25.68/7.83 WORST_CASE(Omega(n^1), O(n^1)) 25.68/7.85 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 25.68/7.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.68/7.85 25.68/7.85 25.68/7.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.68/7.85 25.68/7.85 (0) CpxTRS 25.68/7.85 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 25.68/7.85 (2) CpxTRS 25.68/7.85 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 25.68/7.85 (4) CpxTRS 25.68/7.85 (5) CpxTrsMatchBoundsTAProof [FINISHED, 75 ms] 25.68/7.85 (6) BOUNDS(1, n^1) 25.68/7.85 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 25.68/7.85 (8) CpxTRS 25.68/7.85 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 25.68/7.85 (10) typed CpxTrs 25.68/7.85 (11) OrderProof [LOWER BOUND(ID), 0 ms] 25.68/7.85 (12) typed CpxTrs 25.68/7.85 (13) RewriteLemmaProof [LOWER BOUND(ID), 467 ms] 25.68/7.85 (14) BEST 25.68/7.85 (15) proven lower bound 25.68/7.85 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 25.68/7.85 (17) BOUNDS(n^1, INF) 25.68/7.85 (18) typed CpxTrs 25.68/7.85 (19) RewriteLemmaProof [LOWER BOUND(ID), 125 ms] 25.68/7.85 (20) typed CpxTrs 25.68/7.85 (21) RewriteLemmaProof [LOWER BOUND(ID), 126 ms] 25.68/7.85 (22) typed CpxTrs 25.68/7.85 (23) RewriteLemmaProof [LOWER BOUND(ID), 135 ms] 25.68/7.85 (24) typed CpxTrs 25.68/7.85 (25) RewriteLemmaProof [LOWER BOUND(ID), 54 ms] 25.68/7.85 (26) typed CpxTrs 25.68/7.85 (27) RewriteLemmaProof [LOWER BOUND(ID), 125 ms] 25.68/7.85 (28) typed CpxTrs 25.68/7.85 (29) RewriteLemmaProof [LOWER BOUND(ID), 63 ms] 25.68/7.85 (30) typed CpxTrs 25.68/7.85 (31) RewriteLemmaProof [LOWER BOUND(ID), 134 ms] 25.68/7.85 (32) typed CpxTrs 25.68/7.85 25.68/7.85 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (0) 25.68/7.85 Obligation: 25.68/7.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.68/7.85 25.68/7.85 25.68/7.85 The TRS R consists of the following rules: 25.68/7.85 25.68/7.85 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.85 active(sqr(0)) -> mark(0) 25.68/7.85 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.85 active(dbl(0)) -> mark(0) 25.68/7.85 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.85 active(add(0, X)) -> mark(X) 25.68/7.85 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.85 active(first(0, X)) -> mark(nil) 25.68/7.85 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.85 active(terms(X)) -> terms(active(X)) 25.68/7.85 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.85 active(recip(X)) -> recip(active(X)) 25.68/7.85 active(sqr(X)) -> sqr(active(X)) 25.68/7.85 active(s(X)) -> s(active(X)) 25.68/7.85 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.85 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.85 active(dbl(X)) -> dbl(active(X)) 25.68/7.85 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.85 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.85 terms(mark(X)) -> mark(terms(X)) 25.68/7.85 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.85 recip(mark(X)) -> mark(recip(X)) 25.68/7.85 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.85 s(mark(X)) -> mark(s(X)) 25.68/7.85 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.85 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.85 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.85 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.85 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.85 proper(terms(X)) -> terms(proper(X)) 25.68/7.85 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.85 proper(recip(X)) -> recip(proper(X)) 25.68/7.85 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.85 proper(s(X)) -> s(proper(X)) 25.68/7.85 proper(0) -> ok(0) 25.68/7.85 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.85 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.85 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.85 proper(nil) -> ok(nil) 25.68/7.85 terms(ok(X)) -> ok(terms(X)) 25.68/7.85 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.85 recip(ok(X)) -> ok(recip(X)) 25.68/7.85 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.85 s(ok(X)) -> ok(s(X)) 25.68/7.85 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.85 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.85 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.85 top(mark(X)) -> top(proper(X)) 25.68/7.85 top(ok(X)) -> top(active(X)) 25.68/7.85 25.68/7.85 S is empty. 25.68/7.85 Rewrite Strategy: FULL 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 25.68/7.85 The following defined symbols can occur below the 0th argument of top: proper, active 25.68/7.85 The following defined symbols can occur below the 0th argument of proper: proper, active 25.68/7.85 The following defined symbols can occur below the 0th argument of active: proper, active 25.68/7.85 25.68/7.85 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 25.68/7.85 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.85 active(sqr(0)) -> mark(0) 25.68/7.85 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.85 active(dbl(0)) -> mark(0) 25.68/7.85 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.85 active(add(0, X)) -> mark(X) 25.68/7.85 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.85 active(first(0, X)) -> mark(nil) 25.68/7.85 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.85 active(terms(X)) -> terms(active(X)) 25.68/7.85 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.85 active(recip(X)) -> recip(active(X)) 25.68/7.85 active(sqr(X)) -> sqr(active(X)) 25.68/7.85 active(s(X)) -> s(active(X)) 25.68/7.85 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.85 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.85 active(dbl(X)) -> dbl(active(X)) 25.68/7.85 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.85 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.85 proper(terms(X)) -> terms(proper(X)) 25.68/7.85 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.85 proper(recip(X)) -> recip(proper(X)) 25.68/7.85 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.85 proper(s(X)) -> s(proper(X)) 25.68/7.85 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.85 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.85 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.85 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (2) 25.68/7.85 Obligation: 25.68/7.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 25.68/7.85 25.68/7.85 25.68/7.85 The TRS R consists of the following rules: 25.68/7.85 25.68/7.85 terms(mark(X)) -> mark(terms(X)) 25.68/7.85 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.85 recip(mark(X)) -> mark(recip(X)) 25.68/7.85 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.85 s(mark(X)) -> mark(s(X)) 25.68/7.85 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.85 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.85 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.85 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.85 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.85 proper(0) -> ok(0) 25.68/7.85 proper(nil) -> ok(nil) 25.68/7.85 terms(ok(X)) -> ok(terms(X)) 25.68/7.85 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.85 recip(ok(X)) -> ok(recip(X)) 25.68/7.85 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.85 s(ok(X)) -> ok(s(X)) 25.68/7.85 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.85 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.85 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.85 top(mark(X)) -> top(proper(X)) 25.68/7.85 top(ok(X)) -> top(active(X)) 25.68/7.85 25.68/7.85 S is empty. 25.68/7.85 Rewrite Strategy: FULL 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 25.68/7.85 transformed relative TRS to TRS 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (4) 25.68/7.85 Obligation: 25.68/7.