24.75/8.73 WORST_CASE(Omega(n^1), O(n^1)) 24.93/8.74 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.93/8.74 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.93/8.74 24.93/8.74 24.93/8.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.93/8.74 24.93/8.74 (0) CpxTRS 24.93/8.74 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 24.93/8.74 (2) CpxTRS 24.93/8.74 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 24.93/8.74 (4) CpxTRS 24.93/8.74 (5) CpxTrsMatchBoundsTAProof [FINISHED, 35 ms] 24.93/8.74 (6) BOUNDS(1, n^1) 24.93/8.74 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 24.93/8.74 (8) CpxTRS 24.93/8.74 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 24.93/8.74 (10) typed CpxTrs 24.93/8.74 (11) OrderProof [LOWER BOUND(ID), 0 ms] 24.93/8.74 (12) typed CpxTrs 24.93/8.74 (13) RewriteLemmaProof [LOWER BOUND(ID), 492 ms] 24.93/8.74 (14) BEST 24.93/8.74 (15) proven lower bound 24.93/8.74 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 24.93/8.74 (17) BOUNDS(n^1, INF) 24.93/8.74 (18) typed CpxTrs 24.93/8.74 (19) RewriteLemmaProof [LOWER BOUND(ID), 98 ms] 24.93/8.74 (20) typed CpxTrs 24.93/8.74 (21) RewriteLemmaProof [LOWER BOUND(ID), 102 ms] 24.93/8.74 (22) typed CpxTrs 24.93/8.74 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (0) 24.93/8.74 Obligation: 24.93/8.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.93/8.74 24.93/8.74 24.93/8.74 The TRS R consists of the following rules: 24.93/8.74 24.93/8.74 active(minus(0, Y)) -> mark(0) 24.93/8.74 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 24.93/8.74 active(geq(X, 0)) -> mark(true) 24.93/8.74 active(geq(0, s(Y))) -> mark(false) 24.93/8.74 active(geq(s(X), s(Y))) -> mark(geq(X, Y)) 24.93/8.74 active(div(0, s(Y))) -> mark(0) 24.93/8.74 active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) 24.93/8.74 active(if(true, X, Y)) -> mark(X) 24.93/8.74 active(if(false, X, Y)) -> mark(Y) 24.93/8.74 active(s(X)) -> s(active(X)) 24.93/8.74 active(div(X1, X2)) -> div(active(X1), X2) 24.93/8.74 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.74 s(mark(X)) -> mark(s(X)) 24.93/8.74 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.74 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.74 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 24.93/8.74 proper(0) -> ok(0) 24.93/8.74 proper(s(X)) -> s(proper(X)) 24.93/8.74 proper(geq(X1, X2)) -> geq(proper(X1), proper(X2)) 24.93/8.74 proper(true) -> ok(true) 24.93/8.74 proper(false) -> ok(false) 24.93/8.74 proper(div(X1, X2)) -> div(proper(X1), proper(X2)) 24.93/8.74 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.74 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.74 s(ok(X)) -> ok(s(X)) 24.93/8.74 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.74 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.74 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.74 top(mark(X)) -> top(proper(X)) 24.93/8.74 top(ok(X)) -> top(active(X)) 24.93/8.74 24.93/8.74 S is empty. 24.93/8.74 Rewrite Strategy: FULL 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 24.93/8.74 The following defined symbols can occur below the 0th argument of top: proper, active 24.93/8.74 The following defined symbols can occur below the 0th argument of proper: proper, active 24.93/8.74 The following defined symbols can occur below the 0th argument of active: proper, active 24.93/8.74 24.93/8.74 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 24.93/8.74 active(minus(0, Y)) -> mark(0) 24.93/8.74 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 24.93/8.74 active(geq(X, 0)) -> mark(true) 24.93/8.74 active(geq(0, s(Y))) -> mark(false) 24.93/8.74 active(geq(s(X), s(Y))) -> mark(geq(X, Y)) 24.93/8.74 active(div(0, s(Y))) -> mark(0) 