23.17/8.01 WORST_CASE(Omega(n^1), O(n^1)) 23.17/8.02 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 23.17/8.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 23.17/8.02 23.17/8.02 23.17/8.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.17/8.02 23.17/8.02 (0) CpxTRS 23.17/8.02 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 23.17/8.02 (2) CpxTRS 23.17/8.02 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 23.17/8.02 (4) CpxTRS 23.17/8.02 (5) CpxTrsMatchBoundsTAProof [FINISHED, 33 ms] 23.17/8.02 (6) BOUNDS(1, n^1) 23.17/8.02 (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 23.17/8.02 (8) CpxTRS 23.17/8.02 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 23.17/8.02 (10) typed CpxTrs 23.17/8.02 (11) OrderProof [LOWER BOUND(ID), 0 ms] 23.17/8.02 (12) typed CpxTrs 23.17/8.02 (13) RewriteLemmaProof [LOWER BOUND(ID), 500 ms] 23.17/8.02 (14) BEST 23.17/8.02 (15) proven lower bound 23.17/8.02 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 23.17/8.02 (17) BOUNDS(n^1, INF) 23.17/8.02 (18) typed CpxTrs 23.17/8.02 (19) RewriteLemmaProof [LOWER BOUND(ID), 128 ms] 23.17/8.02 (20) typed CpxTrs 23.17/8.02 (21) RewriteLemmaProof [LOWER BOUND(ID), 85 ms] 23.17/8.02 (22) typed CpxTrs 23.17/8.02 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (0) 23.17/8.02 Obligation: 23.17/8.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.17/8.02 23.17/8.02 23.17/8.02 The TRS R consists of the following rules: 23.17/8.02 23.17/8.02 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 23.17/8.02 active(__(X, nil)) -> mark(X) 23.17/8.02 active(__(nil, X)) -> mark(X) 23.17/8.02 active(and(tt, X)) -> mark(X) 23.17/8.02 active(isNePal(__(I, __(P, I)))) -> mark(tt) 23.17/8.02 active(__(X1, X2)) -> __(active(X1), X2) 23.17/8.02 active(__(X1, X2)) -> __(X1, active(X2)) 23.17/8.02 active(and(X1, X2)) -> and(active(X1), X2) 23.17/8.02 active(isNePal(X)) -> isNePal(active(X)) 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 proper(isNePal(X)) -> isNePal(proper(X)) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 S is empty. 23.17/8.02 Rewrite Strategy: FULL 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 23.17/8.02 The following defined symbols can occur below the 0th argument of top: proper, active 23.17/8.02 The following defined symbols can occur below the 0th argument of proper: proper, active 23.17/8.02 The following defined symbols can occur below the 0th argument of active: proper, active 23.17/8.02 23.17/8.02 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 23.17/8.02 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 23.17/8.02 active(__(X, nil)) -> mark(X) 23.17/8.02 active(__(nil, X)) -> mark(X) 23.17/8.02 active(and(tt, X)) -> mark(X) 23.17/8.02 active(isNePal(__(I, __(P, I)))) -> mark(tt) 23.17/8.02 active(__(X1, X2)) -> __(active(X1), X2) 23.17/8.02 active(__(X1, X2)) -> __(X1, active(X2)) 23.17/8.02 active(and(X1, X2)) -> and(active(X1), X2) 23.17/8.02 active(isNePal(X)) -> isNePal(active(X)) 23.17/8.02 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 23.17/8.02 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.17/8.02 proper(isNePal(X)) -> isNePal(proper(X)) 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (2) 23.17/8.02 Obligation: 23.17/8.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.17/8.02 23.17/8.02 23.17/8.02 The TRS R consists of the following rules: 23.17/8.02 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 S is empty. 23.17/8.02 Rewrite Strategy: FULL 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 23.17/8.02 transformed relative TRS to TRS 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (4) 23.17/8.02 Obligation: 23.17/8.