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 25.68/7.85 25.68/7.85 25.68/7.85 The TRS R consists of the following rules: 25.68/7.85 25.68/7.85 terms(mark(X)) -> mark(terms(X)) 25.68/7.85 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.85 recip(mark(X)) -> mark(recip(X)) 25.68/7.85 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.85 s(mark(X)) -> mark(s(X)) 25.68/7.85 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.85 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.85 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.85 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.85 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.85 proper(0) -> ok(0) 25.68/7.85 proper(nil) -> ok(nil) 25.68/7.85 terms(ok(X)) -> ok(terms(X)) 25.68/7.85 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.85 recip(ok(X)) -> ok(recip(X)) 25.68/7.85 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.85 s(ok(X)) -> ok(s(X)) 25.68/7.85 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.85 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.85 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.85 top(mark(X)) -> top(proper(X)) 25.68/7.85 top(ok(X)) -> top(active(X)) 25.68/7.85 25.68/7.85 S is empty. 25.68/7.85 Rewrite Strategy: FULL 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (5) CpxTrsMatchBoundsTAProof (FINISHED) 25.68/7.85 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. 25.68/7.85 25.68/7.85 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 25.68/7.85 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] 25.68/7.85 transitions: 25.68/7.85 mark0(0) -> 0 25.68/7.85 00() -> 0 25.68/7.85 ok0(0) -> 0 25.68/7.85 nil0() -> 0 25.68/7.85 active0(0) -> 0 25.68/7.85 terms0(0) -> 1 25.68/7.85 cons0(0, 0) -> 2 25.68/7.85 recip0(0) -> 3 25.68/7.85 sqr0(0) -> 4 25.68/7.85 s0(0) -> 5 25.68/7.85 add0(0, 0) -> 6 25.68/7.85 dbl0(0) -> 7 25.68/7.85 first0(0, 0) -> 8 25.68/7.85 proper0(0) -> 9 25.68/7.85 top0(0) -> 10 25.68/7.85 terms1(0) -> 11 25.68/7.85 mark1(11) -> 1 25.68/7.85 cons1(0, 0) -> 12 25.68/7.85 mark1(12) -> 2 25.68/7.85 recip1(0) -> 13 25.68/7.85 mark1(13) -> 3 25.68/7.85 sqr1(0) -> 14 25.68/7.85 mark1(14) -> 4 25.68/7.85 s1(0) -> 15 25.68/7.85 mark1(15) -> 5 25.68/7.85 add1(0, 0) -> 16 25.68/7.85 mark1(16) -> 6 25.68/7.85 dbl1(0) -> 17 25.68/7.85 mark1(17) -> 7 25.68/7.85 first1(0, 0) -> 18 25.68/7.85 mark1(18) -> 8 25.68/7.85 01() -> 19 25.68/7.85 ok1(19) -> 9 25.68/7.85 nil1() -> 20 25.68/7.85 ok1(20) -> 9 25.68/7.85 terms1(0) -> 21 25.68/7.85 ok1(21) -> 1 25.68/7.85 cons1(0, 0) -> 22 25.68/7.85 ok1(22) -> 2 25.68/7.85 recip1(0) -> 23 25.68/7.85 ok1(23) -> 3 25.68/7.85 sqr1(0) -> 24 25.68/7.85 ok1(24) -> 4 25.68/7.85 s1(0) -> 25 25.68/7.85 ok1(25) -> 5 25.68/7.85 add1(0, 0) -> 26 25.68/7.85 ok1(26) -> 6 25.68/7.85 dbl1(0) -> 27 25.68/7.85 ok1(27) -> 7 25.68/7.85 first1(0, 0) -> 28 25.68/7.85 ok1(28) -> 8 25.68/7.85 proper1(0) -> 29 25.68/7.85 top1(29) -> 10 25.68/7.85 active1(0) -> 30 25.68/7.85 top1(30) -> 10 25.68/7.85 mark1(11) -> 11 25.68/7.85 mark1(11) -> 21 25.68/7.85 mark1(12) -> 12 25.68/7.85 mark1(12) -> 22 25.68/7.85 mark1(13) -> 13 25.68/7.85 mark1(13) -> 23 25.68/7.85 mark1(14) -> 14 25.68/7.85 mark1(14) -> 24 25.68/7.85 mark1(15) -> 15 25.68/7.85 mark1(15) -> 25 25.68/7.85 mark1(16) -> 16 25.68/7.85 mark1(16) -> 26 25.68/7.85 mark1(17) -> 17 25.68/7.85 mark1(17) -> 27 25.68/7.85 mark1(18) -> 18 25.68/7.85 mark1(18) -> 28 25.68/7.85 ok1(19) -> 29 25.68/7.85 ok1(20) -> 29 25.68/7.85 ok1(21) -> 11 25.68/7.85 ok1(21) -> 21 25.68/7.85 ok1(22) -> 12 25.68/7.85 ok1(22) -> 22 25.68/7.85 ok1(23) -> 13 25.68/7.85 ok1(23) -> 23 25.68/7.85 ok1(24) -> 14 25.68/7.85 ok1(24) -> 24 25.68/7.85 ok1(25) -> 15 25.68/7.85 ok1(25) -> 25 25.68/7.85 ok1(26) -> 16 25.68/7.85 ok1(26) -> 26 25.68/7.85 ok1(27) -> 17 25.68/7.85 ok1(27) -> 27 25.68/7.85 ok1(28) -> 18 25.68/7.85 ok1(28) -> 28 25.68/7.85 active2(19) -> 31 25.68/7.85 top2(31) -> 10 25.68/7.85 active2(20) -> 31 25.68/7.85 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (6) 25.68/7.85 BOUNDS(1, n^1) 25.68/7.85 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 25.68/7.85 Renamed function symbols to avoid clashes with predefined symbol. 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (8) 25.68/7.85 Obligation: 25.68/7.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 25.68/7.85 25.68/7.85 25.68/7.85 The TRS R consists of the following rules: 25.68/7.85 25.68/7.85 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.85 active(sqr(0')) -> mark(0') 25.68/7.85 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.85 active(dbl(0')) -> mark(0') 25.68/7.85 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.85 active(add(0', X)) -> mark(X) 25.68/7.85 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.85 active(first(0', X)) -> mark(nil) 25.68/7.85 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.85 active(terms(X)) -> terms(active(X)) 25.68/7.85 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.85 active(recip(X)) -> recip(active(X)) 25.68/7.85 active(sqr(X)) -> sqr(active(X)) 25.68/7.85 active(s(X)) -> s(active(X)) 25.68/7.85 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.85 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.85 active(dbl(X)) -> dbl(active(X)) 25.68/7.85 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.85 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.85 terms(mark(X)) -> mark(terms(X)) 25.68/7.85 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.85 recip(mark(X)) -> mark(recip(X)) 25.68/7.85 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.85 s(mark(X)) -> mark(s(X)) 25.68/7.85 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.85 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.85 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.85 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.85 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.85 proper(terms(X)) -> terms(proper(X)) 25.68/7.85 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.85 proper(recip(X)) -> recip(proper(X)) 25.68/7.85 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.85 proper(s(X)) -> s(proper(X)) 25.68/7.85 proper(0') -> ok(0') 25.68/7.85 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.85 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.85 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.85 proper(nil) -> ok(nil) 25.68/7.85 terms(ok(X)) -> ok(terms(X)) 25.68/7.85 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.85 recip(ok(X)) -> ok(recip(X)) 25.68/7.85 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.85 s(ok(X)) -> ok(s(X)) 25.68/7.85 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.85 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.85 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.85 top(mark(X)) -> top(proper(X)) 25.68/7.85 top(ok(X)) -> top(active(X)) 25.68/7.85 25.68/7.85 S is empty. 25.68/7.85 Rewrite Strategy: FULL 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 25.68/7.85 Infered types. 