305.06/291.47 WORST_CASE(Omega(n^2), ?) 305.12/291.51 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 305.12/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 305.12/291.51 305.12/291.51 305.12/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 305.12/291.51 305.12/291.51 (0) CpxTRS 305.12/291.51 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 305.12/291.51 (2) CpxTRS 305.12/291.51 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 305.12/291.51 (4) typed CpxTrs 305.12/291.51 (5) OrderProof [LOWER BOUND(ID), 0 ms] 305.12/291.51 (6) typed CpxTrs 305.12/291.51 (7) RewriteLemmaProof [LOWER BOUND(ID), 299 ms] 305.12/291.51 (8) BEST 305.12/291.51 (9) proven lower bound 305.12/291.51 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 305.12/291.51 (11) BOUNDS(n^1, INF) 305.12/291.51 (12) typed CpxTrs 305.12/291.51 (13) RewriteLemmaProof [LOWER BOUND(ID), 47 ms] 305.12/291.51 (14) typed CpxTrs 305.12/291.51 (15) RewriteLemmaProof [LOWER BOUND(ID), 28 ms] 305.12/291.51 (16) typed CpxTrs 305.12/291.51 (17) RewriteLemmaProof [LOWER BOUND(ID), 74 ms] 305.12/291.51 (18) proven lower bound 305.12/291.51 (19) LowerBoundPropagationProof [FINISHED, 0 ms] 305.12/291.51 (20) BOUNDS(n^2, INF) 305.12/291.51 305.12/291.51 305.12/291.51 ---------------------------------------- 305.12/291.51 305.12/291.51 (0) 305.12/291.51 Obligation: 305.12/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 305.12/291.51 305.12/291.51 305.12/291.51 The TRS R consists of the following rules: 305.12/291.51 305.12/291.51 max(nil) -> 0 305.12/291.51 max(cons(x, nil)) -> x 305.12/291.51 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 305.12/291.51 if1(true, x, y, xs) -> max(cons(x, xs)) 305.12/291.51 if1(false, x, y, xs) -> max(cons(y, xs)) 305.12/291.51 del(x, nil) -> nil 305.12/291.51 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 305.12/291.51 if2(true, x, y, xs) -> xs 305.12/291.51 if2(false, x, y, xs) -> cons(y, del(x, xs)) 305.12/291.51 eq(0, 0) -> true 305.12/291.51 eq(0, s(y)) -> false 305.12/291.51 eq(s(x), 0) -> false 305.12/291.51 eq(s(x), s(y)) -> eq(x, y) 305.12/291.51 sort(nil) -> nil 305.12/291.51 sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 305.12/291.51 ge(x, 0) -> true 305.12/291.51 ge(0, s(x)) -> false 305.12/291.51 ge(s(x), s(y)) -> ge(x, y) 305.12/291.51 305.12/291.51 S is empty. 305.12/291.51 Rewrite Strategy: FULL 305.12/291.51 ---------------------------------------- 305.12/291.51 305.12/291.51 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 305.12/291.51 Renamed function symbols to avoid clashes with predefined symbol. 305.12/291.51 ---------------------------------------- 305.12/291.51 305.12/291.51 (2) 305.12/291.51 Obligation: 305.12/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 305.12/291.52 305.12/291.52 305.12/291.52 The TRS R consists of the following rules: 305.12/291.52 305.12/291.52 max(nil) -> 0' 305.12/291.52 max(cons(x, nil)) -> x 305.12/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 305.12/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 305.12/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 305.12/291.52 del(x, nil) -> nil 305.12/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 305.12/291.52 if2(true, x, y, xs) -> xs 305.12/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 305.12/291.52 eq(0', 0') -> true 305.12/291.52 eq(0', s(y)) -> false 305.12/291.52 eq(s(x), 0') -> false 305.12/291.52 eq(s(x), s(y)) -> eq(x, y) 305.12/291.52 sort(nil) -> nil 305.12/291.52 sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 305.12/291.52 ge(x, 0') -> true 305.12/291.52 ge(0', s(x)) -> false 305.12/291.52 ge(s(x), s(y)) -> ge(x, y) 305.12/291.52 305.12/291.52 S is empty. 305.12/291.52 Rewrite Strategy: FULL 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 305.12/291.52 Infered types. 