28.07/8.14 WORST_CASE(Omega(n^1), O(n^1)) 28.07/8.15 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 28.07/8.15 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.07/8.15 28.07/8.15 28.07/8.15 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.07/8.15 28.07/8.15 (0) CpxTRS 28.07/8.15 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 28.07/8.15 (2) CpxTRS 28.07/8.15 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 28.07/8.15 (4) CpxTRS 28.07/8.15 (5) CpxTrsMatchBoundsTAProof [FINISHED, 102 ms] 28.07/8.15 (6) BOUNDS(1, n^1) 28.07/8.15 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 28.07/8.15 (8) CpxTRS 28.07/8.15 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 28.07/8.15 (10) typed CpxTrs 28.07/8.15 (11) OrderProof [LOWER BOUND(ID), 0 ms] 28.07/8.15 (12) typed CpxTrs 28.07/8.15 (13) RewriteLemmaProof [LOWER BOUND(ID), 484 ms] 28.07/8.15 (14) BEST 28.07/8.15 (15) proven lower bound 28.07/8.15 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 28.07/8.15 (17) BOUNDS(n^1, INF) 28.07/8.15 (18) typed CpxTrs 28.07/8.15 (19) RewriteLemmaProof [LOWER BOUND(ID), 131 ms] 28.07/8.15 (20) typed CpxTrs 28.07/8.15 (21) RewriteLemmaProof [LOWER BOUND(ID), 80 ms] 28.07/8.15 (22) typed CpxTrs 28.07/8.15 (23) RewriteLemmaProof [LOWER BOUND(ID), 94 ms] 28.07/8.15 (24) typed CpxTrs 28.07/8.15 (25) RewriteLemmaProof [LOWER BOUND(ID), 36 ms] 28.07/8.15 (26) typed CpxTrs 28.07/8.15 (27) RewriteLemmaProof [LOWER BOUND(ID), 92 ms] 28.07/8.15 (28) typed CpxTrs 28.07/8.15 (29) RewriteLemmaProof [LOWER BOUND(ID), 88 ms] 28.07/8.15 (30) typed CpxTrs 28.07/8.15 28.07/8.15 28.07/8.15 ---------------------------------------- 28.07/8.15 28.07/8.15 (0) 28.07/8.15 Obligation: 28.07/8.15 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 28.07/8.15 28.07/8.15 28.07/8.15 The TRS R consists of the following rules: 28.07/8.15 28.07/8.15 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.15 active(head(cons(X, XS))) -> mark(X) 28.07/8.15 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.15 active(take(0, XS)) -> mark(nil) 28.07/8.15 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.15 active(sel(0, cons(X, XS))) -> mark(X) 28.07/8.15 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.15 active(from(X)) -> from(active(X)) 28.07/8.15 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.15 active(s(X)) -> s(active(X)) 28.07/8.15 active(head(X)) -> head(active(X)) 28.07/8.15 active(2nd(X)) -> 2nd(active(X)) 28.07/8.15 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.15 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.15 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.15 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.15 from(mark(X)) -> mark(from(X)) 28.07/8.15 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.15 s(mark(X)) -> mark(s(X)) 28.07/8.15 head(mark(X)) -> mark(head(X)) 28.07/8.15 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.15 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.15 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.15 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.15 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.15 proper(from(X)) -> from(proper(X)) 28.07/8.15 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.15 proper(s(X)) -> s(proper(X)) 28.07/8.15 proper(head(X)) -> head(proper(X)) 28.07/8.15 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.15 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.15 proper(0) -> ok(0) 28.07/8.15 proper(nil) -> ok(nil) 28.07/8.15 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.15 from(ok(X)) -> ok(from(X)) 28.07/8.15 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.15 s(ok(X)) -> ok(s(X)) 28.07/8.15 head(ok(X)) -> ok(head(X)) 28.07/8.15 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.15 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.15 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.15 top(mark(X)) -> top(proper(X)) 28.07/8.15 top(ok(X)) -> top(active(X)) 28.07/8.15 28.07/8.15 S is empty. 