23.89/7.12 WORST_CASE(Omega(n^1), O(n^1)) 23.89/7.13 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 23.89/7.13 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 23.89/7.13 23.89/7.13 23.89/7.13 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.89/7.13 23.89/7.13 (0) CpxTRS 23.89/7.13 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 23.89/7.13 (2) CpxTRS 23.89/7.13 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 23.89/7.13 (4) CpxTRS 23.89/7.13 (5) CpxTrsMatchBoundsTAProof [FINISHED, 99 ms] 23.89/7.13 (6) BOUNDS(1, n^1) 23.89/7.13 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 23.89/7.13 (8) CpxTRS 23.89/7.13 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 23.89/7.13 (10) typed CpxTrs 23.89/7.13 (11) OrderProof [LOWER BOUND(ID), 0 ms] 23.89/7.13 (12) typed CpxTrs 23.89/7.13 (13) RewriteLemmaProof [LOWER BOUND(ID), 496 ms] 23.89/7.13 (14) BEST 23.89/7.13 (15) proven lower bound 23.89/7.13 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 23.89/7.13 (17) BOUNDS(n^1, INF) 23.89/7.13 (18) typed CpxTrs 23.89/7.13 (19) RewriteLemmaProof [LOWER BOUND(ID), 109 ms] 23.89/7.13 (20) typed CpxTrs 23.89/7.13 (21) RewriteLemmaProof [LOWER BOUND(ID), 73 ms] 23.89/7.13 (22) typed CpxTrs 23.89/7.13 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (0) 23.89/7.13 Obligation: 23.89/7.13 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.89/7.13 23.89/7.13 23.89/7.13 The TRS R consists of the following rules: 23.89/7.13 23.89/7.13 active(eq(0, 0)) -> mark(true) 23.89/7.13 active(eq(s(X), s(Y))) -> mark(eq(X, Y)) 23.89/7.13 active(eq(X, Y)) -> mark(false) 23.89/7.13 active(inf(X)) -> mark(cons(X, inf(s(X)))) 23.89/7.13 active(take(0, X)) -> mark(nil) 23.89/7.13 active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) 23.89/7.13 active(length(nil)) -> mark(0) 23.89/7.13 active(length(cons(X, L))) -> mark(s(length(L))) 23.89/7.13 active(inf(X)) -> inf(active(X)) 23.89/7.13 active(take(X1, X2)) -> take(active(X1), X2) 23.89/7.13 active(take(X1, X2)) -> take(X1, active(X2)) 23.89/7.13 active(length(X)) -> length(active(X)) 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) 23.89/7.13 proper(0) -> ok(0) 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(s(X)) -> s(proper(X)) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(inf(X)) -> inf(proper(X)) 23.89/7.13 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 23.89/7.13 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 proper(length(X)) -> length(proper(X)) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 S is empty. 23.89/7.13 Rewrite Strategy: FULL 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 23.89/7.13 The following defined symbols can occur below the 0th argument of top: proper, active 23.89/7.13 The following defined symbols can occur below the 0th argument of proper: proper, active 23.89/7.13 The following defined symbols can occur below the 0th argument of active: proper, active 23.89/7.13 23.89/7.13 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 23.89/7.13 active(eq(0, 0)) -> mark(true) 23.89/7.13 active(eq(s(X), s(Y))) -> mark(eq(X, Y)) 23.89/7.13 active(eq(X, Y)) -> mark(false) 23.89/7.13 active(inf(X)) -> mark(cons(X, inf(s(X)))) 23.89/7.13 active(take(0, X)) -> mark(nil) 23.89/7.13 active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) 23.89/7.13 active(length(nil)) -> mark(0) 23.89/7.13 active(length(cons(X, L))) -> mark(s(length(L))) 23.89/7.13 active(inf(X)) -> inf(active(X)) 23.89/7.13 active(take(X1, X2)) -> take(active(X1), X2) 23.89/7.13 active(take(X1, X2)) -> take(X1, active(X2)) 23.89/7.13 active(length(X)) -> length(active(X)) 23.89/7.13 proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) 23.89/7.13 proper(s(X)) -> s(proper(X)) 23.89/7.13 proper(inf(X)) -> inf(proper(X)) 23.89/7.13 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 23.89/7.13 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 23.89/7.13 proper(length(X)) -> length(proper(X)) 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (2) 23.89/7.13 Obligation: 23.89/7.13 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.89/7.13 23.89/7.13 23.89/7.13 The TRS R consists of the following rules: 23.89/7.13 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(0) -> ok(0) 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 S is empty. 