24.12/7.41 WORST_CASE(Omega(n^1), O(n^1)) 24.12/7.42 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 24.12/7.42 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.12/7.42 24.12/7.42 24.12/7.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.12/7.42 24.12/7.42 (0) CpxTRS 24.12/7.42 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 24.12/7.42 (2) CpxTRS 24.12/7.42 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 24.12/7.42 (4) CpxTRS 24.12/7.42 (5) CpxTrsMatchBoundsTAProof [FINISHED, 61 ms] 24.12/7.42 (6) BOUNDS(1, n^1) 24.12/7.42 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 24.12/7.42 (8) CpxTRS 24.12/7.42 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 24.12/7.42 (10) typed CpxTrs 24.12/7.42 (11) OrderProof [LOWER BOUND(ID), 0 ms] 24.12/7.42 (12) typed CpxTrs 24.12/7.42 (13) RewriteLemmaProof [LOWER BOUND(ID), 479 ms] 24.12/7.42 (14) BEST 24.12/7.42 (15) proven lower bound 24.12/7.42 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 24.12/7.42 (17) BOUNDS(n^1, INF) 24.12/7.42 (18) typed CpxTrs 24.12/7.42 (19) RewriteLemmaProof [LOWER BOUND(ID), 75 ms] 24.12/7.42 (20) typed CpxTrs 24.12/7.42 (21) RewriteLemmaProof [LOWER BOUND(ID), 111 ms] 24.12/7.42 (22) typed CpxTrs 24.12/7.42 (23) RewriteLemmaProof [LOWER BOUND(ID), 115 ms] 24.12/7.42 (24) typed CpxTrs 24.12/7.42 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (0) 24.12/7.42 Obligation: 24.12/7.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.12/7.42 24.12/7.42 24.12/7.42 The TRS R consists of the following rules: 24.12/7.42 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0)) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 proper(0) -> ok(0) 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 S is empty. 24.12/7.42 Rewrite Strategy: FULL 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 24.12/7.42 The following defined symbols can occur below the 0th argument of top: proper, active 24.12/7.42 The following defined symbols can occur below the 0th argument of proper: proper, active 24.12/7.42 The following defined symbols can occur below the 0th argument of active: proper, active 24.12/7.42 24.12/7.42 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0)) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (2) 24.12/7.42 Obligation: 24.12/7.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.12/7.42 24.12/7.42 24.12/7.42 The TRS R consists of the following rules: 24.12/7.42 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(0) -> ok(0) 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 S is empty. 24.12/7.42 Rewrite Strategy: FULL 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 24.12/7.42 transformed relative TRS to TRS 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (4) 24.12/7.42 Obligation: 24.12/7.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.12/7.42 24.12/7.42 24.12/7.42 The TRS R consists of the following rules: 24.12/7.42 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(0) -> ok(0) 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 S is empty. 24.12/7.42 Rewrite Strategy: FULL 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (5) CpxTrsMatchBoundsTAProof (FINISHED) 24.12/7.42 A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 24.12/7.42 24.12/7.42 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 24.12/7.42 final states : [1, 2, 3, 4, 5, 6] 24.12/7.42 transitions: 24.12/7.42 mark0(0) -> 0 24.12/7.42 tt0() -> 0 24.12/7.42 ok0(0) -> 0 24.12/7.42 00() -> 0 24.12/7.42 active0(0) -> 0 24.12/7.42 U110(0, 0, 0) -> 1 24.12/7.42 U120(0, 0, 0) -> 2 24.12/7.42 s0(0) -> 3 24.12/7.42 plus0(0, 0) -> 4 24.12/7.42 proper0(0) -> 5 24.12/7.42 top0(0) -> 6 24.12/7.42 U111(0, 0, 0) -> 7 24.12/7.42 mark1(7) -> 1 24.12/7.42 U121(0, 0, 0) -> 8 24.12/7.42 mark1(8) -> 2 24.12/7.42 s1(0) -> 9 24.12/7.42 mark1(9) -> 3 24.12/7.42 plus1(0, 0) -> 10 24.12/7.42 mark1(10) -> 4 24.12/7.42 tt1() -> 11 24.12/7.42 ok1(11) -> 5 24.12/7.42 01() -> 12 24.12/7.42 ok1(12) -> 5 24.12/7.42 U111(0, 0, 0) -> 13 24.12/7.42 ok1(13) -> 1 24.12/7.42 U121(0, 0, 0) -> 