30.17/11.41 WORST_CASE(Omega(n^1), O(n^1)) 30.17/11.42 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 30.17/11.42 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.17/11.42 30.17/11.42 30.17/11.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 30.17/11.42 30.17/11.42 (0) CpxTRS 30.17/11.42 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 10 ms] 30.17/11.42 (2) CpxTRS 30.17/11.42 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 30.17/11.42 (4) CpxTRS 30.17/11.42 (5) CpxTrsMatchBoundsTAProof [FINISHED, 82 ms] 30.17/11.42 (6) BOUNDS(1, n^1) 30.17/11.42 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 30.17/11.42 (8) CpxTRS 30.17/11.42 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 30.17/11.42 (10) typed CpxTrs 30.17/11.42 (11) OrderProof [LOWER BOUND(ID), 0 ms] 30.17/11.42 (12) typed CpxTrs 30.17/11.42 (13) RewriteLemmaProof [LOWER BOUND(ID), 472 ms] 30.17/11.42 (14) BEST 30.17/11.42 (15) proven lower bound 30.17/11.42 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 30.17/11.42 (17) BOUNDS(n^1, INF) 30.17/11.42 (18) typed CpxTrs 30.17/11.42 (19) RewriteLemmaProof [LOWER BOUND(ID), 93 ms] 30.17/11.42 (20) typed CpxTrs 30.17/11.42 (21) RewriteLemmaProof [LOWER BOUND(ID), 74 ms] 30.17/11.42 (22) typed CpxTrs 30.17/11.42 (23) RewriteLemmaProof [LOWER BOUND(ID), 109 ms] 30.17/11.42 (24) typed CpxTrs 30.17/11.42 (25) RewriteLemmaProof [LOWER BOUND(ID), 90 ms] 30.17/11.42 (26) typed CpxTrs 30.17/11.42 30.17/11.42 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (0) 30.17/11.42 Obligation: 30.17/11.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 30.17/11.42 30.17/11.42 30.17/11.42 The TRS R consists of the following rules: 30.17/11.42 30.17/11.42 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.42 active(first(0, Z)) -> mark(nil) 30.17/11.42 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.42 active(sel(0, cons(X, Z))) -> mark(X) 30.17/11.42 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.42 active(from(X)) -> from(active(X)) 30.17/11.42 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.42 active(s(X)) -> s(active(X)) 30.17/11.42 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.42 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.42 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.42 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.42 from(mark(X)) -> mark(from(X)) 30.17/11.42 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.42 s(mark(X)) -> mark(s(X)) 30.17/11.42 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.42 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.42 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.42 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.42 proper(from(X)) -> from(proper(X)) 30.17/11.42 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.42 proper(s(X)) -> s(proper(X)) 30.17/11.42 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.42 proper(0) -> ok(0) 30.17/11.42 proper(nil) -> ok(nil) 30.17/11.42 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.42 from(ok(X)) -> ok(from(X)) 30.17/11.42 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.42 s(ok(X)) -> ok(s(X)) 30.17/11.42 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.42 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.42 top(mark(X)) -> top(proper(X)) 30.17/11.42 top(ok(X)) -> top(active(X)) 30.17/11.42 30.17/11.42 S is empty. 30.17/11.42 Rewrite Strategy: FULL 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 30.17/11.42 The following defined symbols can occur below the 0th argument of top: proper, active 30.17/11.42 The following defined symbols can occur below the 0th argument of proper: proper, active 30.17/11.42 The following defined symbols can occur below the 0th argument of active: proper, active 30.17/11.42 30.17/11.42 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 30.17/11.42 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.42 active(first(0, Z)) -> mark(nil) 30.17/11.42 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.42 active(sel(0, cons(X, Z))) -> mark(X) 30.17/11.42 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.42 active(from(X)) -> from(active(X)) 30.17/11.42 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.42 active(s(X)) -> s(active(X)) 30.17/11.42 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.42 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.42 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.42 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.42 proper(from(X)) -> from(proper(X)) 30.17/11.42 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.42 proper(s(X)) -> s(proper(X)) 30.17/11.42 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.42 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.42 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (2) 30.17/11.42 Obligation: 30.17/11.