24.93/8.74 active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) 24.93/8.74 active(if(true, X, Y)) -> mark(X) 24.93/8.74 active(if(false, X, Y)) -> mark(Y) 24.93/8.74 active(s(X)) -> s(active(X)) 24.93/8.74 active(div(X1, X2)) -> div(active(X1), X2) 24.93/8.74 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.74 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 24.93/8.74 proper(s(X)) -> s(proper(X)) 24.93/8.74 proper(geq(X1, X2)) -> geq(proper(X1), proper(X2)) 24.93/8.74 proper(div(X1, X2)) -> div(proper(X1), proper(X2)) 24.93/8.74 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (2) 24.93/8.74 Obligation: 24.93/8.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.93/8.74 24.93/8.74 24.93/8.74 The TRS R consists of the following rules: 24.93/8.74 24.93/8.74 s(mark(X)) -> mark(s(X)) 24.93/8.74 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.74 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.74 proper(0) -> ok(0) 24.93/8.74 proper(true) -> ok(true) 24.93/8.74 proper(false) -> ok(false) 24.93/8.74 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.74 s(ok(X)) -> ok(s(X)) 24.93/8.74 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.74 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.74 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.74 top(mark(X)) -> top(proper(X)) 24.93/8.74 top(ok(X)) -> top(active(X)) 24.93/8.74 24.93/8.74 S is empty. 24.93/8.74 Rewrite Strategy: FULL 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 24.93/8.74 transformed relative TRS to TRS 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (4) 24.93/8.74 Obligation: 24.93/8.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.93/8.74 24.93/8.74 24.93/8.74 The TRS R consists of the following rules: 24.93/8.74 24.93/8.74 s(mark(X)) -> mark(s(X)) 24.93/8.74 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.74 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.74 proper(0) -> ok(0) 24.93/8.74 proper(true) -> ok(true) 24.93/8.74 proper(false) -> ok(false) 24.93/8.74 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.74 s(ok(X)) -> ok(s(X)) 24.93/8.74 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.74 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.74 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.74 top(mark(X)) -> top(proper(X)) 24.93/8.74 top(ok(X)) -> top(active(X)) 24.93/8.74 24.93/8.74 S is empty. 24.93/8.74 Rewrite Strategy: FULL 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (5) CpxTrsMatchBoundsTAProof (FINISHED) 24.93/8.74 A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 24.93/8.74 24.93/8.74 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 24.93/8.74 final states : [1, 2, 3, 4, 5, 6, 7] 24.93/8.74 transitions: 24.93/8.74 mark0(0) -> 0 24.93/8.74 00() -> 0 24.93/8.74 ok0(0) -> 0 24.93/8.74 true0() -> 0 24.93/8.74 false0() -> 0 24.93/8.74 active0(0) -> 0 24.93/8.74 s0(0) -> 1 24.93/8.74 div0(0, 0) -> 2 24.93/8.74 if0(0, 0, 0) -> 3 24.93/8.74 proper0(0) -> 4 24.93/8.74 minus0(0, 0) -> 5 24.93/8.74 geq0(0, 0) -> 6 24.93/8.74 top0(0) -> 7 24.93/8.74 s1(0) -> 8 24.93/8.74 mark1(8) -> 1 24.93/8.74 div1(0, 0) -> 9 24.93/8.74 mark1(9) -> 2 24.93/8.74 if1(0, 0, 0) -> 10 24.93/8.74 mark1(10) -> 3 24.93/8.74 01() -> 11 24.93/8.74 ok1(11) -> 4 24.93/8.74 true1() -> 12 24.93/8.74 ok1(12) -> 4 24.93/8.74 false1() -> 13 24.93/8.74 ok1(13) -> 4 24.93/8.74 minus1(0, 0) -> 14 24.93/8.74 ok1(14) -> 5 24.93/8.74 s1(0) -> 15 24.93/8.74 ok1(15) -> 1 24.93/8.74 geq1(0, 0) -> 16 24.93/8.74 ok1(16) -> 6 24.93/8.74 div1(0, 0) -> 17 24.93/8.74 ok1(17) -> 2 24.93/8.74 if1(0, 0, 0) -> 18 24.93/8.74 ok1(18) -> 3 24.93/8.74 proper1(0) -> 19 24.93/8.74 top1(19) -> 7 24.93/8.74 active1(0) -> 20 24.93/8.74 