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (10) 25.68/7.85 Obligation: 25.68/7.85 TRS: 25.68/7.85 Rules: 25.68/7.85 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.85 active(sqr(0')) -> mark(0') 25.68/7.85 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.85 active(dbl(0')) -> mark(0') 25.68/7.85 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.85 active(add(0', X)) -> mark(X) 25.68/7.85 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.85 active(first(0', X)) -> mark(nil) 25.68/7.85 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.85 active(terms(X)) -> terms(active(X)) 25.68/7.85 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.85 active(recip(X)) -> recip(active(X)) 25.68/7.85 active(sqr(X)) -> sqr(active(X)) 25.68/7.85 active(s(X)) -> s(active(X)) 25.68/7.85 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.85 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.85 active(dbl(X)) -> dbl(active(X)) 25.68/7.85 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.85 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.85 terms(mark(X)) -> mark(terms(X)) 25.68/7.85 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.85 recip(mark(X)) -> mark(recip(X)) 25.68/7.85 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.85 s(mark(X)) -> mark(s(X)) 25.68/7.85 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.85 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.85 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.85 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.85 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.85 proper(terms(X)) -> terms(proper(X)) 25.68/7.85 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.85 proper(recip(X)) -> recip(proper(X)) 25.68/7.85 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.85 proper(s(X)) -> s(proper(X)) 25.68/7.85 proper(0') -> ok(0') 25.68/7.85 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.85 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.85 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.85 proper(nil) -> ok(nil) 25.68/7.85 terms(ok(X)) -> ok(terms(X)) 25.68/7.85 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.85 recip(ok(X)) -> ok(recip(X)) 25.68/7.85 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.85 s(ok(X)) -> ok(s(X)) 25.68/7.85 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.85 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.85 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.85 top(mark(X)) -> top(proper(X)) 25.68/7.85 top(ok(X)) -> top(active(X)) 25.68/7.85 25.68/7.85 Types: 25.68/7.85 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 0' :: mark:0':nil:ok 25.68/7.85 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 nil :: mark:0':nil:ok 25.68/7.85 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 top :: mark:0':nil:ok -> top 25.68/7.85 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.85 hole_top2_0 :: top 25.68/7.85 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.85 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (11) OrderProof (LOWER BOUND(ID)) 25.68/7.85 Heuristically decided to analyse the following defined symbols: 25.68/7.85 active, cons, recip, sqr, terms, s, add, dbl, first, proper, top 25.68/7.85 25.68/7.85 They will be analysed ascendingly in the following order: 25.68/7.85 cons < active 25.68/7.85 recip < active 25.68/7.85 sqr < active 25.68/7.85 terms < active 25.68/7.85 s < active 25.68/7.85 add < active 25.68/7.85 dbl < active 25.68/7.85 first < active 25.68/7.85 active < top 25.68/7.85 cons < proper 25.68/7.85 recip < proper 25.68/7.85 sqr < proper 25.68/7.85 terms < proper 25.68/7.85 s < proper 25.68/7.85 add < proper 25.68/7.85 dbl < proper 25.68/7.85 first < proper 25.68/7.85 proper < top 25.68/7.85 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (12) 25.68/7.85 Obligation: 25.68/7.85 TRS: 25.68/7.85 Rules: 25.68/7.85 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.85 active(sqr(0')) -> mark(0') 25.68/7.85 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.85 active(dbl(0')) -> mark(0') 25.68/7.85 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.85 active(add(0', X)) -> mark(X) 25.68/7.85 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.85 active(first(0', X)) -> mark(nil) 25.68/7.85 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.85 active(terms(X)) -> terms(active(X)) 25.68/7.85 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.85 active(recip(X)) -> recip(active(X)) 25.68/7.85 active(sqr(X)) -> sqr(active(X)) 25.68/7.85 active(s(X)) -> s(active(X)) 25.68/7.85 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.85 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.85 active(dbl(X)) -> dbl(active(X)) 25.68/7.85 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.85 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.85 terms(mark(X)) -> mark(terms(X)) 25.68/7.85 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.85 recip(mark(X)) -> mark(recip(X)) 25.68/7.85 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.85 s(mark(X)) -> mark(s(X)) 25.68/7.85 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.85 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.85 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.85 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.85 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.85 proper(terms(X)) -> terms(proper(X)) 25.68/7.85 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.85 proper(recip(X)) -> recip(proper(X)) 25.68/7.85 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.85 proper(s(X)) -> s(proper(X)) 25.68/7.85 proper(0') -> ok(0') 25.68/7.85 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.85 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.85 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.85 proper(nil) -> ok(nil) 25.68/7.85 terms(ok(X)) -> ok(terms(X)) 25.68/7.85 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.85 recip(ok(X)) -> ok(recip(X)) 25.68/7.85 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.85 s(ok(X)) -> ok(s(X)) 25.68/7.85 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.85 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.85 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.85 top(mark(X)) -> top(proper(X)) 25.68/7.85 top(ok(X)) -> top(active(X)) 25.68/7.85 25.68/7.85 Types: 25.68/7.85 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 0' :: mark:0':nil:ok 25.68/7.85 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 nil :: mark:0':nil:ok 25.68/7.85 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 top :: mark:0':nil:ok -> top 25.68/7.85 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.85 hole_top2_0 :: top 25.68/7.85 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.85 25.68/7.85 25.68/7.85 Generator Equations: 25.68/7.85 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.85 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.85 25.68/7.85 25.68/7.85 The following defined symbols remain to be analysed: 25.68/7.85 cons, active, recip, sqr, terms, s, add, dbl, first, proper, top 25.68/7.85 25.68/7.85 They will be analysed ascendingly in the following order: 25.68/7.85 cons < active 25.68/7.85 recip < active 25.68/7.85 sqr < active 25.68/7.85 terms < active 25.68/7.85 s < active 25.68/7.85 add < active 25.68/7.85 dbl < active 25.68/7.85 first < active 25.68/7.85 active < top 25.68/7.85 cons < proper 25.68/7.85 recip < proper 25.68/7.85 sqr < proper 25.68/7.85 terms < proper 25.68/7.85 s < proper 25.68/7.85 add < proper 25.68/7.85 dbl < proper 25.68/7.85 first < proper 25.68/7.85 proper < top 25.68/7.85 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (13) RewriteLemmaProof (LOWER BOUND(ID)) 25.68/7.85 Proved the following rewrite lemma: 25.68/7.85 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 25.68/7.85 25.68/7.85 Induction Base: 25.68/7.85 cons(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b)) 25.68/7.85 25.68/7.85 Induction Step: 25.68/7.85 cons(gen_mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1) 25.68/7.85 mark(cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b))) ->_IH 25.68/7.85 mark(*4_0) 25.68/7.85 25.68/7.85 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (14) 25.68/7.85 Complex Obligation (BEST) 25.68/7.85 25.68/7.85 ---------------------------------------- 25.68/7.85 25.68/7.85 (15) 25.68/7.85 Obligation: 25.68/7.85 Proved the lower bound n^1 for the following obligation: 25.68/7.85 25.68/7.85 TRS: 25.68/7.85 Rules: 25.68/7.85 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.85 active(sqr(0')) -> mark(0') 25.68/7.85 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.85 active(dbl(0')) -> mark(0') 25.68/7.85 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.85 active(add(0', X)) -> mark(X) 25.68/7.85 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.85 active(first(0', X)) -> mark(nil) 