24.42/8.42 WORST_CASE(Omega(n^1), O(n^1)) 24.42/8.43 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.42/8.43 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.42/8.43 24.42/8.43 24.42/8.43 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.42/8.43 24.42/8.43 (0) CpxTRS 24.42/8.43 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 24.42/8.43 (2) CpxTRS 24.42/8.43 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 24.42/8.43 (4) CpxTRS 24.42/8.43 (5) CpxTrsMatchBoundsTAProof [FINISHED, 48 ms] 24.42/8.43 (6) BOUNDS(1, n^1) 24.42/8.43 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 24.42/8.43 (8) CpxTRS 24.42/8.43 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 24.42/8.43 (10) typed CpxTrs 24.42/8.43 (11) OrderProof [LOWER BOUND(ID), 7 ms] 24.42/8.43 (12) typed CpxTrs 24.42/8.43 (13) RewriteLemmaProof [LOWER BOUND(ID), 494 ms] 24.42/8.43 (14) BEST 24.42/8.43 (15) proven lower bound 24.42/8.43 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 24.42/8.43 (17) BOUNDS(n^1, INF) 24.42/8.43 (18) typed CpxTrs 24.42/8.43 (19) RewriteLemmaProof [LOWER BOUND(ID), 146 ms] 24.42/8.43 (20) typed CpxTrs 24.42/8.43 (21) RewriteLemmaProof [LOWER BOUND(ID), 89 ms] 24.42/8.43 (22) typed CpxTrs 24.42/8.43 (23) RewriteLemmaProof [LOWER BOUND(ID), 68 ms] 24.42/8.43 (24) typed CpxTrs 24.42/8.43 (25) RewriteLemmaProof [LOWER BOUND(ID), 63 ms] 24.42/8.43 (26) typed CpxTrs 24.42/8.43 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (0) 24.42/8.43 Obligation: 24.42/8.43 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.42/8.43 24.42/8.43 24.42/8.43 The TRS R consists of the following rules: 24.42/8.43 24.42/8.43 active(fst(0, Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0, X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0) 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0) -> ok(0) 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.43 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.43 proper(len(X)) -> len(proper(X)) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 S is empty. 24.42/8.43 Rewrite Strategy: FULL 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 24.42/8.43 The following defined symbols can occur below the 0th argument of top: proper, active 24.42/8.43 The following defined symbols can occur below the 0th argument of proper: proper, active 24.42/8.43 The following defined symbols can occur below the 0th argument of active: proper, active 24.42/8.43 24.42/8.43 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 24.42/8.43 active(fst(0, Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0, X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0) 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.43 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.43 proper(len(X)) -> len(proper(X)) 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (2) 24.42/8.43 Obligation: 24.42/8.43 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.42/8.43 24.42/8.43 24.42/8.43 The TRS R consists of the following rules: 24.42/8.43 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0) -> ok(0) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 S is empty. 24.42/8.43 Rewrite Strategy: FULL 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 24.42/8.43 transformed relative TRS to TRS 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (4) 24.42/8.43 Obligation: 24.42/8.43 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.42/8.43 24.42/8.43 24.42/8.43 The TRS R consists of the following rules: 24.42/8.43 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0) -> ok(0) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 S is empty. 