23.63/8.10 WORST_CASE(Omega(n^1), O(n^1)) 23.63/8.11 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 23.63/8.11 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 23.63/8.11 23.63/8.11 23.63/8.11 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.63/8.11 23.63/8.11 (0) CpxTRS 23.63/8.11 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 23.63/8.11 (2) CpxTRS 23.63/8.11 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 23.63/8.11 (4) CpxTRS 23.63/8.11 (5) CpxTrsMatchBoundsTAProof [FINISHED, 44 ms] 23.63/8.11 (6) BOUNDS(1, n^1) 23.63/8.11 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 23.63/8.11 (8) CpxTRS 23.63/8.11 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 23.63/8.11 (10) typed CpxTrs 23.63/8.11 (11) OrderProof [LOWER BOUND(ID), 0 ms] 23.63/8.11 (12) typed CpxTrs 23.63/8.11 (13) RewriteLemmaProof [LOWER BOUND(ID), 456 ms] 23.63/8.11 (14) BEST 23.63/8.11 (15) proven lower bound 23.63/8.11 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 23.63/8.11 (17) BOUNDS(n^1, INF) 23.63/8.11 (18) typed CpxTrs 23.63/8.11 (19) RewriteLemmaProof [LOWER BOUND(ID), 143 ms] 23.63/8.11 (20) typed CpxTrs 23.63/8.11 (21) RewriteLemmaProof [LOWER BOUND(ID), 118 ms] 23.63/8.11 (22) typed CpxTrs 23.63/8.11 (23) RewriteLemmaProof [LOWER BOUND(ID), 66 ms] 23.63/8.11 (24) typed CpxTrs 23.63/8.11 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (0) 23.63/8.11 Obligation: 23.63/8.11 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.63/8.11 23.63/8.11 23.63/8.11 The TRS R consists of the following rules: 23.63/8.11 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0)) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0)) -> mark(0) 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(0) -> ok(0) 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 S is empty. 23.63/8.11 Rewrite Strategy: FULL 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 23.63/8.11 The following defined symbols can occur below the 0th argument of top: proper, active 23.63/8.11 The following defined symbols can occur below the 0th argument of proper: proper, active 23.63/8.11 The following defined symbols can occur below the 0th argument of active: proper, active 23.63/8.11 23.63/8.11 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0)) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0)) -> mark(0) 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (2) 23.63/8.11 Obligation: 23.63/8.11 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.63/8.11 23.63/8.11 23.63/8.11 The TRS R consists of the following rules: 23.63/8.11 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(0) -> ok(0) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 S is empty. 23.63/8.11 Rewrite Strategy: FULL 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 23.63/8.11 transformed relative TRS to TRS 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (4) 23.63/8.11 Obligation: 23.63/8.11 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.63/8.11 23.63/8.11 23.63/8.11 The TRS R consists of the following rules: 23.63/8.11 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(0) -> ok(0) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 S is empty. 23.63/8.11 Rewrite Strategy: FULL 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (5) CpxTrsMatchBoundsTAProof (FINISHED) 23.63/8.11 A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 23.63/8.11 23.63/8.11 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 23.63/8.11 final states : [1, 2, 3, 4, 5, 6] 23.63/8.11 transitions: 23.63/8.11 mark0(0) -> 0 23.63/8.11 tt0() -> 0 23.63/8.11 ok0(0) -> 0 23.63/8.11 00() -> 0 23.63/8.11 active0(0) -> 0 23.63/8.11 and0(0, 0) -> 1 23.63/8.11 plus0(0, 0) -> 2 23.63/8.11 s0(0) -> 3 23.63/8.11 x0(0, 0) -> 4 23.63/8.11 proper0(0) -> 5 23.63/8.11 top0(0) -> 6 23.63/8.11 and1(0, 0) -> 7 23.63/8.11 mark1(7) -> 1 23.63/8.11 plus1(0, 0) -> 8 23.63/8.11 mark1(8) -> 2 23.63/8.11 s1(0) -> 9 23.63/8.11 mark1(9) -> 3 23.63/8.11 x1(0, 0) -> 10 23.63/8.11 mark1(10) -> 4 23.63/8.11 tt1() -> 11 23.63/8.11 ok1(11) -> 5 23.63/8.11 01() -> 12 23.63/8.11 ok1(12) -> 5 23.63/8.11 and1(0, 0) -> 13 23.63/8.11 ok1(13) -> 1 23.63/8.11 plus1(0, 0) -> 14 23.63/8.11 ok1(14) -> 