top1(20) -> 7 24.93/8.74 mark1(8) -> 8 24.93/8.74 mark1(8) -> 15 24.93/8.74 mark1(9) -> 9 24.93/8.74 mark1(9) -> 17 24.93/8.74 mark1(10) -> 10 24.93/8.74 mark1(10) -> 18 24.93/8.74 ok1(11) -> 19 24.93/8.74 ok1(12) -> 19 24.93/8.74 ok1(13) -> 19 24.93/8.74 ok1(14) -> 14 24.93/8.74 ok1(15) -> 8 24.93/8.74 ok1(15) -> 15 24.93/8.74 ok1(16) -> 16 24.93/8.74 ok1(17) -> 9 24.93/8.74 ok1(17) -> 17 24.93/8.74 ok1(18) -> 10 24.93/8.74 ok1(18) -> 18 24.93/8.74 active2(11) -> 21 24.93/8.74 top2(21) -> 7 24.93/8.74 active2(12) -> 21 24.93/8.74 active2(13) -> 21 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (6) 24.93/8.74 BOUNDS(1, n^1) 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 24.93/8.74 Renamed function symbols to avoid clashes with predefined symbol. 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (8) 24.93/8.74 Obligation: 24.93/8.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 24.93/8.74 24.93/8.74 24.93/8.74 The TRS R consists of the following rules: 24.93/8.74 24.93/8.74 active(minus(0', Y)) -> mark(0') 24.93/8.74 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 24.93/8.74 active(geq(X, 0')) -> mark(true) 24.93/8.74 active(geq(0', s(Y))) -> mark(false) 24.93/8.74 active(geq(s(X), s(Y))) -> mark(geq(X, Y)) 24.93/8.74 active(div(0', s(Y))) -> mark(0') 24.93/8.74 active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')) 24.93/8.74 active(if(true, X, Y)) -> mark(X) 24.93/8.74 active(if(false, X, Y)) -> mark(Y) 24.93/8.74 active(s(X)) -> s(active(X)) 24.93/8.74 active(div(X1, X2)) -> div(active(X1), X2) 24.93/8.74 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.74 s(mark(X)) -> mark(s(X)) 24.93/8.74 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.74 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.74 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 24.93/8.74 proper(0') -> ok(0') 24.93/8.74 proper(s(X)) -> s(proper(X)) 24.93/8.74 proper(geq(X1, X2)) -> geq(proper(X1), proper(X2)) 24.93/8.74 proper(true) -> ok(true) 24.93/8.74 proper(false) -> ok(false) 24.93/8.74 proper(div(X1, X2)) -> div(proper(X1), proper(X2)) 24.93/8.74 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.74 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.74 s(ok(X)) -> ok(s(X)) 24.93/8.74 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.74 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.74 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.74 top(mark(X)) -> top(proper(X)) 24.93/8.74 top(ok(X)) -> top(active(X)) 24.93/8.74 24.93/8.74 S is empty. 24.93/8.74 Rewrite Strategy: FULL 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 24.93/8.74 Infered types. 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (10) 24.93/8.74 Obligation: 24.93/8.74 TRS: 24.93/8.74 Rules: 24.93/8.74 active(minus(0', Y)) -> mark(0') 24.93/8.74 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 24.93/8.74 active(geq(X, 0')) -> mark(true) 24.93/8.74 active(geq(0', s(Y))) -> mark(false) 24.93/8.74 active(geq(s(X), s(Y))) -> mark(geq(X, Y)) 24.93/8.74 active(div(0', s(Y))) -> mark(0') 24.93/8.74 active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')) 24.93/8.74 active(if(true, X, Y)) -> mark(X) 24.93/8.74 active(if(false, X, Y)) -> mark(Y) 24.93/8.74 active(s(X)) -> s(active(X)) 24.93/8.74 active(div(X1, X2)) -> div(active(X1), X2) 24.93/8.74 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.74 s(mark(X)) -> mark(s(X)) 24.93/8.74 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.74 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.74 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 24.93/8.74 proper(0') -> ok(0') 24.93/8.74 proper(s(X)) -> s(proper(X)) 24.93/8.74 proper(geq(X1, X2)) -> geq(proper(X1), proper(X2)) 24.93/8.74 