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.17/8.02 23.17/8.02 23.17/8.02 The TRS R consists of the following rules: 23.17/8.02 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 S is empty. 23.17/8.02 Rewrite Strategy: FULL 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (5) CpxTrsMatchBoundsTAProof (FINISHED) 23.17/8.02 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. 23.17/8.02 23.17/8.02 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 23.17/8.02 final states : [1, 2, 3, 4, 5] 23.17/8.02 transitions: 23.17/8.02 mark0(0) -> 0 23.17/8.02 nil0() -> 0 23.17/8.02 ok0(0) -> 0 23.17/8.02 tt0() -> 0 23.17/8.02 active0(0) -> 0 23.17/8.02 __0(0, 0) -> 1 23.17/8.02 and0(0, 0) -> 2 23.17/8.02 isNePal0(0) -> 3 23.17/8.02 proper0(0) -> 4 23.17/8.02 top0(0) -> 5 23.17/8.02 __1(0, 0) -> 6 23.17/8.02 mark1(6) -> 1 23.17/8.02 and1(0, 0) -> 7 23.17/8.02 mark1(7) -> 2 23.17/8.02 isNePal1(0) -> 8 23.17/8.02 mark1(8) -> 3 23.17/8.02 nil1() -> 9 23.17/8.02 ok1(9) -> 4 23.17/8.02 tt1() -> 10 23.17/8.02 ok1(10) -> 4 23.17/8.02 __1(0, 0) -> 11 23.17/8.02 ok1(11) -> 1 23.17/8.02 and1(0, 0) -> 12 23.17/8.02 ok1(12) -> 2 23.17/8.02 isNePal1(0) -> 13 23.17/8.02 ok1(13) -> 3 23.17/8.02 proper1(0) -> 14 23.17/8.02 top1(14) -> 5 23.17/8.02 active1(0) -> 15 23.17/8.02 top1(15) -> 5 23.17/8.02 mark1(6) -> 6 23.17/8.02 mark1(6) -> 11 23.17/8.02 mark1(7) -> 7 23.17/8.02 mark1(7) -> 12 23.17/8.02 mark1(8) -> 8 23.17/8.02 mark1(8) -> 13 23.17/8.02 ok1(9) -> 14 23.17/8.02 ok1(10) -> 14 23.17/8.02 ok1(11) -> 6 23.17/8.02 ok1(11) -> 11 23.17/8.02 ok1(12) -> 7 23.17/8.02 ok1(12) -> 12 23.17/8.02 ok1(13) -> 8 23.17/8.02 ok1(13) -> 13 23.17/8.02 active2(9) -> 16 23.17/8.02 top2(16) -> 5 23.17/8.02 active2(10) -> 16 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (6) 23.17/8.02 BOUNDS(1, n^1) 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (7) RenamingProof (BOTH BOUNDS(ID, ID)) 23.17/8.02 Renamed function symbols to avoid clashes with predefined symbol. 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (8) 23.17/8.02 Obligation: 23.17/8.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 23.17/8.02 23.17/8.02 23.17/8.02 The TRS R consists of the following rules: 23.17/8.02 23.17/8.02 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 23.17/8.02 active(__(X, nil)) -> mark(X) 23.17/8.02 active(__(nil, X)) -> mark(X) 23.17/8.02 active(and(tt, X)) -> mark(X) 23.17/8.02 active(isNePal(__(I, __(P, I)))) -> mark(tt) 23.17/8.02 active(__(X1, X2)) -> __(active(X1), X2) 23.17/8.02 active(__(X1, X2)) -> __(X1, active(X2)) 23.17/8.02 active(and(X1, X2)) -> and(active(X1), X2) 23.17/8.02 active(isNePal(X)) -> isNePal(active(X)) 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 proper(isNePal(X)) -> isNePal(proper(X)) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 S is empty. 23.17/8.02 Rewrite Strategy: FULL 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 23.17/8.02 Infered types. 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (10) 23.17/8.02 Obligation: 23.17/8.02 TRS: 23.17/8.02 Rules: 23.17/8.02 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 23.17/8.02 active(__(X, nil)) -> mark(X) 23.17/8.02 active(__(nil, X)) -> mark(X) 23.17/8.02 active(and(tt, X)) -> mark(X) 23.17/8.02 active(isNePal(__(I, __(P, I)))) -> mark(tt) 23.17/8.02 active(__(X1, X2)) -> __(active(X1), X2) 23.17/8.02 active(__(X1, X2)) -> __(X1, active(X2)) 23.17/8.02 active(and(X1, X2)) -> and(active(X1), X2) 23.17/8.02 active(isNePal(X)) -> isNePal(active(X)) 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 