14 24.12/7.42 ok1(14) -> 2 24.12/7.42 s1(0) -> 15 24.12/7.42 ok1(15) -> 3 24.12/7.42 plus1(0, 0) -> 16 24.12/7.42 ok1(16) -> 4 24.12/7.42 proper1(0) -> 17 24.12/7.42 top1(17) -> 6 24.12/7.42 active1(0) -> 18 24.12/7.42 top1(18) -> 6 24.12/7.42 mark1(7) -> 7 24.12/7.42 mark1(7) -> 13 24.12/7.42 mark1(8) -> 8 24.12/7.42 mark1(8) -> 14 24.12/7.42 mark1(9) -> 9 24.12/7.42 mark1(9) -> 15 24.12/7.42 mark1(10) -> 10 24.12/7.42 mark1(10) -> 16 24.12/7.42 ok1(11) -> 17 24.12/7.42 ok1(12) -> 17 24.12/7.42 ok1(13) -> 7 24.12/7.42 ok1(13) -> 13 24.12/7.42 ok1(14) -> 8 24.12/7.42 ok1(14) -> 14 24.12/7.42 ok1(15) -> 9 24.12/7.42 ok1(15) -> 15 24.12/7.42 ok1(16) -> 10 24.12/7.42 ok1(16) -> 16 24.12/7.42 active2(11) -> 19 24.12/7.42 top2(19) -> 6 24.12/7.42 active2(12) -> 19 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (6) 24.12/7.42 BOUNDS(1, n^1) 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 24.12/7.42 Renamed function symbols to avoid clashes with predefined symbol. 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (8) 24.12/7.42 Obligation: 24.12/7.42 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 24.12/7.42 24.12/7.42 24.12/7.42 The TRS R consists of the following rules: 24.12/7.42 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0')) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 proper(0') -> ok(0') 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 S is empty. 24.12/7.42 Rewrite Strategy: FULL 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 24.12/7.42 Infered types. 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (10) 24.12/7.42 Obligation: 24.12/7.42 TRS: 24.12/7.42 Rules: 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0')) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 proper(0') -> ok(0') 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 Types: 24.12/7.42 active :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 tt :: tt:mark:0':ok 24.12/7.42 mark :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 s :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 0' :: tt:mark:0':ok 24.12/7.42 proper :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 ok :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 top :: tt:mark:0':ok -> top 24.12/7.42 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 24.12/7.42 hole_top2_0 :: top 24.12/7.42 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (11) OrderProof (LOWER BOUND(ID)) 24.12/7.42 Heuristically decided to analyse the following defined symbols: 24.12/7.42 active, U12, s, plus, U11, proper, top 24.12/7.42 24.12/7.42 They will be analysed ascendingly in the following order: 24.12/7.42 U12 < active 24.12/7.42 s < active 24.12/7.42 plus < active 24.12/7.42 U11 < active 24.12/7.42 active < top 24.12/7.42 U12 < proper 24.12/7.42 s < proper 24.12/7.42 plus < proper 24.12/7.42 U11 < proper 24.12/7.42 proper < top 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (12) 24.12/7.42 Obligation: 24.12/7.42 TRS: 24.12/7.42 Rules: 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0')) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 proper(0') -> ok(0') 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 Types: 24.12/7.42 active :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 tt :: tt:mark:0':ok 24.12/7.42 mark :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 s :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 0' :: tt:mark:0':ok 24.12/7.42 proper :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 ok :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 top :: tt:mark:0':ok -> top 24.12/7.42 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 24.12/7.42 hole_top2_0 :: top 24.12/7.42 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 24.12/7.42 24.12/7.42 24.12/7.42 Generator Equations: 24.12/7.42 gen_tt:mark:0':ok3_0(0) <=> tt 24.12/7.42 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 24.12/7.42 24.12/7.42 24.12/7.42 The following defined symbols remain to be analysed: 24.12/7.42 U12, active, s, plus, U11, proper, top 24.12/7.42 24.12/7.42 They