25.68/7.85 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.85 active(terms(X)) -> terms(active(X)) 25.68/7.85 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.85 active(recip(X)) -> recip(active(X)) 25.68/7.85 active(sqr(X)) -> sqr(active(X)) 25.68/7.85 active(s(X)) -> s(active(X)) 25.68/7.85 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.85 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.85 active(dbl(X)) -> dbl(active(X)) 25.68/7.85 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.85 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.85 terms(mark(X)) -> mark(terms(X)) 25.68/7.85 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.85 recip(mark(X)) -> mark(recip(X)) 25.68/7.85 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.85 s(mark(X)) -> mark(s(X)) 25.68/7.85 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.85 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.85 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.85 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.85 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.85 proper(terms(X)) -> terms(proper(X)) 25.68/7.85 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.85 proper(recip(X)) -> recip(proper(X)) 25.68/7.85 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.85 proper(s(X)) -> s(proper(X)) 25.68/7.85 proper(0') -> ok(0') 25.68/7.85 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.85 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.85 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.85 proper(nil) -> ok(nil) 25.68/7.85 terms(ok(X)) -> ok(terms(X)) 25.68/7.85 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.85 recip(ok(X)) -> ok(recip(X)) 25.68/7.85 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.85 s(ok(X)) -> ok(s(X)) 25.68/7.85 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.85 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.85 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.85 top(mark(X)) -> top(proper(X)) 25.68/7.85 top(ok(X)) -> top(active(X)) 25.68/7.85 25.68/7.85 Types: 25.68/7.85 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 0' :: mark:0':nil:ok 25.68/7.85 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 nil :: mark:0':nil:ok 25.68/7.85 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.85 top :: mark:0':nil:ok -> top 25.68/7.85 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.85 hole_top2_0 :: top 25.68/7.85 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.85 25.68/7.85 25.68/7.85 Generator Equations: 25.68/7.85 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.85 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.85 25.68/7.85 25.68/7.85 The following defined symbols remain to be analysed: 25.68/7.85 cons, active, recip, sqr, terms, s, add, dbl, first, proper, top 25.68/7.85 25.68/7.85 They will be analysed ascendingly in the following order: 25.68/7.86 cons < active 25.68/7.86 recip < active 25.68/7.86 sqr < active 25.68/7.86 terms < active 25.68/7.86 s < active 25.68/7.86 add < active 25.68/7.86 dbl < active 25.68/7.86 first < active 25.68/7.86 active < top 25.68/7.86 cons < proper 25.68/7.86 recip < proper 25.68/7.86 sqr < proper 25.68/7.86 terms < proper 25.68/7.86 s < proper 25.68/7.86 add < proper 25.68/7.86 dbl < proper 25.68/7.86 first < proper 25.68/7.86 proper < top 25.68/7.86 25.68/7.86 ---------------------------------------- 25.68/7.86 25.68/7.86 (16) LowerBoundPropagationProof (FINISHED) 25.68/7.86 Propagated lower bound. 25.68/7.86 ---------------------------------------- 25.68/7.86 25.68/7.86 (17) 25.68/7.86 BOUNDS(n^1, INF) 25.68/7.86 25.68/7.86 ---------------------------------------- 25.68/7.86 25.68/7.86 (18) 25.68/7.86 Obligation: 25.68/7.86 TRS: 25.68/7.86 Rules: 25.68/7.86 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.86 active(sqr(0')) -> mark(0') 25.68/7.86 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.86 active(dbl(0')) -> mark(0') 25.68/7.86 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.86 active(add(0', X)) -> mark(X) 25.68/7.86 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.86 active(first(0', X)) -> mark(nil) 25.68/7.86 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.86 active(terms(X)) -> terms(active(X)) 25.68/7.86 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.86 active(recip(X)) -> recip(active(X)) 25.68/7.86 active(sqr(X)) -> sqr(active(X)) 25.68/7.86 active(s(X)) -> s(active(X)) 25.68/7.86 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.86 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.86 active(dbl(X)) -> dbl(active(X)) 25.68/7.86 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.86 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.86 terms(mark(X)) -> mark(terms(X)) 25.68/7.86 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.86 recip(mark(X)) -> mark(recip(X)) 25.68/7.86 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.86 s(mark(X)) -> mark(s(X)) 25.68/7.86 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.86 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.86 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.86 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.86 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.86 proper(terms(X)) -> terms(proper(X)) 25.68/7.86 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.86 proper(recip(X)) -> recip(proper(X)) 25.68/7.86 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.86 proper(s(X)) -> s(proper(X)) 25.68/7.86 proper(0') -> ok(0') 25.68/7.86 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.86 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.86 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.86 proper(nil) -> ok(nil) 25.68/7.86 terms(ok(X)) -> ok(terms(X)) 25.68/7.86 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.86 recip(ok(X)) -> ok(recip(X)) 25.68/7.86 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.86 s(ok(X)) -> ok(s(X)) 25.68/7.86 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.86 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.86 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.86 top(mark(X)) -> top(proper(X)) 25.68/7.86 top(ok(X)) -> top(active(X)) 25.68/7.86 25.68/7.86 Types: 25.68/7.86 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 0' :: mark:0':nil:ok 25.68/7.86 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 nil :: mark:0':nil:ok 25.68/7.86 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.86 top :: mark:0':nil:ok -> top 25.68/7.86 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.86 hole_top2_0 :: top 25.68/7.86 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.86 25.68/7.86 25.68/7.86 Lemmas: 25.68/7.86 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 25.68/7.87 25.68/7.87 25.68/7.87 Generator Equations: 25.68/7.87 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.87 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.87 25.68/7.87 25.68/7.87 The following defined symbols remain to be analysed: 25.68/7.87 recip, active, sqr, terms, s, add, dbl, first, proper, top 25.68/7.87 25.68/7.87 They will be analysed ascendingly in the following order: 25.68/7.87 recip < active 25.68/7.87 sqr < active 25.68/7.87 terms < active 25.68/7.87 s < active 25.68/7.87 add < active 25.68/7.87 dbl < active 25.68/7.87 first < active 25.68/7.87 active < top 25.68/7.87 recip < proper 25.68/7.87 sqr < proper 25.68/7.87 terms < proper 25.68/7.87 s < proper 25.68/7.87 add < proper 25.68/7.87 dbl < proper 25.68/7.87 first < proper 25.68/7.87 proper < top 25.68/7.87 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (19) RewriteLemmaProof (LOWER BOUND(ID)) 25.68/7.87 Proved the following rewrite lemma: 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0) 25.68/7.87 25.68/7.87 Induction Base: 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, 0))) 25.68/7.87 25.68/7.87 Induction Step: 