306.93/291.48 WORST_CASE(Omega(n^2), ?) 307.04/291.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 307.04/291.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 307.04/291.49 307.04/291.49 307.04/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 307.04/291.49 307.04/291.49 (0) CpxTRS 307.04/291.49 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 307.04/291.49 (2) CpxTRS 307.04/291.49 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 307.04/291.49 (4) typed CpxTrs 307.04/291.49 (5) OrderProof [LOWER BOUND(ID), 0 ms] 307.04/291.49 (6) typed CpxTrs 307.04/291.49 (7) RewriteLemmaProof [LOWER BOUND(ID), 277 ms] 307.04/291.49 (8) BEST 307.04/291.49 (9) proven lower bound 307.04/291.49 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 307.04/291.49 (11) BOUNDS(n^1, INF) 307.04/291.49 (12) typed CpxTrs 307.04/291.49 (13) RewriteLemmaProof [LOWER BOUND(ID), 77 ms] 307.04/291.49 (14) typed CpxTrs 307.04/291.49 (15) RewriteLemmaProof [LOWER BOUND(ID), 0 ms] 307.04/291.49 (16) typed CpxTrs 307.04/291.49 (17) RewriteLemmaProof [LOWER BOUND(ID), 59 ms] 307.04/291.49 (18) proven lower bound 307.04/291.49 (19) LowerBoundPropagationProof [FINISHED, 0 ms] 307.04/291.49 (20) BOUNDS(n^2, INF) 307.04/291.49 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (0) 307.04/291.49 Obligation: 307.04/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 307.04/291.49 307.04/291.49 307.04/291.49 The TRS R consists of the following rules: 307.04/291.49 307.04/291.49 le(0, y) -> true 307.04/291.49 le(s(x), 0) -> false 307.04/291.49 le(s(x), s(y)) -> le(x, y) 307.04/291.49 eq(0, 0) -> true 307.04/291.49 eq(0, s(y)) -> false 307.04/291.49 eq(s(x), 0) -> false 307.04/291.49 eq(s(x), s(y)) -> eq(x, y) 307.04/291.49 if1(true, x, y, xs) -> min(x, xs) 307.04/291.49 if1(false, x, y, xs) -> min(y, xs) 307.04/291.49 if2(true, x, y, xs) -> xs 307.04/291.49 if2(false, x, y, xs) -> cons(y, del(x, xs)) 307.04/291.49 minsort(nil) -> nil 307.04/291.49 minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) 307.04/291.49 min(x, nil) -> x 307.04/291.49 min(x, cons(y, z)) -> if1(le(x, y), x, y, z) 307.04/291.49 del(x, nil) -> nil 307.04/291.49 del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) 307.04/291.49 307.04/291.49 S is empty. 307.04/291.49 Rewrite Strategy: FULL 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 307.04/291.49 Renamed function symbols to avoid clashes with predefined symbol. 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (2) 307.04/291.49 Obligation: 307.04/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 307.04/291.49 307.04/291.49 307.04/291.49 The TRS R consists of the following rules: 307.04/291.49 307.04/291.49 le(0', y) -> true 307.04/291.49 le(s(x), 0') -> false 307.04/291.49 le(s(x), s(y)) -> le(x, y) 307.04/291.49 eq(0', 0') -> true 307.04/291.49 eq(0', s(y)) -> false 307.04/291.49 eq(s(x), 0') -> false 307.04/291.49 eq(s(x), s(y)) -> eq(x, y) 307.04/291.49 if1(true, x, y, xs) -> min(x, xs) 307.04/291.49 if1(false, x, y, xs) -> min(y, xs) 307.04/291.49 if2(true, x, y, xs) -> xs 307.04/291.49 if2(false, x, y, xs) -> cons(y, del(x, xs)) 307.04/291.49 minsort(nil) -> nil 307.04/291.49 minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) 307.04/291.49 min(x, nil) -> x 307.04/291.49 min(x, cons(y, z)) -> if1(le(x, y), x, y, z) 307.04/291.49 del(x, nil) -> nil 307.04/291.49 del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) 307.04/291.49 307.04/291.49 S is empty. 