2 23.63/8.11 s1(0) -> 15 23.63/8.11 ok1(15) -> 3 23.63/8.11 x1(0, 0) -> 16 23.63/8.11 ok1(16) -> 4 23.63/8.11 proper1(0) -> 17 23.63/8.11 top1(17) -> 6 23.63/8.11 active1(0) -> 18 23.63/8.11 top1(18) -> 6 23.63/8.11 mark1(7) -> 7 23.63/8.11 mark1(7) -> 13 23.63/8.11 mark1(8) -> 8 23.63/8.11 mark1(8) -> 14 23.63/8.11 mark1(9) -> 9 23.63/8.11 mark1(9) -> 15 23.63/8.11 mark1(10) -> 10 23.63/8.11 mark1(10) -> 16 23.63/8.11 ok1(11) -> 17 23.63/8.11 ok1(12) -> 17 23.63/8.11 ok1(13) -> 7 23.63/8.11 ok1(13) -> 13 23.63/8.11 ok1(14) -> 8 23.63/8.11 ok1(14) -> 14 23.63/8.11 ok1(15) -> 9 23.63/8.11 ok1(15) -> 15 23.63/8.11 ok1(16) -> 10 23.63/8.11 ok1(16) -> 16 23.63/8.11 active2(11) -> 19 23.63/8.11 top2(19) -> 6 23.63/8.11 active2(12) -> 19 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (6) 23.63/8.11 BOUNDS(1, n^1) 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 23.63/8.11 Renamed function symbols to avoid clashes with predefined symbol. 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (8) 23.63/8.11 Obligation: 23.63/8.11 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 23.63/8.11 23.63/8.11 23.63/8.11 The TRS R consists of the following rules: 23.63/8.11 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0')) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0')) -> mark(0') 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(0') -> ok(0') 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 S is empty. 23.63/8.11 Rewrite Strategy: FULL 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 23.63/8.11 Infered types. 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (10) 23.63/8.11 Obligation: 23.63/8.11 TRS: 23.63/8.11 Rules: 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0')) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0')) -> mark(0') 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(0') -> ok(0') 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 Types: 23.63/8.11 active :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 tt :: tt:mark:0':ok 23.63/8.11 mark :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 0' :: tt:mark:0':ok 23.63/8.11 s :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 proper :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 ok :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 top :: tt:mark:0':ok -> top 23.63/8.11 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 23.63/8.11 hole_top2_0 :: top 23.63/8.11 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (11) OrderProof (LOWER BOUND(ID)) 23.63/8.11 Heuristically decided to analyse the following defined symbols: 23.63/8.11 active, s, plus, x, and, proper, top 23.63/8.11 23.63/8.11 They will be analysed ascendingly in the following order: 23.63/8.11 s < active 23.63/8.11 plus < active 23.63/8.11 x < active 23.63/8.11 and < active 23.63/8.11 active < top 23.63/8.11 s < proper 23.63/8.11 plus < proper 23.63/8.11 x < proper 23.63/8.11 and < proper 23.63/8.11 proper < top 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (12) 23.63/8.11 Obligation: 23.63/8.11 TRS: 23.63/8.11 Rules: 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0')) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0')) -> mark(0') 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(0') -> ok(0') 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 Types: 23.63/8.11 active :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 tt :: tt:mark:0':ok 23.63/8.11 mark :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 0' :: tt:mark:0':ok 23.63/8.11 s :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 proper :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 ok :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 top :: tt:mark:0':ok -> top 23.63/8.11 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 23.63/8.11 hole_top2_0 :: top 23.63/8.11 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 23.63/8.11 23.63/8.11 23.63/8.11 Generator Equations: 23.63/8.11 gen_tt:mark:0':ok3_0(0) <=> tt 23.63/8.11 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 23.63/8.11 23.63/8.11 23.63/8.11 The following defined symbols remain to be analysed: 23.63/8.11 s, active, plus, x, and, proper, top 23.63/8.11 23.63/8.11 