24.42/8.43 Rewrite Strategy: FULL 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (5) CpxTrsMatchBoundsTAProof (FINISHED) 24.42/8.43 A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 24.42/8.43 24.42/8.43 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 24.42/8.43 final states : [1, 2, 3, 4, 5, 6, 7, 8] 24.42/8.43 transitions: 24.42/8.43 mark0(0) -> 0 24.42/8.43 00() -> 0 24.42/8.43 ok0(0) -> 0 24.42/8.43 nil0() -> 0 24.42/8.43 active0(0) -> 0 24.42/8.43 cons0(0, 0) -> 1 24.42/8.43 fst0(0, 0) -> 2 24.42/8.43 from0(0) -> 3 24.42/8.43 add0(0, 0) -> 4 24.42/8.43 len0(0) -> 5 24.42/8.43 proper0(0) -> 6 24.42/8.43 s0(0) -> 7 24.42/8.43 top0(0) -> 8 24.42/8.43 cons1(0, 0) -> 9 24.42/8.43 mark1(9) -> 1 24.42/8.43 fst1(0, 0) -> 10 24.42/8.43 mark1(10) -> 2 24.42/8.43 from1(0) -> 11 24.42/8.43 mark1(11) -> 3 24.42/8.43 add1(0, 0) -> 12 24.42/8.43 mark1(12) -> 4 24.42/8.43 len1(0) -> 13 24.42/8.43 mark1(13) -> 5 24.42/8.43 01() -> 14 24.42/8.43 ok1(14) -> 6 24.42/8.43 nil1() -> 15 24.42/8.43 ok1(15) -> 6 24.42/8.43 s1(0) -> 16 24.42/8.43 ok1(16) -> 7 24.42/8.43 cons1(0, 0) -> 17 24.42/8.43 ok1(17) -> 1 24.42/8.43 fst1(0, 0) -> 18 24.42/8.43 ok1(18) -> 2 24.42/8.43 from1(0) -> 19 24.42/8.43 ok1(19) -> 3 24.42/8.43 add1(0, 0) -> 20 24.42/8.43 ok1(20) -> 4 24.42/8.43 len1(0) -> 21 24.42/8.43 ok1(21) -> 5 24.42/8.43 proper1(0) -> 22 24.42/8.43 top1(22) -> 8 24.42/8.43 active1(0) -> 23 24.42/8.43 top1(23) -> 8 24.42/8.43 mark1(9) -> 9 24.42/8.43 mark1(9) -> 17 24.42/8.43 mark1(10) -> 10 24.42/8.43 mark1(10) -> 18 24.42/8.43 mark1(11) -> 11 24.42/8.43 mark1(11) -> 19 24.42/8.43 mark1(12) -> 12 24.42/8.43 mark1(12) -> 20 24.42/8.43 mark1(13) -> 13 24.42/8.43 mark1(13) -> 21 24.42/8.43 ok1(14) -> 22 24.42/8.43 ok1(15) -> 22 24.42/8.43 ok1(16) -> 16 24.42/8.43 ok1(17) -> 9 24.42/8.43 ok1(17) -> 17 24.42/8.43 ok1(18) -> 10 24.42/8.43 ok1(18) -> 18 24.42/8.43 ok1(19) -> 11 24.42/8.43 ok1(19) -> 19 24.42/8.43 ok1(20) -> 12 24.42/8.43 ok1(20) -> 20 24.42/8.43 ok1(21) -> 13 24.42/8.43 ok1(21) -> 21 24.42/8.43 active2(14) -> 24 24.42/8.43 top2(24) -> 8 24.42/8.43 active2(15) -> 24 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (6) 24.42/8.43 BOUNDS(1, n^1) 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 24.42/8.43 Renamed function symbols to avoid clashes with predefined symbol. 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (8) 24.42/8.43 Obligation: 24.42/8.43 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 24.42/8.43 24.42/8.43 24.42/8.43 The TRS R consists of the following rules: 24.42/8.43 24.42/8.43 active(fst(0', Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0', X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0') 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0') -> ok(0') 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.43 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.43 proper(len(X)) -> len(proper(X)) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 S is empty. 24.42/8.43 Rewrite Strategy: FULL 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 24.42/8.43 Infered types. 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (10) 24.42/8.43 Obligation: 24.42/8.43 TRS: 24.42/8.43 Rules: 24.42/8.43 active(fst(0', Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0', X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0') 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0') -> ok(0') 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.43 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.43 proper(len(X)) -> len(proper(X)) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 Types: 24.42/8.43 active :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 fst :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 0' :: 0':nil:mark:ok 24.42/8.43 mark :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 nil :: 0':nil:mark:ok 24.42/8.43 s :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 cons :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 from :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 add :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 len :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 proper :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 ok :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 top :: 0':nil:mark:ok -> top 24.42/8.43 hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok 24.42/8.43 hole_top2_0 :: top 24.42/8.43 gen_0':nil:mark:ok3_0 :: Nat -> 0':nil:mark:ok 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (11) OrderProof (LOWER BOUND(ID)) 24.42/8.43 Heuristically decided to analyse the following defined symbols: 24.42/8.43 active, cons, fst, from, s, add, len, proper, top 24.42/8.43 24.42/8.43 They will be analysed ascendingly in the following order: 24.42/8.43 cons < active 24.42/8.43 fst < active 24.42/8.43 from < active 24.42/8.43 s < active 24.42/8.43 add < active 24.42/8.43 len < active 24.42/8.43 active < top 24.42/8.43 cons < proper 24.42/8.43 fst < proper 24.42/8.43 from < proper 24.42/8.43 s < proper 24.42/8.43 add < proper 24.42/8.43 len < proper 24.42/8.43 proper < top 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (12) 24.42/8.43 Obligation: 24.42/8.43 TRS: 24.42/8.43 Rules: 24.42/8.43 active(fst(0', Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0', X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0') 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0') -> ok(0') 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.43 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.43 proper(len(X)) -> len(proper(X)) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 Types: 24.42/8.43 active :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 fst :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 0' :: 0':nil:mark:ok 24.42/8.43 mark :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 nil :: 0':nil:mark:ok 24.42/8.43 s :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 cons :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 from :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 add :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 len :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 proper :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 ok :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 top :: 0':nil:mark:ok -> top 24.42/8.43 hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok 24.42/8.43 hole_top2_0 :: top 24.42/8.43 gen_0':nil:mark:ok3_0 :: Nat -> 0':nil:mark:ok 24.42/8.43 24.42/8.43 24.42/8.43 Generator Equations: 24.42/8.43 gen_0':nil:mark:ok3_0(0) <=> 0' 24.42/8.43 gen_0':nil:mark:ok3_0(+(x, 1)) <=> mark(gen_0':nil:mark:ok3_0(x)) 24.42/8.43 24.42/8.43 24.42/8.43 The following defined symbols remain to be analysed: 24.42/8.43 cons, active, fst, from, s, add, len, proper, top 24.42/8.43 24.42/8.43 They will be analysed ascendingly in the following order: 24.42/8.43 cons < active 24.42/8.43 fst < active 24.42/8.43 from < active 24.42/8.43 s < active 24.42/8.43 add < active 24.42/8.43 len < active 24.42/8.43 active < top 24.42/8.43 cons < proper 24.42/8.43 fst < proper 24.42/8.43 from < proper 24.42/8.43 s < proper 24.42/8.43 add < proper 24.42/8.43 len < proper 24.42/8.43 proper < top 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (13) RewriteLemmaProof (LOWER BOUND(ID)) 24.42/8.43 Proved the following rewrite lemma: 24.42/8.43 cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 24.42/8.43 24.42/8.43 Induction Base: 24.42/8.43 cons(gen_0':nil:mark:ok3_0(+(1, 0)), gen_0':nil:mark:ok3_0(b)) 24.42/8.43 24.42/8.43 Induction Step: 24.42/8.43 cons(gen_0':nil:mark:ok3_0(+(1, +(n5_0, 1))), gen_0':nil:mark:ok3_0(b)) ->_R^Omega(1) 24.42/8.43 mark(cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b))) ->_IH 24.42/8.43 mark(*4_0) 24.42/8.43 24.42/8.43 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (14) 24.42/8.43 Complex Obligation (BEST) 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (15) 