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 30.17/11.42 30.17/11.42 30.17/11.42 The TRS R consists of the following rules: 30.17/11.42 30.17/11.42 from(mark(X)) -> mark(from(X)) 30.17/11.42 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.42 s(mark(X)) -> mark(s(X)) 30.17/11.42 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.42 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.42 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.42 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.42 proper(0) -> ok(0) 30.17/11.42 proper(nil) -> ok(nil) 30.17/11.42 from(ok(X)) -> ok(from(X)) 30.17/11.42 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.42 s(ok(X)) -> ok(s(X)) 30.17/11.42 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.42 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.42 top(mark(X)) -> top(proper(X)) 30.17/11.42 top(ok(X)) -> top(active(X)) 30.17/11.42 30.17/11.42 S is empty. 30.17/11.42 Rewrite Strategy: FULL 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 30.17/11.42 transformed relative TRS to TRS 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (4) 30.17/11.42 Obligation: 30.17/11.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 30.17/11.42 30.17/11.42 30.17/11.42 The TRS R consists of the following rules: 30.17/11.42 30.17/11.42 from(mark(X)) -> mark(from(X)) 30.17/11.42 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.42 s(mark(X)) -> mark(s(X)) 30.17/11.42 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.42 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.42 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.42 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.42 proper(0) -> ok(0) 30.17/11.42 proper(nil) -> ok(nil) 30.17/11.42 from(ok(X)) -> ok(from(X)) 30.17/11.42 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.42 s(ok(X)) -> ok(s(X)) 30.17/11.42 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.42 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.42 top(mark(X)) -> top(proper(X)) 30.17/11.42 top(ok(X)) -> top(active(X)) 30.17/11.42 30.17/11.42 S is empty. 30.17/11.42 Rewrite Strategy: FULL 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (5) CpxTrsMatchBoundsTAProof (FINISHED) 30.17/11.42 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. 30.17/11.42 30.17/11.42 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 30.17/11.42 final states : [1, 2, 3, 4, 5, 6, 7] 30.17/11.42 transitions: 30.17/11.42 mark0(0) -> 0 30.17/11.42 00() -> 0 30.17/11.42 ok0(0) -> 0 30.17/11.42 nil0() -> 0 30.17/11.42 active0(0) -> 0 30.17/11.42 from0(0) -> 1 30.17/11.42 cons0(0, 0) -> 2 30.17/11.42 s0(0) -> 3 30.17/11.42 first0(0, 0) -> 4 30.17/11.42 sel0(0, 0) -> 5 30.17/11.42 proper0(0) -> 6 30.17/11.42 top0(0) -> 7 30.17/11.42 from1(0) -> 8 30.17/11.42 mark1(8) -> 1 30.17/11.42 cons1(0, 0) -> 9 30.17/11.42 mark1(9) -> 2 30.17/11.42 s1(0) -> 10 30.17/11.42 mark1(10) -> 3 30.17/11.42 first1(0, 0) -> 11 30.17/11.42 mark1(11) -> 4 30.17/11.42 sel1(0, 0) -> 12 30.17/11.42 mark1(12) -> 5 30.17/11.42 01() -> 13 30.17/11.42 ok1(13) -> 6 30.17/11.42 nil1() -> 14 30.17/11.42 ok1(14) -> 6 30.17/11.42 from1(0) -> 15 30.17/11.42 ok1(15) -> 1 30.17/11.42 cons1(0, 0) -> 16 30.17/11.42 ok1(16) -> 2 30.17/11.42 s1(0) -> 17 30.17/11.42 ok1(17) -> 3 30.17/11.42 first1(0, 0) -> 18 30.17/11.42 ok1(18) -> 4 30.17/11.42 sel1(0, 0) -> 19 30.17/11.42 ok1(19) -> 5 30.17/11.42 proper1(0) -> 20 30.17/11.42 top1(20) -> 7 30.17/11.42 active1(0) -> 21 30.17/11.42 top1(21) -> 7 30.17/11.42 mark1(8) -> 8 30.17/11.42 mark1(8) -> 15 30.17/11.42 mark1(9) -> 9 30.17/11.42 mark1(9) -> 16 30.17/11.42 mark1(10) -> 10 30.17/11.42 mark1(10) -> 17 30.17/11.42 mark1(11) -> 11 30.17/11.42 mark1(11) -> 18 30.17/11.42 mark1(12) -> 12 30.17/11.42 mark1(12) -> 19 30.17/11.42 ok1(13) -> 20 30.17/11.42 ok1(14) -> 20 30.17/11.42 ok1(15) -> 8 30.17/11.42 ok1(15) -> 15 30.17/11.42 ok1(16) -> 9 30.17/11.42 ok1(16) -> 16 30.17/11.42 ok1(17) -> 10 30.17/11.42 ok1(17) -> 17 30.17/11.42 ok1(18) -> 11 30.17/11.42 ok1(18) -> 18 30.17/11.42 ok1(19) -> 12 30.17/11.42 ok1(19) -> 19 30.17/11.42 active2(13) -> 22 30.17/11.42 top2(22) -> 7 30.17/11.42 active2(14) -> 22 30.17/11.42 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (6) 30.17/11.42 BOUNDS(1, n^1) 30.17/11.42 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 30.17/11.42 Renamed function symbols to avoid clashes with predefined symbol. 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (8) 30.17/11.42 Obligation: 30.17/11.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 30.17/11.42 30.17/11.42 30.17/11.42 The TRS R consists of the following rules: 30.17/11.42 30.17/11.42 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.42 active(first(0', Z)) -> mark(nil) 30.17/11.42 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.42 active(sel(0', cons(X, Z))) -> mark(X) 30.17/11.42 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.42 active(from(X)) -> from(active(X)) 30.17/11.42 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.42 active(s(X)) -> s(active(X)) 30.17/11.42 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.42 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.42 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.42 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.42 from(mark(X)) -> mark(from(X)) 30.17/11.42 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.42 s(mark(X)) -> mark(s(X)) 