1112.20/291.50 WORST_CASE(Omega(n^2), ?) 1113.81/291.97 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1113.81/291.97 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1113.81/291.97 1113.81/291.97 1113.81/291.97 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1113.81/291.97 1113.81/291.97 (0) CpxTRS 1113.81/291.97 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1113.81/291.97 (2) CpxTRS 1113.81/291.97 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1113.81/291.97 (4) typed CpxTrs 1113.81/291.97 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1113.81/291.97 (6) typed CpxTrs 1113.81/291.97 (7) RewriteLemmaProof [LOWER BOUND(ID), 358 ms] 1113.81/291.97 (8) BEST 1113.81/291.97 (9) proven lower bound 1113.81/291.97 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1113.81/291.97 (11) BOUNDS(n^1, INF) 1113.81/291.97 (12) typed CpxTrs 1113.81/291.97 (13) RewriteLemmaProof [LOWER BOUND(ID), 40 ms] 1113.81/291.97 (14) BEST 1113.81/291.97 (15) proven lower bound 1113.81/291.97 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 1113.81/291.97 (17) BOUNDS(n^2, INF) 1113.81/291.97 (18) typed CpxTrs 1113.81/291.97 (19) RewriteLemmaProof [LOWER BOUND(ID), 81 ms] 1113.81/291.97 (20) typed CpxTrs 1113.81/291.97 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (0) 1113.81/291.97 Obligation: 1113.81/291.97 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1113.81/291.97 1113.81/291.97 1113.81/291.97 The TRS R consists of the following rules: 1113.81/291.97 1113.81/291.97 a__p(0) -> 0 1113.81/291.97 a__p(s(X)) -> mark(X) 1113.81/291.97 a__leq(0, Y) -> true 1113.81/291.97 a__leq(s(X), 0) -> false 1113.81/291.97 a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y)) 1113.81/291.97 a__if(true, X, Y) -> mark(X) 1113.81/291.97 a__if(false, X, Y) -> mark(Y) 1113.81/291.97 a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0, s(diff(p(X), Y))) 1113.81/291.97 mark(p(X)) -> a__p(mark(X)) 1113.81/291.97 mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2)) 1113.81/291.97 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1113.81/291.97 mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2)) 1113.81/291.97 mark(0) -> 0 1113.81/291.97 mark(s(X)) -> s(mark(X)) 1113.81/291.97 mark(true) -> true 1113.81/291.97 mark(false) -> false 1113.81/291.97 a__p(X) -> p(X) 1113.81/291.97 a__leq(X1, X2) -> leq(X1, X2) 1113.81/291.97 a__if(X1, X2, X3) -> if(X1, X2, X3) 1113.81/291.97 a__diff(X1, X2) -> diff(X1, X2) 1113.81/291.97 1113.81/291.97 S is empty. 1113.81/291.97 Rewrite Strategy: INNERMOST 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1113.81/291.97 Renamed function symbols to avoid clashes with predefined symbol. 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (2) 1113.81/291.97 Obligation: 1113.81/291.97 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1113.81/291.97 1113.81/291.97 1113.81/291.97 The TRS R consists of the following rules: 1113.81/291.97 1113.81/291.97 a__p(0') -> 0' 1113.81/291.97 a__p(s(X)) -> mark(X) 1113.81/291.97 a__leq(0', Y) -> true 1113.81/291.97 a__leq(s(X), 0') -> false 1113.81/291.97 a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y)) 1113.81/291.97 a__if(true, X, Y) -> mark(X) 1113.81/291.97 a__if(false, X, Y) -> mark(Y) 1113.81/291.97 a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y))) 1113.81/291.97 mark(p(X)) -> a__p(mark(X)) 1113.81/291.97 mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2)) 1113.81/291.97 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1113.81/291.97 mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2)) 1113.81/291.97 mark(0') -> 0' 1113.81/291.97 mark(s(X)) -> s(mark(X)) 1113.81/291.97 mark(true) -> true 1113.81/291.97 mark(false) -> false 1113.81/291.97 a__p(X) -> p(X) 1113.81/291.97 a__leq(X1, X2) -> leq(X1, X2) 1113.81/291.97 a__if(X1, X2, X3) -> if(X1, X2, X3) 1113.81/291.97 a__diff(X1, X2) -> diff(X1, X2) 1113.81/291.97 1113.81/291.97 S is empty. 1113.81/291.97 Rewrite Strategy: INNERMOST 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1113.81/291.97 Infered types. 