24.42/8.43 Obligation: 24.42/8.43 Proved the lower bound n^1 for the following obligation: 24.42/8.43 24.42/8.43 TRS: 24.42/8.43 Rules: 24.42/8.43 active(fst(0', Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0', X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0') 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0') -> ok(0') 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.43 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.43 proper(len(X)) -> len(proper(X)) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 Types: 24.42/8.43 active :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 fst :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 0' :: 0':nil:mark:ok 24.42/8.43 mark :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 nil :: 0':nil:mark:ok 24.42/8.43 s :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 cons :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 from :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 add :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 len :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 proper :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 ok :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 top :: 0':nil:mark:ok -> top 24.42/8.43 hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok 24.42/8.43 hole_top2_0 :: top 24.42/8.43 gen_0':nil:mark:ok3_0 :: Nat -> 0':nil:mark:ok 24.42/8.43 24.42/8.43 24.42/8.43 Generator Equations: 24.42/8.43 gen_0':nil:mark:ok3_0(0) <=> 0' 24.42/8.43 gen_0':nil:mark:ok3_0(+(x, 1)) <=> mark(gen_0':nil:mark:ok3_0(x)) 24.42/8.43 24.42/8.43 24.42/8.43 The following defined symbols remain to be analysed: 24.42/8.43 cons, active, fst, from, s, add, len, proper, top 24.42/8.43 24.42/8.43 They will be analysed ascendingly in the following order: 24.42/8.43 cons < active 24.42/8.43 fst < active 24.42/8.43 from < active 24.42/8.43 s < active 24.42/8.43 add < active 24.42/8.43 len < active 24.42/8.43 active < top 24.42/8.43 cons < proper 24.42/8.43 fst < proper 24.42/8.43 from < proper 24.42/8.43 s < proper 24.42/8.43 add < proper 24.42/8.43 len < proper 24.42/8.43 proper < top 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (16) LowerBoundPropagationProof (FINISHED) 24.42/8.43 Propagated lower bound. 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (17) 24.42/8.43 BOUNDS(n^1, INF) 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (18) 24.42/8.43 Obligation: 24.42/8.43 TRS: 24.42/8.43 Rules: 24.42/8.43 active(fst(0', Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0', X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0') 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0') -> ok(0') 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.43 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.43 proper(len(X)) -> len(proper(X)) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 Types: 24.42/8.43 active :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 fst :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 0' :: 0':nil:mark:ok 24.42/8.43 mark :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 nil :: 0':nil:mark:ok 24.42/8.43 s :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 cons :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 from :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 add :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 len :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 proper :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 ok :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 top :: 0':nil:mark:ok -> top 24.42/8.43 hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok 24.42/8.43 hole_top2_0 :: top 24.42/8.43 gen_0':nil:mark:ok3_0 :: Nat -> 0':nil:mark:ok 24.42/8.43 24.42/8.43 24.42/8.43 Lemmas: 24.42/8.43 cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 