309.22/291.50 WORST_CASE(Omega(n^2), ?) 309.22/291.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 309.22/291.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 309.22/291.52 309.22/291.52 309.22/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 309.22/291.52 309.22/291.52 (0) CpxTRS 309.22/291.52 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 309.22/291.52 (2) CpxTRS 309.22/291.52 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 309.22/291.52 (4) typed CpxTrs 309.22/291.52 (5) OrderProof [LOWER BOUND(ID), 0 ms] 309.22/291.52 (6) typed CpxTrs 309.22/291.52 (7) RewriteLemmaProof [LOWER BOUND(ID), 300 ms] 309.22/291.52 (8) BEST 309.22/291.52 (9) proven lower bound 309.22/291.52 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 309.22/291.52 (11) BOUNDS(n^1, INF) 309.22/291.52 (12) typed CpxTrs 309.22/291.52 (13) RewriteLemmaProof [LOWER BOUND(ID), 41 ms] 309.22/291.52 (14) typed CpxTrs 309.22/291.52 (15) RewriteLemmaProof [LOWER BOUND(ID), 42 ms] 309.22/291.52 (16) typed CpxTrs 309.22/291.52 (17) RewriteLemmaProof [LOWER BOUND(ID), 76 ms] 309.22/291.52 (18) proven lower bound 309.22/291.52 (19) LowerBoundPropagationProof [FINISHED, 0 ms] 309.22/291.52 (20) BOUNDS(n^2, INF) 309.22/291.52 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (0) 309.22/291.52 Obligation: 309.22/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 309.22/291.52 309.22/291.52 309.22/291.52 The TRS R consists of the following rules: 309.22/291.52 309.22/291.52 max(nil) -> 0 309.22/291.52 max(cons(x, nil)) -> x 309.22/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 309.22/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 309.22/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 309.22/291.52 del(x, nil) -> nil 309.22/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 309.22/291.52 if2(true, x, y, xs) -> xs 309.22/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 309.22/291.52 eq(0, 0) -> true 309.22/291.52 eq(0, s(y)) -> false 309.22/291.52 eq(s(x), 0) -> false 309.22/291.52 eq(s(x), s(y)) -> eq(x, y) 309.22/291.52 sort(xs) -> if3(empty(xs), xs) 309.22/291.52 if3(true, xs) -> nil 309.22/291.52 if3(false, xs) -> sort(del(max(xs), xs)) 309.22/291.52 empty(nil) -> true 309.22/291.52 empty(cons(x, xs)) -> false 309.22/291.52 ge(x, 0) -> true 309.22/291.52 ge(0, s(x)) -> false 309.22/291.52 ge(s(x), s(y)) -> ge(x, y) 309.22/291.52 309.22/291.52 S is empty. 309.22/291.52 Rewrite Strategy: FULL 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 309.22/291.52 Renamed function symbols to avoid clashes with predefined symbol. 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (2) 309.22/291.52 Obligation: 309.22/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 309.22/291.52 309.22/291.52 309.22/291.52 The TRS R consists of the following rules: 309.22/291.52 309.22/291.52 max(nil) -> 0' 309.22/291.52 max(cons(x, nil)) -> x 309.22/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 309.22/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 309.22/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 309.22/291.52 del(x, nil) -> nil 309.22/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 309.22/291.52 if2(true, x, y, xs) -> xs 309.22/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 309.22/291.52 eq(0', 0') -> true 309.22/291.52 eq(0', s(y)) -> false 309.22/291.52 eq(s(x), 0') -> false 309.22/291.52 eq(s(x), s(y)) -> eq(x, y) 309.22/291.52 sort(xs) -> if3(empty(xs), xs) 309.22/291.52 if3(true, xs) -> nil 309.22/291.52 if3(false, xs) -> sort(del(max(xs), xs)) 309.22/291.52 empty(nil) -> true 309.22/291.52 empty(cons(x, xs)) -> false 309.22/291.52 ge(x, 0') -> true 309.22/291.52 ge(0', s(x)) -> false 309.22/291.52 ge(s(x), s(y)) -> ge(x, y) 309.22/291.52 309.22/291.52 S is empty. 