23.89/7.13 Rewrite Strategy: FULL 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 23.89/7.13 transformed relative TRS to TRS 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (4) 23.89/7.13 Obligation: 23.89/7.13 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.89/7.13 23.89/7.13 23.89/7.13 The TRS R consists of the following rules: 23.89/7.13 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(0) -> ok(0) 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 S is empty. 23.89/7.13 Rewrite Strategy: FULL 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (5) CpxTrsMatchBoundsTAProof (FINISHED) 23.89/7.13 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. 23.89/7.13 23.89/7.13 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 23.89/7.13 final states : [1, 2, 3, 4, 5, 6, 7, 8] 23.89/7.13 transitions: 23.89/7.13 mark0(0) -> 0 23.89/7.13 00() -> 0 23.89/7.13 ok0(0) -> 0 23.89/7.13 true0() -> 0 23.89/7.13 false0() -> 0 23.89/7.13 nil0() -> 0 23.89/7.13 active0(0) -> 0 23.89/7.13 inf0(0) -> 1 23.89/7.13 take0(0, 0) -> 2 23.89/7.13 length0(0) -> 3 23.89/7.13 proper0(0) -> 4 23.89/7.13 eq0(0, 0) -> 5 23.89/7.13 s0(0) -> 6 23.89/7.13 cons0(0, 0) -> 7 23.89/7.13 top0(0) -> 8 23.89/7.13 inf1(0) -> 9 23.89/7.13 mark1(9) -> 1 23.89/7.13 take1(0, 0) -> 10 23.89/7.13 mark1(10) -> 2 23.89/7.13 length1(0) -> 11 23.89/7.13 mark1(11) -> 3 23.89/7.13 01() -> 12 23.89/7.13 ok1(12) -> 4 23.89/7.13 true1() -> 13 23.89/7.13 ok1(13) -> 4 23.89/7.13 false1() -> 14 23.89/7.13 ok1(14) -> 4 23.89/7.13 nil1() -> 15 23.89/7.13 ok1(15) -> 4 23.89/7.13 eq1(0, 0) -> 16 23.89/7.13 ok1(16) -> 5 23.89/7.13 s1(0) -> 17 23.89/7.13 ok1(17) -> 6 23.89/7.13 inf1(0) -> 18 23.89/7.13 ok1(18) -> 1 23.89/7.13 cons1(0, 0) -> 19 23.89/7.13 ok1(19) -> 7 23.89/7.13 take1(0, 0) -> 20 23.89/7.13 ok1(20) -> 2 23.89/7.13 length1(0) -> 21 23.89/7.13 ok1(21) -> 3 23.89/7.13 proper1(0) -> 22 23.89/7.13 top1(22) -> 8 23.89/7.13 active1(0) -> 23 23.89/7.13 top1(23) -> 8 23.89/7.13 mark1(9) -> 9 23.89/7.13 mark1(9) -> 18 23.89/7.13 mark1(10) -> 10 23.89/7.13 mark1(10) -> 20 23.89/7.13 mark1(11) -> 11 23.89/7.13 mark1(11) -> 21 23.89/7.13 ok1(12) -> 22 23.89/7.13 ok1(13) -> 22 23.89/7.13 ok1(14) -> 22 23.89/7.13 ok1(15) -> 22 23.89/7.13 ok1(16) -> 16 23.89/7.13 ok1(17) -> 17 23.89/7.13 ok1(18) -> 9 23.89/7.13 ok1(18) -> 18 23.89/7.13 ok1(19) -> 19 23.89/7.13 ok1(20) -> 10 23.89/7.13 ok1(20) -> 20 23.89/7.13 ok1(21) -> 11 23.89/7.13 ok1(21) -> 21 23.89/7.13 active2(12) -> 24 23.89/7.13 top2(24) -> 8 23.89/7.13 active2(13) -> 24 23.89/7.13 active2(14) -> 24 23.89/7.13 active2(15) -> 24 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (6) 23.89/7.13 BOUNDS(1, n^1) 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 23.89/7.13 Renamed function symbols to avoid clashes with predefined symbol. 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (8) 23.89/7.13 Obligation: 23.89/7.13 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 23.89/7.13 23.89/7.13 23.89/7.13 The TRS R consists of the following rules: 23.89/7.13 23.89/7.13 active(eq(0', 0')) -> mark(true) 23.89/7.13 active(eq(s(X), s(Y))) -> mark(eq(X, Y)) 23.89/7.13 active(eq(X, Y)) -> mark(false) 23.89/7.13 active(inf(X)) -> mark(cons(X, inf(s(X)))) 23.89/7.13 active(take(0', X)) -> mark(nil) 23.89/7.13 active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) 23.89/7.13 active(length(nil)) -> mark(0') 23.89/7.13 active(length(cons(X, L))) -> mark(s(length(L))) 23.89/7.13 active(inf(X)) -> inf(active(X)) 23.89/7.13 