28.07/8.15 Rewrite Strategy: FULL 28.07/8.15 ---------------------------------------- 28.07/8.15 28.07/8.15 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 28.07/8.15 The following defined symbols can occur below the 0th argument of top: proper, active 28.07/8.15 The following defined symbols can occur below the 0th argument of proper: proper, active 28.07/8.15 The following defined symbols can occur below the 0th argument of active: proper, active 28.07/8.15 28.07/8.15 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 28.07/8.15 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.15 active(head(cons(X, XS))) -> mark(X) 28.07/8.15 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.15 active(take(0, XS)) -> mark(nil) 28.07/8.15 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.15 active(sel(0, cons(X, XS))) -> mark(X) 28.07/8.15 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.15 active(from(X)) -> from(active(X)) 28.07/8.15 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.15 active(s(X)) -> s(active(X)) 28.07/8.15 active(head(X)) -> head(active(X)) 28.07/8.15 active(2nd(X)) -> 2nd(active(X)) 28.07/8.15 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.15 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.15 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.15 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.15 proper(from(X)) -> from(proper(X)) 28.07/8.15 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.15 proper(s(X)) -> s(proper(X)) 28.07/8.15 proper(head(X)) -> head(proper(X)) 28.07/8.15 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.15 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.15 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.15 28.07/8.15 ---------------------------------------- 28.07/8.15 28.07/8.15 (2) 28.07/8.15 Obligation: 28.07/8.15 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 28.07/8.15 28.07/8.15 28.07/8.15 The TRS R consists of the following rules: 28.07/8.15 28.07/8.15 from(mark(X)) -> mark(from(X)) 28.07/8.15 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.15 s(mark(X)) -> mark(s(X)) 28.07/8.15 head(mark(X)) -> mark(head(X)) 28.07/8.15 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.15 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.15 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.15 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.15 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.15 proper(0) -> ok(0) 28.07/8.15 proper(nil) -> ok(nil) 28.07/8.15 from(ok(X)) -> ok(from(X)) 28.07/8.15 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.15 s(ok(X)) -> ok(s(X)) 28.07/8.15 head(ok(X)) -> ok(head(X)) 28.07/8.15 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.15 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.15 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.15 top(mark(X)) -> top(proper(X)) 28.07/8.15 top(ok(X)) -> top(active(X)) 28.07/8.15 28.07/8.15 S is empty. 