proper(true) -> ok(true) 24.93/8.74 proper(false) -> ok(false) 24.93/8.74 proper(div(X1, X2)) -> div(proper(X1), proper(X2)) 24.93/8.74 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.74 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.74 s(ok(X)) -> ok(s(X)) 24.93/8.74 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.74 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.74 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.74 top(mark(X)) -> top(proper(X)) 24.93/8.74 top(ok(X)) -> top(active(X)) 24.93/8.74 24.93/8.74 Types: 24.93/8.74 active :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 0' :: 0':mark:true:false:ok 24.93/8.74 mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 s :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 true :: 0':mark:true:false:ok 24.93/8.74 false :: 0':mark:true:false:ok 24.93/8.74 div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 top :: 0':mark:true:false:ok -> top 24.93/8.74 hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok 24.93/8.74 hole_top2_0 :: top 24.93/8.74 gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (11) OrderProof (LOWER BOUND(ID)) 24.93/8.74 Heuristically decided to analyse the following defined symbols: 24.93/8.74 active, minus, geq, if, s, div, proper, top 24.93/8.74 24.93/8.74 They will be analysed ascendingly in the following order: 24.93/8.74 minus < active 24.93/8.74 geq < active 24.93/8.74 if < active 24.93/8.74 s < active 24.93/8.74 div < active 24.93/8.74 active < top 24.93/8.74 minus < proper 24.93/8.74 geq < proper 24.93/8.74 if < proper 24.93/8.74 s < proper 24.93/8.74 div < proper 24.93/8.74 proper < top 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (12) 24.93/8.74 Obligation: 24.93/8.74 TRS: 24.93/8.74 Rules: 24.93/8.74 active(minus(0', Y)) -> mark(0') 24.93/8.74 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 24.93/8.74 active(geq(X, 0')) -> mark(true) 24.93/8.74 active(geq(0', s(Y))) -> mark(false) 24.93/8.74 active(geq(s(X), s(Y))) -> mark(geq(X, Y)) 24.93/8.74 active(div(0', s(Y))) -> mark(0') 24.93/8.74 active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')) 24.93/8.74 active(if(true, X, Y)) -> mark(X) 24.93/8.74 active(if(false, X, Y)) -> mark(Y) 24.93/8.74 active(s(X)) -> s(active(X)) 24.93/8.74 active(div(X1, X2)) -> div(active(X1), X2) 24.93/8.74 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.74 s(mark(X)) -> mark(s(X)) 24.93/8.74 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.74 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.74 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 24.93/8.74 proper(0') -> ok(0') 24.93/8.74 proper(s(X)) -> s(proper(X)) 24.93/8.74 proper(geq(X1, X2)) -> geq(proper(X1), proper(X2)) 24.93/8.74 proper(true) -> ok(true) 24.93/8.74 proper(false) -> ok(false) 24.93/8.74 proper(div(X1, X2)) -> div(proper(X1), proper(X2)) 24.93/8.74 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.74 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.74 s(ok(X)) -> ok(s(X)) 24.93/8.74 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.74 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.74 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.74 top(mark(X)) -> top(proper(X)) 24.93/8.74 top(ok(X)) -> top(active(X)) 24.93/8.74 24.93/8.74 Types: 24.93/8.74 active :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 0' :: 0':mark:true:false:ok 24.93/8.74 mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 s :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 true :: 0':mark:true:false:ok 24.93/8.74 false :: 0':mark:true:false:ok 24.93/8.74 div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 top :: 0':mark:true:false:ok -> top 24.93/8.74 hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok 24.93/8.74 hole_top2_0 :: top 24.93/8.74 gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok 24.93/8.74 24.93/8.74 24.93/8.74 Generator Equations: 24.93/8.74 gen_0':mark:true:false:ok3_0(0) <=> 0' 24.93/8.74 gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x)) 24.93/8.74 24.93/8.74 24.93/8.74 The following defined symbols remain to be analysed: 24.93/8.74 minus, active, geq, if, s, div, proper, top 24.93/8.74 24.93/8.74 They will be analysed ascendingly in the following order: 24.93/8.74 minus < active 24.93/8.74 geq < active 24.93/8.74 if < active 24.93/8.74 s < active 24.93/8.74 div < active 24.93/8.74 active < top 24.93/8.74 minus < proper 24.93/8.74 geq < proper 24.93/8.74 if < proper 24.93/8.74 s < proper 24.93/8.74 div < proper 24.93/8.74 proper < top 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (13) RewriteLemmaProof (LOWER BOUND(ID)) 24.93/8.74 Proved the following rewrite lemma: 24.93/8.74 if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n15_0) 24.93/8.74 24.93/8.74 Induction Base: 24.93/8.74 if(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) 24.93/8.74 24.93/8.74 Induction Step: 24.93/8.74 if(gen_0':mark:true:false:ok3_0(+(1, +(n15_0, 1))), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) ->_R^Omega(1) 24.93/8.74 mark(if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c))) ->_IH 24.93/8.74 mark(*4_0) 24.93/8.74 24.93/8.74 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (14) 24.93/8.74 Complex Obligation (BEST) 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (15) 24.93/8.74 Obligation: 24.93/8.74 Proved the lower bound n^1 for the following obligation: 24.93/8.74 24.93/8.74 TRS: 24.93/8.74 Rules: 24.93/8.74 active(minus(0', Y)) -> mark(0') 24.93/8.74 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 24.93/8.74 active(geq(X, 0')) -> mark(true) 24.93/8.74 active(geq(0', s(Y))) -> mark(false) 24.93/8.74 active(geq(s(X), s(Y))) -> mark(geq(X, Y)) 24.93/8.74 active(div(0', s(Y))) -> mark(0') 24.93/8.74 active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')) 24.93/8.74 active(if(true, X, Y)) -> mark(X) 24.93/8.74 active(if(false, X, Y)) -> mark(Y) 24.93/8.74 active(s(X)) -> s(active(X)) 24.93/8.74 active(div(X1, X2)) -> div(active(X1), X2) 24.93/8.74 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.74 s(mark(X)) -> mark(s(X)) 24.93/8.74 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.74 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.74 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 24.93/8.74 proper(0') -> ok(0') 24.93/8.74 proper(s(X)) -> s(proper(X)) 24.93/8.74 proper(geq(X1, X2)) -> geq(proper(X1), proper(X2)) 24.93/8.74 proper(true) -> ok(true) 24.93/8.74 proper(false) -> ok(false) 24.93/8.74 proper(div(X1, X2)) -> div(proper(X1), proper(X2)) 24.93/8.74 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.74 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.74 s(ok(X)) -> ok(s(X)) 24.93/8.74 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.74 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.74 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.74 top(mark(X)) -> top(proper(X)) 24.93/8.74 top(ok(X)) -> top(active(X)) 24.93/8.74 24.93/8.74 Types: 24.93/8.74 active :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 0' :: 0':mark:true:false:ok 24.93/8.74 mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 s :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 true :: 0':mark:true:false:ok 24.93/8.74 false :: 0':mark:true:false:ok 24.93/8.74 div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 top :: 0':mark:true:false:ok -> top 24.93/8.74 hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok 24.93/8.74 hole_top2_0 :: top 24.93/8.74 gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok 24.93/8.74 24.93/8.74 24.93/8.74 Generator Equations: 24.93/8.74 gen_0':mark:true:false:ok3_0(0) <=> 0' 24.93/8.74 gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x)) 24.93/8.74 24.93/8.74 24.93/8.74 The following defined symbols remain to be analysed: 24.93/8.74 if, active, s, div, proper, top 24.93/8.74 24.93/8.74 They will be analysed ascendingly in the following order: 24.93/8.74 if < active 24.93/8.74 s < active 24.93/8.74 div < active 24.93/8.74 active < top 24.93/8.74 if < proper 24.93/8.74 s < proper 24.93/8.74 div < proper 24.93/8.74 proper < top 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (16) LowerBoundPropagationProof (FINISHED) 24.93/8.74 Propagated lower bound. 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (17) 24.93/8.74 BOUNDS(n^1, INF) 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (18) 24.93/8.74 Obligation: 24.93/8.74 TRS: 24.93/8.74 Rules: 24.93/8.74 active(minus(0', Y)) -> mark(0') 24.93/8.74 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 24.93/8.74 active(geq(X, 0')) -> mark(true) 24.93/8.74 active(geq(0', s(Y))) -> mark(false) 24.93/8.74 active(geq(s(X), s(Y))) -> mark(geq(X, Y)) 24.93/8.74 active(div(0', s(Y))) -> mark(0') 24.93/8.74 active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')) 24.93/8.74 active(if(true, X, Y)) -> mark(X) 24.93/8.74 active(if(false, X, Y)) -> mark(Y) 24.93/8.74 active(s(X)) -> s(active(X)) 24.93/8.74 active(div(X1, X2)) -> div(active(X1), X2) 24.93/8.74 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.74 s(mark(X)) -> mark(s(X)) 24.93/8.74 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.74 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.74 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 24.93/8.74 proper(0') -> ok(0') 24.93/8.74 proper(s(X)) -> s(proper(X)) 24.93/8.74 proper(geq(X1, X2)) -> geq(proper(X1), proper(X2)) 24.93/8.74 proper(true) -> ok(true) 24.93/8.74 proper(false) -> ok(false) 24.93/8.74 proper(div(X1, X2)) -> div(proper(X1), proper(X2)) 24.93/8.74 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.74 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.74 s(ok(X)) -> ok(s(X)) 24.93/8.74 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.74 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.74 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.74 top(mark(X)) -> top(proper(X)) 24.93/8.74 top(ok(X)) -> top(active(X)) 24.93/8.74 24.93/8.74 Types: 24.93/8.74 active :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 0' :: 0':mark:true:false:ok 24.93/8.74 mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 s :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 true :: 0':mark:true:false:ok 24.93/8.74 false :: 0':mark:true:false:ok 24.93/8.74 div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 top :: 0':mark:true:false:ok -> top 24.93/8.74 hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok 24.93/8.74 hole_top2_0 :: top 24.93/8.74 gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok 24.93/8.74 24.93/8.74 24.93/8.74 Lemmas: 24.93/8.74 if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n15_0) 24.93/8.74 24.93/8.74 24.93/8.74 Generator Equations: 24.93/8.74 gen_0':mark:true:false:ok3_0(0) <=> 0' 24.93/8.74 gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x)) 24.93/8.74 24.93/8.74 24.93/8.74 The following defined symbols remain to be analysed: 24.93/8.74 s, active, div, proper, top 24.93/8.74 24.93/8.74 They will be analysed ascendingly in the following order: 24.93/8.74 s < active 24.93/8.74 div < active 24.93/8.74 active < top 24.93/8.74 s < proper 24.93/8.74 div < proper 24.93/8.74 proper < top 