active(take(X1, X2)) -> take(active(X1), X2) 23.89/7.13 active(take(X1, X2)) -> take(X1, active(X2)) 23.89/7.13 active(length(X)) -> length(active(X)) 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) 23.89/7.13 proper(0') -> ok(0') 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(s(X)) -> s(proper(X)) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(inf(X)) -> inf(proper(X)) 23.89/7.13 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 23.89/7.13 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 proper(length(X)) -> length(proper(X)) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 S is empty. 23.89/7.13 Rewrite Strategy: FULL 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 23.89/7.13 Infered types. 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (10) 23.89/7.13 Obligation: 23.89/7.13 TRS: 23.89/7.13 Rules: 23.89/7.13 active(eq(0', 0')) -> mark(true) 23.89/7.13 active(eq(s(X), s(Y))) -> mark(eq(X, Y)) 23.89/7.13 active(eq(X, Y)) -> mark(false) 23.89/7.13 active(inf(X)) -> mark(cons(X, inf(s(X)))) 23.89/7.13 active(take(0', X)) -> mark(nil) 23.89/7.13 active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) 23.89/7.13 active(length(nil)) -> mark(0') 23.89/7.13 active(length(cons(X, L))) -> mark(s(length(L))) 23.89/7.13 active(inf(X)) -> inf(active(X)) 23.89/7.13 active(take(X1, X2)) -> take(active(X1), X2) 23.89/7.13 active(take(X1, X2)) -> take(X1, active(X2)) 23.89/7.13 active(length(X)) -> length(active(X)) 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) 23.89/7.13 proper(0') -> ok(0') 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(s(X)) -> s(proper(X)) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(inf(X)) -> inf(proper(X)) 23.89/7.13 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 23.89/7.13 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 proper(length(X)) -> length(proper(X)) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 Types: 23.89/7.13 active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 0' :: 0':true:mark:false:nil:ok 23.89/7.13 mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 true :: 0':true:mark:false:nil:ok 23.89/7.13 s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 false :: 0':true:mark:false:nil:ok 23.89/7.13 inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 nil :: 0':true:mark:false:nil:ok 23.89/7.13 length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 top :: 0':true:mark:false:nil:ok -> top 23.89/7.13 hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok 23.89/7.13 hole_top2_0 :: top 23.89/7.13 gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (11) OrderProof (LOWER BOUND(ID)) 23.89/7.13 Heuristically decided to analyse the following defined symbols: 23.89/7.13 active, eq, cons, inf, s, take, length, proper, top 23.89/7.13 23.89/7.13 They will be analysed ascendingly in the following order: 23.89/7.13 eq < active 23.89/7.13 cons < active 23.89/7.13 inf < active 23.89/7.13 s < active 23.89/7.13 take < active 23.89/7.13 length < active 23.89/7.13 active < top 23.89/7.13 eq < proper 23.89/7.13 cons < proper 23.89/7.13 inf < proper 23.89/7.13 s < proper 23.89/7.13 take < proper 23.89/7.13 length < proper 23.89/7.13 proper < top 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (12) 23.89/7.13 Obligation: 23.89/7.13 TRS: 23.89/7.13 Rules: 23.89/7.13 active(eq(0', 0')) -> mark(true) 23.89/7.13 active(eq(s(X), s(Y))) -> mark(eq(X, Y)) 23.89/7.13 active(eq(X, Y)) -> mark(false) 23.89/7.13 active(inf(X)) -> mark(cons(X, inf(s(X)))) 23.89/7.13 active(take(0', X)) -> mark(nil) 23.89/7.13 active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) 23.89/7.13 active(length(nil)) -> mark(0') 23.89/7.13 active(length(cons(X, L))) -> mark(s(length(L))) 23.89/7.13 active(inf(X)) -> inf(active(X)) 23.89/7.13 active(take(X1, X2)) -> take(active(X1), X2) 23.89/7.13 active(take(X1, X2)) -> take(X1, active(X2)) 23.89/7.13 