307.04/291.49 Rewrite Strategy: FULL 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 307.04/291.49 Infered types. 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (4) 307.04/291.49 Obligation: 307.04/291.49 TRS: 307.04/291.49 Rules: 307.04/291.49 le(0', y) -> true 307.04/291.49 le(s(x), 0') -> false 307.04/291.49 le(s(x), s(y)) -> le(x, y) 307.04/291.49 eq(0', 0') -> true 307.04/291.49 eq(0', s(y)) -> false 307.04/291.49 eq(s(x), 0') -> false 307.04/291.49 eq(s(x), s(y)) -> eq(x, y) 307.04/291.49 if1(true, x, y, xs) -> min(x, xs) 307.04/291.49 if1(false, x, y, xs) -> min(y, xs) 307.04/291.49 if2(true, x, y, xs) -> xs 307.04/291.49 if2(false, x, y, xs) -> cons(y, del(x, xs)) 307.04/291.49 minsort(nil) -> nil 307.04/291.49 minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) 307.04/291.49 min(x, nil) -> x 307.04/291.49 min(x, cons(y, z)) -> if1(le(x, y), x, y, z) 307.04/291.49 del(x, nil) -> nil 307.04/291.49 del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) 307.04/291.49 307.04/291.49 Types: 307.04/291.49 le :: 0':s -> 0':s -> true:false 307.04/291.49 0' :: 0':s 307.04/291.49 true :: true:false 307.04/291.49 s :: 0':s -> 0':s 307.04/291.49 false :: true:false 307.04/291.49 eq :: 0':s -> 0':s -> true:false 307.04/291.49 if1 :: true:false -> 0':s -> 0':s -> cons:nil -> 0':s 307.04/291.49 min :: 0':s -> cons:nil -> 0':s 307.04/291.49 if2 :: true:false -> 0':s -> 0':s -> cons:nil -> cons:nil 307.04/291.49 cons :: 0':s -> cons:nil -> cons:nil 307.04/291.49 del :: 0':s -> cons:nil -> cons:nil 307.04/291.49 minsort :: cons:nil -> cons:nil 307.04/291.49 nil :: cons:nil 307.04/291.49 hole_true:false1_0 :: true:false 307.04/291.49 hole_0':s2_0 :: 0':s 307.04/291.49 hole_cons:nil3_0 :: cons:nil 307.04/291.49 gen_0':s4_0 :: Nat -> 0':s 307.04/291.49 gen_cons:nil5_0 :: Nat -> cons:nil 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (5) OrderProof (LOWER BOUND(ID)) 307.04/291.49 Heuristically decided to analyse the following defined symbols: 307.04/291.49 le, eq, min, del, minsort 307.04/291.49 307.04/291.49 They will be analysed ascendingly in the following order: 307.04/291.49 le < min 307.04/291.49 eq < del 307.04/291.49 min < minsort 307.04/291.49 del < minsort 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (6) 307.04/291.49 Obligation: 307.04/291.49 TRS: 307.04/291.49 Rules: 307.04/291.49 le(0', y) -> true 307.04/291.49 le(s(x), 0') -> false 307.04/291.49 le(s(x), s(y)) -> le(x, y) 307.04/291.49 eq(0', 0') -> true 307.04/291.49 eq(0', s(y)) -> false 307.04/291.49 eq(s(x), 0') -> false 307.04/291.49 eq(s(x), s(y)) -> eq(x, y) 307.04/291.49 if1(true, x, y, xs) -> min(x, xs) 307.04/291.49 if1(false, x, y, xs) -> min(y, xs) 307.04/291.49 if2(true, x, y, xs) -> xs 307.04/291.49 if2(false, x, y, xs) -> cons(y, del(x, xs)) 307.04/291.49 minsort(nil) -> nil 307.04/291.49 minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) 307.04/291.49 min(x, nil) -> x 307.04/291.49 min(x, cons(y, z)) -> if1(le(x, y), x, y, z) 307.04/291.49 del(x, nil) -> nil 307.04/291.49 del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) 307.04/291.49 307.04/291.49 Types: 307.04/291.49 le :: 0':s -> 0':s -> true:false 307.04/291.49 0' :: 0':s 307.04/291.49 true :: true:false 307.04/291.49 s :: 0':s -> 0':s 307.04/291.49 false :: true:false 307.04/291.49 eq :: 0':s -> 0':s -> true:false 307.04/291.49 if1 :: true:false -> 0':s -> 0':s -> cons:nil -> 0':s 307.04/291.49 min :: 0':s -> cons:nil -> 0':s 307.04/291.49 if2 :: true:false -> 0':s -> 0':s -> cons:nil -> cons:nil 307.04/291.49 cons :: 0':s -> cons:nil -> cons:nil 307.04/291.49 del :: 0':s -> cons:nil -> cons:nil 307.04/291.49 minsort :: cons:nil -> cons:nil 307.04/291.49 nil :: cons:nil 307.04/291.49 hole_true:false1_0 :: true:false 307.04/291.49 hole_0':s2_0 :: 0':s 307.04/291.49 hole_cons:nil3_0 :: cons:nil 307.04/291.49 gen_0':s4_0 :: Nat -> 0':s 307.04/291.49 gen_cons:nil5_0 :: Nat -> cons:nil 307.04/291.49 307.04/291.49 307.04/291.49 Generator Equations: 307.04/291.49 gen_0':s4_0(0) <=> 0' 307.04/291.49 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 307.04/291.49 gen_cons:nil5_0(0) <=> nil 307.04/291.49 gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x)) 307.04/291.49 307.04/291.49 307.04/291.49 The following defined symbols remain to be analysed: 307.04/291.49 le, eq, min, del, minsort 307.04/291.49 307.04/291.49 They will be analysed ascendingly in the following order: 307.04/291.49 le < min 307.04/291.49 eq < del 307.04/291.49 min < minsort 307.04/291.49 del < minsort 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (7) RewriteLemmaProof (LOWER BOUND(ID)) 307.04/291.49 Proved the following rewrite lemma: 307.04/291.49 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 307.04/291.49 307.04/291.49 Induction Base: 307.04/291.49 le(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 307.04/291.49 true 307.04/291.49 307.04/291.49 Induction Step: 307.04/291.49 le(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1) 307.04/291.49 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH 307.04/291.49 true 307.04/291.49 307.04/291.49 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (8) 307.04/291.49 Complex Obligation (BEST) 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (9) 307.04/291.49 Obligation: 307.04/291.49 Proved the lower bound n^1 for the following obligation: 307.04/291.49 307.04/291.49 TRS: 307.04/291.49 Rules: 307.04/291.49 le(0', y) -> true 307.04/291.49 le(s(x), 0') -> false 307.04/291.49 le(s(x), s(y)) -> le(x, y) 307.04/291.49 eq(0', 0') -> true 307.04/291.49 eq(0', s(y)) -> false 307.04/291.49 eq(s(x), 0') -> false 307.04/291.49 eq(s(x), s(y)) -> eq(x, y) 307.04/291.49 if1(true, x, y, xs) -> min(x, xs) 307.04/291.49 if1(false, x, y, xs) -> min(y, xs) 307.04/291.49 if2(true, x, y, xs) -> xs 307.04/291.49 if2(false, x, y, xs) -> cons(y, del(x, xs)) 307.04/291.49 minsort(nil) -> nil 307.04/291.49 minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) 307.04/291.49 min(x, nil) -> x 307.04/291.49 min(x, cons(y, z)) -> if1(le(x, y), x, y, z) 307.04/291.49 del(x, nil) -> nil 307.04/291.49 del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) 307.04/291.49 307.04/291.49 Types: 307.04/291.49 le :: 0':s -> 0':s -> true:false 307.04/291.49 0' :: 0':s 307.04/291.49 true :: true:false 307.04/291.49 s :: 0':s -> 0':s 307.04/291.49 false :: true:false 307.04/291.49 eq :: 0':s -> 0':s -> true:false 307.04/291.49 if1 :: true:false -> 0':s -> 0':s -> cons:nil -> 0':s 307.04/291.49 min :: 0':s -> cons:nil -> 0':s 307.04/291.49 if2 :: true:false -> 0':s -> 0':s -> cons:nil -> cons:nil 307.04/291.49 cons :: 0':s -> cons:nil -> cons:nil 307.04/291.49 del :: 0':s -> cons:nil -> cons:nil 307.04/291.49 minsort :: cons:nil -> cons:nil 307.04/291.49 nil :: cons:nil 307.04/291.49 hole_true:false1_0 :: true:false 307.04/291.49 hole_0':s2_0 :: 0':s 307.04/291.49 hole_cons:nil3_0 :: cons:nil 307.04/291.49 gen_0':s4_0 :: Nat -> 0':s 