24.42/8.43 24.42/8.43 24.42/8.43 Generator Equations: 24.42/8.43 gen_0':nil:mark:ok3_0(0) <=> 0' 24.42/8.43 gen_0':nil:mark:ok3_0(+(x, 1)) <=> mark(gen_0':nil:mark:ok3_0(x)) 24.42/8.43 24.42/8.43 24.42/8.43 The following defined symbols remain to be analysed: 24.42/8.43 fst, active, from, s, add, len, proper, top 24.42/8.43 24.42/8.43 They will be analysed ascendingly in the following order: 24.42/8.43 fst < active 24.42/8.43 from < active 24.42/8.43 s < active 24.42/8.43 add < active 24.42/8.43 len < active 24.42/8.43 active < top 24.42/8.43 fst < proper 24.42/8.43 from < proper 24.42/8.43 s < proper 24.42/8.43 add < proper 24.42/8.43 len < proper 24.42/8.43 proper < top 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (19) RewriteLemmaProof (LOWER BOUND(ID)) 24.42/8.43 Proved the following rewrite lemma: 24.42/8.43 fst(gen_0':nil:mark:ok3_0(+(1, n958_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n958_0) 24.42/8.43 24.42/8.43 Induction Base: 24.42/8.43 fst(gen_0':nil:mark:ok3_0(+(1, 0)), gen_0':nil:mark:ok3_0(b)) 24.42/8.43 24.42/8.43 Induction Step: 24.42/8.43 fst(gen_0':nil:mark:ok3_0(+(1, +(n958_0, 1))), gen_0':nil:mark:ok3_0(b)) ->_R^Omega(1) 24.42/8.43 mark(fst(gen_0':nil:mark:ok3_0(+(1, n958_0)), gen_0':nil:mark:ok3_0(b))) ->_IH 24.42/8.43 mark(*4_0) 24.42/8.43 24.42/8.43 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (20) 24.42/8.43 Obligation: 24.42/8.43 TRS: 24.42/8.43 Rules: 24.42/8.43 active(fst(0', Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0', X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0') 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0') -> ok(0') 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.43 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.43 proper(len(X)) -> len(proper(X)) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 Types: 24.42/8.43 active :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 fst :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 0' :: 0':nil:mark:ok 24.42/8.43 mark :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 nil :: 0':nil:mark:ok 24.42/8.43 s :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 cons :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 from :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 add :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 len :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 proper :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 ok :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 top :: 0':nil:mark:ok -> top 24.42/8.43 hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok 24.42/8.43 hole_top2_0 :: top 24.42/8.43 gen_0':nil:mark:ok3_0 :: Nat -> 0':nil:mark:ok 24.42/8.43 24.42/8.43 24.42/8.43 Lemmas: 24.42/8.43 cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 24.42/8.43 fst(gen_0':nil:mark:ok3_0(+(1, n958_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n958_0) 24.42/8.43 24.42/8.43 24.42/8.43 Generator Equations: 24.42/8.43 gen_0':nil:mark:ok3_0(0) <=> 0' 24.42/8.43 gen_0':nil:mark:ok3_0(+(x, 1)) <=> mark(gen_0':nil:mark:ok3_0(x)) 24.42/8.43 24.42/8.43 24.42/8.43 The following defined symbols remain to be analysed: 24.42/8.43 from, active, s, add, len, proper, top 24.42/8.43 24.42/8.43 They will be analysed ascendingly in the following order: 24.42/8.43 from < active 24.42/8.43 s < active 24.42/8.43 add < active 24.42/8.43 len < active 24.42/8.43 active < top 24.42/8.43 from < proper 24.42/8.43 s < proper 24.42/8.43 add < proper 24.42/8.43 len < proper 24.42/8.43 proper < top 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (21) RewriteLemmaProof (LOWER BOUND(ID)) 24.42/8.43 Proved the following rewrite lemma: 24.42/8.43 from(gen_0':nil:mark:ok3_0(+(1, n2416_0))) -> *4_0, rt in Omega(n2416_0) 24.42/8.43 24.42/8.43 Induction Base: 24.42/8.43 from(gen_0':nil:mark:ok3_0(+(1, 0))) 