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, +(n1126_0, 1)))) ->_R^Omega(1) 25.68/7.87 mark(recip(gen_mark:0':nil:ok3_0(+(1, n1126_0)))) ->_IH 25.68/7.87 mark(*4_0) 25.68/7.87 25.68/7.87 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (20) 25.68/7.87 Obligation: 25.68/7.87 TRS: 25.68/7.87 Rules: 25.68/7.87 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.87 active(sqr(0')) -> mark(0') 25.68/7.87 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.87 active(dbl(0')) -> mark(0') 25.68/7.87 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.87 active(add(0', X)) -> mark(X) 25.68/7.87 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.87 active(first(0', X)) -> mark(nil) 25.68/7.87 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.87 active(terms(X)) -> terms(active(X)) 25.68/7.87 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.87 active(recip(X)) -> recip(active(X)) 25.68/7.87 active(sqr(X)) -> sqr(active(X)) 25.68/7.87 active(s(X)) -> s(active(X)) 25.68/7.87 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.87 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.87 active(dbl(X)) -> dbl(active(X)) 25.68/7.87 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.87 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.87 terms(mark(X)) -> mark(terms(X)) 25.68/7.87 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.87 recip(mark(X)) -> mark(recip(X)) 25.68/7.87 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.87 s(mark(X)) -> mark(s(X)) 25.68/7.87 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.87 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.87 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.87 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.87 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.87 proper(terms(X)) -> terms(proper(X)) 25.68/7.87 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.87 proper(recip(X)) -> recip(proper(X)) 25.68/7.87 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.87 proper(s(X)) -> s(proper(X)) 25.68/7.87 proper(0') -> ok(0') 25.68/7.87 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.87 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.87 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.87 proper(nil) -> ok(nil) 25.68/7.87 terms(ok(X)) -> ok(terms(X)) 25.68/7.87 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.87 recip(ok(X)) -> ok(recip(X)) 25.68/7.87 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.87 s(ok(X)) -> ok(s(X)) 25.68/7.87 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.87 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.87 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.87 top(mark(X)) -> top(proper(X)) 25.68/7.87 top(ok(X)) -> top(active(X)) 25.68/7.87 25.68/7.87 Types: 25.68/7.87 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 0' :: mark:0':nil:ok 25.68/7.87 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 nil :: mark:0':nil:ok 25.68/7.87 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 top :: mark:0':nil:ok -> top 25.68/7.87 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.87 hole_top2_0 :: top 25.68/7.87 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.87 25.68/7.87 25.68/7.87 Lemmas: 25.68/7.87 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0) 25.68/7.87 25.68/7.87 25.68/7.87 Generator Equations: 25.68/7.87 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.87 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.87 25.68/7.87 25.68/7.87 The following defined symbols remain to be analysed: 25.68/7.87 sqr, active, terms, s, add, dbl, first, proper, top 25.68/7.87 25.68/7.87 They will be analysed ascendingly in the following order: 25.68/7.87 sqr < active 25.68/7.87 terms < active 25.68/7.87 s < active 25.68/7.87 add < active 25.68/7.87 dbl < active 25.68/7.87 first < active 25.68/7.87 active < top 25.68/7.87 sqr < proper 25.68/7.87 terms < proper 25.68/7.87 s < proper 25.68/7.87 add < proper 25.68/7.87 dbl < proper 25.68/7.87 first < proper 25.68/7.87 proper < top 25.68/7.87 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (21) RewriteLemmaProof (LOWER BOUND(ID)) 25.68/7.87 Proved the following rewrite lemma: 25.68/7.87 sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0) 25.68/7.87 25.68/7.87 Induction Base: 25.68/7.87 sqr(gen_mark:0':nil:ok3_0(+(1, 0))) 25.68/7.87 25.68/7.87 Induction Step: 25.68/7.87 sqr(gen_mark:0':nil:ok3_0(+(1, +(n1690_0, 1)))) ->_R^Omega(1) 25.68/7.87 mark(sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0)))) ->_IH 25.68/7.87 mark(*4_0) 25.68/7.87 25.68/7.87 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (22) 25.68/7.87 Obligation: 25.68/7.87 TRS: 25.68/7.87 Rules: 25.68/7.87 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.87 active(sqr(0')) -> mark(0') 25.68/7.87 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.87 active(dbl(0')) -> mark(0') 25.68/7.87 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.87 active(add(0', X)) -> mark(X) 25.68/7.87 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.87 active(first(0', X)) -> mark(nil) 25.68/7.87 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.87 active(terms(X)) -> terms(active(X)) 25.68/7.87 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.87 active(recip(X)) -> recip(active(X)) 25.68/7.87 active(sqr(X)) -> sqr(active(X)) 25.68/7.87 active(s(X)) -> s(active(X)) 25.68/7.87 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.87 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.87 active(dbl(X)) -> dbl(active(X)) 25.68/7.87 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.87 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.87 terms(mark(X)) -> mark(terms(X)) 25.68/7.87 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.87 recip(mark(X)) -> mark(recip(X)) 25.68/7.87 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.87 s(mark(X)) -> mark(s(X)) 25.68/7.87 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.87 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.87 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.87 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.87 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.87 proper(terms(X)) -> terms(proper(X)) 25.68/7.87 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.87 proper(recip(X)) -> recip(proper(X)) 25.68/7.87 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.87 proper(s(X)) -> s(proper(X)) 25.68/7.87 proper(0') -> ok(0') 25.68/7.87 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.87 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.87 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.87 proper(nil) -> ok(nil) 25.68/7.87 terms(ok(X)) -> ok(terms(X)) 25.68/7.87 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.87 recip(ok(X)) -> ok(recip(X)) 25.68/7.87 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.87 s(ok(X)) -> ok(s(X)) 25.68/7.87 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.87 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.87 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.87 top(mark(X)) -> top(proper(X)) 25.68/7.87 top(ok(X)) -> top(active(X)) 25.68/7.87 25.68/7.87 Types: 25.68/7.87 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 0' :: mark:0':nil:ok 25.68/7.87 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 nil :: mark:0':nil:ok 25.68/7.87 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 top :: mark:0':nil:ok -> top 25.68/7.87 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.87 hole_top2_0 :: top 25.68/7.87 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.87 25.68/7.87 25.68/7.87 Lemmas: 25.68/7.87 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0) 25.68/7.87 sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0) 25.68/7.87 25.68/7.87 25.68/7.87 Generator Equations: 25.68/7.87 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.87 