30.17/11.42 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.42 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.42 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.42 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.42 proper(from(X)) -> from(proper(X)) 30.17/11.42 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.42 proper(s(X)) -> s(proper(X)) 30.17/11.42 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.42 proper(0') -> ok(0') 30.17/11.42 proper(nil) -> ok(nil) 30.17/11.42 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.42 from(ok(X)) -> ok(from(X)) 30.17/11.42 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.42 s(ok(X)) -> ok(s(X)) 30.17/11.42 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.42 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.42 top(mark(X)) -> top(proper(X)) 30.17/11.42 top(ok(X)) -> top(active(X)) 30.17/11.42 30.17/11.42 S is empty. 30.17/11.42 Rewrite Strategy: FULL 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 30.17/11.42 Infered types. 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (10) 30.17/11.42 Obligation: 30.17/11.42 TRS: 30.17/11.42 Rules: 30.17/11.42 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.42 active(first(0', Z)) -> mark(nil) 30.17/11.42 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.42 active(sel(0', cons(X, Z))) -> mark(X) 30.17/11.42 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.42 active(from(X)) -> from(active(X)) 30.17/11.42 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.42 active(s(X)) -> s(active(X)) 30.17/11.42 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.42 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.42 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.42 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.42 from(mark(X)) -> mark(from(X)) 30.17/11.42 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.42 s(mark(X)) -> mark(s(X)) 30.17/11.42 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.42 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.42 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.42 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.42 proper(from(X)) -> from(proper(X)) 30.17/11.42 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.42 proper(s(X)) -> s(proper(X)) 30.17/11.42 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.42 proper(0') -> ok(0') 30.17/11.42 proper(nil) -> ok(nil) 30.17/11.42 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.42 from(ok(X)) -> ok(from(X)) 30.17/11.42 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.42 s(ok(X)) -> ok(s(X)) 30.17/11.42 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.42 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.42 top(mark(X)) -> top(proper(X)) 30.17/11.42 top(ok(X)) -> top(active(X)) 30.17/11.42 30.17/11.42 Types: 30.17/11.42 active :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.42 from :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.42 mark :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.42 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.42 s :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.42 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.42 0' :: mark:0':nil:ok 30.17/11.42 nil :: mark:0':nil:ok 30.17/11.42 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.42 proper :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.42 ok :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.42 top :: mark:0':nil:ok -> top 30.17/11.42 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 30.17/11.42 hole_top2_0 :: top 30.17/11.42 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 30.17/11.42 30.17/11.42 ---------------------------------------- 30.17/11.42 30.17/11.42 (11) OrderProof (LOWER BOUND(ID)) 30.17/11.42 Heuristically decided to analyse the following defined symbols: 30.17/11.42 active, cons, from, s, first, sel, proper, top 30.17/11.42 30.17/11.42 They will be analysed ascendingly in the following order: 30.17/11.43 cons < active 30.17/11.43 from < active 30.17/11.43 s < active 30.17/11.43 first < active 30.17/11.43 sel < active 30.17/11.43 active < top 30.17/11.43 cons < proper 30.17/11.43 from < proper 30.17/11.43 s < proper 30.17/11.43 first < proper 30.17/11.43 sel < proper 30.17/11.43 proper < top 30.17/11.43 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (12) 30.17/11.43 Obligation: 30.17/11.43 TRS: 30.17/11.43 Rules: 30.17/11.43 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.43 active(first(0', Z)) -> mark(nil) 30.17/11.43 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.43 active(sel(0', cons(X, Z))) -> mark(X) 30.17/11.43 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.43 active(from(X)) -> from(active(X)) 30.17/11.43 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.43 active(s(X)) -> s(active(X)) 30.17/11.43 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.43 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.43 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.43 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.43 from(mark(X)) -> mark(from(X)) 30.17/11.