24.42/8.43 24.42/8.43 Induction Step: 24.42/8.43 from(gen_0':nil:mark:ok3_0(+(1, +(n2416_0, 1)))) ->_R^Omega(1) 24.42/8.43 mark(from(gen_0':nil:mark:ok3_0(+(1, n2416_0)))) ->_IH 24.42/8.43 mark(*4_0) 24.42/8.43 24.42/8.43 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (22) 24.42/8.43 Obligation: 24.42/8.43 TRS: 24.42/8.43 Rules: 24.42/8.43 active(fst(0', Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0', X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0') 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0') -> ok(0') 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.43 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.43 proper(len(X)) -> len(proper(X)) 24.42/8.43 s(ok(X)) -> ok(s(X)) 24.42/8.43 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.43 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.43 from(ok(X)) -> ok(from(X)) 24.42/8.43 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.43 len(ok(X)) -> ok(len(X)) 24.42/8.43 top(mark(X)) -> top(proper(X)) 24.42/8.43 top(ok(X)) -> top(active(X)) 24.42/8.43 24.42/8.43 Types: 24.42/8.43 active :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 fst :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 0' :: 0':nil:mark:ok 24.42/8.43 mark :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 nil :: 0':nil:mark:ok 24.42/8.43 s :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 cons :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 from :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 add :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 len :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 proper :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 ok :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.43 top :: 0':nil:mark:ok -> top 24.42/8.43 hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok 24.42/8.43 hole_top2_0 :: top 24.42/8.43 gen_0':nil:mark:ok3_0 :: Nat -> 0':nil:mark:ok 24.42/8.43 24.42/8.43 24.42/8.43 Lemmas: 24.42/8.43 cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 24.42/8.43 fst(gen_0':nil:mark:ok3_0(+(1, n958_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n958_0) 24.42/8.43 from(gen_0':nil:mark:ok3_0(+(1, n2416_0))) -> *4_0, rt in Omega(n2416_0) 24.42/8.43 24.42/8.43 24.42/8.43 Generator Equations: 24.42/8.43 gen_0':nil:mark:ok3_0(0) <=> 0' 24.42/8.43 gen_0':nil:mark:ok3_0(+(x, 1)) <=> mark(gen_0':nil:mark:ok3_0(x)) 24.42/8.43 24.42/8.43 24.42/8.43 The following defined symbols remain to be analysed: 24.42/8.43 s, active, add, len, proper, top 24.42/8.43 24.42/8.43 They will be analysed ascendingly in the following order: 24.42/8.43 s < active 24.42/8.43 add < active 24.42/8.43 len < active 24.42/8.43 active < top 24.42/8.43 s < proper 24.42/8.43 add < proper 24.42/8.43 len < proper 24.42/8.43 proper < top 24.42/8.43 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (23) RewriteLemmaProof (LOWER BOUND(ID)) 24.42/8.43 Proved the following rewrite lemma: 24.42/8.43 add(gen_0':nil:mark:ok3_0(+(1, n3111_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n3111_0) 24.42/8.43 24.42/8.43 Induction Base: 24.42/8.43 add(gen_0':nil:mark:ok3_0(+(1, 0)), gen_0':nil:mark:ok3_0(b)) 24.42/8.43 24.42/8.43 Induction Step: 24.42/8.43 add(gen_0':nil:mark:ok3_0(+(1, +(n3111_0, 1))), gen_0':nil:mark:ok3_0(b)) ->_R^Omega(1) 24.42/8.43 mark(add(gen_0':nil:mark:ok3_0(+(1, n3111_0)), gen_0':nil:mark:ok3_0(b))) ->_IH 24.42/8.43 mark(*4_0) 24.42/8.43 24.42/8.43 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.42/8.43 ---------------------------------------- 24.42/8.43 24.42/8.43 (24) 24.42/8.43 Obligation: 24.42/8.43 TRS: 24.42/8.43 Rules: 24.42/8.43 active(fst(0', Z)) -> mark(nil) 24.42/8.43 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.43 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.43 active(add(0', X)) -> mark(X) 24.42/8.43 