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (4) 305.12/291.52 Obligation: 305.12/291.52 TRS: 305.12/291.52 Rules: 305.12/291.52 max(nil) -> 0' 305.12/291.52 max(cons(x, nil)) -> x 305.12/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 305.12/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 305.12/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 305.12/291.52 del(x, nil) -> nil 305.12/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 305.12/291.52 if2(true, x, y, xs) -> xs 305.12/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 305.12/291.52 eq(0', 0') -> true 305.12/291.52 eq(0', s(y)) -> false 305.12/291.52 eq(s(x), 0') -> false 305.12/291.52 eq(s(x), s(y)) -> eq(x, y) 305.12/291.52 sort(nil) -> nil 305.12/291.52 sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 305.12/291.52 ge(x, 0') -> true 305.12/291.52 ge(0', s(x)) -> false 305.12/291.52 ge(s(x), s(y)) -> ge(x, y) 305.12/291.52 305.12/291.52 Types: 305.12/291.52 max :: nil:cons -> 0':s 305.12/291.52 nil :: nil:cons 305.12/291.52 0' :: 0':s 305.12/291.52 cons :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 305.12/291.52 ge :: 0':s -> 0':s -> true:false 305.12/291.52 true :: true:false 305.12/291.52 false :: true:false 305.12/291.52 del :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.12/291.52 eq :: 0':s -> 0':s -> true:false 305.12/291.52 s :: 0':s -> 0':s 305.12/291.52 sort :: nil:cons -> nil:cons 305.12/291.52 hole_0':s1_0 :: 0':s 305.12/291.52 hole_nil:cons2_0 :: nil:cons 305.12/291.52 hole_true:false3_0 :: true:false 305.12/291.52 gen_0':s4_0 :: Nat -> 0':s 305.12/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 305.12/291.52 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (5) OrderProof (LOWER BOUND(ID)) 305.12/291.52 Heuristically decided to analyse the following defined symbols: 305.12/291.52 max, ge, del, eq, sort 305.12/291.52 305.12/291.52 They will be analysed ascendingly in the following order: 305.12/291.52 ge < max 305.12/291.52 max < sort 305.12/291.52 eq < del 305.12/291.52 del < sort 305.12/291.52 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (6) 305.12/291.52 Obligation: 305.12/291.52 TRS: 305.12/291.52 Rules: 305.12/291.52 max(nil) -> 0' 305.12/291.52 max(cons(x, nil)) -> x 305.12/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 305.12/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 305.12/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 305.12/291.52 del(x, nil) -> nil 305.12/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 305.12/291.52 if2(true, x, y, xs) -> xs 305.12/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 305.12/291.52 eq(0', 0') -> true 305.12/291.52 eq(0', s(y)) -> false 305.12/291.52 eq(s(x), 0') -> false 305.12/291.52 eq(s(x), s(y)) -> eq(x, y) 305.12/291.52 sort(nil) -> nil 305.12/291.52 sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 305.12/291.52 ge(x, 0') -> true 305.12/291.52 ge(0', s(x)) -> false 305.12/291.52 ge(s(x), s(y)) -> ge(x, y) 305.12/291.52 305.12/291.52 Types: 305.12/291.52 max :: nil:cons -> 0':s 305.12/291.52 nil :: nil:cons 305.12/291.52 0' :: 0':s 305.12/291.52 cons :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 305.12/291.52 ge :: 0':s -> 0':s -> true:false 305.12/291.52 true :: true:false 305.12/291.52 false :: true:false 305.12/291.52 del :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.12/291.52 eq :: 0':s -> 0':s -> true:false 305.12/291.52 s :: 0':s -> 0':s 305.12/291.52 sort :: nil:cons -> nil:cons 305.12/291.52 hole_0':s1_0 :: 0':s 305.12/291.52 hole_nil:cons2_0 :: nil:cons 305.12/291.52 hole_true:false3_0 :: true:false 305.12/291.52 gen_0':s4_0 :: Nat -> 0':s 305.12/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 305.12/291.52 305.12/291.52 305.12/291.52 Generator Equations: 305.12/291.52 gen_0':s4_0(0) <=> 0' 305.12/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.12/291.52 gen_nil:cons5_0(0) <=> nil 305.12/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.12/291.52 305.12/291.52 305.12/291.52 The following defined symbols remain to be analysed: 305.12/291.52 ge, max, del, eq, sort 305.12/291.52 