active(length(X)) -> length(active(X)) 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) 23.89/7.13 proper(0') -> ok(0') 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(s(X)) -> s(proper(X)) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(inf(X)) -> inf(proper(X)) 23.89/7.13 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 23.89/7.13 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 proper(length(X)) -> length(proper(X)) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 Types: 23.89/7.13 active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 0' :: 0':true:mark:false:nil:ok 23.89/7.13 mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 true :: 0':true:mark:false:nil:ok 23.89/7.13 s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 false :: 0':true:mark:false:nil:ok 23.89/7.13 inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 nil :: 0':true:mark:false:nil:ok 23.89/7.13 length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 top :: 0':true:mark:false:nil:ok -> top 23.89/7.13 hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok 23.89/7.13 hole_top2_0 :: top 23.89/7.13 gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok 23.89/7.13 23.89/7.13 23.89/7.13 Generator Equations: 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(0) <=> 0' 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) 23.89/7.13 23.89/7.13 23.89/7.13 The following defined symbols remain to be analysed: 23.89/7.13 eq, active, cons, inf, s, take, length, proper, top 23.89/7.13 23.89/7.13 They will be analysed ascendingly in the following order: 23.89/7.13 eq < active 23.89/7.13 cons < active 23.89/7.13 inf < active 23.89/7.13 s < active 23.89/7.13 take < active 23.89/7.13 length < active 23.89/7.13 active < top 23.89/7.13 eq < proper 23.89/7.13 cons < proper 23.89/7.13 inf < proper 23.89/7.13 s < proper 23.89/7.13 take < proper 23.89/7.13 length < proper 23.89/7.13 proper < top 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (13) RewriteLemmaProof (LOWER BOUND(ID)) 23.89/7.13 Proved the following rewrite lemma: 23.89/7.13 inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0))) -> *4_0, rt in Omega(n15_0) 23.89/7.13 23.89/7.13 Induction Base: 23.89/7.13 inf(gen_0':true:mark:false:nil:ok3_0(+(1, 0))) 23.89/7.13 23.89/7.13 Induction Step: 23.89/7.13 inf(gen_0':true:mark:false:nil:ok3_0(+(1, +(n15_0, 1)))) ->_R^Omega(1) 23.89/7.13 mark(inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0)))) ->_IH 23.89/7.13 mark(*4_0) 23.89/7.13 23.89/7.13 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (14) 23.89/7.13 Complex Obligation (BEST) 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (15) 23.89/7.13 Obligation: 23.89/7.13 Proved the lower bound n^1 for the following obligation: 23.89/7.13 23.89/7.13 TRS: 23.89/7.13 Rules: 23.89/7.13 active(eq(0', 0')) -> mark(true) 23.89/7.13 active(eq(s(X), s(Y))) -> mark(eq(X, Y)) 23.89/7.13 active(eq(X, Y)) -> mark(false) 23.89/7.13 active(inf(X)) -> mark(cons(X, inf(s(X)))) 23.89/7.13 active(take(0', X)) -> mark(nil) 23.89/7.13 active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) 23.89/7.13 active(length(nil)) -> mark(0') 23.89/7.13 active(length(cons(X, L))) -> mark(s(length(L))) 23.89/7.13 active(inf(X)) -> inf(active(X)) 23.89/7.13 active(take(X1, X2)) -> take(active(X1), X2) 23.89/7.13 active(take(X1, X2)) -> take(X1, active(X2)) 23.89/7.13 active(length(X)) -> length(active(X)) 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) 23.89/7.13 proper(0') -> ok(0') 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(s(X)) -> s(proper(X)) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(inf(X)) -> inf(proper(X)) 23.89/7.13 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 23.89/7.13 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 proper(length(X)) -> length(proper(X)) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 Types: 23.89/7.13 active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 