will be analysed ascendingly in the following order: 24.12/7.42 U12 < active 24.12/7.42 s < active 24.12/7.42 plus < active 24.12/7.42 U11 < active 24.12/7.42 active < top 24.12/7.42 U12 < proper 24.12/7.42 s < proper 24.12/7.42 plus < proper 24.12/7.42 U11 < proper 24.12/7.42 proper < top 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (13) RewriteLemmaProof (LOWER BOUND(ID)) 24.12/7.42 Proved the following rewrite lemma: 24.12/7.42 U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) 24.12/7.42 24.12/7.42 Induction Base: 24.12/7.42 U12(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) 24.12/7.42 24.12/7.42 Induction Step: 24.12/7.42 U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) 24.12/7.42 mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH 24.12/7.42 mark(*4_0) 24.12/7.42 24.12/7.42 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (14) 24.12/7.42 Complex Obligation (BEST) 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (15) 24.12/7.42 Obligation: 24.12/7.42 Proved the lower bound n^1 for the following obligation: 24.12/7.42 24.12/7.42 TRS: 24.12/7.42 Rules: 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0')) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 proper(0') -> ok(0') 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 Types: 24.12/7.42 active :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 tt :: tt:mark:0':ok 24.12/7.42 mark :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 s :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 0' :: tt:mark:0':ok 24.12/7.42 proper :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 ok :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 top :: tt:mark:0':ok -> top 24.12/7.42 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 24.12/7.42 hole_top2_0 :: top 24.12/7.42 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 24.12/7.42 24.12/7.42 24.12/7.42 Generator Equations: 24.12/7.42 gen_tt:mark:0':ok3_0(0) <=> tt 24.12/7.42 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 24.12/7.42 24.12/7.42 24.12/7.42 The following defined symbols remain to be analysed: 24.12/7.42 U12, active, s, plus, U11, proper, top 24.12/7.42 24.12/7.42 They will be analysed ascendingly in the following order: 24.12/7.42 U12 < active 24.12/7.42 s < active 24.12/7.42 plus < active 24.12/7.42 U11 < active 24.12/7.42 active < top 24.12/7.42 U12 < proper 24.12/7.42 s < proper 24.12/7.42 plus < proper 24.12/7.42 U11 < proper 24.12/7.42 proper < top 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (16) LowerBoundPropagationProof (FINISHED) 24.12/7.42 Propagated lower bound. 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (17) 24.12/7.42 BOUNDS(n^1, INF) 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (18) 24.12/7.42 Obligation: 24.12/7.42 TRS: 24.12/7.42 Rules: 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0')) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 proper(0') -> ok(0') 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 Types: 24.12/7.42 active :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 tt :: tt:mark:0':ok 24.12/7.42 mark :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 s :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 0' :: tt:mark:0':ok 24.12/7.42 proper :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 ok :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 top :: tt:mark:0':ok -> top 24.12/7.42 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 24.12/7.42 hole_top2_0 :: top 24.12/7.42 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 24.12/7.42 24.12/7.42 24.12/7.42 Lemmas: 24.12/7.42 U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) 24.12/7.42 24.12/7.42 24.12/7.42 Generator Equations: 24.12/7.42 gen_tt:mark:0':ok3_0(0) <=> tt 24.12/7.42 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 24.12/7.42 24.12/7.42 24.12/7.42 The following defined symbols remain to be analysed: 24.12/7.42 s, active, plus, U11, proper, top 24.12/7.42 24.12/7.42 They will be analysed ascendingly in the following order: 24.12/7.42 s < active 