They will be analysed ascendingly in the following order: 23.63/8.11 s < active 23.63/8.11 plus < active 23.63/8.11 x < active 23.63/8.11 and < active 23.63/8.11 active < top 23.63/8.11 s < proper 23.63/8.11 plus < proper 23.63/8.11 x < proper 23.63/8.11 and < proper 23.63/8.11 proper < top 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (13) RewriteLemmaProof (LOWER BOUND(ID)) 23.63/8.11 Proved the following rewrite lemma: 23.63/8.11 s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) 23.63/8.11 23.63/8.11 Induction Base: 23.63/8.11 s(gen_tt:mark:0':ok3_0(+(1, 0))) 23.63/8.11 23.63/8.11 Induction Step: 23.63/8.11 s(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1) 23.63/8.11 mark(s(gen_tt:mark:0':ok3_0(+(1, n5_0)))) ->_IH 23.63/8.11 mark(*4_0) 23.63/8.11 23.63/8.11 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (14) 23.63/8.11 Complex Obligation (BEST) 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (15) 23.63/8.11 Obligation: 23.63/8.11 Proved the lower bound n^1 for the following obligation: 23.63/8.11 23.63/8.11 TRS: 23.63/8.11 Rules: 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0')) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0')) -> mark(0') 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(0') -> ok(0') 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 Types: 23.63/8.11 active :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 tt :: tt:mark:0':ok 23.63/8.11 mark :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 0' :: tt:mark:0':ok 23.63/8.11 s :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 proper :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 ok :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 top :: tt:mark:0':ok -> top 23.63/8.11 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 23.63/8.11 hole_top2_0 :: top 23.63/8.11 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 23.63/8.11 23.63/8.11 23.63/8.11 Generator Equations: 23.63/8.11 gen_tt:mark:0':ok3_0(0) <=> tt 23.63/8.11 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 23.63/8.11 23.63/8.11 23.63/8.11 The following defined symbols remain to be analysed: 23.63/8.11 s, active, plus, x, and, proper, top 23.63/8.11 23.63/8.11 They will be analysed ascendingly in the following order: 23.63/8.11 s < active 23.63/8.11 plus < active 23.63/8.11 x < active 23.63/8.11 and < active 23.63/8.11 active < top 23.63/8.11 s < proper 23.63/8.11 plus < proper 23.63/8.11 x < proper 23.63/8.11 and < proper 23.63/8.11 proper < top 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (16) LowerBoundPropagationProof (FINISHED) 23.63/8.11 Propagated lower bound. 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (17) 23.63/8.11 BOUNDS(n^1, INF) 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (18) 23.63/8.11 Obligation: 23.63/8.11 TRS: 23.63/8.11 Rules: 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0')) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0')) -> mark(0') 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(0') -> ok(0') 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 Types: 23.63/8.11 active :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 tt :: tt:mark:0':ok 23.63/8.11 mark :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 0' :: tt:mark:0':ok 23.63/8.11 s :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 proper :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 ok :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 top :: tt:mark:0':ok -> top 23.63/8.11 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 23.63/8.11 hole_top2_0 :: top 23.63/8.11 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 23.63/8.11 23.63/8.11 23.63/8.11 Lemmas: 23.63/8.11 s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) 23.63/8.11 23.63/8.11 23.63/8.11 Generator Equations: 23.63/8.11 gen_tt:mark:0':ok3_0(0) <=> tt 23.63/8.11 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 23.63/8.11 23.63/8.11 23.63/8.11 The following defined symbols remain to be analysed: 23.63/8.11 plus, active, x, and, proper, top 23.63/8.11 23.63/8.11 They will be analysed ascendingly in the following order: 23.63/8.11 plus < active 23.63/8.11 x < active 23.63/8.11 and < active 