0' :: 0':true:mark:false:nil:ok 23.89/7.13 mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 true :: 0':true:mark:false:nil:ok 23.89/7.13 s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 false :: 0':true:mark:false:nil:ok 23.89/7.13 inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 nil :: 0':true:mark:false:nil:ok 23.89/7.13 length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 top :: 0':true:mark:false:nil:ok -> top 23.89/7.13 hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok 23.89/7.13 hole_top2_0 :: top 23.89/7.13 gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok 23.89/7.13 23.89/7.13 23.89/7.13 Generator Equations: 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(0) <=> 0' 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) 23.89/7.13 23.89/7.13 23.89/7.13 The following defined symbols remain to be analysed: 23.89/7.13 inf, active, s, take, length, proper, top 23.89/7.13 23.89/7.13 They will be analysed ascendingly in the following order: 23.89/7.13 inf < active 23.89/7.13 s < active 23.89/7.13 take < active 23.89/7.13 length < active 23.89/7.13 active < top 23.89/7.13 inf < proper 23.89/7.13 s < proper 23.89/7.13 take < proper 23.89/7.13 length < proper 23.89/7.13 proper < top 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (16) LowerBoundPropagationProof (FINISHED) 23.89/7.13 Propagated lower bound. 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (17) 23.89/7.13 BOUNDS(n^1, INF) 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (18) 23.89/7.13 Obligation: 23.89/7.13 TRS: 23.89/7.13 Rules: 23.89/7.13 active(eq(0', 0')) -> mark(true) 23.89/7.13 active(eq(s(X), s(Y))) -> mark(eq(X, Y)) 23.89/7.13 active(eq(X, Y)) -> mark(false) 23.89/7.13 active(inf(X)) -> mark(cons(X, inf(s(X)))) 23.89/7.13 active(take(0', X)) -> mark(nil) 23.89/7.13 active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) 23.89/7.13 active(length(nil)) -> mark(0') 23.89/7.13 active(length(cons(X, L))) -> mark(s(length(L))) 23.89/7.13 active(inf(X)) -> inf(active(X)) 23.89/7.13 active(take(X1, X2)) -> take(active(X1), X2) 23.89/7.13 active(take(X1, X2)) -> take(X1, active(X2)) 23.89/7.13 active(length(X)) -> length(active(X)) 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) 23.89/7.13 proper(0') -> ok(0') 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(s(X)) -> s(proper(X)) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(inf(X)) -> inf(proper(X)) 23.89/7.13 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 23.89/7.13 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 proper(length(X)) -> length(proper(X)) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 Types: 23.89/7.13 active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 0' :: 0':true:mark:false:nil:ok 23.89/7.13 mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 true :: 0':true:mark:false:nil:ok 23.89/7.13 s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 false :: 0':true:mark:false:nil:ok 23.89/7.13 inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 nil :: 0':true:mark:false:nil:ok 23.89/7.13 length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 top :: 0':true:mark:false:nil:ok -> top 23.89/7.13 hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok 23.89/7.13 hole_top2_0 :: top 23.89/7.13 gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok 23.89/7.13 23.89/7.13 23.89/7.13 Lemmas: 23.89/7.13 inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0))) -> *4_0, rt in Omega(n15_0) 23.89/7.13 23.89/7.13 23.89/7.13 Generator Equations: 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(0) <=> 0' 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) 23.89/7.13 23.89/7.13 23.89/7.13 The following defined symbols remain to be analysed: 23.89/7.13 s, active, take, length, proper, top 23.89/7.13 23.89/7.13 They will be analysed ascendingly in the following order: 23.89/7.13 s < active 