proper(isNePal(X)) -> isNePal(proper(X)) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 Types: 23.17/8.02 active :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 __ :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 mark :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 nil :: mark:nil:tt:ok 23.17/8.02 and :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 tt :: mark:nil:tt:ok 23.17/8.02 isNePal :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 proper :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 ok :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 top :: mark:nil:tt:ok -> top 23.17/8.02 hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok 23.17/8.02 hole_top2_0 :: top 23.17/8.02 gen_mark:nil:tt:ok3_0 :: Nat -> mark:nil:tt:ok 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (11) OrderProof (LOWER BOUND(ID)) 23.17/8.02 Heuristically decided to analyse the following defined symbols: 23.17/8.02 active, __, and, isNePal, proper, top 23.17/8.02 23.17/8.02 They will be analysed ascendingly in the following order: 23.17/8.02 __ < active 23.17/8.02 and < active 23.17/8.02 isNePal < active 23.17/8.02 active < top 23.17/8.02 __ < proper 23.17/8.02 and < proper 23.17/8.02 isNePal < proper 23.17/8.02 proper < top 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (12) 23.17/8.02 Obligation: 23.17/8.02 TRS: 23.17/8.02 Rules: 23.17/8.02 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 23.17/8.02 active(__(X, nil)) -> mark(X) 23.17/8.02 active(__(nil, X)) -> mark(X) 23.17/8.02 active(and(tt, X)) -> mark(X) 23.17/8.02 active(isNePal(__(I, __(P, I)))) -> mark(tt) 23.17/8.02 active(__(X1, X2)) -> __(active(X1), X2) 23.17/8.02 active(__(X1, X2)) -> __(X1, active(X2)) 23.17/8.02 active(and(X1, X2)) -> and(active(X1), X2) 23.17/8.02 active(isNePal(X)) -> isNePal(active(X)) 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 proper(isNePal(X)) -> isNePal(proper(X)) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 Types: 23.17/8.02 active :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 __ :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 mark :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 nil :: mark:nil:tt:ok 23.17/8.02 and :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 tt :: mark:nil:tt:ok 23.17/8.02 isNePal :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 proper :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 ok :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 top :: mark:nil:tt:ok -> top 23.17/8.02 hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok 23.17/8.02 hole_top2_0 :: top 23.17/8.02 gen_mark:nil:tt:ok3_0 :: Nat -> mark:nil:tt:ok 23.17/8.02 23.17/8.02 23.17/8.02 Generator Equations: 23.17/8.02 gen_mark:nil:tt:ok3_0(0) <=> nil 23.17/8.02 gen_mark:nil:tt:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:ok3_0(x)) 23.17/8.02 23.17/8.02 23.17/8.02 The following defined symbols remain to be analysed: 23.17/8.02 __, active, and, isNePal, proper, top 23.17/8.02 23.17/8.02 They will be analysed ascendingly in the following order: 23.17/8.02 __ < active 23.17/8.02 and < active 23.17/8.02 isNePal < active 23.17/8.02 active < top 23.17/8.02 __ < proper 23.17/8.02 and < proper 23.17/8.02 isNePal < proper 23.17/8.02 proper < top 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (13) RewriteLemmaProof (LOWER BOUND(ID)) 23.17/8.02 Proved the following rewrite lemma: 23.17/8.02 __(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 23.17/8.02 23.17/8.02 Induction Base: 23.17/8.02 __(gen_mark:nil:tt:ok3_0(+(1, 0)), gen_mark:nil:tt:ok3_0(b)) 23.17/8.02 