305.12/291.52 They will be analysed ascendingly in the following order: 305.12/291.52 ge < max 305.12/291.52 max < sort 305.12/291.52 eq < del 305.12/291.52 del < sort 305.12/291.52 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (7) RewriteLemmaProof (LOWER BOUND(ID)) 305.12/291.52 Proved the following rewrite lemma: 305.12/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.12/291.52 305.12/291.52 Induction Base: 305.12/291.52 ge(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 305.12/291.52 true 305.12/291.52 305.12/291.52 Induction Step: 305.12/291.52 ge(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1) 305.12/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH 305.12/291.52 true 305.12/291.52 305.12/291.52 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (8) 305.12/291.52 Complex Obligation (BEST) 305.12/291.52 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (9) 305.12/291.52 Obligation: 305.12/291.52 Proved the lower bound n^1 for the following obligation: 305.12/291.52 305.12/291.52 TRS: 305.12/291.52 Rules: 305.12/291.52 max(nil) -> 0' 305.12/291.52 max(cons(x, nil)) -> x 305.12/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 305.12/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 305.12/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 305.12/291.52 del(x, nil) -> nil 305.12/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 305.12/291.52 if2(true, x, y, xs) -> xs 305.12/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 305.12/291.52 eq(0', 0') -> true 305.12/291.52 eq(0', s(y)) -> false 305.12/291.52 eq(s(x), 0') -> false 305.12/291.52 eq(s(x), s(y)) -> eq(x, y) 305.12/291.52 sort(nil) -> nil 305.12/291.52 sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 305.12/291.52 ge(x, 0') -> true 305.12/291.52 ge(0', s(x)) -> false 305.12/291.52 ge(s(x), s(y)) -> ge(x, y) 305.12/291.52 305.12/291.52 Types: 305.12/291.52 max :: nil:cons -> 0':s 305.12/291.52 nil :: nil:cons 305.12/291.52 0' :: 0':s 305.12/291.52 cons :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 305.12/291.52 ge :: 0':s -> 0':s -> true:false 305.12/291.52 true :: true:false 305.12/291.52 false :: true:false 305.12/291.52 del :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.12/291.52 eq :: 0':s -> 0':s -> true:false 305.12/291.52 s :: 0':s -> 0':s 305.12/291.52 sort :: nil:cons -> nil:cons 305.12/291.52 hole_0':s1_0 :: 0':s 305.12/291.52 hole_nil:cons2_0 :: nil:cons 305.12/291.52 hole_true:false3_0 :: true:false 305.12/291.52 gen_0':s4_0 :: Nat -> 0':s 305.12/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 305.12/291.52 305.12/291.52 305.12/291.52 Generator Equations: 305.12/291.52 gen_0':s4_0(0) <=> 0' 305.12/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.12/291.52 gen_nil:cons5_0(0) <=> nil 305.12/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.12/291.52 305.12/291.52 305.12/291.52 The following defined symbols remain to be analysed: 305.12/291.52 ge, max, del, eq, sort 305.12/291.52 305.12/291.52 They will be analysed ascendingly in the following order: 305.12/291.52 ge < max 305.12/291.52 max < sort 305.12/291.52 eq < del 305.12/291.52 del < sort 305.12/291.52 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (10) LowerBoundPropagationProof (FINISHED) 305.12/291.52 Propagated lower bound. 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (11) 305.12/291.52 BOUNDS(n^1, INF) 305.12/291.52 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (12) 305.12/291.52 Obligation: 305.12/291.52 TRS: 305.12/291.52 Rules: 305.12/291.52 max(nil) -> 0' 305.12/291.52 max(cons(x, nil)) -> x 305.12/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 305.12/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 305.12/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 305.12/291.52 del(x, nil) -> nil 305.12/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 305.12/291.52 if2(true, x, y, xs) -> xs 305.12/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 