310.81/291.50 WORST_CASE(Omega(n^2), ?) 310.90/291.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 310.90/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 310.90/291.51 310.90/291.51 310.90/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 310.90/291.51 310.90/291.51 (0) CpxTRS 310.90/291.51 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 310.90/291.51 (2) CpxTRS 310.90/291.51 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 310.90/291.51 (4) typed CpxTrs 310.90/291.51 (5) OrderProof [LOWER BOUND(ID), 0 ms] 310.90/291.51 (6) typed CpxTrs 310.90/291.51 (7) RewriteLemmaProof [LOWER BOUND(ID), 253 ms] 310.90/291.51 (8) BEST 310.90/291.51 (9) proven lower bound 310.90/291.51 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 310.90/291.51 (11) BOUNDS(n^1, INF) 310.90/291.51 (12) typed CpxTrs 310.90/291.51 (13) RewriteLemmaProof [LOWER BOUND(ID), 57 ms] 310.90/291.51 (14) typed CpxTrs 310.90/291.51 (15) RewriteLemmaProof [LOWER BOUND(ID), 45 ms] 310.90/291.51 (16) proven lower bound 310.90/291.51 (17) LowerBoundPropagationProof [FINISHED, 0 ms] 310.90/291.51 (18) BOUNDS(n^2, INF) 310.90/291.51 310.90/291.51 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (0) 310.90/291.51 Obligation: 310.90/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 310.90/291.51 310.90/291.51 310.90/291.51 The TRS R consists of the following rules: 310.90/291.51 310.90/291.51 last(nil) -> 0 310.90/291.51 last(cons(x, nil)) -> x 310.90/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 310.90/291.51 del(x, nil) -> nil 310.90/291.51 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 310.90/291.51 if(true, x, y, xs) -> xs 310.90/291.51 if(false, x, y, xs) -> cons(y, del(x, xs)) 310.90/291.51 eq(0, 0) -> true 310.90/291.51 eq(0, s(y)) -> false 310.90/291.51 eq(s(x), 0) -> false 310.90/291.51 eq(s(x), s(y)) -> eq(x, y) 310.90/291.51 reverse(nil) -> nil 310.90/291.51 reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) 310.90/291.51 310.90/291.51 S is empty. 310.90/291.51 Rewrite Strategy: FULL 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 310.90/291.51 Renamed function symbols to avoid clashes with predefined symbol. 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (2) 310.90/291.51 Obligation: 310.90/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 310.90/291.51 310.90/291.51 310.90/291.51 The TRS R consists of the following rules: 310.90/291.51 310.90/291.51 last(nil) -> 0' 310.90/291.51 last(cons(x, nil)) -> x 310.90/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 310.90/291.51 del(x, nil) -> nil 310.90/291.51 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 310.90/291.51 if(true, x, y, xs) -> xs 310.90/291.51 if(false, x, y, xs) -> cons(y, del(x, xs)) 310.90/291.51 eq(0', 0') -> true 310.90/291.51 eq(0', s(y)) -> false 310.90/291.51 eq(s(x), 0') -> false 310.90/291.51 eq(s(x), s(y)) -> eq(x, y) 310.90/291.51 reverse(nil) -> nil 310.90/291.51 reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) 310.90/291.51 310.90/291.51 S is empty. 310.90/291.51 Rewrite Strategy: FULL 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 310.90/291.51 Infered types. 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (4) 310.90/291.51 Obligation: 310.90/291.51 TRS: 310.90/291.51 Rules: 310.90/291.51 last(nil) -> 0' 310.90/291.51 last(cons(x, nil)) -> x 310.90/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 310.90/291.51 del(x, nil) -> nil 310.90/291.51 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 310.90/291.51 if(true, x, y, xs) -> xs 310.90/291.51 if(false, x, y, xs) -> cons(y, del(x, xs)) 310.90/291.51 eq(0', 0') -> true 310.90/291.51 eq(0', s(y)) -> false 310.90/291.51 eq(s(x), 0') -> false 310.90/291.51 eq(s(x), s(y)) -> eq(x, y) 310.90/291.51 reverse(nil) -> nil 310.90/291.51 reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) 310.90/291.51 310.90/291.51 Types: 310.90/291.51 last :: nil:cons -> 0':s 310.90/291.51 nil :: nil:cons 310.90/291.51 0' :: 0':s 310.90/291.51 cons :: 0':s -> nil:cons -> nil:cons 310.90/291.51 del :: 0':s -> nil:cons -> nil:cons 