28.07/8.15 Rewrite Strategy: FULL 28.07/8.15 ---------------------------------------- 28.07/8.15 28.07/8.15 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 28.07/8.15 transformed relative TRS to TRS 28.07/8.15 ---------------------------------------- 28.07/8.15 28.07/8.15 (4) 28.07/8.15 Obligation: 28.07/8.15 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 28.07/8.15 28.07/8.15 28.07/8.15 The TRS R consists of the following rules: 28.07/8.15 28.07/8.15 from(mark(X)) -> mark(from(X)) 28.07/8.15 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.15 s(mark(X)) -> mark(s(X)) 28.07/8.15 head(mark(X)) -> mark(head(X)) 28.07/8.15 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.15 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.15 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.15 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.15 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.15 proper(0) -> ok(0) 28.07/8.15 proper(nil) -> ok(nil) 28.07/8.15 from(ok(X)) -> ok(from(X)) 28.07/8.15 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.15 s(ok(X)) -> ok(s(X)) 28.07/8.15 head(ok(X)) -> ok(head(X)) 28.07/8.15 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.15 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.15 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.15 top(mark(X)) -> top(proper(X)) 28.07/8.15 top(ok(X)) -> top(active(X)) 28.07/8.15 28.07/8.15 S is empty. 28.07/8.15 Rewrite Strategy: FULL 28.07/8.15 ---------------------------------------- 28.07/8.15 28.07/8.15 (5) CpxTrsMatchBoundsTAProof (FINISHED) 28.07/8.15 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. 28.07/8.15 28.07/8.15 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 28.07/8.15 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9] 28.07/8.15 transitions: 28.07/8.15 mark0(0) -> 0 28.07/8.15 00() -> 0 28.07/8.15 ok0(0) -> 0 28.07/8.15 nil0() -> 0 28.07/8.15 active0(0) -> 0 28.07/8.15 from0(0) -> 1 28.07/8.15 cons0(0, 0) -> 2 28.07/8.15 s0(0) -> 3 28.07/8.15 head0(0) -> 4 28.07/8.15 2nd0(0) -> 5 28.07/8.15 take0(0, 0) -> 6 28.07/8.15 sel0(0, 0) -> 7 28.07/8.15 proper0(0) -> 8 28.07/8.15 top0(0) -> 9 28.07/8.15 from1(0) -> 10 28.07/8.15 mark1(10) -> 1 28.07/8.15 cons1(0, 0) -> 11 28.07/8.15 mark1(11) -> 2 28.07/8.15 s1(0) -> 12 28.07/8.15 mark1(12) -> 3 28.07/8.15 head1(0) -> 13 28.07/8.15 mark1(13) -> 4 28.07/8.15 2nd1(0) -> 14 28.07/8.15 mark1(14) -> 5 28.07/8.15 take1(0, 0) -> 15 28.07/8.15 mark1(15) -> 6 28.07/8.15 sel1(0, 0) -> 16 28.07/8.15 mark1(16) -> 7 28.07/8.15 01() -> 17 28.07/8.15 ok1(17) -> 8 28.07/8.15 nil1() -> 18 28.07/8.15 ok1(18) -> 8 28.07/8.15 from1(0) -> 19 28.07/8.15 ok1(19) -> 1 28.07/8.15 cons1(0, 0) -> 20 28.07/8.15 ok1(20) -> 2 28.07/8.15 s1(0) -> 21 28.07/8.15 ok1(21) -> 3 28.07/8.15 head1(0) -> 22 28.07/8.15 ok1(22) -> 4 28.07/8.15 2nd1(0) -> 23 28.07/8.15 ok1(23) -> 5 28.07/8.15 take1(0, 0) -> 24 28.07/8.15 ok1(24) -> 6 28.07/8.15 sel1(0, 0) -> 25 28.07/8.15 ok1(25) -> 7 28.07/8.15 proper1(0) -> 26 28.07/8.15 top1(26) -> 9 28.07/8.15 active1(0) -> 27 28.07/8.15 top1(27) -> 9 28.07/8.15 mark1(10) -> 10 28.07/8.15 mark1(10) -> 19 28.07/8.15 mark1(11) -> 11 28.07/8.15 mark1(11) -> 20 28.07/8.15 mark1(12) -> 12 28.07/8.15 mark1(12) -> 21 28.07/8.15 mark1(13) -> 13 28.07/8.15 mark1(13) -> 22 28.07/8.15 mark1(14) -> 14 28.07/8.15 mark1(14) -> 23 28.07/8.15 mark1(15) -> 15 28.07/8.15 mark1(15) -> 24 28.07/8.15 mark1(16) -> 16 28.07/8.15 mark1(16) -> 25 28.07/8.15 ok1(17) -> 26 28.07/8.15 ok1(18) -> 26 28.07/8.15 ok1(19) -> 10 28.07/8.15 ok1(19) -> 19 28.07/8.15 ok1(20) -> 11 28.07/8.16 ok1(20) -> 20 28.07/8.16 