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.43 active(len(nil)) -> mark(0') 24.42/8.43 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.43 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.43 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.43 active(from(X)) -> from(active(X)) 24.42/8.43 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.43 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.43 active(len(X)) -> len(active(X)) 24.42/8.43 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.43 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.43 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.43 from(mark(X)) -> mark(from(X)) 24.42/8.43 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.43 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.43 len(mark(X)) -> mark(len(X)) 24.42/8.43 proper(0') -> ok(0') 24.42/8.43 proper(s(X)) -> s(proper(X)) 24.42/8.43 proper(nil) -> ok(nil) 24.42/8.43 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.43 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.43 proper(from(X)) -> from(proper(X)) 24.42/8.44 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.44 proper(len(X)) -> len(proper(X)) 24.42/8.44 s(ok(X)) -> ok(s(X)) 24.42/8.44 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.44 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.44 from(ok(X)) -> ok(from(X)) 24.42/8.44 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.44 len(ok(X)) -> ok(len(X)) 24.42/8.44 top(mark(X)) -> top(proper(X)) 24.42/8.44 top(ok(X)) -> top(active(X)) 24.42/8.44 24.42/8.44 Types: 24.42/8.44 active :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 fst :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 0' :: 0':nil:mark:ok 24.42/8.44 mark :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 nil :: 0':nil:mark:ok 24.42/8.44 s :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 cons :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 from :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 add :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 len :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 proper :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 ok :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 top :: 0':nil:mark:ok -> top 24.42/8.44 hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok 24.42/8.44 hole_top2_0 :: top 24.42/8.44 gen_0':nil:mark:ok3_0 :: Nat -> 0':nil:mark:ok 24.42/8.44 24.42/8.44 24.42/8.44 Lemmas: 24.42/8.44 cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 24.42/8.44 fst(gen_0':nil:mark:ok3_0(+(1, n958_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n958_0) 24.42/8.44 from(gen_0':nil:mark:ok3_0(+(1, n2416_0))) -> *4_0, rt in Omega(n2416_0) 24.42/8.44 add(gen_0':nil:mark:ok3_0(+(1, n3111_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n3111_0) 24.42/8.44 24.42/8.44 24.42/8.44 Generator Equations: 24.42/8.44 gen_0':nil:mark:ok3_0(0) <=> 0' 24.42/8.44 gen_0':nil:mark:ok3_0(+(x, 1)) <=> mark(gen_0':nil:mark:ok3_0(x)) 24.42/8.44 24.42/8.44 24.42/8.44 The following defined symbols remain to be analysed: 24.42/8.44 len, active, proper, top 24.42/8.44 24.42/8.44 They will be analysed ascendingly in the following order: 24.42/8.44 len < active 24.42/8.44 active < top 24.42/8.44 len < proper 24.42/8.44 proper < top 24.42/8.44 24.42/8.44 ---------------------------------------- 24.42/8.44 24.42/8.44 (25) RewriteLemmaProof (LOWER BOUND(ID)) 24.42/8.44 Proved the following rewrite lemma: 24.42/8.44 len(gen_0':nil:mark:ok3_0(+(1, n5083_0))) -> *4_0, rt in Omega(n5083_0) 24.42/8.44 24.42/8.44 Induction Base: 24.42/8.44 len(gen_0':nil:mark:ok3_0(+(1, 0))) 24.42/8.44 24.42/8.44 Induction Step: 24.42/8.44 len(gen_0':nil:mark:ok3_0(+(1, +(n5083_0, 1)))) ->_R^Omega(1) 24.42/8.44 mark(len(gen_0':nil:mark:ok3_0(+(1, n5083_0)))) ->_IH 24.42/8.44 mark(*4_0) 24.42/8.44 24.42/8.44 