23.17/8.02 Induction Step: 23.17/8.02 __(gen_mark:nil:tt:ok3_0(+(1, +(n5_0, 1))), gen_mark:nil:tt:ok3_0(b)) ->_R^Omega(1) 23.17/8.02 mark(__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b))) ->_IH 23.17/8.02 mark(*4_0) 23.17/8.02 23.17/8.02 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (14) 23.17/8.02 Complex Obligation (BEST) 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (15) 23.17/8.02 Obligation: 23.17/8.02 Proved the lower bound n^1 for the following obligation: 23.17/8.02 23.17/8.02 TRS: 23.17/8.02 Rules: 23.17/8.02 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 23.17/8.02 active(__(X, nil)) -> mark(X) 23.17/8.02 active(__(nil, X)) -> mark(X) 23.17/8.02 active(and(tt, X)) -> mark(X) 23.17/8.02 active(isNePal(__(I, __(P, I)))) -> mark(tt) 23.17/8.02 active(__(X1, X2)) -> __(active(X1), X2) 23.17/8.02 active(__(X1, X2)) -> __(X1, active(X2)) 23.17/8.02 active(and(X1, X2)) -> and(active(X1), X2) 23.17/8.02 active(isNePal(X)) -> isNePal(active(X)) 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 proper(isNePal(X)) -> isNePal(proper(X)) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 Types: 23.17/8.02 active :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 __ :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 mark :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 nil :: mark:nil:tt:ok 23.17/8.02 and :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 tt :: mark:nil:tt:ok 23.17/8.02 isNePal :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 proper :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 ok :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 top :: mark:nil:tt:ok -> top 23.17/8.02 hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok 23.17/8.02 hole_top2_0 :: top 23.17/8.02 gen_mark:nil:tt:ok3_0 :: Nat -> mark:nil:tt:ok 23.17/8.02 23.17/8.02 23.17/8.02 Generator Equations: 23.17/8.02 gen_mark:nil:tt:ok3_0(0) <=> nil 23.17/8.02 gen_mark:nil:tt:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:ok3_0(x)) 23.17/8.02 23.17/8.02 23.17/8.02 The following defined symbols remain to be analysed: 23.17/8.02 __, active, and, isNePal, proper, top 23.17/8.02 23.17/8.02 They will be analysed ascendingly in the following order: 23.17/8.02 __ < active 23.17/8.02 and < active 23.17/8.02 isNePal < active 23.17/8.02 active < top 23.17/8.02 __ < proper 23.17/8.02 and < proper 23.17/8.02 isNePal < proper 23.17/8.02 proper < top 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (16) LowerBoundPropagationProof (FINISHED) 23.17/8.02 Propagated lower bound. 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (17) 23.17/8.02 BOUNDS(n^1, INF) 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (18) 23.17/8.02 Obligation: 23.17/8.02 TRS: 23.17/8.02 Rules: 23.17/8.02 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 23.17/8.02 active(__(X, nil)) -> mark(X) 23.17/8.02 active(__(nil, X)) -> mark(X) 23.17/8.02 active(and(tt, X)) -> mark(X) 23.17/8.02 active(isNePal(__(I, __(P, I)))) -> mark(tt) 23.17/8.02 active(__(X1, X2)) -> __(active(X1), X2) 23.17/8.02 active(__(X1, X2)) -> __(X1, active(X2)) 23.17/8.02 active(and(X1, X2)) -> and(active(X1), X2) 23.17/8.02 active(isNePal(X)) -> isNePal(active(X)) 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 proper(isNePal(X)) -> isNePal(proper(X)) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 Types: 23.17/8.02 active :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 __ :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 mark :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 nil :: mark:nil:tt:ok 23.17/8.02 and :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 tt :: mark:nil:tt:ok 23.17/8.02 isNePal :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 proper :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 ok :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 top :: mark:nil:tt:ok -> top 23.17/8.02 hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok 23.17/8.02 hole_top2_0 :: top 23.17/8.02 gen_mark:nil:tt:ok3_0 :: Nat -> mark:nil:tt:ok 23.17/8.02 23.17/8.02 23.17/8.02 Lemmas: 23.17/8.02 __(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 23.17/8.02 23.17/8.02 23.17/8.02 Generator Equations: 23.17/8.02 gen_mark:nil:tt:ok3_0(0) <=> nil 23.17/8.02 gen_mark:nil:tt:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:ok3_0(x)) 23.17/8.02 23.17/8.02 23.17/8.02 The following defined symbols remain to be analysed: 23.17/8.02 and, active, isNePal, proper, top 23.17/8.02 23.17/8.02 They will be analysed ascendingly in the following order: 23.17/8.02 and < active 23.17/8.02 isNePal < active 23.17/8.02 active < top 23.17/8.02 and < proper 23.17/8.02 isNePal < proper 23.17/8.02 proper < top 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (19) RewriteLemmaProof (LOWER BOUND(ID)) 23.17/8.02 Proved the following rewrite lemma: 23.17/8.02 and(gen_mark:nil:tt:ok3_0(+(1, n961_0)), gen_mark:nil:tt:ok3_0(b)) -> *4_0, rt in Omega(n961_0) 23.17/8.02 23.17/8.02 Induction Base: 23.17/8.02 and(gen_mark:nil:tt:ok3_0(+(1, 0)), gen_mark:nil:tt:ok3_0(b)) 23.17/8.02 23.17/8.02 Induction Step: 23.17/8.02 and(gen_mark:nil:tt:ok3_0(+(1, +(n961_0, 1))), gen_mark:nil:tt:ok3_0(b)) ->_R^Omega(1) 23.17/8.02 mark(and(gen_mark:nil:tt:ok3_0(+(1, n961_0)), gen_mark:nil:tt:ok3_0(b))) ->_IH 23.17/8.02 mark(*4_0) 23.17/8.02 23.17/8.02 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (20) 23.17/8.02 Obligation: 23.17/8.02 TRS: 23.17/8.02 Rules: 23.17/8.02 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 23.17/8.02 active(__(X, nil)) -> mark(X) 23.17/8.02 active(__(nil, X)) -> mark(X) 23.17/8.02 active(and(tt, X)) -> mark(X) 23.17/8.02 active(isNePal(__(I, __(P, I)))) -> mark(tt) 23.17/8.02 active(__(X1, X2)) -> __(active(X1), X2) 23.17/8.02 active(__(X1, X2)) -> __(X1, active(X2)) 23.17/8.02 active(and(X1, X2)) -> and(active(X1), X2) 23.17/8.02 active(isNePal(X)) -> isNePal(active(X)) 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 proper(isNePal(X)) -> isNePal(proper(X)) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 Types: 23.17/8.02 active :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 __ :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 mark :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 nil :: mark:nil:tt:ok 23.17/8.02 and :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 tt :: mark:nil:tt:ok 23.17/8.02 isNePal :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 proper :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 ok :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 top :: mark:nil:tt:ok -> top 23.17/8.02 hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok 23.17/8.02 hole_top2_0 :: top 23.17/8.02 gen_mark:nil:tt:ok3_0 :: Nat -> mark:nil:tt:ok 23.17/8.02 23.17/8.02 23.17/8.02 Lemmas: 23.17/8.02 __(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 23.17/8.02 and(gen_mark:nil:tt:ok3_0(+(1, n961_0)), gen_mark:nil:tt:ok3_0(b)) -> *4_0, rt in Omega(n961_0) 23.17/8.02 23.17/8.02 23.17/8.02 Generator Equations: 23.17/8.02 gen_mark:nil:tt:ok3_0(0) <=> nil 23.17/8.02 gen_mark:nil:tt:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:ok3_0(x)) 23.17/8.02 23.17/8.02 23.17/8.02 The following defined symbols remain to be analysed: 