ok1(21) -> 12 28.07/8.16 ok1(21) -> 21 28.07/8.16 ok1(22) -> 13 28.07/8.16 ok1(22) -> 22 28.07/8.16 ok1(23) -> 14 28.07/8.16 ok1(23) -> 23 28.07/8.16 ok1(24) -> 15 28.07/8.16 ok1(24) -> 24 28.07/8.16 ok1(25) -> 16 28.07/8.16 ok1(25) -> 25 28.07/8.16 active2(17) -> 28 28.07/8.16 top2(28) -> 9 28.07/8.16 active2(18) -> 28 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (6) 28.07/8.16 BOUNDS(1, n^1) 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 28.07/8.16 Renamed function symbols to avoid clashes with predefined symbol. 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (8) 28.07/8.16 Obligation: 28.07/8.16 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 28.07/8.16 28.07/8.16 28.07/8.16 The TRS R consists of the following rules: 28.07/8.16 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 S is empty. 28.07/8.16 Rewrite Strategy: FULL 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 28.07/8.16 Infered types. 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (10) 28.07/8.16 Obligation: 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (11) OrderProof (LOWER BOUND(ID)) 28.07/8.16 Heuristically decided to analyse the following defined symbols: 28.07/8.16 active, cons, from, s, head, take, sel, 2nd, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 cons < active 28.07/8.16 from < active 28.07/8.16 s < active 28.07/8.16 head < active 28.07/8.16 take < active 28.07/8.16 sel < active 28.07/8.16 2nd < active 28.07/8.16 active < top 28.07/8.16 cons < proper 28.07/8.16 from < proper 28.07/8.16 s < proper 28.07/8.16 head < proper 28.07/8.16 take < proper 28.07/8.16 sel < proper 28.07/8.16 2nd < proper 28.07/8.16 proper < top 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (12) 28.07/8.16 Obligation: 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 28.07/8.16 Generator Equations: 28.07/8.16 gen_mark:0':nil:ok3_0(0) <=> 0' 28.07/8.16 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 28.07/8.16 28.07/8.16 28.07/8.16 The following defined symbols remain to be analysed: 28.07/8.16 cons, active, from, s, head, take, sel, 2nd, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 cons < active 28.07/8.16 from < active 28.07/8.16 s < active 28.07/8.16 head < active 28.07/8.16 take < active 28.07/8.16 sel < active 28.07/8.16 2nd < active 28.07/8.16 active < top 28.07/8.16 cons < proper 28.07/8.16 from < proper 28.07/8.16 s < proper 28.07/8.16 head < proper 28.07/8.16 take < proper 28.07/8.16 sel < proper 28.07/8.16 2nd < proper 28.07/8.16 proper < top 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (13) RewriteLemmaProof (LOWER BOUND(ID)) 28.07/8.16 Proved the following rewrite lemma: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 28.07/8.16 28.07/8.16 Induction Base: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b)) 28.07/8.16 28.07/8.16 Induction Step: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1) 28.07/8.16 mark(cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b))) ->_IH 28.07/8.16 mark(*4_0) 28.07/8.16 28.07/8.16 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (14) 28.07/8.16 Complex Obligation (BEST) 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (15) 28.07/8.16 Obligation: 28.07/8.16 Proved the lower bound n^1 for the following obligation: 28.07/8.16 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 28.07/8.16 Generator Equations: 28.07/8.16 gen_mark:0':nil:ok3_0(0) <=> 0' 28.07/8.16 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 28.07/8.16 28.07/8.16 28.07/8.16 The following defined symbols