310.90/291.51 if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 310.90/291.51 eq :: 0':s -> 0':s -> true:false 310.90/291.51 true :: true:false 310.90/291.51 false :: true:false 310.90/291.51 s :: 0':s -> 0':s 310.90/291.51 reverse :: nil:cons -> nil:cons 310.90/291.51 hole_0':s1_0 :: 0':s 310.90/291.51 hole_nil:cons2_0 :: nil:cons 310.90/291.51 hole_true:false3_0 :: true:false 310.90/291.51 gen_0':s4_0 :: Nat -> 0':s 310.90/291.51 gen_nil:cons5_0 :: Nat -> nil:cons 310.90/291.51 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (5) OrderProof (LOWER BOUND(ID)) 310.90/291.51 Heuristically decided to analyse the following defined symbols: 310.90/291.51 last, del, eq, reverse 310.90/291.51 310.90/291.51 They will be analysed ascendingly in the following order: 310.90/291.51 last < reverse 310.90/291.51 eq < del 310.90/291.51 del < reverse 310.90/291.51 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (6) 310.90/291.51 Obligation: 310.90/291.51 TRS: 310.90/291.51 Rules: 310.90/291.51 last(nil) -> 0' 310.90/291.51 last(cons(x, nil)) -> x 310.90/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 310.90/291.51 del(x, nil) -> nil 310.90/291.51 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 310.90/291.51 if(true, x, y, xs) -> xs 310.90/291.51 if(false, x, y, xs) -> cons(y, del(x, xs)) 310.90/291.51 eq(0', 0') -> true 310.90/291.51 eq(0', s(y)) -> false 310.90/291.51 eq(s(x), 0') -> false 310.90/291.51 eq(s(x), s(y)) -> eq(x, y) 310.90/291.51 reverse(nil) -> nil 310.90/291.51 reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) 310.90/291.51 310.90/291.51 Types: 310.90/291.51 last :: nil:cons -> 0':s 310.90/291.51 nil :: nil:cons 310.90/291.51 0' :: 0':s 310.90/291.51 cons :: 0':s -> nil:cons -> nil:cons 310.90/291.51 del :: 0':s -> nil:cons -> nil:cons 310.90/291.51 if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 310.90/291.51 eq :: 0':s -> 0':s -> true:false 310.90/291.51 true :: true:false 310.90/291.51 false :: true:false 310.90/291.51 s :: 0':s -> 0':s 310.90/291.51 reverse :: nil:cons -> nil:cons 310.90/291.51 hole_0':s1_0 :: 0':s 310.90/291.51 hole_nil:cons2_0 :: nil:cons 310.90/291.51 hole_true:false3_0 :: true:false 310.90/291.51 gen_0':s4_0 :: Nat -> 0':s 310.90/291.51 gen_nil:cons5_0 :: Nat -> nil:cons 310.90/291.51 310.90/291.51 310.90/291.51 Generator Equations: 310.90/291.51 gen_0':s4_0(0) <=> 0' 310.90/291.51 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 310.90/291.51 gen_nil:cons5_0(0) <=> nil 310.90/291.51 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 310.90/291.51 310.90/291.51 310.90/291.51 The following defined symbols remain to be analysed: 310.90/291.51 last, del, eq, reverse 310.90/291.51 310.90/291.51 They will be analysed ascendingly in the following order: 310.90/291.51 last < reverse 310.90/291.51 eq < del 310.90/291.51 del < reverse 310.90/291.51 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (7) RewriteLemmaProof (LOWER BOUND(ID)) 310.90/291.51 Proved the following rewrite lemma: 310.90/291.51 last(gen_nil:cons5_0(+(1, n7_0))) -> gen_0':s4_0(0), rt in Omega(1 + n7_0) 310.90/291.51 310.90/291.51 Induction Base: 310.90/291.51 last(gen_nil:cons5_0(+(1, 0))) ->_R^Omega(1) 310.90/291.51 0' 310.90/291.51 310.90/291.51 Induction Step: 310.90/291.51 last(gen_nil:cons5_0(+(1, +(n7_0, 1)))) ->_R^Omega(1) 310.90/291.51 last(cons(0', gen_nil:cons5_0(n7_0))) ->_IH 310.90/291.51 gen_0':s4_0(0) 310.90/291.51 310.90/291.51 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (8) 310.90/291.51 Complex Obligation (BEST) 310.90/291.51 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (9) 310.90/291.51 Obligation: 310.90/291.51 Proved the lower bound n^1 for the following obligation: 310.90/291.51 310.90/291.51 TRS: 310.90/291.51 Rules: 310.90/291.51 last(nil) -> 0' 310.90/291.51 last(cons(x, nil)) -> x 310.90/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 310.90/291.51 del(x, nil) -> nil 310.90/291.51 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 310.90/291.51 if(true, x, y, xs) -> xs 310.90/291.51 if(false, x, y, xs) -> cons(y, del(x, xs)) 