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.87 25.68/7.87 25.68/7.87 The following defined symbols remain to be analysed: 25.68/7.87 terms, active, s, add, dbl, first, proper, top 25.68/7.87 25.68/7.87 They will be analysed ascendingly in the following order: 25.68/7.87 terms < active 25.68/7.87 s < active 25.68/7.87 add < active 25.68/7.87 dbl < active 25.68/7.87 first < active 25.68/7.87 active < top 25.68/7.87 terms < proper 25.68/7.87 s < proper 25.68/7.87 add < proper 25.68/7.87 dbl < proper 25.68/7.87 first < proper 25.68/7.87 proper < top 25.68/7.87 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (23) RewriteLemmaProof (LOWER BOUND(ID)) 25.68/7.87 Proved the following rewrite lemma: 25.68/7.87 terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0) 25.68/7.87 25.68/7.87 Induction Base: 25.68/7.87 terms(gen_mark:0':nil:ok3_0(+(1, 0))) 25.68/7.87 25.68/7.87 Induction Step: 25.68/7.87 terms(gen_mark:0':nil:ok3_0(+(1, +(n2355_0, 1)))) ->_R^Omega(1) 25.68/7.87 mark(terms(gen_mark:0':nil:ok3_0(+(1, n2355_0)))) ->_IH 25.68/7.87 mark(*4_0) 25.68/7.87 25.68/7.87 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (24) 25.68/7.87 Obligation: 25.68/7.87 TRS: 25.68/7.87 Rules: 25.68/7.87 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.87 active(sqr(0')) -> mark(0') 25.68/7.87 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.87 active(dbl(0')) -> mark(0') 25.68/7.87 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.87 active(add(0', X)) -> mark(X) 25.68/7.87 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.87 active(first(0', X)) -> mark(nil) 25.68/7.87 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.87 active(terms(X)) -> terms(active(X)) 25.68/7.87 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.87 active(recip(X)) -> recip(active(X)) 25.68/7.87 active(sqr(X)) -> sqr(active(X)) 25.68/7.87 active(s(X)) -> s(active(X)) 25.68/7.87 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.87 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.87 active(dbl(X)) -> dbl(active(X)) 25.68/7.87 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.87 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.87 terms(mark(X)) -> mark(terms(X)) 25.68/7.87 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.87 recip(mark(X)) -> mark(recip(X)) 25.68/7.87 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.87 s(mark(X)) -> mark(s(X)) 25.68/7.87 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.87 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.87 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.87 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.87 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.87 proper(terms(X)) -> terms(proper(X)) 25.68/7.87 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.87 proper(recip(X)) -> recip(proper(X)) 25.68/7.87 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.87 proper(s(X)) -> s(proper(X)) 25.68/7.87 proper(0') -> ok(0') 25.68/7.87 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.87 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.87 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.87 proper(nil) -> ok(nil) 25.68/7.87 terms(ok(X)) -> ok(terms(X)) 25.68/7.87 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.87 recip(ok(X)) -> ok(recip(X)) 25.68/7.87 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.87 s(ok(X)) -> ok(s(X)) 25.68/7.87 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.87 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.87 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.87 top(mark(X)) -> top(proper(X)) 25.68/7.87 top(ok(X)) -> top(active(X)) 25.68/7.87 25.68/7.87 Types: 25.68/7.87 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 0' :: mark:0':nil:ok 25.68/7.87 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 nil :: mark:0':nil:ok 25.68/7.87 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 top :: mark:0':nil:ok -> top 25.68/7.87 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.87 hole_top2_0 :: top 25.68/7.87 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.87 25.68/7.87 25.68/7.87 Lemmas: 25.68/7.87 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0) 25.68/7.87 sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0) 25.68/7.87 terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0) 25.68/7.87 25.68/7.87 25.68/7.87 Generator Equations: 25.68/7.87 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.87 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.87 25.68/7.87 25.68/7.87 The following defined symbols remain to be analysed: 25.68/7.87 s, active, add, dbl, first, proper, top 25.68/7.87 25.68/7.87 They will be analysed ascendingly in the following order: 25.68/7.87 s < active 25.68/7.87 add < active 25.68/7.87 dbl < active 25.68/7.87 first < active 25.68/7.87 active < top 25.68/7.87 s < proper 25.68/7.87 add < proper 25.68/7.87 dbl < proper 25.68/7.87 first < proper 25.68/7.87 proper < top 25.68/7.87 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (25) RewriteLemmaProof (LOWER BOUND(ID)) 25.68/7.87 Proved the following rewrite lemma: 25.68/7.87 s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0) 25.68/7.87 25.68/7.87 Induction Base: 25.68/7.87 s(gen_mark:0':nil:ok3_0(+(1, 0))) 25.68/7.87 25.68/7.87 Induction Step: 25.68/7.87 s(gen_mark:0':nil:ok3_0(+(1, +(n3121_0, 1)))) ->_R^Omega(1) 25.68/7.87 mark(s(gen_mark:0':nil:ok3_0(+(1, n3121_0)))) ->_IH 25.68/7.87 mark(*4_0) 25.68/7.87 25.68/7.87 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (26) 25.68/7.87 Obligation: 25.68/7.87 TRS: 25.68/7.87 Rules: 25.68/7.87 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.87 active(sqr(0')) -> mark(0') 25.68/7.87 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.87 active(dbl(0')) -> mark(0') 25.68/7.87 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.87 active(add(0', X)) -> mark(X) 25.68/7.87 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.87 active(first(0', X)) -> mark(nil) 25.68/7.87 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.87 active(terms(X)) -> terms(active(X)) 25.68/7.87 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.87 active(recip(X)) -> recip(active(X)) 25.68/7.87 active(sqr(X)) -> sqr(active(X)) 25.68/7.87 active(s(X)) -> s(active(X)) 25.68/7.87 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.87 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.87 active(dbl(X)) -> dbl(active(X)) 25.68/7.87 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.87 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.87 terms(mark(X)) -> mark(terms(X)) 25.68/7.87 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.87 recip(mark(X)) -> mark(recip(X)) 25.68/7.87 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.87 s(mark(X)) -> mark(s(X)) 25.68/7.87 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.87 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.87 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.87 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.87 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.87 proper(terms(X)) -> terms(proper(X)) 25.68/7.87 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.87 proper(recip(X)) -> recip(proper(X)) 25.68/7.87 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.87 proper(s(X)) -> s(proper(X)) 25.68/7.87 proper(0') -> ok(0') 25.68/7.87 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.87 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.87 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.87 proper(nil) -> ok(nil) 25.68/7.87 terms(ok(X)) -> ok(terms(X)) 25.68/7.87 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.87 recip(ok(X)) -> ok(recip(X)) 25.68/7.87 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.87 s(ok(X)) -> ok(s(X)) 25.68/7.87 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.87 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.87 