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.42/8.44 ---------------------------------------- 24.42/8.44 24.42/8.44 (26) 24.42/8.44 Obligation: 24.42/8.44 TRS: 24.42/8.44 Rules: 24.42/8.44 active(fst(0', Z)) -> mark(nil) 24.42/8.44 active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z))) 24.42/8.44 active(from(X)) -> mark(cons(X, from(s(X)))) 24.42/8.44 active(add(0', X)) -> mark(X) 24.42/8.44 active(add(s(X), Y)) -> mark(s(add(X, Y))) 24.42/8.44 active(len(nil)) -> mark(0') 24.42/8.44 active(len(cons(X, Z))) -> mark(s(len(Z))) 24.42/8.44 active(cons(X1, X2)) -> cons(active(X1), X2) 24.42/8.44 active(fst(X1, X2)) -> fst(active(X1), X2) 24.42/8.44 active(fst(X1, X2)) -> fst(X1, active(X2)) 24.42/8.44 active(from(X)) -> from(active(X)) 24.42/8.44 active(add(X1, X2)) -> add(active(X1), X2) 24.42/8.44 active(add(X1, X2)) -> add(X1, active(X2)) 24.42/8.44 active(len(X)) -> len(active(X)) 24.42/8.44 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.42/8.44 fst(mark(X1), X2) -> mark(fst(X1, X2)) 24.42/8.44 fst(X1, mark(X2)) -> mark(fst(X1, X2)) 24.42/8.44 from(mark(X)) -> mark(from(X)) 24.42/8.44 add(mark(X1), X2) -> mark(add(X1, X2)) 24.42/8.44 add(X1, mark(X2)) -> mark(add(X1, X2)) 24.42/8.44 len(mark(X)) -> mark(len(X)) 24.42/8.44 proper(0') -> ok(0') 24.42/8.44 proper(s(X)) -> s(proper(X)) 24.42/8.44 proper(nil) -> ok(nil) 24.42/8.44 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.42/8.44 proper(fst(X1, X2)) -> fst(proper(X1), proper(X2)) 24.42/8.44 proper(from(X)) -> from(proper(X)) 24.42/8.44 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 24.42/8.44 proper(len(X)) -> len(proper(X)) 24.42/8.44 s(ok(X)) -> ok(s(X)) 24.42/8.44 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.42/8.44 fst(ok(X1), ok(X2)) -> ok(fst(X1, X2)) 24.42/8.44 from(ok(X)) -> ok(from(X)) 24.42/8.44 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 24.42/8.44 len(ok(X)) -> ok(len(X)) 24.42/8.44 top(mark(X)) -> top(proper(X)) 24.42/8.44 top(ok(X)) -> top(active(X)) 24.42/8.44 24.42/8.44 Types: 24.42/8.44 active :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 fst :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 0' :: 0':nil:mark:ok 24.42/8.44 mark :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 nil :: 0':nil:mark:ok 24.42/8.44 s :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 cons :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 from :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 add :: 0':nil:mark:ok -> 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 len :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 proper :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 ok :: 0':nil:mark:ok -> 0':nil:mark:ok 24.42/8.44 top :: 0':nil:mark:ok -> top 24.42/8.44 hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok 24.42/8.44 hole_top2_0 :: top 24.42/8.44 gen_0':nil:mark:ok3_0 :: Nat -> 0':nil:mark:ok 24.42/8.44 24.42/8.44 24.42/8.44 Lemmas: 24.42/8.44 cons(gen_0':nil:mark:ok3_0(+(1, n5_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 24.42/8.44 fst(gen_0':nil:mark:ok3_0(+(1, n958_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n958_0) 24.42/8.44 from(gen_0':nil:mark:ok3_0(+(1, n2416_0))) -> *4_0, rt in Omega(n2416_0) 24.42/8.44 add(gen_0':nil:mark:ok3_0(+(1, n3111_0)), gen_0':nil:mark:ok3_0(b)) -> *4_0, rt in Omega(n3111_0) 24.42/8.44 len(gen_0':nil:mark:ok3_0(+(1, n5083_0))) -> *4_0, rt in Omega(n5083_0) 24.42/8.44 24.42/8.44 24.42/8.44 Generator Equations: 24.42/8.44 gen_0':nil:mark:ok3_0(0) <=> 0' 24.42/8.44 gen_0':nil:mark:ok3_0(+(x, 1)) <=> mark(gen_0':nil:mark:ok3_0(x)) 24.42/8.44 24.42/8.44 24.42/8.44 The following defined symbols remain to be analysed: 24.42/8.44 active, proper, top 24.42/8.44 24.42/8.44 They will be analysed ascendingly in the following order: 24.42/8.44 active < top 24.42/8.44 proper < top 24.65/8.47 EOF