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (4) 1113.81/291.97 Obligation: 1113.81/291.97 Innermost TRS: 1113.81/291.97 Rules: 1113.81/291.97 a__p(0') -> 0' 1113.81/291.97 a__p(s(X)) -> mark(X) 1113.81/291.97 a__leq(0', Y) -> true 1113.81/291.97 a__leq(s(X), 0') -> false 1113.81/291.97 a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y)) 1113.81/291.97 a__if(true, X, Y) -> mark(X) 1113.81/291.97 a__if(false, X, Y) -> mark(Y) 1113.81/291.97 a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y))) 1113.81/291.97 mark(p(X)) -> a__p(mark(X)) 1113.81/291.97 mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2)) 1113.81/291.97 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1113.81/291.97 mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2)) 1113.81/291.97 mark(0') -> 0' 1113.81/291.97 mark(s(X)) -> s(mark(X)) 1113.81/291.97 mark(true) -> true 1113.81/291.97 mark(false) -> false 1113.81/291.97 a__p(X) -> p(X) 1113.81/291.97 a__leq(X1, X2) -> leq(X1, X2) 1113.81/291.97 a__if(X1, X2, X3) -> if(X1, X2, X3) 1113.81/291.97 a__diff(X1, X2) -> diff(X1, X2) 1113.81/291.97 1113.81/291.97 Types: 1113.81/291.97 a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 0' :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 true :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 false :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (5) OrderProof (LOWER BOUND(ID)) 1113.81/291.97 Heuristically decided to analyse the following defined symbols: 1113.81/291.97 a__p, mark, a__leq, a__if 1113.81/291.97 1113.81/291.97 They will be analysed ascendingly in the following order: 1113.81/291.97 a__p = mark 1113.81/291.97 a__p = a__leq 1113.81/291.97 a__p = a__if 1113.81/291.97 mark = a__leq 1113.81/291.97 mark = a__if 1113.81/291.97 a__leq = a__if 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (6) 1113.81/291.97 Obligation: 1113.81/291.97 Innermost TRS: 1113.81/291.97 Rules: 1113.81/291.97 a__p(0') -> 0' 1113.81/291.97 a__p(s(X)) -> mark(X) 1113.81/291.97 a__leq(0', Y) -> true 1113.81/291.97 a__leq(s(X), 0') -> false 1113.81/291.97 a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y)) 1113.81/291.97 a__if(true, X, Y) -> mark(X) 1113.81/291.97 a__if(false, X, Y) -> mark(Y) 1113.81/291.97 a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y))) 1113.81/291.97 mark(p(X)) -> a__p(mark(X)) 1113.81/291.97 mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2)) 1113.81/291.97 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1113.81/291.97 mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2)) 1113.81/291.97 mark(0') -> 0' 1113.81/291.97 mark(s(X)) -> s(mark(X)) 1113.81/291.97 mark(true) -> true 1113.81/291.97 mark(false) -> false 1113.81/291.97 a__p(X) -> p(X) 1113.81/291.97 a__leq(X1, X2) -> leq(X1, X2) 1113.81/291.97 a__if(X1, X2, X3) -> if(X1, X2, X3) 1113.81/291.97 a__diff(X1, X2) -> diff(X1, X2) 1113.81/291.97 1113.81/291.97 Types: 1113.81/291.97 a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 0' :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 true :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 false :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 1113.81/291.97 1113.81/291.97 Generator Equations: 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0' 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x)) 1113.81/291.97 1113.81/291.97 1113.81/291.97 The following defined symbols remain to be analysed: 1113.81/291.97 mark, a__p, a__leq, a__if 1113.81/291.97 1113.81/291.97 They will be analysed ascendingly in the following order: 1113.81/291.97 a__p = mark 1113.81/291.97 a__p = a__leq 1113.81/291.97 a__p = a__if 1113.81/291.97 mark = a__leq 1113.81/291.97 mark = a__if 1113.81/291.97 a__leq = a__if 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1113.81/291.97 Proved the following rewrite lemma: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n4_0), rt in Omega(1 + n4_0) 1113.81/291.97 1113.81/291.97 Induction Base: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(0)) ->_R^Omega(1) 1113.81/291.97 0' 1113.81/291.97 1113.81/291.97 Induction Step: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(+(n4_0, 1))) ->_R^Omega(1) 1113.81/291.97 s(mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0))) ->_IH 1113.81/291.97 s(gen_0':s:true:false:p:diff:leq:if2_0(c5_0)) 1113.81/291.97 1113.81/291.97 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (8) 1113.81/291.97 Complex Obligation (BEST) 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (9) 1113.81/291.97 Obligation: 1113.81/291.97 Proved the lower bound n^1 for the following obligation: 