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.87 top(mark(X)) -> top(proper(X)) 25.68/7.87 top(ok(X)) -> top(active(X)) 25.68/7.87 25.68/7.87 Types: 25.68/7.87 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 0' :: mark:0':nil:ok 25.68/7.87 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 nil :: mark:0':nil:ok 25.68/7.87 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 top :: mark:0':nil:ok -> top 25.68/7.87 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.87 hole_top2_0 :: top 25.68/7.87 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.87 25.68/7.87 25.68/7.87 Lemmas: 25.68/7.87 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0) 25.68/7.87 sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0) 25.68/7.87 terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0) 25.68/7.87 s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0) 25.68/7.87 25.68/7.87 25.68/7.87 Generator Equations: 25.68/7.87 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.87 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.87 25.68/7.87 25.68/7.87 The following defined symbols remain to be analysed: 25.68/7.87 add, active, dbl, first, proper, top 25.68/7.87 25.68/7.87 They will be analysed ascendingly in the following order: 25.68/7.87 add < active 25.68/7.87 dbl < active 25.68/7.87 first < active 25.68/7.87 active < top 25.68/7.87 add < proper 25.68/7.87 dbl < proper 25.68/7.87 first < proper 25.68/7.87 proper < top 25.68/7.87 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (27) RewriteLemmaProof (LOWER BOUND(ID)) 25.68/7.87 Proved the following rewrite lemma: 25.68/7.87 add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3988_0) 25.68/7.87 25.68/7.87 Induction Base: 25.68/7.87 add(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b)) 25.68/7.87 25.68/7.87 Induction Step: 25.68/7.87 add(gen_mark:0':nil:ok3_0(+(1, +(n3988_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1) 25.68/7.87 mark(add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark:0':nil:ok3_0(b))) ->_IH 25.68/7.87 mark(*4_0) 25.68/7.87 25.68/7.87 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (28) 25.68/7.87 Obligation: 25.68/7.87 TRS: 25.68/7.87 Rules: 25.68/7.87 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.87 active(sqr(0')) -> mark(0') 25.68/7.87 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.87 active(dbl(0')) -> mark(0') 25.68/7.87 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.87 active(add(0', X)) -> mark(X) 25.68/7.87 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.87 active(first(0', X)) -> mark(nil) 25.68/7.87 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.87 active(terms(X)) -> terms(active(X)) 25.68/7.87 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.87 active(recip(X)) -> recip(active(X)) 25.68/7.87 active(sqr(X)) -> sqr(active(X)) 25.68/7.87 active(s(X)) -> s(active(X)) 25.68/7.87 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.87 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.87 active(dbl(X)) -> dbl(active(X)) 25.68/7.87 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.87 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.87 terms(mark(X)) -> mark(terms(X)) 25.68/7.87 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.87 recip(mark(X)) -> mark(recip(X)) 25.68/7.87 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.87 s(mark(X)) -> mark(s(X)) 25.68/7.87 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.87 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.87 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.87 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.87 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.87 proper(terms(X)) -> terms(proper(X)) 25.68/7.87 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.87 proper(recip(X)) -> recip(proper(X)) 25.68/7.87 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.87 proper(s(X)) -> s(proper(X)) 25.68/7.87 proper(0') -> ok(0') 25.68/7.87 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.87 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.87 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.87 proper(nil) -> ok(nil) 25.68/7.87 terms(ok(X)) -> ok(terms(X)) 25.68/7.87 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.87 recip(ok(X)) -> ok(recip(X)) 25.68/7.87 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.87 s(ok(X)) -> ok(s(X)) 25.68/7.87 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.87 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.87 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.87 top(mark(X)) -> top(proper(X)) 25.68/7.87 top(ok(X)) -> top(active(X)) 25.68/7.87 25.68/7.87 Types: 25.68/7.87 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 0' :: mark:0':nil:ok 25.68/7.87 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 nil :: mark:0':nil:ok 25.68/7.87 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 top :: mark:0':nil:ok -> top 25.68/7.87 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.87 hole_top2_0 :: top 25.68/7.87 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.87 25.68/7.87 25.68/7.87 Lemmas: 25.68/7.87 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0) 25.68/7.87 sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0) 25.68/7.87 terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0) 25.68/7.87 s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0) 25.68/7.87 add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3988_0) 25.68/7.87 25.68/7.87 25.68/7.87 Generator Equations: 25.68/7.87 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.87 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.87 25.68/7.87 25.68/7.87 The following defined symbols remain to be analysed: 25.68/7.87 dbl, active, first, proper, top 25.68/7.87 25.68/7.87 They will be analysed ascendingly in the following order: 25.68/7.87 dbl < active 25.68/7.87 first < active 25.68/7.87 active < top 25.68/7.87 dbl < proper 25.68/7.87 first < proper 25.68/7.87 proper < top 25.68/7.87 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (29) RewriteLemmaProof (LOWER BOUND(ID)) 25.68/7.87 Proved the following rewrite lemma: 25.68/7.87 dbl(gen_mark:0':nil:ok3_0(+(1, n6446_0))) -> *4_0, rt in Omega(n6446_0) 25.68/7.87 25.68/7.87 Induction Base: 25.68/7.87 dbl(gen_mark:0':nil:ok3_0(+(1, 0))) 25.68/7.87 25.68/7.87 Induction Step: 25.68/7.87 dbl(gen_mark:0':nil:ok3_0(+(1, +(n6446_0, 1)))) ->_R^Omega(1) 25.68/7.87 mark(dbl(gen_mark:0':nil:ok3_0(+(1, n6446_0)))) ->_IH 25.68/7.87 mark(*4_0) 25.68/7.87 25.68/7.87 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (30) 25.68/7.87 Obligation: 25.68/7.87 TRS: 25.68/7.87 Rules: 25.68/7.87 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.87 active(sqr(0')) -> mark(0') 25.68/7.87 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.87 active(dbl(0')) -> mark(0') 25.68/7.87 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.87 active(add(0', X)) -> mark(X) 25.68/7.87 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.87 active(first(0', X)) -> mark(nil) 25.68/7.87 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.87 active(terms(X)) -> terms(active(X)) 25.68/7.87 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.87 active(recip(X)) -> recip(active(X)) 25.68/7.87 active(sqr(X)) -> sqr(active(X)) 25.68/7.87 active(s(X)) -> s(active(X)) 25.68/7.87 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.87 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.87 active(dbl(X)) -> dbl(active(X)) 25.68/7.87 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.87 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.87 terms(mark(X)) -> mark(terms(X)) 25.68/7.87 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.87 recip(mark(X)) -> mark(recip(X)) 25.68/7.87 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.87 s(mark(X)) -> mark(s(X)) 25.68/7.87 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.87 