305.12/291.52 eq(0', 0') -> true 305.12/291.52 eq(0', s(y)) -> false 305.12/291.52 eq(s(x), 0') -> false 305.12/291.52 eq(s(x), s(y)) -> eq(x, y) 305.12/291.52 sort(nil) -> nil 305.12/291.52 sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 305.12/291.52 ge(x, 0') -> true 305.12/291.52 ge(0', s(x)) -> false 305.12/291.52 ge(s(x), s(y)) -> ge(x, y) 305.12/291.52 305.12/291.52 Types: 305.12/291.52 max :: nil:cons -> 0':s 305.12/291.52 nil :: nil:cons 305.12/291.52 0' :: 0':s 305.12/291.52 cons :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 305.12/291.52 ge :: 0':s -> 0':s -> true:false 305.12/291.52 true :: true:false 305.12/291.52 false :: true:false 305.12/291.52 del :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.12/291.52 eq :: 0':s -> 0':s -> true:false 305.12/291.52 s :: 0':s -> 0':s 305.12/291.52 sort :: nil:cons -> nil:cons 305.12/291.52 hole_0':s1_0 :: 0':s 305.12/291.52 hole_nil:cons2_0 :: nil:cons 305.12/291.52 hole_true:false3_0 :: true:false 305.12/291.52 gen_0':s4_0 :: Nat -> 0':s 305.12/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 305.12/291.52 305.12/291.52 305.12/291.52 Lemmas: 305.12/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.12/291.52 305.12/291.52 305.12/291.52 Generator Equations: 305.12/291.52 gen_0':s4_0(0) <=> 0' 305.12/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.12/291.52 gen_nil:cons5_0(0) <=> nil 305.12/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.12/291.52 305.12/291.52 305.12/291.52 The following defined symbols remain to be analysed: 305.12/291.52 max, del, eq, sort 305.12/291.52 305.12/291.52 They will be analysed ascendingly in the following order: 305.12/291.52 max < sort 305.12/291.52 eq < del 305.12/291.52 del < sort 305.12/291.52 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (13) RewriteLemmaProof (LOWER BOUND(ID)) 305.12/291.52 Proved the following rewrite lemma: 305.12/291.52 max(gen_nil:cons5_0(+(1, n305_0))) -> gen_0':s4_0(0), rt in Omega(1 + n305_0) 305.12/291.52 305.12/291.52 Induction Base: 305.12/291.52 max(gen_nil:cons5_0(+(1, 0))) ->_R^Omega(1) 305.12/291.52 0' 305.12/291.52 305.12/291.52 Induction Step: 305.12/291.52 max(gen_nil:cons5_0(+(1, +(n305_0, 1)))) ->_R^Omega(1) 305.12/291.52 if1(ge(0', 0'), 0', 0', gen_nil:cons5_0(n305_0)) ->_L^Omega(1) 305.12/291.52 if1(true, 0', 0', gen_nil:cons5_0(n305_0)) ->_R^Omega(1) 305.12/291.52 max(cons(0', gen_nil:cons5_0(n305_0))) ->_IH 305.12/291.52 gen_0':s4_0(0) 305.12/291.52 305.12/291.52 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (14) 305.12/291.52 Obligation: 305.12/291.52 TRS: 305.12/291.52 Rules: 305.12/291.52 max(nil) -> 0' 305.12/291.52 max(cons(x, nil)) -> x 305.12/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 305.12/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 305.12/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 305.12/291.52 del(x, nil) -> nil 305.12/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 305.12/291.52 if2(true, x, y, xs) -> xs 305.12/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 305.12/291.52 eq(0', 0') -> true 305.12/291.52 eq(0', s(y)) -> false 305.12/291.52 eq(s(x), 0') -> false 305.12/291.52 eq(s(x), s(y)) -> eq(x, y) 305.12/291.52 sort(nil) -> nil 305.12/291.52 sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 305.12/291.52 ge(x, 0') -> true 305.12/291.52 ge(0', s(x)) -> false 305.12/291.52 ge(s(x), s(y)) -> ge(x, y) 305.12/291.52 305.12/291.52 Types: 305.12/291.52 max :: nil:cons -> 0':s 305.12/291.52 nil :: nil:cons 305.12/291.52 0' :: 0':s 305.12/291.52 cons :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 305.12/291.52 ge :: 0':s -> 0':s -> true:false 305.12/291.52 true :: true:false 305.12/291.52 false :: true:false 305.12/291.52 del :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.12/291.52 eq :: 0':s -> 0':s -> true:false 305.12/291.52 s :: 0':s -> 0':s 305.12/291.52 sort :: nil:cons -> nil:cons 305.12/291.52 hole_0':s1_0 :: 0':s 305.12/291.52 hole_nil:cons2_0 :: nil:cons 305.12/291.52 hole_true:false3_0 :: true:false 305.12/291.52 gen_0':s4_0 :: Nat -> 0':s 305.12/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 305.12/291.52 305.12/291.52 305.12/291.52 Lemmas: 305.12/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.12/291.52 max(gen_nil:cons5_0(+(1, n305_0))) -> gen_0':s4_0(0), rt in Omega(1 + n305_0) 305.12/291.52 305.12/291.52 305.12/291.52 Generator Equations: 305.12/291.52 gen_0':s4_0(0) <=> 0' 305.12/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.12/291.52 gen_nil:cons5_0(0) <=> nil 305.12/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.12/291.52 305.12/291.52 305.12/291.52 The following defined symbols remain to be analysed: 305.12/291.52 eq, del, sort 305.12/291.52 305.12/291.52 They will be analysed ascendingly in the following order: 305.12/291.52 eq < del 305.12/291.52 del < sort 305.12/291.52 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (15) RewriteLemmaProof (LOWER BOUND(ID)) 305.12/291.52 Proved the following rewrite lemma: 305.12/291.52 eq(gen_0':s4_0(n727_0), gen_0':s4_0(n727_0)) -> true, rt in Omega(1 + n727_0) 305.12/291.52 305.12/291.52 Induction Base: 305.12/291.52 eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 305.12/291.52 true 305.12/291.52 305.12/291.52 Induction Step: 305.12/291.52 eq(gen_0':s4_0(+(n727_0, 1)), gen_0':s4_0(+(n727_0, 1))) ->_R^Omega(1) 305.12/291.52 eq(gen_0':s4_0(n727_0), gen_0':s4_0(n727_0)) ->_IH 305.12/291.52 true 305.12/291.52 305.12/291.52 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (16) 305.12/291.52 Obligation: 305.12/291.52 TRS: 305.12/291.52 Rules: 305.12/291.52 max(nil) -> 0' 305.12/291.52 max(cons(x, nil)) -> x 305.12/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 305.12/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 305.12/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 305.12/291.52 del(x, nil) -> nil 305.12/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 305.12/291.52 if2(true, x, y, xs) -> xs 305.12/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 305.12/291.52 eq(0', 0') -> true 305.12/291.52 eq(0', s(y)) -> false 305.12/291.52 eq(s(x), 0') -> false 305.12/291.52 eq(s(x), s(y)) -> eq(x, y) 305.12/291.52 sort(nil) -> nil 305.12/291.52 sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 305.12/291.52 ge(x, 0') -> true 305.12/291.52 ge(0', s(x)) -> false 305.12/291.52 ge(s(x), s(y)) -> ge(x, y) 305.12/291.52 305.12/291.52 Types: 305.12/291.52 max :: nil:cons -> 0':s 305.12/291.52 nil :: nil:cons 305.12/291.52 0' :: 0':s 305.12/291.52 cons :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 305.12/291.52 ge :: 0':s -> 0':s -> true:false 305.12/291.52 true :: true:false 305.12/291.52 false :: true:false 305.12/291.52 del :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.12/291.52 eq :: 0':s -> 0':s -> true:false 305.12/291.52 s :: 0':s -> 0':s 305.12/291.52 sort :: nil:cons -> nil:cons 305.12/291.52 hole_0':s1_0 :: 0':s 305.12/291.52 hole_nil:cons2_0 :: nil:cons 305.12/291.52 hole_true:false3_0 :: true:false 305.12/291.52 gen_0':s4_0 :: Nat -> 0':s 305.12/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 305.12/291.52 305.12/291.52 305.12/291.52 Lemmas: 305.12/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.12/291.52 max(gen_nil:cons5_0(+(1, n305_0))) -> gen_0':s4_0(0), rt in Omega(1 + n305_0) 305.12/291.52 eq(gen_0':s4_0(n727_0), gen_0':s4_0(n727_0)) -> true, rt in Omega(1 + n727_0) 305.12/291.52 305.12/291.52 305.12/291.52 Generator Equations: 305.12/291.52 gen_0':s4_0(0) <=> 0' 305.12/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.12/291.52 gen_nil:cons5_0(0) <=> nil 305.12/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.12/291.52 305.12/291.52 305.12/291.52 The following defined symbols remain to be analysed: 305.12/291.52 del, sort 305.12/291.52 305.12/291.52 They will be analysed ascendingly in the following order: 305.12/291.52 del < sort 305.12/291.52 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (17) RewriteLemmaProof (LOWER BOUND(ID)) 305.12/291.52 Proved the following rewrite lemma: 305.12/291.52 sort(gen_nil:cons5_0(n1412_0)) -> gen_nil:cons5_0(n1412_0), rt in Omega(1 + n1412_0 + n1412_0^2) 305.12/291.52 305.12/291.52 Induction Base: 305.12/291.52 sort(gen_nil:cons5_0(0)) ->_R^Omega(1) 305.12/291.52 nil 305.12/291.52 305.12/291.52 Induction Step: 305.12/291.52 sort(gen_nil:cons5_0(+(n1412_0, 1))) ->_R^Omega(1) 305.12/291.52 cons(max(cons(0', gen_nil:cons5_0(n1412_0))), sort(del(max(cons(0', gen_nil:cons5_0(n1412_0))), cons(0', gen_nil:cons5_0(n1412_0))))) ->_L^Omega(1 + n1412_0) 305.12/291.52 cons(gen_0':s4_0(0), sort(del(max(cons(0', gen_nil:cons5_0(n1412_0))), cons(0', gen_nil:cons5_0(n1412_0))))) ->_L^Omega(1 + n1412_0) 305.12/291.52 cons(gen_0':s4_0(0), sort(del(gen_0':s4_0(0), cons(0', gen_nil:cons5_0(n1412_0))))) ->_R^Omega(1) 305.12/291.52 cons(gen_0':s4_0(0), sort(if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_nil:cons5_0(n1412_0)))) ->_L^Omega(1) 305.12/291.52 cons(gen_0':s4_0(0), sort(if2(true, gen_0':s4_0(0), 0', gen_nil:cons5_0(n1412_0)))) ->_R^Omega(1) 305.12/291.52 cons(gen_0':s4_0(0), sort(gen_nil:cons5_0(n1412_0))) ->_IH 305.12/291.52 cons(gen_0':s4_0(0), gen_nil:cons5_0(c1413_0)) 305.12/291.52 305.12/291.52 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (18) 305.12/291.52 Obligation: 305.12/291.52 Proved the lower bound n^2 for the following obligation: 305.12/291.52 305.12/291.52 TRS: 305.12/291.52 Rules: 305.12/291.52 max(nil) -> 0' 305.12/291.52 max(cons(x, nil)) -> x 305.12/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 305.12/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 305.12/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 305.12/291.52 del(x, nil) -> nil 305.12/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 305.12/291.52 if2(true, x, y, xs) -> xs 305.12/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 305.12/291.52 eq(0', 0') -> true 305.12/291.52 eq(0', s(y)) -> false 305.12/291.52 eq(s(x), 0') -> false 305.12/291.52 eq(s(x), s(y)) -> eq(x, y) 305.12/291.52 sort(nil) -> nil 305.12/291.52 sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 305.12/291.52 ge(x, 0') -> true 305.12/291.52 ge(0', s(x)) -> false 305.12/291.52 ge(s(x), s(y)) -> ge(x, y) 305.12/291.52 305.12/291.52 Types: 305.12/291.52 max :: nil:cons -> 0':s 305.12/291.52 nil :: nil:cons 305.12/291.52 0' :: 0':s 305.12/291.52 cons :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 305.12/291.52 ge :: 0':s -> 0':s -> true:false 305.12/291.52 true :: true:false 305.12/291.52 false :: true:false 305.12/291.52 del :: 0':s -> nil:cons -> nil:cons 305.12/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.12/291.52 eq :: 0':s -> 0':s -> true:false 305.12/291.52 s :: 0':s -> 0':s 305.12/291.52 sort :: nil:cons -> nil:cons 305.12/291.52 hole_0':s1_0 :: 0':s 305.12/291.52 hole_nil:cons2_0 :: nil:cons 305.12/291.52 hole_true:false3_0 :: true:false 305.12/291.52 gen_0':s4_0 :: Nat -> 0':s 305.12/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 305.12/291.52 305.12/291.52 305.12/291.52 Lemmas: 305.12/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.12/291.52 max(gen_nil:cons5_0(+(1, n305_0))) -> gen_0':s4_0(0), rt in Omega(1 + n305_0) 305.12/291.52 eq(gen_0':s4_0(n727_0), gen_0':s4_0(n727_0)) -> true, rt in Omega(1 + n727_0) 305.12/291.52 305.12/291.52 305.12/291.52 Generator Equations: 305.12/291.52 gen_0':s4_0(0) <=> 0' 305.12/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.12/291.52 gen_nil:cons5_0(0) <=> nil 305.12/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.12/291.52 305.12/291.52 305.12/291.52 The following defined symbols remain to be analysed: 305.12/291.52 sort 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (19) LowerBoundPropagationProof (FINISHED) 305.12/291.52 Propagated lower bound. 305.12/291.52 ---------------------------------------- 305.12/291.52 305.12/291.52 (20) 305.12/291.52 BOUNDS(n^2, INF) 305.22/291.54 EOF