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.43 s(mark(X)) -> mark(s(X)) 30.17/11.43 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.43 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.43 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.43 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.43 proper(from(X)) -> from(proper(X)) 30.17/11.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.43 proper(s(X)) -> s(proper(X)) 30.17/11.43 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.43 proper(0') -> ok(0') 30.17/11.43 proper(nil) -> ok(nil) 30.17/11.43 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.43 from(ok(X)) -> ok(from(X)) 30.17/11.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.43 s(ok(X)) -> ok(s(X)) 30.17/11.43 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.43 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.43 top(mark(X)) -> top(proper(X)) 30.17/11.43 top(ok(X)) -> top(active(X)) 30.17/11.43 30.17/11.43 Types: 30.17/11.43 active :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 from :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 mark :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 s :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 0' :: mark:0':nil:ok 30.17/11.43 nil :: mark:0':nil:ok 30.17/11.43 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 proper :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 ok :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 top :: mark:0':nil:ok -> top 30.17/11.43 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 30.17/11.43 hole_top2_0 :: top 30.17/11.43 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 30.17/11.43 30.17/11.43 30.17/11.43 Generator Equations: 30.17/11.43 gen_mark:0':nil:ok3_0(0) <=> 0' 30.17/11.43 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 30.17/11.43 30.17/11.43 30.17/11.43 The following defined symbols remain to be analysed: 30.17/11.43 cons, active, from, s, first, sel, proper, top 30.17/11.43 30.17/11.43 They will be analysed ascendingly in the following order: 30.17/11.43 cons < active 30.17/11.43 from < active 30.17/11.43 s < active 30.17/11.43 first < active 30.17/11.43 sel < active 30.17/11.43 active < top 30.17/11.43 cons < proper 30.17/11.43 from < proper 30.17/11.43 s < proper 30.17/11.43 first < proper 30.17/11.43 sel < proper 30.17/11.43 proper < top 30.17/11.43 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (13) RewriteLemmaProof (LOWER BOUND(ID)) 30.17/11.43 Proved the following rewrite lemma: 30.17/11.43 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 30.17/11.43 30.17/11.43 Induction Base: 30.17/11.43 cons(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b)) 30.17/11.43 30.17/11.43 Induction Step: 30.17/11.43 cons(gen_mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1) 30.17/11.43 mark(cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b))) ->_IH 30.17/11.43 mark(*4_0) 30.17/11.43 30.17/11.43 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (14) 30.17/11.43 Complex Obligation (BEST) 30.17/11.43 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (15) 30.17/11.43 Obligation: 30.17/11.43 Proved the lower bound n^1 for the following obligation: 30.17/11.43 30.17/11.43 TRS: 30.17/11.43 Rules: 30.17/11.43 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.43 active(first(0', Z)) -> mark(nil) 30.17/11.43 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.43 active(sel(0', cons(X, Z))) -> mark(X) 30.17/11.43 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.43 active(from(X)) -> from(active(X)) 30.17/11.43 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.43 active(s(X)) -> s(active(X)) 30.17/11.43 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.43 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.43 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.43 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.43 from(mark(X)) -> mark(from(X)) 30.17/11.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.43 s(mark(X)) -> mark(s(X)) 30.17/11.43 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.43 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.43 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.43 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.43 proper(from(X)) -> from(proper(X)) 30.17/11.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.43 proper(s(X)) -> s(proper(X)) 30.17/11.43 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.43 proper(0') -> ok(0') 30.17/11.43 proper(nil) -> ok(nil) 30.17/11.43 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.43 from(ok(X)) -> ok(from(X)) 30.17/11.