23.17/8.02 isNePal, active, proper, top 23.17/8.02 23.17/8.02 They will be analysed ascendingly in the following order: 23.17/8.02 isNePal < active 23.17/8.02 active < top 23.17/8.02 isNePal < proper 23.17/8.02 proper < top 23.17/8.02 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (21) RewriteLemmaProof (LOWER BOUND(ID)) 23.17/8.02 Proved the following rewrite lemma: 23.17/8.02 isNePal(gen_mark:nil:tt:ok3_0(+(1, n2024_0))) -> *4_0, rt in Omega(n2024_0) 23.17/8.02 23.17/8.02 Induction Base: 23.17/8.02 isNePal(gen_mark:nil:tt:ok3_0(+(1, 0))) 23.17/8.02 23.17/8.02 Induction Step: 23.17/8.02 isNePal(gen_mark:nil:tt:ok3_0(+(1, +(n2024_0, 1)))) ->_R^Omega(1) 23.17/8.02 mark(isNePal(gen_mark:nil:tt:ok3_0(+(1, n2024_0)))) ->_IH 23.17/8.02 mark(*4_0) 23.17/8.02 23.17/8.02 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 23.17/8.02 ---------------------------------------- 23.17/8.02 23.17/8.02 (22) 23.17/8.02 Obligation: 23.17/8.02 TRS: 23.17/8.02 Rules: 23.17/8.02 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 23.17/8.02 active(__(X, nil)) -> mark(X) 23.17/8.02 active(__(nil, X)) -> mark(X) 23.17/8.02 active(and(tt, X)) -> mark(X) 23.17/8.02 active(isNePal(__(I, __(P, I)))) -> mark(tt) 23.17/8.02 active(__(X1, X2)) -> __(active(X1), X2) 23.17/8.02 active(__(X1, X2)) -> __(X1, active(X2)) 23.17/8.02 active(and(X1, X2)) -> and(active(X1), X2) 23.17/8.02 active(isNePal(X)) -> isNePal(active(X)) 23.17/8.02 __(mark(X1), X2) -> mark(__(X1, X2)) 23.17/8.02 __(X1, mark(X2)) -> mark(__(X1, X2)) 23.17/8.02 and(mark(X1), X2) -> mark(and(X1, X2)) 23.17/8.02 isNePal(mark(X)) -> mark(isNePal(X)) 23.17/8.02 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 23.17/8.02 proper(nil) -> ok(nil) 23.17/8.02 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.17/8.02 proper(tt) -> ok(tt) 23.17/8.02 proper(isNePal(X)) -> isNePal(proper(X)) 23.17/8.02 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 23.17/8.02 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.17/8.02 isNePal(ok(X)) -> ok(isNePal(X)) 23.17/8.02 top(mark(X)) -> top(proper(X)) 23.17/8.02 top(ok(X)) -> top(active(X)) 23.17/8.02 23.17/8.02 Types: 23.17/8.02 active :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 __ :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 mark :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 nil :: mark:nil:tt:ok 23.17/8.02 and :: mark:nil:tt:ok -> mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 tt :: mark:nil:tt:ok 23.17/8.02 isNePal :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 proper :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 ok :: mark:nil:tt:ok -> mark:nil:tt:ok 23.17/8.02 top :: mark:nil:tt:ok -> top 23.17/8.02 hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok 23.17/8.02 hole_top2_0 :: top 23.17/8.02 gen_mark:nil:tt:ok3_0 :: Nat -> mark:nil:tt:ok 23.17/8.02 23.17/8.02 23.17/8.02 Lemmas: 23.17/8.02 __(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) -> *4_0, rt in Omega(n5_0) 23.17/8.02 and(gen_mark:nil:tt:ok3_0(+(1, n961_0)), gen_mark:nil:tt:ok3_0(b)) -> *4_0, rt in Omega(n961_0) 23.17/8.02 isNePal(gen_mark:nil:tt:ok3_0(+(1, n2024_0))) -> *4_0, rt in Omega(n2024_0) 23.17/8.02 23.17/8.02 23.17/8.02 Generator Equations: 23.17/8.02 gen_mark:nil:tt:ok3_0(0) <=> nil 23.17/8.02 gen_mark:nil:tt:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:ok3_0(x)) 23.17/8.02 23.17/8.02 23.17/8.02 The following defined symbols remain to be analysed: 23.17/8.02 active, proper, top 23.17/8.02 23.17/8.02 They will be analysed ascendingly in the following order: 23.17/8.02 active < top 23.17/8.02 proper < top 23.17/8.06 EOF