307.04/291.49 gen_cons:nil5_0 :: Nat -> cons:nil 307.04/291.49 307.04/291.49 307.04/291.49 Generator Equations: 307.04/291.49 gen_0':s4_0(0) <=> 0' 307.04/291.49 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 307.04/291.49 gen_cons:nil5_0(0) <=> nil 307.04/291.49 gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x)) 307.04/291.49 307.04/291.49 307.04/291.49 The following defined symbols remain to be analysed: 307.04/291.49 le, eq, min, del, minsort 307.04/291.49 307.04/291.49 They will be analysed ascendingly in the following order: 307.04/291.49 le < min 307.04/291.49 eq < del 307.04/291.49 min < minsort 307.04/291.49 del < minsort 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (10) LowerBoundPropagationProof (FINISHED) 307.04/291.49 Propagated lower bound. 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (11) 307.04/291.49 BOUNDS(n^1, INF) 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (12) 307.04/291.49 Obligation: 307.04/291.49 TRS: 307.04/291.49 Rules: 307.04/291.49 le(0', y) -> true 307.04/291.49 le(s(x), 0') -> false 307.04/291.49 le(s(x), s(y)) -> le(x, y) 307.04/291.49 eq(0', 0') -> true 307.04/291.49 eq(0', s(y)) -> false 307.04/291.49 eq(s(x), 0') -> false 307.04/291.49 eq(s(x), s(y)) -> eq(x, y) 307.04/291.49 if1(true, x, y, xs) -> min(x, xs) 307.04/291.49 if1(false, x, y, xs) -> min(y, xs) 307.04/291.49 if2(true, x, y, xs) -> xs 307.04/291.49 if2(false, x, y, xs) -> cons(y, del(x, xs)) 307.04/291.49 minsort(nil) -> nil 307.04/291.49 minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) 307.04/291.49 min(x, nil) -> x 307.04/291.49 min(x, cons(y, z)) -> if1(le(x, y), x, y, z) 307.04/291.49 del(x, nil) -> nil 307.04/291.49 del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) 307.04/291.49 307.04/291.49 Types: 307.04/291.49 le :: 0':s -> 0':s -> true:false 307.04/291.49 0' :: 0':s 307.04/291.49 true :: true:false 307.04/291.49 s :: 0':s -> 0':s 307.04/291.49 false :: true:false 307.04/291.49 eq :: 0':s -> 0':s -> true:false 307.04/291.49 if1 :: true:false -> 0':s -> 0':s -> cons:nil -> 0':s 307.04/291.49 min :: 0':s -> cons:nil -> 0':s 307.04/291.49 if2 :: true:false -> 0':s -> 0':s -> cons:nil -> cons:nil 307.04/291.49 cons :: 0':s -> cons:nil -> cons:nil 307.04/291.49 del :: 0':s -> cons:nil -> cons:nil 307.04/291.49 minsort :: cons:nil -> cons:nil 307.04/291.49 nil :: cons:nil 307.04/291.49 hole_true:false1_0 :: true:false 307.04/291.49 hole_0':s2_0 :: 0':s 307.04/291.49 hole_cons:nil3_0 :: cons:nil 307.04/291.49 gen_0':s4_0 :: Nat -> 0':s 307.04/291.49 gen_cons:nil5_0 :: Nat -> cons:nil 307.04/291.49 307.04/291.49 307.04/291.49 Lemmas: 307.04/291.49 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 307.04/291.49 307.04/291.49 307.04/291.49 Generator Equations: 307.04/291.49 gen_0':s4_0(0) <=> 0' 307.04/291.49 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 307.04/291.49 gen_cons:nil5_0(0) <=> nil 307.04/291.49 gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x)) 307.04/291.49 307.04/291.49 307.04/291.49 The following defined symbols remain to be analysed: 307.04/291.49 eq, min, del, minsort 307.04/291.49 307.04/291.49 They will be analysed ascendingly in the following order: 307.04/291.49 eq < del 307.04/291.49 min < minsort 307.04/291.49 del < minsort 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (13) RewriteLemmaProof (LOWER BOUND(ID)) 307.04/291.49 Proved the following rewrite lemma: 307.04/291.49 eq(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) -> true, rt