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.43 s(ok(X)) -> ok(s(X)) 30.17/11.43 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.43 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.43 top(mark(X)) -> top(proper(X)) 30.17/11.43 top(ok(X)) -> top(active(X)) 30.17/11.43 30.17/11.43 Types: 30.17/11.43 active :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 from :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 mark :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 s :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 0' :: mark:0':nil:ok 30.17/11.43 nil :: mark:0':nil:ok 30.17/11.43 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 proper :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 ok :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 top :: mark:0':nil:ok -> top 30.17/11.43 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 30.17/11.43 hole_top2_0 :: top 30.17/11.43 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 30.17/11.43 30.17/11.43 30.17/11.43 Generator Equations: 30.17/11.43 gen_mark:0':nil:ok3_0(0) <=> 0' 30.17/11.43 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 30.17/11.43 30.17/11.43 30.17/11.43 The following defined symbols remain to be analysed: 30.17/11.43 cons, active, from, s, first, sel, proper, top 30.17/11.43 30.17/11.43 They will be analysed ascendingly in the following order: 30.17/11.43 cons < active 30.17/11.43 from < active 30.17/11.43 s < active 30.17/11.43 first < active 30.17/11.43 sel < active 30.17/11.43 active < top 30.17/11.43 cons < proper 30.17/11.43 from < proper 30.17/11.43 s < proper 30.17/11.43 first < proper 30.17/11.43 sel < proper 30.17/11.43 proper < top 30.17/11.43 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (16) LowerBoundPropagationProof (FINISHED) 30.17/11.43 Propagated lower bound. 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (17) 30.17/11.43 BOUNDS(n^1, INF) 30.17/11.43 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (18) 30.17/11.43 Obligation: 30.17/11.43 TRS: 30.17/11.43 Rules: 30.17/11.43 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.43 active(first(0', Z)) -> mark(nil) 30.17/11.43 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.43 active(sel(0', cons(X, Z))) -> mark(X) 30.17/11.43 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.43 active(from(X)) -> from(active(X)) 30.17/11.43 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.43 active(s(X)) -> s(active(X)) 30.17/11.43 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.43 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.43 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.43 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.43 from(mark(X)) -> mark(from(X)) 30.17/11.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.43 s(mark(X)) -> mark(s(X)) 30.17/11.43 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.43 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.43 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.43 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.43 proper(from(X)) -> from(proper(X)) 30.17/11.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.43 proper(s(X)) -> s(proper(X)) 30.17/11.43 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.43 proper(0') -> ok(0') 30.17/11.43 proper(nil) -> ok(nil) 30.17/11.43 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.43 from(ok(X)) -> ok(from(X)) 30.17/11.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.43 s(ok(X)) -> ok(s(X)) 30.17/11.43 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.43 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.43 top(mark(X)) -> top(proper(X)) 30.17/11.43 top(ok(X)) -> top(active(X)) 30.17/11.43 30.17/11.43 Types: 30.17/11.43 active :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 from :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 mark :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 s :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 0' :: mark:0':nil:ok 30.17/11.43 nil :: mark:0':nil:ok 30.17/11.43 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 proper :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 ok :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 top :: mark:0':nil:ok -> top 30.17/11.43 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 30.17/11.43 hole_top2_0 :: top 30.17/11.43 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 30.17/11.43 30.17/11.43 30.17/11.43 Lemmas: 30.17/11.43 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 30.17/11.43 30.17/11.43 30.17/11.43 Generator Equations: 30.17/11.43 gen_mark:0':nil:ok3_0(0) <=> 0' 30.17/11.43 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 30.17/11.43 30.17/11.43 30.17/11.43 The following defined symbols remain to be analysed: 30.17/11.43 from, active, s, first, sel, proper, top 30.17/11.43 30.17/11.43 They will be analysed ascendingly in the following order: 30.17/11.43 from < active 30.17/11.43 s < active 30.17/11.43 first < active 30.17/11.43 sel < active 30.17/11.43 active < top 30.17/11.43 from < proper 30.17/11.43 s < proper 30.17/11.43 first < proper 30.17/11.43 sel < proper 30.17/11.43 proper < top 30.17/11.43 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (19) RewriteLemmaProof (LOWER BOUND(ID)) 30.17/11.43 Proved the following rewrite lemma: 30.17/11.43 from(gen_mark:0':nil:ok3_0(+(1, n902_0))) -> *4_0, rt in Omega(n902_0) 30.17/11.43 30.17/11.43 Induction Base: 30.17/11.43 from(gen_mark:0':nil:ok3_0(+(1, 0))) 30.17/11.43 30.17/11.43 Induction Step: 30.17/11.43 from(gen_mark:0':nil:ok3_0(+(1, +(n902_0, 1)))) ->_R^Omega(1) 30.17/11.43 