1118.60/291.53 WORST_CASE(Omega(n^1), ?) 1123.52/292.79 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1123.52/292.79 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1123.52/292.79 1123.52/292.79 1123.52/292.79 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.52/292.79 1123.52/292.79 (0) CpxTRS 1123.52/292.79 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1123.52/292.79 (2) CpxTRS 1123.52/292.79 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1123.52/292.79 (4) typed CpxTrs 1123.52/292.79 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1123.52/292.79 (6) typed CpxTrs 1123.52/292.79 (7) RewriteLemmaProof [LOWER BOUND(ID), 333 ms] 1123.52/292.79 (8) BEST 1123.52/292.79 (9) proven lower bound 1123.52/292.79 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1123.52/292.79 (11) BOUNDS(n^1, INF) 1123.52/292.79 (12) typed CpxTrs 1123.52/292.79 (13) RewriteLemmaProof [LOWER BOUND(ID), 19 ms] 1123.52/292.79 (14) typed CpxTrs 1123.52/292.79 (15) RewriteLemmaProof [LOWER BOUND(ID), 142 ms] 1123.52/292.79 (16) BOUNDS(1, INF) 1123.52/292.79 1123.52/292.79 1123.52/292.79 ---------------------------------------- 1123.52/292.79 1123.52/292.79 (0) 1123.52/292.79 Obligation: 1123.52/292.79 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.52/292.79 1123.52/292.79 1123.52/292.79 The TRS R consists of the following rules: 1123.52/292.79 1123.52/292.79 a__minus(0, Y) -> 0 1123.52/292.79 a__minus(s(X), s(Y)) -> a__minus(X, Y) 1123.52/292.79 a__geq(X, 0) -> true 1123.52/292.79 a__geq(0, s(Y)) -> false 1123.52/292.79 a__geq(s(X), s(Y)) -> a__geq(X, Y) 1123.52/292.79 a__div(0, s(Y)) -> 0 1123.52/292.79 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) 1123.52/292.79 a__if(true, X, Y) -> mark(X) 1123.52/292.79 a__if(false, X, Y) -> mark(Y) 1123.52/292.79 mark(minus(X1, X2)) -> a__minus(X1, X2) 1123.52/292.79 mark(geq(X1, X2)) -> a__geq(X1, X2) 1123.52/292.79 mark(div(X1, X2)) -> a__div(mark(X1), X2) 1123.52/292.79 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1123.52/292.79 mark(0) -> 0 1123.52/292.79 mark(s(X)) -> s(mark(X)) 1123.52/292.79 mark(true) -> true 1123.52/292.79 mark(false) -> false 1123.52/292.79 a__minus(X1, X2) -> minus(X1, X2) 1123.52/292.79 a__geq(X1, X2) -> geq(X1, X2) 1123.52/292.79 a__div(X1, X2) -> div(X1, X2) 1123.52/292.79 a__if(X1, X2, X3) -> if(X1, X2, X3) 1123.52/292.79 1123.52/292.79 S is empty. 1123.52/292.79 Rewrite Strategy: INNERMOST 1123.52/292.79 ---------------------------------------- 1123.52/292.79 1123.52/292.79 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1123.52/292.79 Renamed function symbols to avoid clashes with predefined symbol. 1123.52/292.79 ---------------------------------------- 1123.52/292.79 1123.52/292.79 (2) 1123.52/292.79 Obligation: 1123.52/292.79 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.52/292.79 1123.52/292.79 1123.52/292.79 The TRS R consists of the following rules: 1123.52/292.79 1123.52/292.79 a__minus(0', Y) -> 0' 1123.52/292.79 a__minus(s(X), s(Y)) -> a__minus(X, Y) 1123.52/292.79 a__geq(X, 0') -> true 1123.52/292.79 a__geq(0', s(Y)) -> false 1123.52/292.79 a__geq(s(X), s(Y)) -> a__geq(X, Y) 1123.52/292.79 a__div(0', s(Y)) -> 0' 1123.52/292.79 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0') 1123.52/292.79 a__if(true, X, Y) -> mark(X) 1123.52/292.79 a__if(false, X, Y) -> mark(Y) 1123.52/292.79 