in Omega(1 + n300_0) 307.04/291.49 307.04/291.49 Induction Base: 307.04/291.49 eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 307.04/291.49 true 307.04/291.49 307.04/291.49 Induction Step: 307.04/291.49 eq(gen_0':s4_0(+(n300_0, 1)), gen_0':s4_0(+(n300_0, 1))) ->_R^Omega(1) 307.04/291.49 eq(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) ->_IH 307.04/291.49 true 307.04/291.49 307.04/291.49 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (14) 307.04/291.49 Obligation: 307.04/291.49 TRS: 307.04/291.49 Rules: 307.04/291.49 le(0', y) -> true 307.04/291.49 le(s(x), 0') -> false 307.04/291.49 le(s(x), s(y)) -> le(x, y) 307.04/291.49 eq(0', 0') -> true 307.04/291.49 eq(0', s(y)) -> false 307.04/291.49 eq(s(x), 0') -> false 307.04/291.49 eq(s(x), s(y)) -> eq(x, y) 307.04/291.49 if1(true, x, y, xs) -> min(x, xs) 307.04/291.49 if1(false, x, y, xs) -> min(y, xs) 307.04/291.49 if2(true, x, y, xs) -> xs 307.04/291.49 if2(false, x, y, xs) -> cons(y, del(x, xs)) 307.04/291.49 minsort(nil) -> nil 307.04/291.49 minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) 307.04/291.49 min(x, nil) -> x 307.04/291.49 min(x, cons(y, z)) -> if1(le(x, y), x, y, z) 307.04/291.49 del(x, nil) -> nil 307.04/291.49 del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) 307.04/291.49 307.04/291.49 Types: 307.04/291.49 le :: 0':s -> 0':s -> true:false 307.04/291.49 0' :: 0':s 307.04/291.49 true :: true:false 307.04/291.49 s :: 0':s -> 0':s 307.04/291.49 false :: true:false 307.04/291.49 eq :: 0':s -> 0':s -> true:false 307.04/291.49 if1 :: true:false -> 0':s -> 0':s -> cons:nil -> 0':s 307.04/291.49 min :: 0':s -> cons:nil -> 0':s 307.04/291.49 if2 :: true:false -> 0':s -> 0':s -> cons:nil -> cons:nil 307.04/291.49 cons :: 0':s -> cons:nil -> cons:nil 307.04/291.49 del :: 0':s -> cons:nil -> cons:nil 307.04/291.49 minsort :: cons:nil -> cons:nil 307.04/291.49 nil :: cons:nil 307.04/291.49 hole_true:false1_0 :: true:false 307.04/291.49 hole_0':s2_0 :: 0':s 307.04/291.49 hole_cons:nil3_0 :: cons:nil 307.04/291.49 gen_0':s4_0 :: Nat -> 0':s 307.04/291.49 gen_cons:nil5_0 :: Nat -> cons:nil 307.04/291.49 307.04/291.49 307.04/291.49 Lemmas: 307.04/291.49 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 307.04/291.49 eq(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) -> true, rt in Omega(1 + n300_0) 307.04/291.49 307.04/291.49 307.04/291.49 Generator Equations: 307.04/291.49 gen_0':s4_0(0) <=> 0' 307.04/291.49 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 307.04/291.49 gen_cons:nil5_0(0) <=> nil 307.04/291.49 gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x)) 307.04/291.49 307.04/291.49 307.04/291.49 The following defined symbols remain to be analysed: 307.04/291.49 min, del, minsort 307.04/291.49 307.04/291.49 They will be analysed ascendingly in the following order: 307.04/291.49 min < minsort 307.04/291.49 del < minsort 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (15) RewriteLemmaProof (LOWER BOUND(ID)) 307.04/291.49 Proved the following rewrite lemma: 307.04/291.49 min(gen_0':s4_0(0), gen_cons:nil5_0(n811_0)) -> gen_0':s4_0(0), rt in Omega(1 + n811_0) 307.04/291.49 307.04/291.49 Induction Base: 307.04/291.49 min(gen_0':s4_0(0), gen_cons:nil5_0(0)) ->_R^Omega(1) 307.04/291.49 gen_0':s4_0(0) 307.04/291.49 307.04/291.49 Induction Step: 307.04/291.49 min(gen_0':s4_0(0), gen_cons:nil5_0(+(n811_0, 1))) ->_R^Omega(1) 307.04/291.49 