309.22/291.52 Rewrite Strategy: FULL 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 309.22/291.52 Infered types. 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (4) 309.22/291.52 Obligation: 309.22/291.52 TRS: 309.22/291.52 Rules: 309.22/291.52 max(nil) -> 0' 309.22/291.52 max(cons(x, nil)) -> x 309.22/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 309.22/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 309.22/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 309.22/291.52 del(x, nil) -> nil 309.22/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 309.22/291.52 if2(true, x, y, xs) -> xs 309.22/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 309.22/291.52 eq(0', 0') -> true 309.22/291.52 eq(0', s(y)) -> false 309.22/291.52 eq(s(x), 0') -> false 309.22/291.52 eq(s(x), s(y)) -> eq(x, y) 309.22/291.52 sort(xs) -> if3(empty(xs), xs) 309.22/291.52 if3(true, xs) -> nil 309.22/291.52 if3(false, xs) -> sort(del(max(xs), xs)) 309.22/291.52 empty(nil) -> true 309.22/291.52 empty(cons(x, xs)) -> false 309.22/291.52 ge(x, 0') -> true 309.22/291.52 ge(0', s(x)) -> false 309.22/291.52 ge(s(x), s(y)) -> ge(x, y) 309.22/291.52 309.22/291.52 Types: 309.22/291.52 max :: nil:cons -> 0':s 309.22/291.52 nil :: nil:cons 309.22/291.52 0' :: 0':s 309.22/291.52 cons :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 309.22/291.52 ge :: 0':s -> 0':s -> true:false 309.22/291.52 true :: true:false 309.22/291.52 false :: true:false 309.22/291.52 del :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 309.22/291.52 eq :: 0':s -> 0':s -> true:false 309.22/291.52 s :: 0':s -> 0':s 309.22/291.52 sort :: nil:cons -> nil:cons 309.22/291.52 if3 :: true:false -> nil:cons -> nil:cons 309.22/291.52 empty :: nil:cons -> true:false 309.22/291.52 hole_0':s1_0 :: 0':s 309.22/291.52 hole_nil:cons2_0 :: nil:cons 309.22/291.52 hole_true:false3_0 :: true:false 309.22/291.52 gen_0':s4_0 :: Nat -> 0':s 309.22/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (5) OrderProof (LOWER BOUND(ID)) 309.22/291.52 Heuristically decided to analyse the following defined symbols: 309.22/291.52 max, ge, del, eq, sort 309.22/291.52 309.22/291.52 They will be analysed ascendingly in the following order: 309.22/291.52 ge < max 309.22/291.52 max < sort 309.22/291.52 eq < del 309.22/291.52 del < sort 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (6) 309.22/291.52 Obligation: 309.22/291.52 TRS: 309.22/291.52 Rules: 309.22/291.52 max(nil) -> 0' 309.22/291.52 max(cons(x, nil)) -> x 309.22/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 309.22/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 309.22/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 309.22/291.52 del(x, nil) -> nil 309.22/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 309.22/291.52 if2(true, x, y, xs) -> xs 309.22/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 309.22/291.52 eq(0', 0') -> true 309.22/291.52 eq(0', s(y)) -> false 309.22/291.52 eq(s(x), 0') -> false 309.22/291.52 eq(s(x), s(y)) -> eq(x, y) 309.22/291.52 sort(xs) -> if3(empty(xs), xs) 309.22/291.52 if3(true, xs) -> nil 309.22/291.52 if3(false, xs) -> sort(del(max(xs), xs)) 309.22/291.52 empty(nil) -> true 309.22/291.52 empty(cons(x, xs)) -> false 309.22/291.52 ge(x, 0') -> true 309.22/291.52 ge(0', s(x)) -> false 309.22/291.52 ge(s(x), s(y)) -> ge(x, y) 309.22/291.52 309.22/291.52 Types: 309.22/291.52 max :: nil:cons -> 0':s 309.22/291.52 nil :: nil:cons 309.22/291.52 