mark(from(gen_mark:0':nil:ok3_0(+(1, n902_0)))) ->_IH 30.17/11.43 mark(*4_0) 30.17/11.43 30.17/11.43 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (20) 30.17/11.43 Obligation: 30.17/11.43 TRS: 30.17/11.43 Rules: 30.17/11.43 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.43 active(first(0', Z)) -> mark(nil) 30.17/11.43 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.43 active(sel(0', cons(X, Z))) -> mark(X) 30.17/11.43 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.43 active(from(X)) -> from(active(X)) 30.17/11.43 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.43 active(s(X)) -> s(active(X)) 30.17/11.43 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.43 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.43 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.43 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.43 from(mark(X)) -> mark(from(X)) 30.17/11.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.43 s(mark(X)) -> mark(s(X)) 30.17/11.43 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.43 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.43 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.43 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.43 proper(from(X)) -> from(proper(X)) 30.17/11.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.43 proper(s(X)) -> s(proper(X)) 30.17/11.43 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.43 proper(0') -> ok(0') 30.17/11.43 proper(nil) -> ok(nil) 30.17/11.43 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.43 from(ok(X)) -> ok(from(X)) 30.17/11.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.43 s(ok(X)) -> ok(s(X)) 30.17/11.43 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.43 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.43 top(mark(X)) -> top(proper(X)) 30.17/11.43 top(ok(X)) -> top(active(X)) 30.17/11.43 30.17/11.43 Types: 30.17/11.43 active :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 from :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 mark :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 s :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 0' :: mark:0':nil:ok 30.17/11.43 nil :: mark:0':nil:ok 30.17/11.43 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 proper :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 ok :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 top :: mark:0':nil:ok -> top 30.17/11.43 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 30.17/11.43 hole_top2_0 :: top 30.17/11.43 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 30.17/11.43 30.17/11.43 30.17/11.43 Lemmas: 30.17/11.43 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 30.17/11.43 from(gen_mark:0':nil:ok3_0(+(1, n902_0))) -> *4_0, rt in Omega(n902_0) 30.17/11.43 30.17/11.43 30.17/11.43 Generator Equations: 30.17/11.43 gen_mark:0':nil:ok3_0(0) <=> 0' 30.17/11.43 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 30.17/11.43 30.17/11.43 30.17/11.43 The following defined symbols remain to be analysed: 30.17/11.43 s, active, first, sel, proper, top 30.17/11.43 30.17/11.43 They will be analysed ascendingly in the following order: 30.17/11.43 s < active 30.17/11.43 first < active 30.17/11.43 sel < active 30.17/11.43 active < top 30.17/11.43 s < proper 30.17/11.43 first < proper 30.17/11.43 sel < proper 30.17/11.43 proper < top 30.17/11.43 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (21) RewriteLemmaProof (LOWER BOUND(ID)) 30.17/11.43 Proved the following rewrite lemma: 30.17/11.43 s(gen_mark:0':nil:ok3_0(+(1, n1418_0))) -> *4_0, rt in Omega(n1418_0) 30.17/11.43 30.17/11.43 Induction Base: 30.17/11.43 s(gen_mark:0':nil:ok3_0(+(1, 0))) 30.17/11.43 30.17/11.43 Induction Step: 30.17/11.43 s(gen_mark:0':nil:ok3_0(+(1, +(n1418_0, 1)))) ->_R^Omega(1) 30.17/11.43 mark(s(gen_mark:0':nil:ok3_0(+(1, n1418_0)))) ->_IH 30.17/11.43 mark(*4_0) 30.17/11.43 30.17/11.43 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (22) 30.17/11.43 Obligation: 30.17/11.43 TRS: 30.17/11.43 Rules: 30.17/11.43 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.43 active(first(0', Z)) -> mark(nil) 30.17/11.43 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.43 active(sel(0', cons(X, Z))) -> mark(X) 30.17/11.43 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.43 active(from(X)) -> from(active(X)) 30.17/11.43 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.43 active(s(X)) -> s(active(X)) 30.17/11.43 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.43 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.43 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.43 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.43 from(mark(X)) -> mark(from(X)) 30.17/11.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.43 s(mark(X)) -> mark(s(X)) 30.17/11.43 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.43 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.43 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.43 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.43 proper(from(X)) -> from(proper(X)) 30.17/11.