(ge(x, y)) -> ge_active(x, y) 1116.97/291.61 ge_active(s(x), s(y)) -> ge_active(x, y) 1116.97/291.61 mark(div(x, y)) -> div_active(mark(x), y) 1116.97/291.61 div_active(0', s(y)) -> 0' 1116.97/291.61 mark(if(x, y, z)) -> if_active(mark(x), y, z) 1116.97/291.61 div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0') 1116.97/291.61 if_active(true, x, y) -> mark(x) 1116.97/291.61 minus_active(x, y) -> minus(x, y) 1116.97/291.61 if_active(false, x, y) -> mark(y) 1116.97/291.61 ge_active(x, y) -> ge(x, y) 1116.97/291.61 if_active(x, y, z) -> if(x, y, z) 1116.97/291.61 div_active(x, y) -> div(x, y) 1116.97/291.61 1116.97/291.61 Types: 1116.97/291.61 minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 0' :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 true :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 false :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 1116.97/291.61 1116.97/291.61 Lemmas: 1116.97/291.61 minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) -> gen_0':s:true:minus:false:ge:div:if2_0(0), rt in Omega(1 + n4_0) 1116.97/291.61 1116.97/291.61 1116.97/291.61 Generator Equations: 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0(0) <=> 0' 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if2_0(x)) 1116.97/291.61 1116.97/291.61 1116.97/291.61 The following defined symbols remain to be analysed: 1116.97/291.61 ge_active, mark 1116.97/291.61 1116.97/291.61 They will be analysed ascendingly in the following order: 1116.97/291.61 ge_active < mark 1116.97/291.61 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1116.97/291.61 Proved the following rewrite lemma: 1116.97/291.61 ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n700_0), gen_0':s:true:minus:false:ge:div:if2_0(n700_0)) -> true, rt in Omega(1 + n700_0) 1116.97/291.61 1116.97/291.61 Induction Base: 1116.97/291.61 ge_active(gen_0':s:true:minus:false:ge:div:if2_0(0), gen_0':s:true:minus:false:ge:div:if2_0(0)) ->_R^Omega(1) 1116.97/291.61 true 1116.97/291.61 1116.97/291.61 Induction Step: 1116.97/291.61 ge_active(gen_0':s:true:minus:false:ge:div:if2_0(+(n700_0, 1)), gen_0':s:true:minus:false:ge:div:if2_0(+(n700_0, 1))) ->_R^Omega(1) 1116.97/291.61 ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n700_0), gen_0':s:true:minus:false:ge:div:if2_0(n700_0)) ->_IH 1116.97/291.61 true 1116.97/291.61 1116.97/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (14) 1116.97/291.61 Obligation: 1116.97/291.61 Innermost TRS: 1116.97/291.61 Rules: 1116.97/291.61 minus_active(0', y) -> 0' 1116.97/291.61 mark(0') -> 0' 1116.97/291.61 minus_active(s(x), s(y)) -> minus_active(x, y) 1116.97/291.61 mark(s(x)) -> s(mark(x)) 1116.97/291.61 ge_active(x, 0') -> true 1116.97/291.61 mark(minus(x, y)) -> minus_active(x, y) 1116.97/291.61 ge_active(0', s(y)) -> false 1116.97/291.61 mark(ge(x, y)) -> ge_active(x, y) 1116.97/291.61 ge_active(s(x), s(y)) -> ge_active(x, y) 1116.97/291.61 mark(div(x, y)) -> div_active(mark(x), y) 1116.97/291.61 div_active(0', s(y)) -> 0' 1116.97/291.61 mark(if(x, y, z)) -> if_active(mark(x), y, z) 1116.97/291.61 div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0') 1116.97/291.61 if_active(true, x, y) -> mark(x) 1116.97/291.61 minus_active(x, y) -> minus(x, y) 1116.97/291.61 if_active(false, x, y) -> mark(y) 1116.97/291.61 ge_active(x, y) -> ge(x, y) 1116.97/291.61 if_active(x, y, z) -> if(x, y, z) 1116.97/291.61 div_active(x, y) -> div(x, y) 1116.97/291.61 1116.97/291.61 Types: 1116.97/291.61 minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 0' :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 true :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 false :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if 1116.97/291.61 1116.97/291.61 1116.97/291.61 Lemmas: 1116.97/291.61 minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) -> gen_0':s:true:minus:false:ge:div:if2_0(0), rt in Omega(1 + n4_0) 1116.97/291.61 ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n700_0), gen_0':s:true:minus:false:ge:div:if2_0(n700_0)) -> true, rt in Omega(1 + n700_0) 1116.97/291.61 1116.97/291.61 1116.97/291.61 Generator Equations: 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0(0) <=> 0' 1116.97/291.61 gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if2_0(x)) 1116.97/291.61 1116.97/291.61 1116.97/291.61 The following defined symbols remain to be analysed: 1116.97/291.61 mark 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1116.97/291.61 Proved the following rewrite lemma: 1116.97/291.61 mark(gen_0':s:true:minus:false:ge:div:if2_0(n1492_0)) -> gen_0':s:true:minus:false:ge:div:if2_0(n1492_0), rt in Omega(1 + n1492_0) 1116.97/291.61 1116.97/291.61 Induction Base: 1116.97/291.61 mark(gen_0':s:true:minus:false:ge:div:if2_0(0)) ->_R^Omega(1) 1116.97/291.61 0' 1116.97/291.61 1116.97/291.61 Induction Step: 1116.97/291.61 mark(gen_0':s:true:minus:false:ge:div:if2_0(+(n1492_0, 1))) ->_R^Omega(1) 1116.97/291.61 s(mark(gen_0':s:true:minus:false:ge:div:if2_0(n1492_0))) ->_IH 1116.97/291.61 s(gen_0':s:true:minus:false:ge:div:if2_0(c1493_0)) 1116.97/291.61 1116.97/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1116.97/291.61 ---------------------------------------- 1116.97/291.61 1116.97/291.61 (16) 1116.97/291.61 BOUNDS(1, INF) 1117.27/291.68 EOF