0' :: 0':s 309.22/291.52 cons :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 309.22/291.52 ge :: 0':s -> 0':s -> true:false 309.22/291.52 true :: true:false 309.22/291.52 false :: true:false 309.22/291.52 del :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 309.22/291.52 eq :: 0':s -> 0':s -> true:false 309.22/291.52 s :: 0':s -> 0':s 309.22/291.52 sort :: nil:cons -> nil:cons 309.22/291.52 if3 :: true:false -> nil:cons -> nil:cons 309.22/291.52 empty :: nil:cons -> true:false 309.22/291.52 hole_0':s1_0 :: 0':s 309.22/291.52 hole_nil:cons2_0 :: nil:cons 309.22/291.52 hole_true:false3_0 :: true:false 309.22/291.52 gen_0':s4_0 :: Nat -> 0':s 309.22/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 309.22/291.52 309.22/291.52 309.22/291.52 Generator Equations: 309.22/291.52 gen_0':s4_0(0) <=> 0' 309.22/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 309.22/291.52 gen_nil:cons5_0(0) <=> nil 309.22/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 309.22/291.52 309.22/291.52 309.22/291.52 The following defined symbols remain to be analysed: 309.22/291.52 ge, max, del, eq, sort 309.22/291.52 309.22/291.52 They will be analysed ascendingly in the following order: 309.22/291.52 ge < max 309.22/291.52 max < sort 309.22/291.52 eq < del 309.22/291.52 del < sort 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (7) RewriteLemmaProof (LOWER BOUND(ID)) 309.22/291.52 Proved the following rewrite lemma: 309.22/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 309.22/291.52 309.22/291.52 Induction Base: 309.22/291.52 ge(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 309.22/291.52 true 309.22/291.52 309.22/291.52 Induction Step: 309.22/291.52 ge(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1) 309.22/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH 309.22/291.52 true 309.22/291.52 309.22/291.52 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (8) 309.22/291.52 Complex Obligation (BEST) 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (9) 309.22/291.52 Obligation: 309.22/291.52 Proved the lower bound n^1 for the following obligation: 309.22/291.52 309.22/291.52 TRS: 309.22/291.52 Rules: 309.22/291.52 max(nil) -> 0' 309.22/291.52 max(cons(x, nil)) -> x 309.22/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 309.22/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 309.22/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 309.22/291.52 del(x, nil) -> nil 309.22/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 309.22/291.52 if2(true, x, y, xs) -> xs 309.22/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 309.22/291.52 eq(0', 0') -> true 309.22/291.52 eq(0', s(y)) -> false 309.22/291.52 eq(s(x), 0') -> false 309.22/291.52 eq(s(x), s(y)) -> eq(x, y) 309.22/291.52 sort(xs) -> if3(empty(xs), xs) 309.22/291.52 if3(true, xs) -> nil 309.22/291.52 if3(false, xs) -> sort(del(max(xs), xs)) 309.22/291.52 empty(nil) -> true 309.22/291.52 empty(cons(x, xs)) -> false 309.22/291.52 ge(x, 0') -> true 309.22/291.52 ge(0', s(x)) -> false 309.22/291.52 ge(s(x), s(y)) -> ge(x, y) 309.22/291.52 309.22/291.52 Types: 309.22/291.52 max :: nil:cons -> 0':s 309.22/291.52 nil :: nil:cons 309.22/291.52 0' :: 0':s 309.22/291.52 cons :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 309.22/291.52 ge :: 0':s -> 0':s -> true:false 309.22/291.52 true :: true:false 309.22/291.52 false :: true:false 309.22/291.52 del :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 309.22/291.52 eq :: 0':s -> 0':s -> true:false 309.22/291.52 s :: 0':s -> 0':s 309.22/291.52 sort :: nil:cons -> nil:cons 309.22/291.52 if3 :: true:false -> nil:cons -> nil:cons 309.22/291.52 empty :: nil:cons -> true:false 309.22/291.52 hole_0':s1_0 :: 0':s 309.22/291.52 hole_nil:cons2_0 :: nil:cons 309.22/291.52 hole_true:false3_0 :: true:false 309.22/291.52 gen_0':s4_0 :: Nat -> 0':s 309.22/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 309.22/291.52 309.22/291.52 309.22/291.52 Generator Equations: 309.22/291.52 gen_0':s4_0(0) <=> 0' 309.22/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 309.22/291.52 gen_nil:cons5_0(0) <=> nil 309.22/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 309.22/291.52 309.22/291.52 309.22/291.52 The following defined symbols remain to be analysed: 309.22/291.52 ge, max, del, eq, sort 309.22/291.52 309.22/291.52 They will be analysed ascendingly in the following order: 309.22/291.52 ge < max 309.22/291.52 max < sort 309.22/291.52 eq < del 309.22/291.52 del < sort 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (10) LowerBoundPropagationProof (FINISHED) 309.22/291.52 Propagated lower bound. 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (11) 309.22/291.52 BOUNDS(n^1, INF) 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (12) 309.22/291.52 Obligation: 309.22/291.52 TRS: 309.22/291.52 Rules: 309.22/291.52 max(nil) -> 0' 309.22/291.52 max(cons(x, nil)) -> x 309.22/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 309.22/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 309.22/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 309.22/291.52 del(x, nil) -> nil 309.22/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 309.22/291.52 if2(true, x, y, xs) -> xs 309.22/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 309.22/291.52 eq(0', 0') -> true 309.22/291.52 eq(0', s(y)) -> false 309.22/291.52 eq(s(x), 0') -> false 309.22/291.52 eq(s(x), s(y)) -> eq(x, y) 309.22/291.52 sort(xs) -> if3(empty(xs), xs) 309.22/291.52 if3(true, xs) -> nil 309.22/291.52 if3(false, xs) -> sort(del(max(xs), xs)) 309.22/291.52 empty(nil) -> true 309.22/291.52 empty(cons(x, xs)) -> false 309.22/291.52 ge(x, 0') -> true 309.22/291.52 ge(0', s(x)) -> false 309.22/291.52 ge(s(x), s(y)) -> ge(x, y) 309.22/291.52 309.22/291.52 Types: 309.22/291.52 max :: nil:cons -> 0':s 309.22/291.52 nil :: nil:cons 309.22/291.52 0' :: 0':s 309.22/291.52 cons :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 309.22/291.52 ge :: 0':s -> 0':s -> true:false 309.22/291.52 true :: true:false 309.22/291.52 false :: true:false 309.22/291.52 del :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 309.22/291.52 eq :: 0':s -> 0':s -> true:false 309.22/291.52 s :: 0':s -> 0':s 309.22/291.52 sort :: nil:cons -> nil:cons 309.22/291.52 if3 :: true:false -> nil:cons -> nil:cons 309.22/291.52 empty :: nil:cons -> true:false 309.22/291.52 hole_0':s1_0 :: 0':s 309.22/291.52 hole_nil:cons2_0 :: nil:cons 309.22/291.52 hole_true:false3_0 :: true:false 309.22/291.52 gen_0':s4_0 :: Nat -> 0':s 309.22/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 309.22/291.52 309.22/291.52 309.22/291.52 Lemmas: 309.22/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 309.22/291.52 309.22/291.52 309.22/291.52 Generator Equations: 309.22/291.52 gen_0':s4_0(0) <=> 0' 309.22/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 309.22/291.52 gen_nil:cons5_0(0) <=> nil 309.22/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 309.22/291.52 309.22/291.52 309.22/291.52 The following defined symbols remain to be analysed: 309.22/291.52 max, del, eq, sort 309.22/291.52 309.22/291.52 They will be analysed ascendingly in the following order: 309.22/291.52 max < sort 309.22/291.52 eq < del 309.22/291.52 del < sort 