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.43 proper(s(X)) -> s(proper(X)) 30.17/11.43 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.43 proper(0') -> ok(0') 30.17/11.43 proper(nil) -> ok(nil) 30.17/11.43 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.43 from(ok(X)) -> ok(from(X)) 30.17/11.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.43 s(ok(X)) -> ok(s(X)) 30.17/11.43 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.43 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.43 top(mark(X)) -> top(proper(X)) 30.17/11.43 top(ok(X)) -> top(active(X)) 30.17/11.43 30.17/11.43 Types: 30.17/11.43 active :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 from :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 mark :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 s :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 0' :: mark:0':nil:ok 30.17/11.43 nil :: mark:0':nil:ok 30.17/11.43 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 proper :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 ok :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 top :: mark:0':nil:ok -> top 30.17/11.43 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 30.17/11.43 hole_top2_0 :: top 30.17/11.43 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 30.17/11.43 30.17/11.43 30.17/11.43 Lemmas: 30.17/11.43 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 30.17/11.43 from(gen_mark:0':nil:ok3_0(+(1, n902_0))) -> *4_0, rt in Omega(n902_0) 30.17/11.43 s(gen_mark:0':nil:ok3_0(+(1, n1418_0))) -> *4_0, rt in Omega(n1418_0) 30.17/11.43 30.17/11.43 30.17/11.43 Generator Equations: 30.17/11.43 gen_mark:0':nil:ok3_0(0) <=> 0' 30.17/11.43 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 30.17/11.43 30.17/11.43 30.17/11.43 The following defined symbols remain to be analysed: 30.17/11.43 first, active, sel, proper, top 30.17/11.43 30.17/11.43 They will be analysed ascendingly in the following order: 30.17/11.43 first < active 30.17/11.43 sel < active 30.17/11.43 active < top 30.17/11.43 first < proper 30.17/11.43 sel < proper 30.17/11.43 proper < top 30.17/11.43 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (23) RewriteLemmaProof (LOWER BOUND(ID)) 30.17/11.43 Proved the following rewrite lemma: 30.17/11.43 first(gen_mark:0':nil:ok3_0(+(1, n2035_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2035_0) 30.17/11.43 30.17/11.43 Induction Base: 30.17/11.43 first(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b)) 30.17/11.43 30.17/11.43 Induction Step: 30.17/11.43 first(gen_mark:0':nil:ok3_0(+(1, +(n2035_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1) 30.17/11.43 mark(first(gen_mark:0':nil:ok3_0(+(1, n2035_0)), gen_mark:0':nil:ok3_0(b))) ->_IH 30.17/11.43 mark(*4_0) 30.17/11.43 30.17/11.43 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (24) 30.17/11.43 Obligation: 30.17/11.43 TRS: 30.17/11.43 Rules: 30.17/11.43 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.43 active(first(0', Z)) -> mark(nil) 30.17/11.43 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.43 active(sel(0', cons(X, Z))) -> mark(X) 30.17/11.43 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.43 active(from(X)) -> from(active(X)) 30.17/11.43 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.43 active(s(X)) -> s(active(X)) 30.17/11.43 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.43 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.43 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.43 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.43 from(mark(X)) -> mark(from(X)) 30.17/11.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.43 s(mark(X)) -> mark(s(X)) 30.17/11.43 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.43 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.43 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.43 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.43 proper(from(X)) -> from(proper(X)) 30.17/11.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.43 proper(s(X)) -> s(proper(X)) 30.17/11.43 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.43 proper(0') -> ok(0') 30.17/11.43 proper(nil) -> ok(nil) 30.17/11.43 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.43 from(ok(X)) -> ok(from(X)) 30.17/11.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.43 s(ok(X)) -> ok(s(X)) 30.17/11.43 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.43 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.43 top(mark(X)) -> top(proper(X)) 30.17/11.43 top(ok(X)) -> top(active(X)) 30.17/11.43 30.17/11.43 Types: 30.17/11.43 active :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 from :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 mark :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 s :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 0' :: mark:0':nil:ok 30.17/11.43 nil :: mark:0':nil:ok 30.17/11.43 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 proper :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 ok :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 top :: mark:0':nil:ok -> top 30.17/11.43 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 30.17/11.43 hole_top2_0 :: top 30.17/11.43 