1113.81/291.97 1113.81/291.97 Innermost TRS: 1113.81/291.97 Rules: 1113.81/291.97 a__p(0') -> 0' 1113.81/291.97 a__p(s(X)) -> mark(X) 1113.81/291.97 a__leq(0', Y) -> true 1113.81/291.97 a__leq(s(X), 0') -> false 1113.81/291.97 a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y)) 1113.81/291.97 a__if(true, X, Y) -> mark(X) 1113.81/291.97 a__if(false, X, Y) -> mark(Y) 1113.81/291.97 a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y))) 1113.81/291.97 mark(p(X)) -> a__p(mark(X)) 1113.81/291.97 mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2)) 1113.81/291.97 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1113.81/291.97 mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2)) 1113.81/291.97 mark(0') -> 0' 1113.81/291.97 mark(s(X)) -> s(mark(X)) 1113.81/291.97 mark(true) -> true 1113.81/291.97 mark(false) -> false 1113.81/291.97 a__p(X) -> p(X) 1113.81/291.97 a__leq(X1, X2) -> leq(X1, X2) 1113.81/291.97 a__if(X1, X2, X3) -> if(X1, X2, X3) 1113.81/291.97 a__diff(X1, X2) -> diff(X1, X2) 1113.81/291.97 1113.81/291.97 Types: 1113.81/291.97 a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 0' :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 true :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 false :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 1113.81/291.97 1113.81/291.97 Generator Equations: 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0' 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x)) 1113.81/291.97 1113.81/291.97 1113.81/291.97 The following defined symbols remain to be analysed: 1113.81/291.97 mark, a__p, a__leq, a__if 1113.81/291.97 1113.81/291.97 They will be analysed ascendingly in the following order: 1113.81/291.97 a__p = mark 1113.81/291.97 a__p = a__leq 1113.81/291.97 a__p = a__if 1113.81/291.97 mark = a__leq 1113.81/291.97 mark = a__if 1113.81/291.97 a__leq = a__if 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (10) LowerBoundPropagationProof (FINISHED) 1113.81/291.97 Propagated lower bound. 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (11) 1113.81/291.97 BOUNDS(n^1, INF) 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (12) 1113.81/291.97 Obligation: 1113.81/291.97 Innermost TRS: 1113.81/291.97 Rules: 1113.81/291.97 a__p(0') -> 0' 1113.81/291.97 a__p(s(X)) -> mark(X) 1113.81/291.97 a__leq(0', Y) -> true 1113.81/291.97 a__leq(s(X), 0') -> false 1113.81/291.97 a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y)) 1113.81/291.97 a__if(true, X, Y) -> mark(X) 1113.81/291.97 a__if(false, X, Y) -> mark(Y) 1113.81/291.97 a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y))) 1113.81/291.97 mark(p(X)) -> a__p(mark(X)) 1113.81/291.97 mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2)) 1113.81/291.97 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1113.81/291.97 mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2)) 1113.81/291.97 mark(0') -> 0' 1113.81/291.97 mark(s(X)) -> s(mark(X)) 1113.81/291.97 mark(true) -> true 1113.81/291.97 mark(false) -> false 1113.81/291.97 a__p(X) -> p(X) 1113.81/291.97 a__leq(X1, X2) -> leq(X1, X2) 1113.81/291.97 a__if(X1, X2, X3) -> if(X1, X2, X3) 1113.81/291.97 a__diff(X1, X2) -> diff(X1, X2) 1113.81/291.97 1113.81/291.97 Types: 1113.81/291.97 a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 0' :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 true :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 false :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 1113.81/291.97 1113.81/291.97 Lemmas: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n4_0), rt in Omega(1 + n4_0) 1113.81/291.97 1113.81/291.97 1113.81/291.97 Generator Equations: 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0' 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x)) 1113.81/291.97 1113.81/291.97 1113.81/291.97 The following defined symbols remain to be analysed: 1113.81/291.97 a__p, a__leq, a__if 1113.81/291.97 1113.81/291.97 They will be analysed ascendingly in the following order: 1113.81/291.97 a__p = mark 1113.81/291.97 a__p = a__leq 1113.81/291.97 a__p = a__if 1113.81/291.97 mark = a__leq 1113.81/291.97 mark = a__if 1113.81/291.97 a__leq = a__if 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1113.81/291.97 Proved the following rewrite lemma: 1113.81/291.97 a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)) -> true, rt