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (19) RewriteLemmaProof (LOWER BOUND(ID)) 24.93/8.74 Proved the following rewrite lemma: 24.93/8.74 s(gen_0':mark:true:false:ok3_0(+(1, n1368_0))) -> *4_0, rt in Omega(n1368_0) 24.93/8.74 24.93/8.74 Induction Base: 24.93/8.74 s(gen_0':mark:true:false:ok3_0(+(1, 0))) 24.93/8.74 24.93/8.74 Induction Step: 24.93/8.74 s(gen_0':mark:true:false:ok3_0(+(1, +(n1368_0, 1)))) ->_R^Omega(1) 24.93/8.74 mark(s(gen_0':mark:true:false:ok3_0(+(1, n1368_0)))) ->_IH 24.93/8.74 mark(*4_0) 24.93/8.74 24.93/8.74 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (20) 24.93/8.74 Obligation: 24.93/8.74 TRS: 24.93/8.74 Rules: 24.93/8.74 active(minus(0', Y)) -> mark(0') 24.93/8.74 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 24.93/8.74 active(geq(X, 0')) -> mark(true) 24.93/8.74 active(geq(0', s(Y))) -> mark(false) 24.93/8.74 active(geq(s(X), s(Y))) -> mark(geq(X, Y)) 24.93/8.74 active(div(0', s(Y))) -> mark(0') 24.93/8.74 active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')) 24.93/8.74 active(if(true, X, Y)) -> mark(X) 24.93/8.74 active(if(false, X, Y)) -> mark(Y) 24.93/8.74 active(s(X)) -> s(active(X)) 24.93/8.74 active(div(X1, X2)) -> div(active(X1), X2) 24.93/8.74 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.74 s(mark(X)) -> mark(s(X)) 24.93/8.74 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.74 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.74 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 24.93/8.74 proper(0') -> ok(0') 24.93/8.74 proper(s(X)) -> s(proper(X)) 24.93/8.74 proper(geq(X1, X2)) -> geq(proper(X1), proper(X2)) 24.93/8.74 proper(true) -> ok(true) 24.93/8.74 proper(false) -> ok(false) 24.93/8.74 proper(div(X1, X2)) -> div(proper(X1), proper(X2)) 24.93/8.74 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.74 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.74 s(ok(X)) -> ok(s(X)) 24.93/8.74 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.74 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.74 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.74 top(mark(X)) -> top(proper(X)) 24.93/8.74 top(ok(X)) -> top(active(X)) 24.93/8.74 24.93/8.74 Types: 24.93/8.74 active :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 0' :: 0':mark:true:false:ok 24.93/8.74 mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 s :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 true :: 0':mark:true:false:ok 24.93/8.74 false :: 0':mark:true:false:ok 24.93/8.74 div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.74 top :: 0':mark:true:false:ok -> top 24.93/8.74 hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok 24.93/8.74 hole_top2_0 :: top 24.93/8.74 gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok 24.93/8.74 24.93/8.74 24.93/8.74 Lemmas: 24.93/8.74 if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n15_0) 24.93/8.74 s(gen_0':mark:true:false:ok3_0(+(1, n1368_0))) -> *4_0, rt in Omega(n1368_0) 24.93/8.74 24.93/8.74 24.93/8.74 Generator Equations: 24.93/8.74 gen_0':mark:true:false:ok3_0(0) <=> 0' 24.93/8.74 gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x)) 24.93/8.74 24.93/8.74 24.93/8.74 The following defined symbols remain to be analysed: 24.93/8.74 div, active, proper, top 24.93/8.74 24.93/8.74 They will be analysed ascendingly in the following order: 24.93/8.74 div < active 24.93/8.74 active < top 24.93/8.74 div < proper 24.93/8.74 proper < top 24.93/8.74 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (21) RewriteLemmaProof (LOWER BOUND(ID)) 24.93/8.74 Proved the following rewrite lemma: 24.93/8.74 div(gen_0':mark:true:false:ok3_0(+(1, n1924_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n1924_0) 24.93/8.74 24.93/8.74 Induction Base: 24.93/8.74 