mark(minus(X1, X2)) -> a__minus(X1, X2) 1123.52/292.79 mark(geq(X1, X2)) -> a__geq(X1, X2) 1123.52/292.79 mark(div(X1, X2)) -> a__div(mark(X1), X2) 1123.52/292.79 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1123.52/292.79 mark(0') -> 0' 1123.52/292.79 mark(s(X)) -> s(mark(X)) 1123.52/292.79 mark(true) -> true 1123.52/292.79 mark(false) -> false 1123.52/292.79 a__minus(X1, X2) -> minus(X1, X2) 1123.52/292.79 a__geq(X1, X2) -> geq(X1, X2) 1123.52/292.79 a__div(X1, X2) -> div(X1, X2) 1123.52/292.79 a__if(X1, X2, X3) -> if(X1, X2, X3) 1123.52/292.79 1123.52/292.79 S is empty. 1123.52/292.79 Rewrite Strategy: INNERMOST 1123.52/292.79 ---------------------------------------- 1123.52/292.79 1123.52/292.79 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1123.52/292.79 Infered types. 1123.52/292.79 ---------------------------------------- 1123.52/292.79 1123.52/292.79 (4) 1123.52/292.79 Obligation: 1123.52/292.79 Innermost TRS: 1123.52/292.79 Rules: 1123.52/292.79 a__minus(0', Y) -> 0' 1123.52/292.79 a__minus(s(X), s(Y)) -> a__minus(X, Y) 1123.52/292.79 a__geq(X, 0') -> true 1123.52/292.79 a__geq(0', s(Y)) -> false 1123.52/292.79 a__geq(s(X), s(Y)) -> a__geq(X, Y) 1123.52/292.79 a__div(0', s(Y)) -> 0' 1123.52/292.79 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0') 1123.52/292.79 a__if(true, X, Y) -> mark(X) 1123.52/292.79 a__if(false, X, Y) -> mark(Y) 1123.52/292.79 mark(minus(X1, X2)) -> a__minus(X1, X2) 1123.52/292.79 mark(geq(X1, X2)) -> a__geq(X1, X2) 1123.52/292.79 mark(div(X1, X2)) -> a__div(mark(X1), X2) 1123.52/292.79 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1123.52/292.79 mark(0') -> 0' 1123.52/292.79 mark(s(X)) -> s(mark(X)) 1123.52/292.79 mark(true) -> true 1123.52/292.79 mark(false) -> false 1123.52/292.79 a__minus(X1, X2) -> minus(X1, X2) 1123.52/292.79 a__geq(X1, X2) -> geq(X1, X2) 1123.52/292.79 a__div(X1, X2) -> div(X1, X2) 1123.52/292.79 a__if(X1, X2, X3) -> if(X1, X2, X3) 1123.52/292.79 1123.52/292.79 Types: 1123.52/292.79 a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.79 0' :: 0':s:true:false:minus:div:geq:if 1123.52/292.79 s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.79 a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 true :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 false :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (5) OrderProof (LOWER BOUND(ID)) 1123.52/292.80 Heuristically decided to analyse the following defined symbols: 1123.52/292.80 a__minus, a__geq, mark 1123.52/292.80 1123.52/292.80 They will be analysed ascendingly in the following order: 1123.52/292.80 a__minus < mark 1123.52/292.80 a__geq < mark 1123.52/292.80 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (6) 1123.52/292.80 Obligation: 1123.52/292.80 Innermost TRS: 1123.52/292.80 Rules: 1123.52/292.80 a__minus(0', Y) -> 0' 1123.52/292.80 a__minus(s(X), s(Y)) -> a__minus(X, Y) 1123.52/292.80 a__geq(X, 0') -> true 1123.52/292.80 a__geq(0', s(Y)) -> false 1123.52/292.80 a__geq(s(X), s(Y)) -> a__geq(X, Y) 1123.52/292.80 a__div(0', s(Y)) -> 0' 1123.52/292.80 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0') 1123.52/292.80 a__if(true, X, Y) -> mark(X) 1123.52/292.80 a__if(false, X, Y) -> mark(Y) 1123.52/292.80 mark(minus(X1, X2)) -> a__minus(X1, X2) 1123.52/292.80 mark(geq(X1, X2)) -> a__geq(X1, X2) 1123.52/292.80 mark(div(X1, X2)) -> a__div(mark(X1), X2) 1123.52/292.80 