in Omega(1 + n1537_0 + n1537_0^2) 1113.81/291.97 1113.81/291.97 Induction Base: 1113.81/291.97 a__leq(gen_0':s:true:false:p:diff:leq:if2_0(0), gen_0':s:true:false:p:diff:leq:if2_0(0)) ->_R^Omega(1) 1113.81/291.97 true 1113.81/291.97 1113.81/291.97 Induction Step: 1113.81/291.97 a__leq(gen_0':s:true:false:p:diff:leq:if2_0(+(n1537_0, 1)), gen_0':s:true:false:p:diff:leq:if2_0(+(n1537_0, 1))) ->_R^Omega(1) 1113.81/291.97 a__leq(mark(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)), mark(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0))) ->_L^Omega(1 + n1537_0) 1113.81/291.97 a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), mark(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0))) ->_L^Omega(1 + n1537_0) 1113.81/291.97 a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)) ->_IH 1113.81/291.97 true 1113.81/291.97 1113.81/291.97 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (14) 1113.81/291.97 Complex Obligation (BEST) 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (15) 1113.81/291.97 Obligation: 1113.81/291.97 Proved the lower bound n^2 for the following obligation: 1113.81/291.97 1113.81/291.97 Innermost TRS: 1113.81/291.97 Rules: 1113.81/291.97 a__p(0') -> 0' 1113.81/291.97 a__p(s(X)) -> mark(X) 1113.81/291.97 a__leq(0', Y) -> true 1113.81/291.97 a__leq(s(X), 0') -> false 1113.81/291.97 a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y)) 1113.81/291.97 a__if(true, X, Y) -> mark(X) 1113.81/291.97 a__if(false, X, Y) -> mark(Y) 1113.81/291.97 a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y))) 1113.81/291.97 mark(p(X)) -> a__p(mark(X)) 1113.81/291.97 mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2)) 1113.81/291.97 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1113.81/291.97 mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2)) 1113.81/291.97 mark(0') -> 0' 1113.81/291.97 mark(s(X)) -> s(mark(X)) 1113.81/291.97 mark(true) -> true 1113.81/291.97 mark(false) -> false 1113.81/291.97 a__p(X) -> p(X) 1113.81/291.97 a__leq(X1, X2) -> leq(X1, X2) 1113.81/291.97 a__if(X1, X2, X3) -> if(X1, X2, X3) 1113.81/291.97 a__diff(X1, X2) -> diff(X1, X2) 1113.81/291.97 1113.81/291.97 Types: 1113.81/291.97 a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 0' :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 true :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 false :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 1113.81/291.97 1113.81/291.97 Lemmas: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n4_0), rt in Omega(1 + n4_0) 1113.81/291.97 1113.81/291.97 1113.81/291.97 Generator Equations: 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0' 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x)) 1113.81/291.97 1113.81/291.97 1113.81/291.97 The following defined symbols remain to be analysed: 1113.81/291.97 a__leq, a__if 1113.81/291.97 1113.81/291.97 They will be analysed ascendingly in the following order: 1113.81/291.97 a__p = mark 1113.81/291.97 a__p = a__leq 1113.81/291.97 a__p = a__if 1113.81/291.97 mark = a__leq 1113.81/291.97 mark = a__if 1113.81/291.97 a__leq = a__if 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (16) LowerBoundPropagationProof (FINISHED) 1113.81/291.97 Propagated lower bound. 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (17) 1113.81/291.97 BOUNDS(n^2, INF) 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (18) 1113.81/291.97 Obligation: 1113.81/291.97 Innermost TRS: 1113.81/291.97 Rules: 1113.81/291.97 a__p(0') -> 0' 1113.81/291.97 a__p(s(X)) -> mark(X) 1113.81/291.97 a__leq(0', Y) -> true 1113.81/291.97 a__leq(s(X), 0') -> false 1113.81/291.97 a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y)) 1113.81/291.97 a__if(true, X, Y) -> mark(X) 1113.81/291.97 a__if(false, X, Y) -> mark(Y) 1113.81/291.97 a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y))) 1113.81/291.97 mark(p(X)) -> a__p(mark(X)) 1113.81/291.97 mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2)) 1113.81/291.97 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1113.81/291.97 mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2)) 1113.81/291.97 mark(0') -> 0' 1113.81/291.97 mark(s(X)) -> s(mark(X)) 1113.81/291.97 mark(true) -> true 1113.81/291.97 mark(false) -> false 1113.81/291.97 a__p(X) -> p(X) 1113.81/291.97 