remain to be analysed: 28.07/8.16 cons, active, from, s, head, take, sel, 2nd, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 cons < active 28.07/8.16 from < active 28.07/8.16 s < active 28.07/8.16 head < active 28.07/8.16 take < active 28.07/8.16 sel < active 28.07/8.16 2nd < active 28.07/8.16 active < top 28.07/8.16 cons < proper 28.07/8.16 from < proper 28.07/8.16 s < proper 28.07/8.16 head < proper 28.07/8.16 take < proper 28.07/8.16 sel < proper 28.07/8.16 2nd < proper 28.07/8.16 proper < top 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (16) LowerBoundPropagationProof (FINISHED) 28.07/8.16 Propagated lower bound. 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (17) 28.07/8.16 BOUNDS(n^1, INF) 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (18) 28.07/8.16 Obligation: 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 28.07/8.16 Lemmas: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 28.07/8.16 28.07/8.16 28.07/8.16 Generator Equations: 28.07/8.16 gen_mark:0':nil:ok3_0(0) <=> 0' 28.07/8.16 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 28.07/8.16 28.07/8.16 28.07/8.16 The following defined symbols remain to be analysed: 28.07/8.16 from, active, s, head, take, sel, 2nd, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 from < active 28.07/8.16 s < active 28.07/8.16 head < active 28.07/8.16 take < active 28.07/8.16 sel < active 28.07/8.16 2nd < active 28.07/8.16 active < top 28.07/8.16 from < proper 28.07/8.16 s < proper 28.07/8.16 head < proper 28.07/8.16 take < proper 28.07/8.16 sel < proper 28.07/8.16 2nd < proper 28.07/8.16 proper < top 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (19) RewriteLemmaProof (LOWER BOUND(ID)) 28.07/8.16 Proved the following rewrite lemma: 28.07/8.16 from(gen_mark:0':nil:ok3_0(+(1, n1042_0))) -> *4_0, rt in Omega(n1042_0) 28.07/8.16 28.07/8.16 Induction Base: 28.07/8.16 from(gen_mark:0':nil:ok3_0(+(1, 0))) 28.07/8.16 28.07/8.16 Induction Step: 28.07/8.16 from(gen_mark:0':nil:ok3_0(+(1, +(n1042_0, 1)))) ->_R^Omega(1) 28.07/8.16 mark(from(gen_mark:0':nil:ok3_0(+(1, n1042_0)))) ->_IH 28.07/8.16 mark(*4_0) 28.07/8.16 28.07/8.16 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (20) 28.07/8.16 Obligation: 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 28.07/8.16 Lemmas: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 28.07/8.16 from(gen_mark:0':nil:ok3_0(+(1, n1042_0))) -> *4_0, rt in Omega(n1042_0) 28.07/8.16 28.07/8.16 28.07/8.16 Generator Equations: 28.07/8.16 gen_mark:0':nil:ok3_0(0) <=> 0' 28.07/8.16 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 28.07/8.16 28.07/8.16 28.07/8.16 The following defined symbols remain to be analysed: 28.07/8.16 s, active, head, take, sel, 2nd, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 s < active 28.07/8.16 head < active 28.07/8.16 take < active 28.07/8.16 sel < active 28.07/8.16 2nd < active 28.07/8.16 active < top 28.07/8.16 s < proper 28.07/8.16 head < proper 28.07/8.16 take < proper 28.07/8.16 sel < proper 28.07/8.16 2nd < proper 28.07/8.16 proper < top 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (21) RewriteLemmaProof (LOWER BOUND(ID)) 28.07/8.16 Proved the following rewrite lemma: 28.07/8.16 s(gen_mark:0':nil:ok3_0(+(1, n1588_0))) -> *4_0, rt in Omega(n1588_0) 28.07/8.16 28.07/8.16 Induction Base: 28.07/8.16 s(gen_mark:0':nil:ok3_0(+(1, 0))) 28.07/8.16 28.07/8.16 Induction Step: 28.07/8.16 s(gen_mark:0':nil:ok3_0(+(1, +(n1588_0, 1)))) ->_R^Omega(1) 28.07/8.16 mark(s(gen_mark:0':nil:ok3_0(+(1, n1588_0)))) ->_IH 28.07/8.16 