23.63/8.11 active < top 23.63/8.11 plus < proper 23.63/8.11 x < proper 23.63/8.11 and < proper 23.63/8.11 proper < top 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (19) RewriteLemmaProof (LOWER BOUND(ID)) 23.63/8.11 Proved the following rewrite lemma: 23.63/8.11 plus(gen_tt:mark:0':ok3_0(+(1, n359_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n359_0) 23.63/8.11 23.63/8.11 Induction Base: 23.63/8.11 plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) 23.63/8.11 23.63/8.11 Induction Step: 23.63/8.11 plus(gen_tt:mark:0':ok3_0(+(1, +(n359_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) 23.63/8.11 mark(plus(gen_tt:mark:0':ok3_0(+(1, n359_0)), gen_tt:mark:0':ok3_0(b))) ->_IH 23.63/8.11 mark(*4_0) 23.63/8.11 23.63/8.11 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (20) 23.63/8.11 Obligation: 23.63/8.11 TRS: 23.63/8.11 Rules: 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0')) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0')) -> mark(0') 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(0') -> ok(0') 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 Types: 23.63/8.11 active :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 tt :: tt:mark:0':ok 23.63/8.11 mark :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 0' :: tt:mark:0':ok 23.63/8.11 s :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 proper :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 ok :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 top :: tt:mark:0':ok -> top 23.63/8.11 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 23.63/8.11 hole_top2_0 :: top 23.63/8.11 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 23.63/8.11 23.63/8.11 23.63/8.11 Lemmas: 23.63/8.11 s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) 23.63/8.11 plus(gen_tt:mark:0':ok3_0(+(1, n359_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n359_0) 23.63/8.11 23.63/8.11 23.63/8.11 Generator Equations: 23.63/8.11 gen_tt:mark:0':ok3_0(0) <=> tt 23.63/8.11 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 23.63/8.11 23.63/8.11 23.63/8.11 The following defined symbols remain to be analysed: 23.63/8.11 x, active, and, proper, top 23.63/8.11 23.63/8.11 They will be analysed ascendingly in the following order: 23.63/8.11 x < active 23.63/8.11 and < active 23.63/8.11 active < top 23.63/8.11 x < proper 23.63/8.11 and < proper 23.63/8.11 proper < top 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (21) RewriteLemmaProof (LOWER BOUND(ID)) 23.63/8.11 Proved the following rewrite lemma: 23.63/8.11 x(gen_tt:mark:0':ok3_0(+(1, n1607_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1607_0) 23.63/8.11 23.63/8.11 Induction Base: 23.63/8.11 x(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) 23.63/8.11 23.63/8.11 Induction Step: 23.63/8.11 x(gen_tt:mark:0':ok3_0(+(1, +(n1607_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) 23.63/8.11 mark(x(gen_tt:mark:0':ok3_0(+(1, n1607_0)), gen_tt:mark:0':ok3_0(b))) ->_IH 23.63/8.11 mark(*4_0) 23.63/8.11 23.63/8.11 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (22) 23.63/8.11 Obligation: 23.63/8.11 TRS: 23.63/8.11 Rules: 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0')) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0')) -> mark(0') 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(0') -> ok(0') 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 Types: 23.63/8.11 active :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 tt :: tt:mark:0':ok 23.63/8.11 mark :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 0' :: tt:mark:0':ok 23.63/8.11 s :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 proper :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 ok :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 top :: tt:mark:0':ok -> top 23.63/8.11 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 23.63/8.11 hole_top2_0 :: top 23.63/8.11 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 23.63/8.11 23.63/8.11 23.63/8.11 Lemmas: 23.63/8.11 s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) 23.63/8.11 plus(gen_tt:mark:0':ok3_0(+(1, n359_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n359_0) 23.63/8.11 