WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 291 ms] (6) BOUNDS(1, n^1) (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 438 ms] (14) BEST (15) proven lower bound (16) LowerBoundPropagationProof [FINISHED, 0 ms] (17) BOUNDS(n^1, INF) (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 111 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 100 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 70 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 95 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 84 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 98 ms] (30) typed CpxTrs (31) RewriteLemmaProof [LOWER BOUND(ID), 100 ms] (32) typed CpxTrs (33) RewriteLemmaProof [LOWER BOUND(ID), 108 ms] (34) typed CpxTrs (35) RewriteLemmaProof [LOWER BOUND(ID), 142 ms] (36) typed CpxTrs (37) RewriteLemmaProof [LOWER BOUND(ID), 120 ms] (38) typed CpxTrs (39) RewriteLemmaProof [LOWER BOUND(ID), 97 ms] (40) typed CpxTrs (41) RewriteLemmaProof [LOWER BOUND(ID), 132 ms] (42) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0, Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0, Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0))) active(plus(0, Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0, Y)) -> mark(0) active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0) -> ok(0) proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0, Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0, Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0))) active(plus(0, Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0, Y)) -> mark(0) active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0) -> ok(0) proper(nil) -> ok(nil) proper(rnil) -> ok(rnil) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0) -> ok(0) proper(nil) -> ok(nil) proper(rnil) -> ok(rnil) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (5) CpxTrsMatchBoundsTAProof (FINISHED) 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. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] transitions: mark0(0) -> 0 00() -> 0 ok0(0) -> 0 nil0() -> 0 rnil0() -> 0 active0(0) -> 0 s0(0) -> 1 posrecip0(0) -> 2 negrecip0(0) -> 3 cons0(0, 0) -> 4 cons20(0, 0) -> 5 rcons0(0, 0) -> 6 from0(0) -> 7 2ndspos0(0, 0) -> 8 2ndsneg0(0, 0) -> 9 pi0(0) -> 10 plus0(0, 0) -> 11 times0(0, 0) -> 12 square0(0) -> 13 proper0(0) -> 14 top0(0) -> 15 s1(0) -> 16 mark1(16) -> 1 posrecip1(0) -> 17 mark1(17) -> 2 negrecip1(0) -> 18 mark1(18) -> 3 cons1(0, 0) -> 19 mark1(19) -> 4 cons21(0, 0) -> 20 mark1(20) -> 5 rcons1(0, 0) -> 21 mark1(21) -> 6 from1(0) -> 22 mark1(22) -> 7 2ndspos1(0, 0) -> 23 mark1(23) -> 8 2ndsneg1(0, 0) -> 24 mark1(24) -> 9 pi1(0) -> 25 mark1(25) -> 10 plus1(0, 0) -> 26 mark1(26) -> 11 times1(0, 0) -> 27 mark1(27) -> 12 square1(0) -> 28 mark1(28) -> 13 01() -> 29 ok1(29) -> 14 nil1() -> 30 ok1(30) -> 14 rnil1() -> 31 ok1(31) -> 14 s1(0) -> 32 ok1(32) -> 1 posrecip1(0) -> 33 ok1(33) -> 2 negrecip1(0) -> 34 ok1(34) -> 3 cons1(0, 0) -> 35 ok1(35) -> 4 cons21(0, 0) -> 36 ok1(36) -> 5 rcons1(0, 0) -> 37 ok1(37) -> 6 from1(0) -> 38 ok1(38) -> 7 2ndspos1(0, 0) -> 39 ok1(39) -> 8 2ndsneg1(0, 0) -> 40 ok1(40) -> 9 pi1(0) -> 41 ok1(41) -> 10 plus1(0, 0) -> 42 ok1(42) -> 11 times1(0, 0) -> 43 ok1(43) -> 12 square1(0) -> 44 ok1(44) -> 13 proper1(0) -> 45 top1(45) -> 15 active1(0) -> 46 top1(46) -> 15 mark1(16) -> 16 mark1(16) -> 32 mark1(17) -> 17 mark1(17) -> 33 mark1(18) -> 18 mark1(18) -> 34 mark1(19) -> 19 mark1(19) -> 35 mark1(20) -> 20 mark1(20) -> 36 mark1(21) -> 21 mark1(21) -> 37 mark1(22) -> 22 mark1(22) -> 38 mark1(23) -> 23 mark1(23) -> 39 mark1(24) -> 24 mark1(24) -> 40 mark1(25) -> 25 mark1(25) -> 41 mark1(26) -> 26 mark1(26) -> 42 mark1(27) -> 27 mark1(27) -> 43 mark1(28) -> 28 mark1(28) -> 44 ok1(29) -> 45 ok1(30) -> 45 ok1(31) -> 45 ok1(32) -> 16 ok1(32) -> 32 ok1(33) -> 17 ok1(33) -> 33 ok1(34) -> 18 ok1(34) -> 34 ok1(35) -> 19 ok1(35) -> 35 ok1(36) -> 20 ok1(36) -> 36 ok1(37) -> 21 ok1(37) -> 37 ok1(38) -> 22 ok1(38) -> 38 ok1(39) -> 23 ok1(39) -> 39 ok1(40) -> 24 ok1(40) -> 40 ok1(41) -> 25 ok1(41) -> 41 ok1(42) -> 26 ok1(42) -> 42 ok1(43) -> 27 ok1(43) -> 43 ok1(44) -> 28 ok1(44) -> 44 active2(29) -> 47 top2(47) -> 15 active2(30) -> 47 active2(31) -> 47 ---------------------------------------- (6) BOUNDS(1, n^1) ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, cons, from, s, 2ndspos, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: cons < active from < active s < active 2ndspos < active cons2 < active rcons < active posrecip < active 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top cons < proper from < proper s < proper 2ndspos < proper cons2 < proper rcons < proper posrecip < proper 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (12) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: cons, active, from, s, 2ndspos, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: cons < active from < active s < active 2ndspos < active cons2 < active rcons < active posrecip < active 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top cons < proper from < proper s < proper 2ndspos < proper cons2 < proper rcons < proper posrecip < proper 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) Induction Base: cons(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b)) Induction Step: cons(gen_mark:0':rnil:ok:nil3_0(+(1, +(n5_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) ->_R^Omega(1) mark(cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Complex Obligation (BEST) ---------------------------------------- (15) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: cons, active, from, s, 2ndspos, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: cons < active from < active s < active 2ndspos < active cons2 < active rcons < active posrecip < active 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top cons < proper from < proper s < proper 2ndspos < proper cons2 < proper rcons < proper posrecip < proper 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (16) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (17) BOUNDS(n^1, INF) ---------------------------------------- (18) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: from, active, s, 2ndspos, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: from < active s < active 2ndspos < active cons2 < active rcons < active posrecip < active 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top from < proper s < proper 2ndspos < proper cons2 < proper rcons < proper posrecip < proper 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) Induction Base: from(gen_mark:0':rnil:ok:nil3_0(+(1, 0))) Induction Step: from(gen_mark:0':rnil:ok:nil3_0(+(1, +(n1560_0, 1)))) ->_R^Omega(1) mark(from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: s, active, 2ndspos, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: s < active 2ndspos < active cons2 < active rcons < active posrecip < active 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top s < proper 2ndspos < proper cons2 < proper rcons < proper posrecip < proper 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) Induction Base: s(gen_mark:0':rnil:ok:nil3_0(+(1, 0))) Induction Step: s(gen_mark:0':rnil:ok:nil3_0(+(1, +(n2217_0, 1)))) ->_R^Omega(1) mark(s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: 2ndspos, active, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: 2ndspos < active cons2 < active rcons < active posrecip < active 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top 2ndspos < proper cons2 < proper rcons < proper posrecip < proper 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) Induction Base: 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b)) Induction Step: 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, +(n2975_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) ->_R^Omega(1) mark(2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: cons2, active, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: cons2 < active rcons < active posrecip < active 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top cons2 < proper rcons < proper posrecip < proper 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) Induction Base: cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, 0))) Induction Step: cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, +(n5451_0, 1)))) ->_R^Omega(1) mark(cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (26) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: rcons, active, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: rcons < active posrecip < active 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top rcons < proper posrecip < proper 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n8034_0) Induction Base: rcons(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b)) Induction Step: rcons(gen_mark:0':rnil:ok:nil3_0(+(1, +(n8034_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) ->_R^Omega(1) mark(rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (28) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n8034_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: posrecip, active, 2ndsneg, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: posrecip < active 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top posrecip < proper 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n11122_0))) -> *4_0, rt in Omega(n11122_0) Induction Base: posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, 0))) Induction Step: posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, +(n11122_0, 1)))) ->_R^Omega(1) mark(posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n11122_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (30) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n8034_0) posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n11122_0))) -> *4_0, rt in Omega(n11122_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: 2ndsneg, active, negrecip, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: 2ndsneg < active negrecip < active plus < active times < active pi < active square < active active < top 2ndsneg < proper negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (31) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: 2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12431_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n12431_0) Induction Base: 2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b)) Induction Step: 2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, +(n12431_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) ->_R^Omega(1) mark(2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12431_0)), gen_mark:0':rnil:ok:nil3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (32) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n8034_0) posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n11122_0))) -> *4_0, rt in Omega(n11122_0) 2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12431_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n12431_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: negrecip, active, plus, times, pi, square, proper, top They will be analysed ascendingly in the