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (13) RewriteLemmaProof (LOWER BOUND(ID)) 309.22/291.52 Proved the following rewrite lemma: 309.22/291.52 max(gen_nil:cons5_0(+(1, n323_0))) -> gen_0':s4_0(0), rt in Omega(1 + n323_0) 309.22/291.52 309.22/291.52 Induction Base: 309.22/291.52 max(gen_nil:cons5_0(+(1, 0))) ->_R^Omega(1) 309.22/291.52 0' 309.22/291.52 309.22/291.52 Induction Step: 309.22/291.52 max(gen_nil:cons5_0(+(1, +(n323_0, 1)))) ->_R^Omega(1) 309.22/291.52 if1(ge(0', 0'), 0', 0', gen_nil:cons5_0(n323_0)) ->_L^Omega(1) 309.22/291.52 if1(true, 0', 0', gen_nil:cons5_0(n323_0)) ->_R^Omega(1) 309.22/291.52 max(cons(0', gen_nil:cons5_0(n323_0))) ->_IH 309.22/291.52 gen_0':s4_0(0) 309.22/291.52 309.22/291.52 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (14) 309.22/291.52 Obligation: 309.22/291.52 TRS: 309.22/291.52 Rules: 309.22/291.52 max(nil) -> 0' 309.22/291.52 max(cons(x, nil)) -> x 309.22/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 309.22/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 309.22/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 309.22/291.52 del(x, nil) -> nil 309.22/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 309.22/291.52 if2(true, x, y, xs) -> xs 309.22/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 309.22/291.52 eq(0', 0') -> true 309.22/291.52 eq(0', s(y)) -> false 309.22/291.52 eq(s(x), 0') -> false 309.22/291.52 eq(s(x), s(y)) -> eq(x, y) 309.22/291.52 sort(xs) -> if3(empty(xs), xs) 309.22/291.52 if3(true, xs) -> nil 309.22/291.52 if3(false, xs) -> sort(del(max(xs), xs)) 309.22/291.52 empty(nil) -> true 309.22/291.52 empty(cons(x, xs)) -> false 309.22/291.52 ge(x, 0') -> true 309.22/291.52 ge(0', s(x)) -> false 309.22/291.52 ge(s(x), s(y)) -> ge(x, y) 309.22/291.52 309.22/291.52 Types: 309.22/291.52 max :: nil:cons -> 0':s 309.22/291.52 nil :: nil:cons 309.22/291.52 0' :: 0':s 309.22/291.52 cons :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 309.22/291.52 ge :: 0':s -> 0':s -> true:false 309.22/291.52 true :: true:false 309.22/291.52 false :: true:false 309.22/291.52 del :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 309.22/291.52 eq :: 0':s -> 0':s -> true:false 309.22/291.52 s :: 0':s -> 0':s 309.22/291.52 sort :: nil:cons -> nil:cons 309.22/291.52 if3 :: true:false -> nil:cons -> nil:cons 309.22/291.52 empty :: nil:cons -> true:false 309.22/291.52 hole_0':s1_0 :: 0':s 309.22/291.52 hole_nil:cons2_0 :: nil:cons 309.22/291.52 hole_true:false3_0 :: true:false 309.22/291.52 gen_0':s4_0 :: Nat -> 0':s 309.22/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 309.22/291.52 309.22/291.52 309.22/291.52 Lemmas: 309.22/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 309.22/291.52 max(gen_nil:cons5_0(+(1, n323_0))) -> gen_0':s4_0(0), rt in Omega(1 + n323_0) 309.22/291.52 309.22/291.52 309.22/291.52 Generator Equations: 309.22/291.52 gen_0':s4_0(0) <=> 0' 309.22/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 309.22/291.52 gen_nil:cons5_0(0) <=> nil 309.22/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 309.22/291.52 309.22/291.52 309.22/291.52 The following defined symbols remain to be analysed: 309.22/291.52 eq, del, sort 309.22/291.52 309.22/291.52 They will be analysed ascendingly in the following order: 309.22/291.52 eq < del 309.22/291.52 del < sort 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (15) RewriteLemmaProof (LOWER BOUND(ID)) 309.22/291.52 Proved the following rewrite lemma: 309.22/291.52 eq(gen_0':s4_0(n772_0), gen_0':s4_0(n772_0)) -> true, rt in Omega(1 + n772_0) 309.22/291.52 309.22/291.52 Induction Base: 309.22/291.52 eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 309.22/291.52 true 309.22/291.52 309.22/291.52 Induction Step: 309.22/291.52 eq(gen_0':s4_0(+(n772_0, 1)), gen_0':s4_0(+(n772_0, 1))) ->_R^Omega(1) 309.22/291.52 eq(gen_0':s4_0(n772_0), gen_0':s4_0(n772_0)) ->_IH 309.22/291.52 true 309.22/291.52 309.22/291.52 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (16) 309.22/291.52 Obligation: 309.22/291.52 TRS: 309.22/291.52 Rules: 309.22/291.52 max(nil) -> 0' 309.22/291.52 max(cons(x, nil)) -> x 309.22/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 309.22/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 309.22/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 309.22/291.52 del(x, nil) -> nil 309.22/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 309.22/291.52 if2(true, x, y, xs) -> xs 309.22/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 309.22/291.52 eq(0', 0') -> true 309.22/291.52 eq(0', s(y)) -> false 309.22/291.52 eq(s(x), 0') -> false 309.22/291.52 eq(s(x), s(y)) -> eq(x, y) 309.22/291.52 sort(xs) -> if3(empty(xs), xs) 309.22/291.52 if3(true, xs) -> nil 309.22/291.52 if3(false, xs) -> sort(del(max(xs), xs)) 309.22/291.52 empty(nil) -> true 309.22/291.52 empty(cons(x, xs)) -> false 309.22/291.52 ge(x, 0') -> true 309.22/291.52 ge(0', s(x)) -> false 309.22/291.52 ge(s(x), s(y)) -> ge(x, y) 309.22/291.52 309.22/291.52 Types: 309.22/291.52 max :: nil:cons -> 0':s 309.22/291.52 nil :: nil:cons 309.22/291.52 0' :: 0':s 309.22/291.52 cons :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 309.22/291.52 ge :: 0':s -> 0':s -> true:false 309.22/291.52 true :: true:false 309.22/291.52 false :: true:false 309.22/291.52 del :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 309.22/291.52 eq :: 0':s -> 0':s -> true:false 309.22/291.52 s :: 0':s -> 0':s 309.22/291.52 sort :: nil:cons -> nil:cons 309.22/291.52 if3 :: true:false -> nil:cons -> nil:cons 309.22/291.52 empty :: nil:cons -> true:false 309.22/291.52 hole_0':s1_0 :: 0':s 309.22/291.52 hole_nil:cons2_0 :: nil:cons 309.22/291.52 hole_true:false3_0 :: true:false 309.22/291.52 gen_0':s4_0 :: Nat -> 0':s 309.22/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 309.22/291.52 309.22/291.52 309.22/291.52 Lemmas: 309.22/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 309.22/291.52 max(gen_nil:cons5_0(+(1, n323_0))) -> gen_0':s4_0(0), rt in Omega(1 + n323_0) 309.22/291.52 eq(gen_0':s4_0(n772_0), gen_0':s4_0(n772_0)) -> true, rt in Omega(1 + n772_0) 309.22/291.52 309.22/291.52 309.22/291.52 Generator Equations: 309.22/291.52 gen_0':s4_0(0) <=> 0' 309.22/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 309.22/291.52 gen_nil:cons5_0(0) <=> nil 309.22/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 309.22/291.52 309.22/291.52 309.22/291.52 The following defined symbols remain to be analysed: 309.22/291.52 del, sort 309.22/291.52 309.22/291.52 They will be analysed ascendingly in the following order: 309.22/291.52 del < sort 309.22/291.52 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (17) RewriteLemmaProof (LOWER BOUND(ID)) 309.22/291.52 Proved the following rewrite lemma: 309.22/291.52 sort(gen_nil:cons5_0(n1475_0)) -> gen_nil:cons5_0(0), rt in Omega(1 + n1475_0 + n1475_0^2) 309.22/291.52 309.22/291.52 Induction Base: 309.22/291.52 sort(gen_nil:cons5_0(0)) ->_R^Omega(1) 309.22/291.52 if3(empty(gen_nil:cons5_0(0)), gen_nil:cons5_0(0)) ->_R^Omega(1) 309.22/291.52 if3(true, gen_nil:cons5_0(0)) ->_R^Omega(1) 309.22/291.52 nil 309.22/291.52 309.22/291.52 Induction Step: 309.22/291.52 sort(gen_nil:cons5_0(+(n1475_0, 1))) ->_R^Omega(1) 309.22/291.52 