mark(*4_0) 28.07/8.16 28.07/8.16 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (22) 28.07/8.16 Obligation: 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 28.07/8.16 Lemmas: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 28.07/8.16 from(gen_mark:0':nil:ok3_0(+(1, n1042_0))) -> *4_0, rt in Omega(n1042_0) 28.07/8.16 s(gen_mark:0':nil:ok3_0(+(1, n1588_0))) -> *4_0, rt in Omega(n1588_0) 28.07/8.16 28.07/8.16 28.07/8.16 Generator Equations: 28.07/8.16 gen_mark:0':nil:ok3_0(0) <=> 0' 28.07/8.16 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 28.07/8.16 28.07/8.16 28.07/8.16 The following defined symbols remain to be analysed: 28.07/8.16 head, active, take, sel, 2nd, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 head < active 28.07/8.16 take < active 28.07/8.16 sel < active 28.07/8.16 2nd < active 28.07/8.16 active < top 28.07/8.16 head < proper 28.07/8.16 take < proper 28.07/8.16 sel < proper 28.07/8.16 2nd < proper 28.07/8.16 proper < top 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (23) RewriteLemmaProof (LOWER BOUND(ID)) 28.07/8.16 Proved the following rewrite lemma: 28.07/8.16 head(gen_mark:0':nil:ok3_0(+(1, n2235_0))) -> *4_0, rt in Omega(n2235_0) 28.07/8.16 28.07/8.16 Induction Base: 28.07/8.16 head(gen_mark:0':nil:ok3_0(+(1, 0))) 28.07/8.16 28.07/8.16 Induction Step: 28.07/8.16 head(gen_mark:0':nil:ok3_0(+(1, +(n2235_0, 1)))) ->_R^Omega(1) 28.07/8.16 mark(head(gen_mark:0':nil:ok3_0(+(1, n2235_0)))) ->_IH 28.07/8.16 mark(*4_0) 28.07/8.16 28.07/8.16 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (24) 28.07/8.16 Obligation: 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 28.07/8.16 Lemmas: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 28.07/8.16 from(gen_mark:0':nil:ok3_0(+(1, n1042_0))) -> *4_0, rt in Omega(n1042_0) 28.07/8.16 s(gen_mark:0':nil:ok3_0(+(1, n1588_0))) -> *4_0, rt in Omega(n1588_0) 28.07/8.16 head(gen_mark:0':nil:ok3_0(+(1, n2235_0))) -> *4_0, rt in Omega(n2235_0) 28.07/8.16 28.07/8.16 28.07/8.16 Generator Equations: 28.07/8.16 gen_mark:0':nil:ok3_0(0) <=> 0' 28.07/8.16 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 28.07/8.16 28.07/8.16 28.07/8.16 The following defined symbols remain to be analysed: 28.07/8.16 take, active, sel, 2nd, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 take < active 28.07/8.16 sel < active 28.07/8.16 2nd < active 28.07/8.16 active < top 28.07/8.16 take < proper 28.07/8.16 sel < proper 28.07/8.16 2nd < proper 28.07/8.16 proper < top 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (25) RewriteLemmaProof (LOWER BOUND(ID)) 28.07/8.16 Proved the following rewrite lemma: 28.07/8.16 take(gen_mark:0':nil:ok3_0(+(1, n2983_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2983_0) 28.07/8.16 28.07/8.16 Induction Base: 28.07/8.16 take(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b)) 28.07/8.16 28.07/8.16 Induction Step: 28.07/8.16 take(gen_mark:0':nil:ok3_0(+(1, +(n2983_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1) 28.07/8.16 mark(take(gen_mark:0':nil:ok3_0(+(1, n2983_0)), gen_mark:0':nil:ok3_0(b))) ->_IH 28.07/8.16 mark(*4_0) 28.07/8.16 28.07/8.16 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (26) 28.07/8.16 Obligation: 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 28.07/8.16 Lemmas: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 28.07/8.16 from(gen_mark:0':nil:ok3_0(+(1, n1042_0))) -> *4_0, rt in Omega(n1042_0) 28.07/8.16 s(gen_mark:0':nil:ok3_0(+(1, n1588_0))) -> *4_0, rt in Omega(n1588_0) 28.07/8.16 