div(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b)) 24.93/8.74 24.93/8.74 Induction Step: 24.93/8.74 div(gen_0':mark:true:false:ok3_0(+(1, +(n1924_0, 1))), gen_0':mark:true:false:ok3_0(b)) ->_R^Omega(1) 24.93/8.74 mark(div(gen_0':mark:true:false:ok3_0(+(1, n1924_0)), gen_0':mark:true:false:ok3_0(b))) ->_IH 24.93/8.74 mark(*4_0) 24.93/8.74 24.93/8.74 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.93/8.74 ---------------------------------------- 24.93/8.74 24.93/8.74 (22) 24.93/8.74 Obligation: 24.93/8.75 TRS: 24.93/8.75 Rules: 24.93/8.75 active(minus(0', Y)) -> mark(0') 24.93/8.75 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 24.93/8.75 active(geq(X, 0')) -> mark(true) 24.93/8.75 active(geq(0', s(Y))) -> mark(false) 24.93/8.75 active(geq(s(X), s(Y))) -> mark(geq(X, Y)) 24.93/8.75 active(div(0', s(Y))) -> mark(0') 24.93/8.75 active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')) 24.93/8.75 active(if(true, X, Y)) -> mark(X) 24.93/8.75 active(if(false, X, Y)) -> mark(Y) 24.93/8.75 active(s(X)) -> s(active(X)) 24.93/8.75 active(div(X1, X2)) -> div(active(X1), X2) 24.93/8.75 active(if(X1, X2, X3)) -> if(active(X1), X2, X3) 24.93/8.75 s(mark(X)) -> mark(s(X)) 24.93/8.75 div(mark(X1), X2) -> mark(div(X1, X2)) 24.93/8.75 if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) 24.93/8.75 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 24.93/8.75 proper(0') -> ok(0') 24.93/8.75 proper(s(X)) -> s(proper(X)) 24.93/8.75 proper(geq(X1, X2)) -> geq(proper(X1), proper(X2)) 24.93/8.75 proper(true) -> ok(true) 24.93/8.75 proper(false) -> ok(false) 24.93/8.75 proper(div(X1, X2)) -> div(proper(X1), proper(X2)) 24.93/8.75 proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) 24.93/8.75 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 24.93/8.75 s(ok(X)) -> ok(s(X)) 24.93/8.75 geq(ok(X1), ok(X2)) -> ok(geq(X1, X2)) 24.93/8.75 div(ok(X1), ok(X2)) -> ok(div(X1, X2)) 24.93/8.75 if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 24.93/8.75 top(mark(X)) -> top(proper(X)) 24.93/8.75 top(ok(X)) -> top(active(X)) 24.93/8.75 24.93/8.75 Types: 24.93/8.75 active :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.75 minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.75 0' :: 0':mark:true:false:ok 24.93/8.75 mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.75 s :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.75 geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.75 true :: 0':mark:true:false:ok 24.93/8.75 false :: 0':mark:true:false:ok 24.93/8.75 div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.75 if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.75 proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.75 ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok 24.93/8.75 top :: 0':mark:true:false:ok -> top 24.93/8.75 hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok 24.93/8.75 hole_top2_0 :: top 24.93/8.75 gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok 24.93/8.75 24.93/8.75 24.93/8.75 Lemmas: 24.93/8.75 if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n15_0) 24.93/8.75 s(gen_0':mark:true:false:ok3_0(+(1, n1368_0))) -> *4_0, rt in Omega(n1368_0) 24.93/8.75 div(gen_0':mark:true:false:ok3_0(+(1, n1924_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n1924_0) 24.93/8.75 24.93/8.75 24.93/8.75 Generator Equations: 24.93/8.75 gen_0':mark:true:false:ok3_0(0) <=> 0' 24.93/8.75 gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x)) 24.93/8.75 24.93/8.75 24.93/8.75 The following defined symbols remain to be analysed: 24.93/8.75 active, proper, top 24.93/8.75 24.93/8.75 They will be analysed ascendingly in the following order: 24.93/8.75 active < top 24.93/8.75 proper < top 24.93/8.78 EOF