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1123.52/292.80 mark(0') -> 0' 1123.52/292.80 mark(s(X)) -> s(mark(X)) 1123.52/292.80 mark(true) -> true 1123.52/292.80 mark(false) -> false 1123.52/292.80 a__minus(X1, X2) -> minus(X1, X2) 1123.52/292.80 a__geq(X1, X2) -> geq(X1, X2) 1123.52/292.80 a__div(X1, X2) -> div(X1, X2) 1123.52/292.80 a__if(X1, X2, X3) -> if(X1, X2, X3) 1123.52/292.80 1123.52/292.80 Types: 1123.52/292.80 a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 0' :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 true :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 false :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 1123.52/292.80 1123.52/292.80 Generator Equations: 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0(0) <=> 0' 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:minus:div:geq:if2_0(x)) 1123.52/292.80 1123.52/292.80 1123.52/292.80 The following defined symbols remain to be analysed: 1123.52/292.80 a__minus, a__geq, mark 1123.52/292.80 1123.52/292.80 They will be analysed ascendingly in the following order: 1123.52/292.80 a__minus < mark 1123.52/292.80 a__geq < mark 1123.52/292.80 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1123.52/292.80 Proved the following rewrite lemma: 1123.52/292.80 a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) -> gen_0':s:true:false:minus:div:geq:if2_0(0), rt in Omega(1 + n4_0) 1123.52/292.80 1123.52/292.80 Induction Base: 1123.52/292.80 a__minus(gen_0':s:true:false:minus:div:geq:if2_0(0), gen_0':s:true:false:minus:div:geq:if2_0(0)) ->_R^Omega(1) 1123.52/292.80 0' 1123.52/292.80 1123.52/292.80 Induction Step: 1123.52/292.80 a__minus(gen_0':s:true:false:minus:div:geq:if2_0(+(n4_0, 1)), gen_0':s:true:false:minus:div:geq:if2_0(+(n4_0, 1))) ->_R^Omega(1) 1123.52/292.80 a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) ->_IH 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0(0) 1123.52/292.80 1123.52/292.80 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (8) 1123.52/292.80 Complex Obligation (BEST) 1123.52/292.80 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (9) 1123.52/292.80 Obligation: 1123.52/292.80 Proved the lower bound n^1 for the following obligation: 1123.52/292.80 1123.52/292.80 Innermost TRS: 1123.52/292.80 Rules: 1123.52/292.80 a__minus(0', Y) -> 0' 1123.52/292.80 a__minus(s(X), s(Y)) -> a__minus(X, Y) 1123.52/292.80 a__geq(X, 0') -> true 1123.52/292.80 a__geq(0', s(Y)) -> false 1123.52/292.80 a__geq(s(X), s(Y)) -> a__geq(X, Y) 1123.52/292.80 a__div(0', s(Y)) -> 0' 1123.52/292.80 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0') 1123.52/292.80 a__if(true, X, Y) -> mark(X) 1123.52/292.80 a__if(false, X, Y) -> mark(Y) 1123.52/292.80 mark(minus(X1, X2)) -> a__minus(X1, X2) 1123.52/292.80 mark(geq(X1, X2)) -> a__geq(X1, X2) 1123.52/292.80 mark(div(X1, X2)) -> a__div(mark(X1), X2) 1123.52/292.80 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1123.52/292.80 mark(0') -> 0' 1123.52/292.80 mark(s(X)) -> s(mark(X)) 1123.52/292.80 mark(true) -> true 1123.52/292.80 mark(false) -> false 1123.52/292.80 a__minus(X1, X2) -> minus(X1, X2) 1123.52/292.80 a__geq(X1, X2) -> geq(X1, X2) 1123.52/292.80 a__div(X1, X2) -> div(X1, X2) 1123.52/292.80 a__if(X1, X2, X3) -> if(X1, X2, X3) 1123.52/292.80 1123.52/292.80 Types: 1123.52/292.80 a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 