head(gen_mark:0':nil:ok3_0(+(1, n2235_0))) -> *4_0, rt in Omega(n2235_0) 28.07/8.16 take(gen_mark:0':nil:ok3_0(+(1, n2983_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2983_0) 28.07/8.16 28.07/8.16 28.07/8.16 Generator Equations: 28.07/8.16 gen_mark:0':nil:ok3_0(0) <=> 0' 28.07/8.16 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 28.07/8.16 28.07/8.16 28.07/8.16 The following defined symbols remain to be analysed: 28.07/8.16 sel, active, 2nd, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 sel < active 28.07/8.16 2nd < active 28.07/8.16 active < top 28.07/8.16 sel < proper 28.07/8.16 2nd < proper 28.07/8.16 proper < top 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (27) RewriteLemmaProof (LOWER BOUND(ID)) 28.07/8.16 Proved the following rewrite lemma: 28.07/8.16 sel(gen_mark:0':nil:ok3_0(+(1, n5149_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5149_0) 28.07/8.16 28.07/8.16 Induction Base: 28.07/8.16 sel(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b)) 28.07/8.16 28.07/8.16 Induction Step: 28.07/8.16 sel(gen_mark:0':nil:ok3_0(+(1, +(n5149_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1) 28.07/8.16 mark(sel(gen_mark:0':nil:ok3_0(+(1, n5149_0)), gen_mark:0':nil:ok3_0(b))) ->_IH 28.07/8.16 mark(*4_0) 28.07/8.16 28.07/8.16 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (28) 28.07/8.16 Obligation: 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 28.07/8.16 Lemmas: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 28.07/8.16 from(gen_mark:0':nil:ok3_0(+(1, n1042_0))) -> *4_0, rt in Omega(n1042_0) 28.07/8.16 s(gen_mark:0':nil:ok3_0(+(1, n1588_0))) -> *4_0, rt in Omega(n1588_0) 28.07/8.16 head(gen_mark:0':nil:ok3_0(+(1, n2235_0))) -> *4_0, rt in Omega(n2235_0) 28.07/8.16 take(gen_mark:0':nil:ok3_0(+(1, n2983_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2983_0) 28.07/8.16 sel(gen_mark:0':nil:ok3_0(+(1, n5149_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5149_0) 28.07/8.16 28.07/8.16 28.07/8.16 Generator Equations: 28.07/8.16 gen_mark:0':nil:ok3_0(0) <=> 0' 28.07/8.16 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 28.07/8.16 28.07/8.16 28.07/8.16 The following defined symbols remain to be analysed: 28.07/8.16 2nd, active, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 2nd < active 28.07/8.16 active < top 28.07/8.16 2nd < proper 28.07/8.16 proper < top 28.07/8.16 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (29) RewriteLemmaProof (LOWER BOUND(ID)) 28.07/8.16 Proved the following rewrite lemma: 28.07/8.16 2nd(gen_mark:0':nil:ok3_0(+(1, n7621_0))) -> *4_0, rt in Omega(n7621_0) 28.07/8.16 28.07/8.16 Induction Base: 28.07/8.16 2nd(gen_mark:0':nil:ok3_0(+(1, 0))) 28.07/8.16 28.07/8.16 Induction Step: 28.07/8.16 2nd(gen_mark:0':nil:ok3_0(+(1, +(n7621_0, 1)))) ->_R^Omega(1) 28.07/8.16 mark(2nd(gen_mark:0':nil:ok3_0(+(1, n7621_0)))) ->_IH 28.07/8.16 mark(*4_0) 28.07/8.16 28.07/8.16 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 28.07/8.16 ---------------------------------------- 28.07/8.16 28.07/8.16 (30) 28.07/8.16 Obligation: 28.07/8.16 TRS: 28.07/8.16 Rules: 28.07/8.16 active(from(X)) -> mark(cons(X, from(s(X)))) 28.07/8.16 active(head(cons(X, XS))) -> mark(X) 28.07/8.16 active(2nd(cons(X, XS))) -> mark(head(XS)) 28.07/8.16 active(take(0', XS)) -> mark(nil) 28.07/8.16 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 28.07/8.16 active(sel(0', cons(X, XS))) -> mark(X) 28.07/8.16 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 28.07/8.16 active(from(X)) -> from(active(X)) 28.07/8.16 