x(gen_tt:mark:0':ok3_0(+(1, n1607_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1607_0) 23.63/8.11 23.63/8.11 23.63/8.11 Generator Equations: 23.63/8.11 gen_tt:mark:0':ok3_0(0) <=> tt 23.63/8.11 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 23.63/8.11 23.63/8.11 23.63/8.11 The following defined symbols remain to be analysed: 23.63/8.11 and, active, proper, top 23.63/8.11 23.63/8.11 They will be analysed ascendingly in the following order: 23.63/8.11 and < active 23.63/8.11 active < top 23.63/8.11 and < proper 23.63/8.11 proper < top 23.63/8.11 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (23) RewriteLemmaProof (LOWER BOUND(ID)) 23.63/8.11 Proved the following rewrite lemma: 23.63/8.11 and(gen_tt:mark:0':ok3_0(+(1, n3161_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3161_0) 23.63/8.11 23.63/8.11 Induction Base: 23.63/8.11 and(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) 23.63/8.11 23.63/8.11 Induction Step: 23.63/8.11 and(gen_tt:mark:0':ok3_0(+(1, +(n3161_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) 23.63/8.11 mark(and(gen_tt:mark:0':ok3_0(+(1, n3161_0)), gen_tt:mark:0':ok3_0(b))) ->_IH 23.63/8.11 mark(*4_0) 23.63/8.11 23.63/8.11 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.63/8.11 ---------------------------------------- 23.63/8.11 23.63/8.11 (24) 23.63/8.11 Obligation: 23.63/8.11 TRS: 23.63/8.11 Rules: 23.63/8.11 active(and(tt, X)) -> mark(X) 23.63/8.11 active(plus(N, 0')) -> mark(N) 23.63/8.11 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.63/8.11 active(x(N, 0')) -> mark(0') 23.63/8.11 active(x(N, s(M))) -> mark(plus(x(N, M), N)) 23.63/8.11 active(and(X1, X2)) -> and(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(active(X1), X2) 23.63/8.11 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.63/8.11 active(s(X)) -> s(active(X)) 23.63/8.11 active(x(X1, X2)) -> x(active(X1), X2) 23.63/8.11 active(x(X1, X2)) -> x(X1, active(X2)) 23.63/8.11 and(mark(X1), X2) -> mark(and(X1, X2)) 23.63/8.11 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.63/8.11 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.63/8.11 s(mark(X)) -> mark(s(X)) 23.63/8.11 x(mark(X1), X2) -> mark(x(X1, X2)) 23.63/8.11 x(X1, mark(X2)) -> mark(x(X1, X2)) 23.63/8.11 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.63/8.11 proper(tt) -> ok(tt) 23.63/8.11 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.63/8.11 proper(0') -> ok(0') 23.63/8.11 proper(s(X)) -> s(proper(X)) 23.63/8.11 proper(x(X1, X2)) -> x(proper(X1), proper(X2)) 23.63/8.11 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.63/8.11 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.63/8.11 s(ok(X)) -> ok(s(X)) 23.63/8.11 x(ok(X1), ok(X2)) -> ok(x(X1, X2)) 23.63/8.11 top(mark(X)) -> top(proper(X)) 23.63/8.11 top(ok(X)) -> top(active(X)) 23.63/8.11 23.63/8.11 Types: 23.63/8.11 active :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 tt :: tt:mark:0':ok 23.63/8.11 mark :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 0' :: tt:mark:0':ok 23.63/8.11 s :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 proper :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 ok :: tt:mark:0':ok -> tt:mark:0':ok 23.63/8.11 top :: tt:mark:0':ok -> top 23.63/8.11 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 23.63/8.11 hole_top2_0 :: top 23.63/8.11 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 23.63/8.11 23.63/8.11 23.63/8.11 Lemmas: 23.63/8.11 s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) 23.63/8.11 plus(gen_tt:mark:0':ok3_0(+(1, n359_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n359_0) 23.63/8.11 x(gen_tt:mark:0':ok3_0(+(1, n1607_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1607_0) 23.63/8.11 and(gen_tt:mark:0':ok3_0(+(1, n3161_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3161_0) 23.63/8.11 23.63/8.11 23.63/8.11 Generator Equations: 23.63/8.11 gen_tt:mark:0':ok3_0(0) <=> tt 23.63/8.11 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 23.63/8.11 23.63/8.11 23.63/8.11 The following defined symbols remain to be analysed: 23.63/8.11 active, proper, top 23.63/8.11 23.63/8.11 They will be analysed ascendingly in the following order: 23.63/8.11 active < top 23.63/8.11 proper < top 23.63/8.15 EOF