24.12/7.42 plus < active 24.12/7.42 U11 < active 24.12/7.42 active < top 24.12/7.42 s < proper 24.12/7.42 plus < proper 24.12/7.42 U11 < proper 24.12/7.42 proper < top 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (19) RewriteLemmaProof (LOWER BOUND(ID)) 24.12/7.42 Proved the following rewrite lemma: 24.12/7.42 s(gen_tt:mark:0':ok3_0(+(1, n1258_0))) -> *4_0, rt in Omega(n1258_0) 24.12/7.42 24.12/7.42 Induction Base: 24.12/7.42 s(gen_tt:mark:0':ok3_0(+(1, 0))) 24.12/7.42 24.12/7.42 Induction Step: 24.12/7.42 s(gen_tt:mark:0':ok3_0(+(1, +(n1258_0, 1)))) ->_R^Omega(1) 24.12/7.42 mark(s(gen_tt:mark:0':ok3_0(+(1, n1258_0)))) ->_IH 24.12/7.42 mark(*4_0) 24.12/7.42 24.12/7.42 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (20) 24.12/7.42 Obligation: 24.12/7.42 TRS: 24.12/7.42 Rules: 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0')) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 proper(0') -> ok(0') 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 Types: 24.12/7.42 active :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 tt :: tt:mark:0':ok 24.12/7.42 mark :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 s :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 0' :: tt:mark:0':ok 24.12/7.42 proper :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 ok :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 top :: tt:mark:0':ok -> top 24.12/7.42 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 24.12/7.42 hole_top2_0 :: top 24.12/7.42 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 24.12/7.42 24.12/7.42 24.12/7.42 Lemmas: 24.12/7.42 U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) 24.12/7.42 s(gen_tt:mark:0':ok3_0(+(1, n1258_0))) -> *4_0, rt in Omega(n1258_0) 24.12/7.42 24.12/7.42 24.12/7.42 Generator Equations: 24.12/7.42 gen_tt:mark:0':ok3_0(0) <=> tt 24.12/7.42 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 24.12/7.42 24.12/7.42 24.12/7.42 The following defined symbols remain to be analysed: 24.12/7.42 plus, active, U11, proper, top 24.12/7.42 24.12/7.42 They will be analysed ascendingly in the following order: 24.12/7.42 plus < active 24.12/7.42 U11 < active 24.12/7.42 active < top 24.12/7.42 plus < proper 24.12/7.42 U11 < proper 24.12/7.42 proper < top 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (21) RewriteLemmaProof (LOWER BOUND(ID)) 24.12/7.42 Proved the following rewrite lemma: 24.12/7.42 plus(gen_tt:mark:0':ok3_0(+(1, n1802_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1802_0) 24.12/7.42 24.12/7.42 Induction Base: 24.12/7.42 plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) 24.12/7.42 24.12/7.42 Induction Step: 24.12/7.42 plus(gen_tt:mark:0':ok3_0(+(1, +(n1802_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) 24.12/7.42 mark(plus(gen_tt:mark:0':ok3_0(+(1, n1802_0)), gen_tt:mark:0':ok3_0(b))) ->_IH 24.12/7.42 mark(*4_0) 24.12/7.42 24.12/7.42 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (22) 24.12/7.42 Obligation: 24.12/7.42 TRS: 24.12/7.42 Rules: 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0')) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 proper(0') -> ok(0') 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 Types: 24.12/7.42 active :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 tt :: tt:mark:0':ok 24.12/7.42 mark :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 s :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 0' :: tt:mark:0':ok 24.12/7.42 proper :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 ok :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 top :: tt:mark:0':ok -> top 24.12/7.42 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 24.12/7.42 hole_top2_0 :: top 24.12/7.42 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 24.12/7.42 24.12/7.42 24.12/7.42 Lemmas: 24.12/7.42 U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) 24.12/7.42 s(gen_tt:mark:0':ok3_0(+(1, n1258_0))) -> *4_0, rt in Omega(n1258_0) 24.12/7.42 plus(gen_tt:mark:0':ok3_0(+(1, n1802_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1802_0) 24.12/7.42 