active(cons(X1, X2)) -> cons(active(X1), X2) 28.07/8.16 active(s(X)) -> s(active(X)) 28.07/8.16 active(head(X)) -> head(active(X)) 28.07/8.16 active(2nd(X)) -> 2nd(active(X)) 28.07/8.16 active(take(X1, X2)) -> take(active(X1), X2) 28.07/8.16 active(take(X1, X2)) -> take(X1, active(X2)) 28.07/8.16 active(sel(X1, X2)) -> sel(active(X1), X2) 28.07/8.16 active(sel(X1, X2)) -> sel(X1, active(X2)) 28.07/8.16 from(mark(X)) -> mark(from(X)) 28.07/8.16 cons(mark(X1), X2) -> mark(cons(X1, X2)) 28.07/8.16 s(mark(X)) -> mark(s(X)) 28.07/8.16 head(mark(X)) -> mark(head(X)) 28.07/8.16 2nd(mark(X)) -> mark(2nd(X)) 28.07/8.16 take(mark(X1), X2) -> mark(take(X1, X2)) 28.07/8.16 take(X1, mark(X2)) -> mark(take(X1, X2)) 28.07/8.16 sel(mark(X1), X2) -> mark(sel(X1, X2)) 28.07/8.16 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 28.07/8.16 proper(from(X)) -> from(proper(X)) 28.07/8.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 28.07/8.16 proper(s(X)) -> s(proper(X)) 28.07/8.16 proper(head(X)) -> head(proper(X)) 28.07/8.16 proper(2nd(X)) -> 2nd(proper(X)) 28.07/8.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 28.07/8.16 proper(0') -> ok(0') 28.07/8.16 proper(nil) -> ok(nil) 28.07/8.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 28.07/8.16 from(ok(X)) -> ok(from(X)) 28.07/8.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 28.07/8.16 s(ok(X)) -> ok(s(X)) 28.07/8.16 head(ok(X)) -> ok(head(X)) 28.07/8.16 2nd(ok(X)) -> ok(2nd(X)) 28.07/8.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 28.07/8.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 28.07/8.16 top(mark(X)) -> top(proper(X)) 28.07/8.16 top(ok(X)) -> top(active(X)) 28.07/8.16 28.07/8.16 Types: 28.07/8.16 active :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 from :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 mark :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 s :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 head :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 2nd :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 0' :: mark:0':nil:ok 28.07/8.16 nil :: mark:0':nil:ok 28.07/8.16 sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 proper :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 ok :: mark:0':nil:ok -> mark:0':nil:ok 28.07/8.16 top :: mark:0':nil:ok -> top 28.07/8.16 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 28.07/8.16 hole_top2_0 :: top 28.07/8.16 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 28.07/8.16 28.07/8.16 28.07/8.16 Lemmas: 28.07/8.16 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 28.07/8.16 from(gen_mark:0':nil:ok3_0(+(1, n1042_0))) -> *4_0, rt in Omega(n1042_0) 28.07/8.16 s(gen_mark:0':nil:ok3_0(+(1, n1588_0))) -> *4_0, rt in Omega(n1588_0) 28.07/8.16 head(gen_mark:0':nil:ok3_0(+(1, n2235_0))) -> *4_0, rt in Omega(n2235_0) 28.07/8.16 take(gen_mark:0':nil:ok3_0(+(1, n2983_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2983_0) 28.07/8.16 sel(gen_mark:0':nil:ok3_0(+(1, n5149_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5149_0) 28.07/8.16 2nd(gen_mark:0':nil:ok3_0(+(1, n7621_0))) -> *4_0, rt in Omega(n7621_0) 28.07/8.16 28.07/8.16 28.07/8.16 Generator Equations: 28.07/8.16 gen_mark:0':nil:ok3_0(0) <=> 0' 28.07/8.16 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 28.07/8.16 28.07/8.16 28.07/8.16 The following defined symbols remain to be analysed: 28.07/8.16 active, proper, top 28.07/8.16 28.07/8.16 They will be analysed ascendingly in the following order: 28.07/8.16 active < top 28.07/8.16 proper < top 28.48/8.36 EOF