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.87 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.87 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.87 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.87 proper(terms(X)) -> terms(proper(X)) 25.68/7.87 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.87 proper(recip(X)) -> recip(proper(X)) 25.68/7.87 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.87 proper(s(X)) -> s(proper(X)) 25.68/7.87 proper(0') -> ok(0') 25.68/7.87 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.87 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.87 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.87 proper(nil) -> ok(nil) 25.68/7.87 terms(ok(X)) -> ok(terms(X)) 25.68/7.87 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.87 recip(ok(X)) -> ok(recip(X)) 25.68/7.87 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.87 s(ok(X)) -> ok(s(X)) 25.68/7.87 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.87 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.87 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.87 top(mark(X)) -> top(proper(X)) 25.68/7.87 top(ok(X)) -> top(active(X)) 25.68/7.87 25.68/7.87 Types: 25.68/7.87 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 0' :: mark:0':nil:ok 25.68/7.87 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 nil :: mark:0':nil:ok 25.68/7.87 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 top :: mark:0':nil:ok -> top 25.68/7.87 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.87 hole_top2_0 :: top 25.68/7.87 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.87 25.68/7.87 25.68/7.87 Lemmas: 25.68/7.87 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0) 25.68/7.87 sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0) 25.68/7.87 terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0) 25.68/7.87 s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0) 25.68/7.87 add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3988_0) 25.68/7.87 dbl(gen_mark:0':nil:ok3_0(+(1, n6446_0))) -> *4_0, rt in Omega(n6446_0) 25.68/7.87 25.68/7.87 25.68/7.87 Generator Equations: 25.68/7.87 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.87 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.87 25.68/7.87 25.68/7.87 The following defined symbols remain to be analysed: 25.68/7.87 first, active, proper, top 25.68/7.87 25.68/7.87 They will be analysed ascendingly in the following order: 25.68/7.87 first < active 25.68/7.87 active < top 25.68/7.87 first < proper 25.68/7.87 proper < top 25.68/7.87 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (31) RewriteLemmaProof (LOWER BOUND(ID)) 25.68/7.87 Proved the following rewrite lemma: 25.68/7.87 first(gen_mark:0':nil:ok3_0(+(1, n7564_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n7564_0) 25.68/7.87 25.68/7.87 Induction Base: 25.68/7.87 first(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b)) 25.68/7.87 25.68/7.87 Induction Step: 25.68/7.87 first(gen_mark:0':nil:ok3_0(+(1, +(n7564_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1) 25.68/7.87 mark(first(gen_mark:0':nil:ok3_0(+(1, n7564_0)), gen_mark:0':nil:ok3_0(b))) ->_IH 25.68/7.87 mark(*4_0) 25.68/7.87 25.68/7.87 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 25.68/7.87 ---------------------------------------- 25.68/7.87 25.68/7.87 (32) 25.68/7.87 Obligation: 25.68/7.87 TRS: 25.68/7.87 Rules: 25.68/7.87 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 25.68/7.87 active(sqr(0')) -> mark(0') 25.68/7.87 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 25.68/7.87 active(dbl(0')) -> mark(0') 25.68/7.87 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 25.68/7.87 active(add(0', X)) -> mark(X) 25.68/7.87 active(add(s(X), Y)) -> mark(s(add(X, Y))) 25.68/7.87 active(first(0', X)) -> mark(nil) 25.68/7.87 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 25.68/7.87 active(terms(X)) -> terms(active(X)) 25.68/7.87 active(cons(X1, X2)) -> cons(active(X1), X2) 25.68/7.87 active(recip(X)) -> recip(active(X)) 25.68/7.87 active(sqr(X)) -> sqr(active(X)) 25.68/7.87 active(s(X)) -> s(active(X)) 25.68/7.87 active(add(X1, X2)) -> add(active(X1), X2) 25.68/7.87 active(add(X1, X2)) -> add(X1, active(X2)) 25.68/7.87 active(dbl(X)) -> dbl(active(X)) 25.68/7.87 active(first(X1, X2)) -> first(active(X1), X2) 25.68/7.87 active(first(X1, X2)) -> first(X1, active(X2)) 25.68/7.87 terms(mark(X)) -> mark(terms(X)) 25.68/7.87 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.68/7.87 recip(mark(X)) -> mark(recip(X)) 25.68/7.87 sqr(mark(X)) -> mark(sqr(X)) 25.68/7.87 s(mark(X)) -> mark(s(X)) 25.68/7.87 add(mark(X1), X2) -> mark(add(X1, X2)) 25.68/7.87 add(X1, mark(X2)) -> mark(add(X1, X2)) 25.68/7.87 dbl(mark(X)) -> mark(dbl(X)) 25.68/7.87 first(mark(X1), X2) -> mark(first(X1, X2)) 25.68/7.87 first(X1, mark(X2)) -> mark(first(X1, X2)) 25.68/7.87 proper(terms(X)) -> terms(proper(X)) 25.68/7.87 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.68/7.87 proper(recip(X)) -> recip(proper(X)) 25.68/7.87 proper(sqr(X)) -> sqr(proper(X)) 25.68/7.87 proper(s(X)) -> s(proper(X)) 25.68/7.87 proper(0') -> ok(0') 25.68/7.87 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 25.68/7.87 proper(dbl(X)) -> dbl(proper(X)) 25.68/7.87 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 25.68/7.87 proper(nil) -> ok(nil) 25.68/7.87 terms(ok(X)) -> ok(terms(X)) 25.68/7.87 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.68/7.87 recip(ok(X)) -> ok(recip(X)) 25.68/7.87 sqr(ok(X)) -> ok(sqr(X)) 25.68/7.87 s(ok(X)) -> ok(s(X)) 25.68/7.87 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 25.68/7.87 dbl(ok(X)) -> ok(dbl(X)) 25.68/7.87 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 25.68/7.87 top(mark(X)) -> top(proper(X)) 25.68/7.87 top(ok(X)) -> top(active(X)) 25.68/7.87 25.68/7.87 Types: 25.68/7.87 active :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 terms :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 mark :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 recip :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 sqr :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 s :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 0' :: mark:0':nil:ok 25.68/7.87 add :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 dbl :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 first :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 nil :: mark:0':nil:ok 25.68/7.87 proper :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 ok :: mark:0':nil:ok -> mark:0':nil:ok 25.68/7.87 top :: mark:0':nil:ok -> top 25.68/7.87 hole_mark:0':nil:ok1_0 :: mark:0':nil:ok 25.68/7.87 hole_top2_0 :: top 25.68/7.87 gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok 25.68/7.87 25.68/7.87 25.68/7.87 Lemmas: 25.68/7.87 cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 25.68/7.87 recip(gen_mark:0':nil:ok3_0(+(1, n1126_0))) -> *4_0, rt in Omega(n1126_0) 25.68/7.87 sqr(gen_mark:0':nil:ok3_0(+(1, n1690_0))) -> *4_0, rt in Omega(n1690_0) 25.68/7.87 terms(gen_mark:0':nil:ok3_0(+(1, n2355_0))) -> *4_0, rt in Omega(n2355_0) 25.68/7.87 s(gen_mark:0':nil:ok3_0(+(1, n3121_0))) -> *4_0, rt in Omega(n3121_0) 25.68/7.87 add(gen_mark:0':nil:ok3_0(+(1, n3988_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3988_0) 25.68/7.87 dbl(gen_mark:0':nil:ok3_0(+(1, n6446_0))) -> *4_0, rt in Omega(n6446_0) 25.68/7.87 first(gen_mark:0':nil:ok3_0(+(1, n7564_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n7564_0) 25.68/7.87 25.68/7.87 25.68/7.87 Generator Equations: 25.68/7.87 gen_mark:0':nil:ok3_0(0) <=> 0' 25.68/7.87 gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x)) 25.68/7.87 25.68/7.87 25.68/7.87 The following defined symbols remain to be analysed: 25.68/7.87 active, proper, top 25.68/7.87 25.68/7.87 They will be analysed ascendingly in the following order: 25.68/7.87 active < top 25.68/7.87 proper < top 26.00/7.91 EOF