23.89/7.13 take < active 23.89/7.13 length < active 23.89/7.13 active < top 23.89/7.13 s < proper 23.89/7.13 take < proper 23.89/7.13 length < proper 23.89/7.13 proper < top 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (19) RewriteLemmaProof (LOWER BOUND(ID)) 23.89/7.13 Proved the following rewrite lemma: 23.89/7.13 take(gen_0':true:mark:false:nil:ok3_0(+(1, n391_0)), gen_0':true:mark:false:nil:ok3_0(b)) -> *4_0, rt in Omega(n391_0) 23.89/7.13 23.89/7.13 Induction Base: 23.89/7.13 take(gen_0':true:mark:false:nil:ok3_0(+(1, 0)), gen_0':true:mark:false:nil:ok3_0(b)) 23.89/7.13 23.89/7.13 Induction Step: 23.89/7.13 take(gen_0':true:mark:false:nil:ok3_0(+(1, +(n391_0, 1))), gen_0':true:mark:false:nil:ok3_0(b)) ->_R^Omega(1) 23.89/7.13 mark(take(gen_0':true:mark:false:nil:ok3_0(+(1, n391_0)), gen_0':true:mark:false:nil:ok3_0(b))) ->_IH 23.89/7.13 mark(*4_0) 23.89/7.13 23.89/7.13 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (20) 23.89/7.13 Obligation: 23.89/7.13 TRS: 23.89/7.13 Rules: 23.89/7.13 active(eq(0', 0')) -> mark(true) 23.89/7.13 active(eq(s(X), s(Y))) -> mark(eq(X, Y)) 23.89/7.13 active(eq(X, Y)) -> mark(false) 23.89/7.13 active(inf(X)) -> mark(cons(X, inf(s(X)))) 23.89/7.13 active(take(0', X)) -> mark(nil) 23.89/7.13 active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) 23.89/7.13 active(length(nil)) -> mark(0') 23.89/7.13 active(length(cons(X, L))) -> mark(s(length(L))) 23.89/7.13 active(inf(X)) -> inf(active(X)) 23.89/7.13 active(take(X1, X2)) -> take(active(X1), X2) 23.89/7.13 active(take(X1, X2)) -> take(X1, active(X2)) 23.89/7.13 active(length(X)) -> length(active(X)) 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) 23.89/7.13 proper(0') -> ok(0') 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(s(X)) -> s(proper(X)) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(inf(X)) -> inf(proper(X)) 23.89/7.13 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 23.89/7.13 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 proper(length(X)) -> length(proper(X)) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 Types: 23.89/7.13 active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 0' :: 0':true:mark:false:nil:ok 23.89/7.13 mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 true :: 0':true:mark:false:nil:ok 23.89/7.13 s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 false :: 0':true:mark:false:nil:ok 23.89/7.13 inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 nil :: 0':true:mark:false:nil:ok 23.89/7.13 length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 top :: 0':true:mark:false:nil:ok -> top 23.89/7.13 hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok 23.89/7.13 hole_top2_0 :: top 23.89/7.13 gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok 23.89/7.13 23.89/7.13 23.89/7.13 Lemmas: 23.89/7.13 inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0))) -> *4_0, rt in Omega(n15_0) 23.89/7.13 take(gen_0':true:mark:false:nil:ok3_0(+(1, n391_0)), gen_0':true:mark:false:nil:ok3_0(b)) -> *4_0, rt in Omega(n391_0) 23.89/7.13 23.89/7.13 23.89/7.13 Generator Equations: 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(0) <=> 0' 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) 23.89/7.13 23.89/7.13 23.89/7.13 The following defined symbols remain to be analysed: 23.89/7.13 length, active, proper, top 23.89/7.13 23.89/7.13 They will be analysed ascendingly in the following order: 23.89/7.13 length < active 23.89/7.13 active < top 23.89/7.13 length < proper 23.89/7.13 proper < top 23.89/7.13 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (21) RewriteLemmaProof (LOWER BOUND(ID)) 23.89/7.13 Proved the following rewrite lemma: 23.89/7.13 length(gen_0':true:mark:false:nil:ok3_0(+(1, n1709_0))) -> *4_0, rt in Omega(n1709_0) 23.89/7.13 23.89/7.13 Induction Base: 23.89/7.13 length(gen_0':true:mark:false:nil:ok3_0(+(1, 0))) 23.89/7.13 23.89/7.13 Induction Step: 