if3(empty(gen_nil:cons5_0(+(n1475_0, 1))), gen_nil:cons5_0(+(n1475_0, 1))) ->_R^Omega(1) 309.22/291.52 if3(false, gen_nil:cons5_0(+(1, n1475_0))) ->_R^Omega(1) 309.22/291.52 sort(del(max(gen_nil:cons5_0(+(1, n1475_0))), gen_nil:cons5_0(+(1, n1475_0)))) ->_L^Omega(1 + n1475_0) 309.22/291.52 sort(del(gen_0':s4_0(0), gen_nil:cons5_0(+(1, n1475_0)))) ->_R^Omega(1) 309.22/291.52 sort(if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_nil:cons5_0(n1475_0))) ->_L^Omega(1) 309.22/291.52 sort(if2(true, gen_0':s4_0(0), 0', gen_nil:cons5_0(n1475_0))) ->_R^Omega(1) 309.22/291.52 sort(gen_nil:cons5_0(n1475_0)) ->_IH 309.22/291.52 gen_nil:cons5_0(0) 309.22/291.52 309.22/291.52 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (18) 309.22/291.52 Obligation: 309.22/291.52 Proved the lower bound n^2 for the following obligation: 309.22/291.52 309.22/291.52 TRS: 309.22/291.52 Rules: 309.22/291.52 max(nil) -> 0' 309.22/291.52 max(cons(x, nil)) -> x 309.22/291.52 max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) 309.22/291.52 if1(true, x, y, xs) -> max(cons(x, xs)) 309.22/291.52 if1(false, x, y, xs) -> max(cons(y, xs)) 309.22/291.52 del(x, nil) -> nil 309.22/291.52 del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 309.22/291.52 if2(true, x, y, xs) -> xs 309.22/291.52 if2(false, x, y, xs) -> cons(y, del(x, xs)) 309.22/291.52 eq(0', 0') -> true 309.22/291.52 eq(0', s(y)) -> false 309.22/291.52 eq(s(x), 0') -> false 309.22/291.52 eq(s(x), s(y)) -> eq(x, y) 309.22/291.52 sort(xs) -> if3(empty(xs), xs) 309.22/291.52 if3(true, xs) -> nil 309.22/291.52 if3(false, xs) -> sort(del(max(xs), xs)) 309.22/291.52 empty(nil) -> true 309.22/291.52 empty(cons(x, xs)) -> false 309.22/291.52 ge(x, 0') -> true 309.22/291.52 ge(0', s(x)) -> false 309.22/291.52 ge(s(x), s(y)) -> ge(x, y) 309.22/291.52 309.22/291.52 Types: 309.22/291.52 max :: nil:cons -> 0':s 309.22/291.52 nil :: nil:cons 309.22/291.52 0' :: 0':s 309.22/291.52 cons :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 309.22/291.52 ge :: 0':s -> 0':s -> true:false 309.22/291.52 true :: true:false 309.22/291.52 false :: true:false 309.22/291.52 del :: 0':s -> nil:cons -> nil:cons 309.22/291.52 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 309.22/291.52 eq :: 0':s -> 0':s -> true:false 309.22/291.52 s :: 0':s -> 0':s 309.22/291.52 sort :: nil:cons -> nil:cons 309.22/291.52 if3 :: true:false -> nil:cons -> nil:cons 309.22/291.52 empty :: nil:cons -> true:false 309.22/291.52 hole_0':s1_0 :: 0':s 309.22/291.52 hole_nil:cons2_0 :: nil:cons 309.22/291.52 hole_true:false3_0 :: true:false 309.22/291.52 gen_0':s4_0 :: Nat -> 0':s 309.22/291.52 gen_nil:cons5_0 :: Nat -> nil:cons 309.22/291.52 309.22/291.52 309.22/291.52 Lemmas: 309.22/291.52 ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 309.22/291.52 max(gen_nil:cons5_0(+(1, n323_0))) -> gen_0':s4_0(0), rt in Omega(1 + n323_0) 309.22/291.52 eq(gen_0':s4_0(n772_0), gen_0':s4_0(n772_0)) -> true, rt in Omega(1 + n772_0) 309.22/291.52 309.22/291.52 309.22/291.52 Generator Equations: 309.22/291.52 gen_0':s4_0(0) <=> 0' 309.22/291.52 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 309.22/291.52 gen_nil:cons5_0(0) <=> nil 309.22/291.52 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 309.22/291.52 309.22/291.52 309.22/291.52 The following defined symbols remain to be analysed: 309.22/291.52 sort 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (19) LowerBoundPropagationProof (FINISHED) 309.22/291.52 Propagated lower bound. 309.22/291.52 ---------------------------------------- 309.22/291.52 309.22/291.52 (20) 309.22/291.52 BOUNDS(n^2, INF) 309.32/291.55 EOF