if1(le(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_cons:nil5_0(n811_0)) ->_L^Omega(1) 307.04/291.49 if1(true, gen_0':s4_0(0), 0', gen_cons:nil5_0(n811_0)) ->_R^Omega(1) 307.04/291.49 min(gen_0':s4_0(0), gen_cons:nil5_0(n811_0)) ->_IH 307.04/291.49 gen_0':s4_0(0) 307.04/291.49 307.04/291.49 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (16) 307.04/291.49 Obligation: 307.04/291.49 TRS: 307.04/291.49 Rules: 307.04/291.49 le(0', y) -> true 307.04/291.49 le(s(x), 0') -> false 307.04/291.49 le(s(x), s(y)) -> le(x, y) 307.04/291.49 eq(0', 0') -> true 307.04/291.49 eq(0', s(y)) -> false 307.04/291.49 eq(s(x), 0') -> false 307.04/291.49 eq(s(x), s(y)) -> eq(x, y) 307.04/291.49 if1(true, x, y, xs) -> min(x, xs) 307.04/291.49 if1(false, x, y, xs) -> min(y, xs) 307.04/291.49 if2(true, x, y, xs) -> xs 307.04/291.49 if2(false, x, y, xs) -> cons(y, del(x, xs)) 307.04/291.49 minsort(nil) -> nil 307.04/291.49 minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) 307.04/291.49 min(x, nil) -> x 307.04/291.49 min(x, cons(y, z)) -> if1(le(x, y), x, y, z) 307.04/291.49 del(x, nil) -> nil 307.04/291.49 del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) 307.04/291.49 307.04/291.49 Types: 307.04/291.49 le :: 0':s -> 0':s -> true:false 307.04/291.49 0' :: 0':s 307.04/291.49 true :: true:false 307.04/291.49 s :: 0':s -> 0':s 307.04/291.49 false :: true:false 307.04/291.49 eq :: 0':s -> 0':s -> true:false 307.04/291.49 if1 :: true:false -> 0':s -> 0':s -> cons:nil -> 0':s 307.04/291.49 min :: 0':s -> cons:nil -> 0':s 307.04/291.49 if2 :: true:false -> 0':s -> 0':s -> cons:nil -> cons:nil 307.04/291.49 cons :: 0':s -> cons:nil -> cons:nil 307.04/291.49 del :: 0':s -> cons:nil -> cons:nil 307.04/291.49 minsort :: cons:nil -> cons:nil 307.04/291.49 nil :: cons:nil 307.04/291.49 hole_true:false1_0 :: true:false 307.04/291.49 hole_0':s2_0 :: 0':s 307.04/291.49 hole_cons:nil3_0 :: cons:nil 307.04/291.49 gen_0':s4_0 :: Nat -> 0':s 307.04/291.49 gen_cons:nil5_0 :: Nat -> cons:nil 307.04/291.49 307.04/291.49 307.04/291.49 Lemmas: 307.04/291.49 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 307.04/291.49 eq(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) -> true, rt in Omega(1 + n300_0) 307.04/291.49 min(gen_0':s4_0(0), gen_cons:nil5_0(n811_0)) -> gen_0':s4_0(0), rt in Omega(1 + n811_0) 307.04/291.49 307.04/291.49 307.04/291.49 Generator Equations: 307.04/291.49 gen_0':s4_0(0) <=> 0' 307.04/291.49 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 307.04/291.49 gen_cons:nil5_0(0) <=> nil 307.04/291.49 gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x)) 307.04/291.49 307.04/291.49 307.04/291.49 The following defined symbols remain to be analysed: 307.04/291.49 del, minsort 307.04/291.49 307.04/291.49 They will be analysed ascendingly in the following order: 307.04/291.49 del < minsort 307.04/291.49 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (17) RewriteLemmaProof (LOWER BOUND(ID)) 307.04/291.49 Proved the following rewrite lemma: 307.04/291.49 minsort(gen_cons:nil5_0(n1533_0)) -> gen_cons:nil5_0(n1533_0), rt in Omega(1 + n1533_0 + n1533_0^2) 307.04/291.49 307.04/291.49 Induction Base: 307.04/291.49 minsort(gen_cons:nil5_0(0)) ->_R^Omega(1) 307.04/291.49 nil 307.04/291.49 307.04/291.49 Induction Step: 307.04/291.49 minsort(gen_cons:nil5_0(+(n1533_0, 1))) ->_R^Omega(1) 307.04/291.49 cons(min(0', gen_cons:nil5_0(n1533_0)), minsort(del(min(0', gen_cons:nil5_0(n1533_0)), cons(0', gen_cons:nil5_0(n1533_0))))) ->_L^Omega(1 + n1533_0) 307.04/291.49 cons(gen_0':s4_0(0), minsort(del(min(0', gen_cons:nil5_0(n1533_0)), cons(0', gen_cons:nil5_0(n1533_0))))) ->_L^Omega(1 + n1533_0) 307.04/291.49 cons(gen_0':s4_0(0), minsort(del(gen_0':s4_0(0), cons(0', gen_cons:nil5_0(n1533_0))))) ->_R^Omega(1) 307.04/291.49 cons(gen_0':s4_0(0), minsort(if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_cons:nil5_0(n1533_0)))) ->_L^Omega(1) 307.04/291.49 cons(gen_0':s4_0(0), minsort(if2(true, gen_0':s4_0(0), 0', gen_cons:nil5_0(n1533_0)))) ->_R^Omega(1) 307.04/291.49 cons(gen_0':s4_0(0), minsort(gen_cons:nil5_0(n1533_0))) ->_IH 307.04/291.49 cons(gen_0':s4_0(0), gen_cons:nil5_0(c1534_0)) 307.04/291.49 307.04/291.49 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (18) 307.04/291.49 Obligation: 307.04/291.49 Proved the lower bound n^2 for the following obligation: 307.04/291.49 307.04/291.49 TRS: 307.04/291.49 Rules: 307.04/291.49 le(0', y) -> true 307.04/291.49 le(s(x), 0') -> false 307.04/291.49 le(s(x), s(y)) -> le(x, y) 307.04/291.49 eq(0', 0') -> true 307.04/291.49 eq(0', s(y)) -> false 307.04/291.49 eq(s(x), 0') -> false 307.04/291.49 eq(s(x), s(y)) -> eq(x, y) 307.04/291.49 if1(true, x, y, xs) -> min(x, xs) 307.04/291.49 if1(false, x, y, xs) -> min(y, xs) 307.04/291.49 if2(true, x, y, xs) -> xs 307.04/291.49 if2(false, x, y, xs) -> cons(y, del(x, xs)) 307.04/291.49 minsort(nil) -> nil 307.04/291.49 minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) 307.04/291.49 min(x, nil) -> x 307.04/291.49 min(x, cons(y, z)) -> if1(le(x, y), x, y, z) 307.04/291.49 del(x, nil) -> nil 307.04/291.49 del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) 307.04/291.49 307.04/291.49 Types: 307.04/291.49 le :: 0':s -> 0':s -> true:false 307.04/291.49 0' :: 0':s 307.04/291.49 true :: true:false 307.04/291.49 s :: 0':s -> 0':s 307.04/291.49 false :: true:false 307.04/291.49 eq :: 0':s -> 0':s -> true:false 307.04/291.49 if1 :: true:false -> 0':s -> 0':s -> cons:nil -> 0':s 307.04/291.49 min :: 0':s -> cons:nil -> 0':s 307.04/291.49 if2 :: true:false -> 0':s -> 0':s -> cons:nil -> cons:nil 307.04/291.49 cons :: 0':s -> cons:nil -> cons:nil 307.04/291.49 del :: 0':s -> cons:nil -> cons:nil 307.04/291.49 minsort :: cons:nil -> cons:nil 307.04/291.49 nil :: cons:nil 307.04/291.49 hole_true:false1_0 :: true:false 307.04/291.49 hole_0':s2_0 :: 0':s 307.04/291.49 hole_cons:nil3_0 :: cons:nil 307.04/291.49 gen_0':s4_0 :: Nat -> 0':s 307.04/291.49 gen_cons:nil5_0 :: Nat -> cons:nil 307.04/291.49 307.04/291.49 307.04/291.49 Lemmas: 307.04/291.49 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 307.04/291.49 eq(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) -> true, rt in Omega(1 + n300_0) 307.04/291.49 min(gen_0':s4_0(0), gen_cons:nil5_0(n811_0)) -> gen_0':s4_0(0), rt in Omega(1 + n811_0) 307.04/291.49 307.04/291.49 307.04/291.49 Generator Equations: 307.04/291.49 gen_0':s4_0(0) <=> 0' 307.04/291.49 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 307.04/291.49 gen_cons:nil5_0(0) <=> nil 307.04/291.49 gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x)) 307.04/291.49 307.04/291.49 307.04/291.49 The following defined symbols remain to be analysed: 307.04/291.49 minsort 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (19) LowerBoundPropagationProof (FINISHED) 307.04/291.49 Propagated lower bound. 307.04/291.49 ---------------------------------------- 307.04/291.49 307.04/291.49 (20) 307.04/291.49 BOUNDS(n^2, INF) 307.04/291.52 EOF