23.89/7.13 length(gen_0':true:mark:false:nil:ok3_0(+(1, +(n1709_0, 1)))) ->_R^Omega(1) 23.89/7.13 mark(length(gen_0':true:mark:false:nil:ok3_0(+(1, n1709_0)))) ->_IH 23.89/7.13 mark(*4_0) 23.89/7.13 23.89/7.13 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.89/7.13 ---------------------------------------- 23.89/7.13 23.89/7.13 (22) 23.89/7.13 Obligation: 23.89/7.13 TRS: 23.89/7.13 Rules: 23.89/7.13 active(eq(0', 0')) -> mark(true) 23.89/7.13 active(eq(s(X), s(Y))) -> mark(eq(X, Y)) 23.89/7.13 active(eq(X, Y)) -> mark(false) 23.89/7.13 active(inf(X)) -> mark(cons(X, inf(s(X)))) 23.89/7.13 active(take(0', X)) -> mark(nil) 23.89/7.13 active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L))) 23.89/7.13 active(length(nil)) -> mark(0') 23.89/7.13 active(length(cons(X, L))) -> mark(s(length(L))) 23.89/7.13 active(inf(X)) -> inf(active(X)) 23.89/7.13 active(take(X1, X2)) -> take(active(X1), X2) 23.89/7.13 active(take(X1, X2)) -> take(X1, active(X2)) 23.89/7.13 active(length(X)) -> length(active(X)) 23.89/7.13 inf(mark(X)) -> mark(inf(X)) 23.89/7.13 take(mark(X1), X2) -> mark(take(X1, X2)) 23.89/7.13 take(X1, mark(X2)) -> mark(take(X1, X2)) 23.89/7.13 length(mark(X)) -> mark(length(X)) 23.89/7.13 proper(eq(X1, X2)) -> eq(proper(X1), proper(X2)) 23.89/7.13 proper(0') -> ok(0') 23.89/7.13 proper(true) -> ok(true) 23.89/7.13 proper(s(X)) -> s(proper(X)) 23.89/7.13 proper(false) -> ok(false) 23.89/7.13 proper(inf(X)) -> inf(proper(X)) 23.89/7.13 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 23.89/7.13 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 23.89/7.13 proper(nil) -> ok(nil) 23.89/7.13 proper(length(X)) -> length(proper(X)) 23.89/7.13 eq(ok(X1), ok(X2)) -> ok(eq(X1, X2)) 23.89/7.13 s(ok(X)) -> ok(s(X)) 23.89/7.13 inf(ok(X)) -> ok(inf(X)) 23.89/7.13 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 23.89/7.13 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 23.89/7.13 length(ok(X)) -> ok(length(X)) 23.89/7.13 top(mark(X)) -> top(proper(X)) 23.89/7.13 top(ok(X)) -> top(active(X)) 23.89/7.13 23.89/7.13 Types: 23.89/7.13 active :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 eq :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 0' :: 0':true:mark:false:nil:ok 23.89/7.13 mark :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 true :: 0':true:mark:false:nil:ok 23.89/7.13 s :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 false :: 0':true:mark:false:nil:ok 23.89/7.13 inf :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 cons :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 take :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 nil :: 0':true:mark:false:nil:ok 23.89/7.13 length :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 proper :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 ok :: 0':true:mark:false:nil:ok -> 0':true:mark:false:nil:ok 23.89/7.13 top :: 0':true:mark:false:nil:ok -> top 23.89/7.13 hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok 23.89/7.13 hole_top2_0 :: top 23.89/7.13 gen_0':true:mark:false:nil:ok3_0 :: Nat -> 0':true:mark:false:nil:ok 23.89/7.13 23.89/7.13 23.89/7.13 Lemmas: 23.89/7.13 inf(gen_0':true:mark:false:nil:ok3_0(+(1, n15_0))) -> *4_0, rt in Omega(n15_0) 23.89/7.13 take(gen_0':true:mark:false:nil:ok3_0(+(1, n391_0)), gen_0':true:mark:false:nil:ok3_0(b)) -> *4_0, rt in Omega(n391_0) 23.89/7.13 length(gen_0':true:mark:false:nil:ok3_0(+(1, n1709_0))) -> *4_0, rt in Omega(n1709_0) 23.89/7.13 23.89/7.13 23.89/7.13 Generator Equations: 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(0) <=> 0' 23.89/7.13 gen_0':true:mark:false:nil:ok3_0(+(x, 1)) <=> mark(gen_0':true:mark:false:nil:ok3_0(x)) 23.89/7.13 23.89/7.13 23.89/7.13 The following defined symbols remain to be analysed: 23.89/7.13 active, proper, top 23.89/7.13 23.89/7.13 They will be analysed ascendingly in the following order: 23.89/7.13 active < top 23.89/7.13 proper < top 25.36/9.81 EOF