a__leq(X1, X2) -> leq(X1, X2) 1113.81/291.97 a__if(X1, X2, X3) -> if(X1, X2, X3) 1113.81/291.97 a__diff(X1, X2) -> diff(X1, X2) 1113.81/291.97 1113.81/291.97 Types: 1113.81/291.97 a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 0' :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 true :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 false :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 1113.81/291.97 1113.81/291.97 Lemmas: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n4_0), rt in Omega(1 + n4_0) 1113.81/291.97 a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)) -> true, rt in Omega(1 + n1537_0 + n1537_0^2) 1113.81/291.97 1113.81/291.97 1113.81/291.97 Generator Equations: 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0' 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x)) 1113.81/291.97 1113.81/291.97 1113.81/291.97 The following defined symbols remain to be analysed: 1113.81/291.97 a__if, a__p, mark 1113.81/291.97 1113.81/291.97 They will be analysed ascendingly in the following order: 1113.81/291.97 a__p = mark 1113.81/291.97 a__p = a__leq 1113.81/291.97 a__p = a__if 1113.81/291.97 mark = a__leq 1113.81/291.97 mark = a__if 1113.81/291.97 a__leq = a__if 1113.81/291.97 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (19) RewriteLemmaProof (LOWER BOUND(ID)) 1113.81/291.97 Proved the following rewrite lemma: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(n2399_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n2399_0), rt in Omega(1 + n2399_0) 1113.81/291.97 1113.81/291.97 Induction Base: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(0)) ->_R^Omega(1) 1113.81/291.97 0' 1113.81/291.97 1113.81/291.97 Induction Step: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(+(n2399_0, 1))) ->_R^Omega(1) 1113.81/291.97 s(mark(gen_0':s:true:false:p:diff:leq:if2_0(n2399_0))) ->_IH 1113.81/291.97 s(gen_0':s:true:false:p:diff:leq:if2_0(c2400_0)) 1113.81/291.97 1113.81/291.97 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1113.81/291.97 ---------------------------------------- 1113.81/291.97 1113.81/291.97 (20) 1113.81/291.97 Obligation: 1113.81/291.97 Innermost TRS: 1113.81/291.97 Rules: 1113.81/291.97 a__p(0') -> 0' 1113.81/291.97 a__p(s(X)) -> mark(X) 1113.81/291.97 a__leq(0', Y) -> true 1113.81/291.97 a__leq(s(X), 0') -> false 1113.81/291.97 a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y)) 1113.81/291.97 a__if(true, X, Y) -> mark(X) 1113.81/291.97 a__if(false, X, Y) -> mark(Y) 1113.81/291.97 a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y))) 1113.81/291.97 mark(p(X)) -> a__p(mark(X)) 1113.81/291.97 mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2)) 1113.81/291.97 mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1113.81/291.97 mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2)) 1113.81/291.97 mark(0') -> 0' 1113.81/291.97 mark(s(X)) -> s(mark(X)) 1113.81/291.97 mark(true) -> true 1113.81/291.97 mark(false) -> false 1113.81/291.97 a__p(X) -> p(X) 1113.81/291.97 a__leq(X1, X2) -> leq(X1, X2) 1113.81/291.97 a__if(X1, X2, X3) -> if(X1, X2, X3) 1113.81/291.97 a__diff(X1, X2) -> diff(X1, X2) 1113.81/291.97 1113.81/291.97 Types: 1113.81/291.97 a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 0' :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 true :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 false :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if 1113.81/291.97 1113.81/291.97 1113.81/291.97 Lemmas: 1113.81/291.97 mark(gen_0':s:true:false:p:diff:leq:if2_0(n2399_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n2399_0), rt in Omega(1 + n2399_0) 1113.81/291.97 a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)) -> true, rt in Omega(1 + n1537_0 + n1537_0^2) 1113.81/291.97 1113.81/291.97 1113.81/291.97 Generator Equations: 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0' 1113.81/291.97 gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x)) 1113.81/291.97 1113.81/291.97 1113.81/291.97 The following defined symbols remain to be analysed: 1113.81/291.97 a__p 1113.81/291.97 1113.81/291.97 They will be analysed ascendingly in the following order: 1113.81/291.97 a__p = mark 1113.81/291.97 a__p = a__leq 1113.81/291.97 a__p = a__if 1113.81/291.97 mark = a__leq 1113.81/291.97 mark = a__if 1113.81/291.97 a__leq = a__if 1114.13/292.05 EOF