24.12/7.42 24.12/7.42 Generator Equations: 24.12/7.42 gen_tt:mark:0':ok3_0(0) <=> tt 24.12/7.42 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 24.12/7.42 24.12/7.42 24.12/7.42 The following defined symbols remain to be analysed: 24.12/7.42 U11, active, proper, top 24.12/7.42 24.12/7.42 They will be analysed ascendingly in the following order: 24.12/7.42 U11 < active 24.12/7.42 active < top 24.12/7.42 U11 < proper 24.12/7.42 proper < top 24.12/7.42 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (23) RewriteLemmaProof (LOWER BOUND(ID)) 24.12/7.42 Proved the following rewrite lemma: 24.12/7.42 U11(gen_tt:mark:0':ok3_0(+(1, n3412_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3412_0) 24.12/7.42 24.12/7.42 Induction Base: 24.12/7.42 U11(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) 24.12/7.42 24.12/7.42 Induction Step: 24.12/7.42 U11(gen_tt:mark:0':ok3_0(+(1, +(n3412_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) 24.12/7.42 mark(U11(gen_tt:mark:0':ok3_0(+(1, n3412_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH 24.12/7.42 mark(*4_0) 24.12/7.42 24.12/7.42 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 24.12/7.42 ---------------------------------------- 24.12/7.42 24.12/7.42 (24) 24.12/7.42 Obligation: 24.12/7.42 TRS: 24.12/7.42 Rules: 24.12/7.42 active(U11(tt, M, N)) -> mark(U12(tt, M, N)) 24.12/7.42 active(U12(tt, M, N)) -> mark(s(plus(N, M))) 24.12/7.42 active(plus(N, 0')) -> mark(N) 24.12/7.42 active(plus(N, s(M))) -> mark(U11(tt, M, N)) 24.12/7.42 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 24.12/7.42 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 24.12/7.42 active(s(X)) -> s(active(X)) 24.12/7.42 active(plus(X1, X2)) -> plus(active(X1), X2) 24.12/7.42 active(plus(X1, X2)) -> plus(X1, active(X2)) 24.12/7.42 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 24.12/7.42 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 24.12/7.42 s(mark(X)) -> mark(s(X)) 24.12/7.42 plus(mark(X1), X2) -> mark(plus(X1, X2)) 24.12/7.42 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 24.12/7.42 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(tt) -> ok(tt) 24.12/7.42 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 24.12/7.42 proper(s(X)) -> s(proper(X)) 24.12/7.42 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 24.12/7.42 proper(0') -> ok(0') 24.12/7.42 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 24.12/7.42 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 24.12/7.42 s(ok(X)) -> ok(s(X)) 24.12/7.42 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 24.12/7.42 top(mark(X)) -> top(proper(X)) 24.12/7.42 top(ok(X)) -> top(active(X)) 24.12/7.42 24.12/7.42 Types: 24.12/7.42 active :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 tt :: tt:mark:0':ok 24.12/7.42 mark :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 s :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 0' :: tt:mark:0':ok 24.12/7.42 proper :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 ok :: tt:mark:0':ok -> tt:mark:0':ok 24.12/7.42 top :: tt:mark:0':ok -> top 24.12/7.42 hole_tt:mark:0':ok1_0 :: tt:mark:0':ok 24.12/7.42 hole_top2_0 :: top 24.12/7.42 gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok 24.12/7.42 24.12/7.42 24.12/7.42 Lemmas: 24.12/7.42 U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) 24.12/7.42 s(gen_tt:mark:0':ok3_0(+(1, n1258_0))) -> *4_0, rt in Omega(n1258_0) 24.12/7.42 plus(gen_tt:mark:0':ok3_0(+(1, n1802_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1802_0) 24.12/7.42 U11(gen_tt:mark:0':ok3_0(+(1, n3412_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3412_0) 24.12/7.42 24.12/7.42 24.12/7.42 Generator Equations: 24.12/7.42 gen_tt:mark:0':ok3_0(0) <=> tt 24.12/7.42 gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) 24.12/7.42 24.12/7.42 24.12/7.42 The following defined symbols remain to be analysed: 24.12/7.42 active, proper, top 24.12/7.42 24.12/7.42 They will be analysed ascendingly in the following order: 24.12/7.42 active < top 24.12/7.42 proper < top 24.51/7.62 EOF