following order: negrecip < active plus < active times < active pi < active square < active active < top negrecip < proper plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (33) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n16033_0))) -> *4_0, rt in Omega(n16033_0) Induction Base: negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, 0))) Induction Step: negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, +(n16033_0, 1)))) ->_R^Omega(1) mark(negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n16033_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (34) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n8034_0) posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n11122_0))) -> *4_0, rt in Omega(n11122_0) 2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12431_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n12431_0) negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n16033_0))) -> *4_0, rt in Omega(n16033_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: plus, active, times, pi, square, proper, top They will be analysed ascendingly in the following order: plus < active times < active pi < active square < active active < top plus < proper times < proper pi < proper square < proper proper < top ---------------------------------------- (35) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17593_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n17593_0) Induction Base: plus(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b)) Induction Step: plus(gen_mark:0':rnil:ok:nil3_0(+(1, +(n17593_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) ->_R^Omega(1) mark(plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17593_0)), gen_mark:0':rnil:ok:nil3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (36) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n8034_0) posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n11122_0))) -> *4_0, rt in Omega(n11122_0) 2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12431_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n12431_0) negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n16033_0))) -> *4_0, rt in Omega(n16033_0) plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17593_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n17593_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: times, active, pi, square, proper, top They will be analysed ascendingly in the following order: times < active pi < active square < active active < top times < proper pi < proper square < proper proper < top ---------------------------------------- (37) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: times(gen_mark:0':rnil:ok:nil3_0(+(1, n21709_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n21709_0) Induction Base: times(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b)) Induction Step: times(gen_mark:0':rnil:ok:nil3_0(+(1, +(n21709_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) ->_R^Omega(1) mark(times(gen_mark:0':rnil:ok:nil3_0(+(1, n21709_0)), gen_mark:0':rnil:ok:nil3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (38) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n8034_0) posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n11122_0))) -> *4_0, rt in Omega(n11122_0) 2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12431_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n12431_0) negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n16033_0))) -> *4_0, rt in Omega(n16033_0) plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17593_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n17593_0) times(gen_mark:0':rnil:ok:nil3_0(+(1, n21709_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n21709_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: pi, active, square, proper, top They will be analysed ascendingly in the following order: pi < active square < active active < top pi < proper square < proper proper < top ---------------------------------------- (39) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: pi(gen_mark:0':rnil:ok:nil3_0(+(1, n26131_0))) -> *4_0, rt in Omega(n26131_0) Induction Base: pi(gen_mark:0':rnil:ok:nil3_0(+(1, 0))) Induction Step: pi(gen_mark:0':rnil:ok:nil3_0(+(1, +(n26131_0, 1)))) ->_R^Omega(1) mark(pi(gen_mark:0':rnil:ok:nil3_0(+(1, n26131_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (40) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n8034_0) posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n11122_0))) -> *4_0, rt in Omega(n11122_0) 2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12431_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n12431_0) negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n16033_0))) -> *4_0, rt in Omega(n16033_0) plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17593_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n17593_0) times(gen_mark:0':rnil:ok:nil3_0(+(1, n21709_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n21709_0) pi(gen_mark:0':rnil:ok:nil3_0(+(1, n26131_0))) -> *4_0, rt in Omega(n26131_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: square, active, proper, top They will be analysed ascendingly in the following order: square < active active < top square < proper proper < top ---------------------------------------- (41) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: square(gen_mark:0':rnil:ok:nil3_0(+(1, n28092_0))) -> *4_0, rt in Omega(n28092_0) Induction Base: square(gen_mark:0':rnil:ok:nil3_0(+(1, 0))) Induction Step: square(gen_mark:0':rnil:ok:nil3_0(+(1, +(n28092_0, 