0' :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 true :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 false :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 1123.52/292.80 1123.52/292.80 Generator Equations: 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0(0) <=> 0' 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:minus:div:geq:if2_0(x)) 1123.52/292.80 1123.52/292.80 1123.52/292.80 The following defined symbols remain to be analysed: 1123.52/292.80 a__minus, a__geq, mark 1123.52/292.80 1123.52/292.80 They will be analysed ascendingly in the following order: 1123.52/292.80 a__minus < mark 1123.52/292.80 a__geq < mark 1123.52/292.80 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (10) LowerBoundPropagationProof (FINISHED) 1123.52/292.80 Propagated lower bound. 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (11) 1123.52/292.80 BOUNDS(n^1, INF) 1123.52/292.80 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (12) 1123.52/292.80 Obligation: 1123.52/292.80 Innermost TRS: 1123.52/292.80 Rules: 1123.52/292.80 a__minus(0', Y) -> 0' 1123.52/292.80 a__minus(s(X), s(Y)) -> a__minus(X, Y) 1123.52/292.80 a__geq(X, 0') -> true 1123.52/292.80 a__geq(0', s(Y)) -> false 1123.52/292.80 a__geq(s(X), s(Y)) -> a__geq(X, Y) 1123.52/292.80 a__div(0', s(Y)) -> 0' 1123.52/292.80 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0') 1123.52/292.80 a__if(true, X, Y) -> mark(X) 1123.52/292.80 a__if(false, X, Y) -> mark(Y) 1123.52/292.80 mark(minus(X1, X2)) -> a__minus(X1, X2) 1123.52/292.80 mark(geq(X1, X2)) -> a__geq(X1, X2) 1123.52/292.80 mark(div(X1, X2)) -> a__div(mark(X1), X2) 1123.52/292.80 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1123.52/292.80 mark(0') -> 0' 1123.52/292.80 mark(s(X)) -> s(mark(X)) 1123.52/292.80 mark(true) -> true 1123.52/292.80 mark(false) -> false 1123.52/292.80 a__minus(X1, X2) -> minus(X1, X2) 1123.52/292.80 a__geq(X1, X2) -> geq(X1, X2) 1123.52/292.80 a__div(X1, X2) -> div(X1, X2) 1123.52/292.80 a__if(X1, X2, X3) -> if(X1, X2, X3) 1123.52/292.80 1123.52/292.80 Types: 1123.52/292.80 a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 0' :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 true :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 false :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 1123.52/292.80 1123.52/292.80 Lemmas: 1123.52/292.80 a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) -> gen_0':s:true:false:minus:div:geq:if2_0(0), rt in Omega(1 + n4_0) 1123.52/292.80 1123.52/292.80 1123.52/292.80 Generator Equations: 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0(0) <=> 0' 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:minus:div:geq:if2_0(x)) 1123.52/292.80 1123.52/292.80 1123.52/292.80 The following defined symbols remain to be analysed: 1123.52/292.80 a__geq, mark 1123.52/292.80 1123.52/292.80 They will be analysed ascendingly in the following order: 1123.52/292.80 a__geq < mark 1123.52/292.80 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1123.52/292.80 Proved the following rewrite lemma: 1123.52/292.80 a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n752_0), gen_0':s:true:false:minus:div:geq:if2_0(n752_0)) -> true, rt in Omega(1 + n752_0) 1123.52/292.80 1123.52/292.80 Induction Base: 1123.52/292.80 a__geq(gen_0':s:true:false:minus:div:geq:if2_0(0), gen_0':s:true:false:minus:div:geq:if2_0(0)) ->_R^Omega(1) 1123.52/292.80 