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 30.17/11.43 30.17/11.43 30.17/11.43 Lemmas: 30.17/11.43 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 30.17/11.43 from(gen_mark:0':nil:ok3_0(+(1, n902_0))) -> *4_0, rt in Omega(n902_0) 30.17/11.43 s(gen_mark:0':nil:ok3_0(+(1, n1418_0))) -> *4_0, rt in Omega(n1418_0) 30.17/11.43 first(gen_mark:0':nil:ok3_0(+(1, n2035_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2035_0) 30.17/11.43 30.17/11.43 30.17/11.43 Generator Equations: 30.17/11.43 gen_mark:0':nil:ok3_0(0) <=> 0' 30.17/11.43 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 30.17/11.43 30.17/11.43 30.17/11.43 The following defined symbols remain to be analysed: 30.17/11.43 sel, active, proper, top 30.17/11.43 30.17/11.43 They will be analysed ascendingly in the following order: 30.17/11.43 sel < active 30.17/11.43 active < top 30.17/11.43 sel < proper 30.17/11.43 proper < top 30.17/11.43 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (25) RewriteLemmaProof (LOWER BOUND(ID)) 30.17/11.43 Proved the following rewrite lemma: 30.17/11.43 sel(gen_mark:0':nil:ok3_0(+(1, n3853_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3853_0) 30.17/11.43 30.17/11.43 Induction Base: 30.17/11.43 sel(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b)) 30.17/11.43 30.17/11.43 Induction Step: 30.17/11.43 sel(gen_mark:0':nil:ok3_0(+(1, +(n3853_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1) 30.17/11.43 mark(sel(gen_mark:0':nil:ok3_0(+(1, n3853_0)), gen_mark:0':nil:ok3_0(b))) ->_IH 30.17/11.43 mark(*4_0) 30.17/11.43 30.17/11.43 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 30.17/11.43 ---------------------------------------- 30.17/11.43 30.17/11.43 (26) 30.17/11.43 Obligation: 30.17/11.43 TRS: 30.17/11.43 Rules: 30.17/11.43 active(from(X)) -> mark(cons(X, from(s(X)))) 30.17/11.43 active(first(0', Z)) -> mark(nil) 30.17/11.43 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 30.17/11.43 active(sel(0', cons(X, Z))) -> mark(X) 30.17/11.43 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 30.17/11.43 active(from(X)) -> from(active(X)) 30.17/11.43 active(cons(X1, X2)) -> cons(active(X1), X2) 30.17/11.43 active(s(X)) -> s(active(X)) 30.17/11.43 active(first(X1, X2)) -> first(active(X1), X2) 30.17/11.43 active(first(X1, X2)) -> first(X1, active(X2)) 30.17/11.43 active(sel(X1, X2)) -> sel(active(X1), X2) 30.17/11.43 active(sel(X1, X2)) -> sel(X1, active(X2)) 30.17/11.43 from(mark(X)) -> mark(from(X)) 30.17/11.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 30.17/11.43 s(mark(X)) -> mark(s(X)) 30.17/11.43 first(mark(X1), X2) -> mark(first(X1, X2)) 30.17/11.43 first(X1, mark(X2)) -> mark(first(X1, X2)) 30.17/11.43 sel(mark(X1), X2) -> mark(sel(X1, X2)) 30.17/11.43 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 30.17/11.43 proper(from(X)) -> from(proper(X)) 30.17/11.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 30.17/11.43 proper(s(X)) -> s(proper(X)) 30.17/11.43 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 30.17/11.43 proper(0') -> ok(0') 30.17/11.43 proper(nil) -> ok(nil) 30.17/11.43 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 30.17/11.43 from(ok(X)) -> ok(from(X)) 30.17/11.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 30.17/11.43 s(ok(X)) -> ok(s(X)) 30.17/11.43 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 30.17/11.43 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 30.17/11.43 top(mark(X)) -> top(proper(X)) 30.17/11.43 top(ok(X)) -> top(active(X)) 30.17/11.43 30.17/11.43 Types: 30.17/11.43 active :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 from :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 mark :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 s :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 0' :: mark:0':nil:ok 30.17/11.43 nil :: mark:0':nil:ok 30.17/11.43 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 proper :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 ok :: mark:0':nil:ok -> mark:0':nil:ok 30.17/11.43 top :: mark:0':nil:ok -> top 30.17/11.43 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 30.17/11.43 hole_top2_0 :: top 30.17/11.43 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 30.17/11.43 30.17/11.43 30.17/11.43 Lemmas: 30.17/11.43 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 30.17/11.43 from(gen_mark:0':nil:ok3_0(+(1, n902_0))) -> *4_0, rt in Omega(n902_0) 30.17/11.43 s(gen_mark:0':nil:ok3_0(+(1, n1418_0))) -> *4_0, rt in Omega(n1418_0) 30.17/11.43 first(gen_mark:0':nil:ok3_0(+(1, n2035_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2035_0) 30.17/11.43 sel(gen_mark:0':nil:ok3_0(+(1, n3853_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3853_0) 30.17/11.43 30.17/11.43 30.17/11.43 Generator Equations: 30.17/11.43 gen_mark:0':nil:ok3_0(0) <=> 0' 30.17/11.43 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 30.17/11.43 30.17/11.43 30.17/11.43 The following defined symbols remain to be analysed: 30.17/11.43 active, proper, top 30.17/11.43 30.17/11.43 They will be analysed ascendingly in the following order: 30.17/11.43 active < top 30.17/11.43 proper < top 30.42/11.48 EOF