1)))) ->_R^Omega(1) mark(square(gen_mark:0':rnil:ok:nil3_0(+(1, n28092_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (42) Obligation: TRS: Rules: active(from(X)) -> mark(cons(X, from(s(X)))) active(2ndspos(0', Z)) -> mark(rnil) active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z))) active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) active(2ndsneg(0', Z)) -> mark(rnil) active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z))) active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) active(pi(X)) -> mark(2ndspos(X, from(0'))) active(plus(0', Y)) -> mark(Y) active(plus(s(X), Y)) -> mark(s(plus(X, Y))) active(times(0', Y)) -> mark(0') active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) active(square(X)) -> mark(times(X, X)) active(s(X)) -> s(active(X)) active(posrecip(X)) -> posrecip(active(X)) active(negrecip(X)) -> negrecip(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(cons2(X1, X2)) -> cons2(X1, active(X2)) active(rcons(X1, X2)) -> rcons(active(X1), X2) active(rcons(X1, X2)) -> rcons(X1, active(X2)) active(from(X)) -> from(active(X)) active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) active(pi(X)) -> pi(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(times(X1, X2)) -> times(active(X1), X2) active(times(X1, X2)) -> times(X1, active(X2)) active(square(X)) -> square(active(X)) s(mark(X)) -> mark(s(X)) posrecip(mark(X)) -> mark(posrecip(X)) negrecip(mark(X)) -> mark(negrecip(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) cons2(X1, mark(X2)) -> mark(cons2(X1, X2)) rcons(mark(X1), X2) -> mark(rcons(X1, X2)) rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) from(mark(X)) -> mark(from(X)) 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) pi(mark(X)) -> mark(pi(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) times(mark(X1), X2) -> mark(times(X1, X2)) times(X1, mark(X2)) -> mark(times(X1, X2)) square(mark(X)) -> mark(square(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(posrecip(X)) -> posrecip(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2)) proper(rnil) -> ok(rnil) proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) proper(pi(X)) -> pi(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(times(X1, X2)) -> times(proper(X1), proper(X2)) proper(square(X)) -> square(proper(X)) s(ok(X)) -> ok(s(X)) posrecip(ok(X)) -> ok(posrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2)) rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) from(ok(X)) -> ok(from(X)) 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) pi(ok(X)) -> ok(pi(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) times(ok(X1), ok(X2)) -> ok(times(X1, X2)) square(ok(X)) -> ok(square(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil from :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil mark :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil cons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil s :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndspos :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 0' :: mark:0':rnil:ok:nil rnil :: mark:0':rnil:ok:nil cons2 :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil rcons :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil posrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil 2ndsneg :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil negrecip :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil pi :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil plus :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil times :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil square :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil proper :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil ok :: mark:0':rnil:ok:nil -> mark:0':rnil:ok:nil nil :: mark:0':rnil:ok:nil top :: mark:0':rnil:ok:nil -> top hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil hole_top2_0 :: top gen_mark:0':rnil:ok:nil3_0 :: Nat -> mark:0':rnil:ok:nil Lemmas: cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n5_0) from(gen_mark:0':rnil:ok:nil3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0) s(gen_mark:0':rnil:ok:nil3_0(+(1, n2217_0))) -> *4_0, rt in Omega(n2217_0) 2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2975_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n2975_0) cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5451_0))) -> *4_0, rt in Omega(n5451_0) rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n8034_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n8034_0) posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n11122_0))) -> *4_0, rt in Omega(n11122_0) 2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12431_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n12431_0) negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n16033_0))) -> *4_0, rt in Omega(n16033_0) plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17593_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n17593_0) times(gen_mark:0':rnil:ok:nil3_0(+(1, n21709_0)), gen_mark:0':rnil:ok:nil3_0(b)) -> *4_0, rt in Omega(n21709_0) pi(gen_mark:0':rnil:ok:nil3_0(+(1, n26131_0))) -> *4_0, rt in Omega(n26131_0) square(gen_mark:0':rnil:ok:nil3_0(+(1, n28092_0))) -> *4_0, rt in Omega(n28092_0) Generator Equations: gen_mark:0':rnil:ok:nil3_0(0) <=> 0' gen_mark:0':rnil:ok:nil3_0(+(x, 1)) <=> mark(gen_mark:0':rnil:ok:nil3_0(x)) The following defined symbols remain to be analysed: active, proper, top They will be analysed ascendingly in the following order: active < top proper < top