true 1123.52/292.80 1123.52/292.80 Induction Step: 1123.52/292.80 a__geq(gen_0':s:true:false:minus:div:geq:if2_0(+(n752_0, 1)), gen_0':s:true:false:minus:div:geq:if2_0(+(n752_0, 1))) ->_R^Omega(1) 1123.52/292.80 a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n752_0), gen_0':s:true:false:minus:div:geq:if2_0(n752_0)) ->_IH 1123.52/292.80 true 1123.52/292.80 1123.52/292.80 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (14) 1123.52/292.80 Obligation: 1123.52/292.80 Innermost TRS: 1123.52/292.80 Rules: 1123.52/292.80 a__minus(0', Y) -> 0' 1123.52/292.80 a__minus(s(X), s(Y)) -> a__minus(X, Y) 1123.52/292.80 a__geq(X, 0') -> true 1123.52/292.80 a__geq(0', s(Y)) -> false 1123.52/292.80 a__geq(s(X), s(Y)) -> a__geq(X, Y) 1123.52/292.80 a__div(0', s(Y)) -> 0' 1123.52/292.80 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0') 1123.52/292.80 a__if(true, X, Y) -> mark(X) 1123.52/292.80 a__if(false, X, Y) -> mark(Y) 1123.52/292.80 mark(minus(X1, X2)) -> a__minus(X1, X2) 1123.52/292.80 mark(geq(X1, X2)) -> a__geq(X1, X2) 1123.52/292.80 mark(div(X1, X2)) -> a__div(mark(X1), X2) 1123.52/292.80 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1123.52/292.80 mark(0') -> 0' 1123.52/292.80 mark(s(X)) -> s(mark(X)) 1123.52/292.80 mark(true) -> true 1123.52/292.80 mark(false) -> false 1123.52/292.80 a__minus(X1, X2) -> minus(X1, X2) 1123.52/292.80 a__geq(X1, X2) -> geq(X1, X2) 1123.52/292.80 a__div(X1, X2) -> div(X1, X2) 1123.52/292.80 a__if(X1, X2, X3) -> if(X1, X2, X3) 1123.52/292.80 1123.52/292.80 Types: 1123.52/292.80 a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 0' :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 true :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 false :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if 1123.52/292.80 1123.52/292.80 1123.52/292.80 Lemmas: 1123.52/292.80 a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) -> gen_0':s:true:false:minus:div:geq:if2_0(0), rt in Omega(1 + n4_0) 1123.52/292.80 a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n752_0), gen_0':s:true:false:minus:div:geq:if2_0(n752_0)) -> true, rt in Omega(1 + n752_0) 1123.52/292.80 1123.52/292.80 1123.52/292.80 Generator Equations: 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0(0) <=> 0' 1123.52/292.80 gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:minus:div:geq:if2_0(x)) 1123.52/292.80 1123.52/292.80 1123.52/292.80 The following defined symbols remain to be analysed: 1123.52/292.80 mark 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1123.52/292.80 Proved the following rewrite lemma: 1123.52/292.80 mark(gen_0':s:true:false:minus:div:geq:if2_0(n1592_0)) -> gen_0':s:true:false:minus:div:geq:if2_0(n1592_0), rt in Omega(1 + n1592_0) 1123.52/292.80 1123.52/292.80 Induction Base: 1123.52/292.80 mark(gen_0':s:true:false:minus:div:geq:if2_0(0)) ->_R^Omega(1) 1123.52/292.80 0' 1123.52/292.80 1123.52/292.80 Induction Step: 1123.52/292.80 mark(gen_0':s:true:false:minus:div:geq:if2_0(+(n1592_0, 1))) ->_R^Omega(1) 1123.52/292.80 s(mark(gen_0':s:true:false:minus:div:geq:if2_0(n1592_0))) ->_IH 1123.52/292.80 s(gen_0':s:true:false:minus:div:geq:if2_0(c1593_0)) 1123.52/292.80 1123.52/292.80 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1123.52/292.80 ---------------------------------------- 1123.52/292.80 1123.52/292.80 (16) 1123.52/292.80 BOUNDS(1, INF) 1123.63/292.86 EOF