310.90/291.51 eq(0', 0') -> true 310.90/291.51 eq(0', s(y)) -> false 310.90/291.51 eq(s(x), 0') -> false 310.90/291.51 eq(s(x), s(y)) -> eq(x, y) 310.90/291.51 reverse(nil) -> nil 310.90/291.51 reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) 310.90/291.51 310.90/291.51 Types: 310.90/291.51 last :: nil:cons -> 0':s 310.90/291.51 nil :: nil:cons 310.90/291.51 0' :: 0':s 310.90/291.51 cons :: 0':s -> nil:cons -> nil:cons 310.90/291.51 del :: 0':s -> nil:cons -> nil:cons 310.90/291.51 if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 310.90/291.51 eq :: 0':s -> 0':s -> true:false 310.90/291.51 true :: true:false 310.90/291.51 false :: true:false 310.90/291.51 s :: 0':s -> 0':s 310.90/291.51 reverse :: nil:cons -> nil:cons 310.90/291.51 hole_0':s1_0 :: 0':s 310.90/291.51 hole_nil:cons2_0 :: nil:cons 310.90/291.51 hole_true:false3_0 :: true:false 310.90/291.51 gen_0':s4_0 :: Nat -> 0':s 310.90/291.51 gen_nil:cons5_0 :: Nat -> nil:cons 310.90/291.51 310.90/291.51 310.90/291.51 Generator Equations: 310.90/291.51 gen_0':s4_0(0) <=> 0' 310.90/291.51 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 310.90/291.51 gen_nil:cons5_0(0) <=> nil 310.90/291.51 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 310.90/291.51 310.90/291.51 310.90/291.51 The following defined symbols remain to be analysed: 310.90/291.51 last, del, eq, reverse 310.90/291.51 310.90/291.51 They will be analysed ascendingly in the following order: 310.90/291.51 last < reverse 310.90/291.51 eq < del 310.90/291.51 del < reverse 310.90/291.51 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (10) LowerBoundPropagationProof (FINISHED) 310.90/291.51 Propagated lower bound. 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (11) 310.90/291.51 BOUNDS(n^1, INF) 310.90/291.51 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (12) 310.90/291.51 Obligation: 310.90/291.51 TRS: 310.90/291.51 Rules: 310.90/291.51 last(nil) -> 0' 310.90/291.51 last(cons(x, nil)) -> x 310.90/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 310.90/291.51 del(x, nil) -> nil 310.90/291.51 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 310.90/291.51 if(true, x, y, xs) -> xs 310.90/291.51 if(false, x, y, xs) -> cons(y, del(x, xs)) 310.90/291.51 eq(0', 0') -> true 310.90/291.51 eq(0', s(y)) -> false 310.90/291.51 eq(s(x), 0') -> false 310.90/291.51 eq(s(x), s(y)) -> eq(x, y) 310.90/291.51 reverse(nil) -> nil 310.90/291.51 reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) 310.90/291.51 310.90/291.51 Types: 310.90/291.51 last :: nil:cons -> 0':s 310.90/291.51 nil :: nil:cons 310.90/291.51 0' :: 0':s 310.90/291.51 cons :: 0':s -> nil:cons -> nil:cons 310.90/291.51 del :: 0':s -> nil:cons -> nil:cons 310.90/291.51 if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 310.90/291.51 eq :: 0':s -> 0':s -> true:false 310.90/291.51 true :: true:false 310.90/291.51 false :: true:false 310.90/291.51 s :: 0':s -> 0':s 310.90/291.51 reverse :: nil:cons -> nil:cons 310.90/291.51 hole_0':s1_0 :: 0':s 310.90/291.51 hole_nil:cons2_0 :: nil:cons 310.90/291.51 hole_true:false3_0 :: true:false 310.90/291.51 gen_0':s4_0 :: Nat -> 0':s 310.90/291.51 gen_nil:cons5_0 :: Nat -> nil:cons 310.90/291.51 310.90/291.51 310.90/291.51 Lemmas: 310.90/291.51 last(gen_nil:cons5_0(+(1, n7_0))) -> gen_0':s4_0(0), rt in Omega(1 + n7_0) 310.90/291.51 310.90/291.51 310.90/291.51 Generator Equations: 310.90/291.51 gen_0':s4_0(0) <=> 0' 310.90/291.51 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 310.90/291.51 gen_nil:cons5_0(0) <=> nil 310.90/291.51 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 310.90/291.51 310.90/291.51 310.90/291.51 The following defined symbols remain to be analysed: 310.90/291.51 eq, del, reverse 310.90/291.51 310.90/291.51 They will be analysed ascendingly in the following order: 310.90/291.51 eq < del 310.90/291.51 del < reverse 310.90/291.51 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (13) RewriteLemmaProof (LOWER BOUND(ID)) 310.90/291.51 Proved the following rewrite lemma: 310.90/291.51 eq(gen_0':s4_0(n335_0), gen_0':s4_0(n335_0)) -> true, rt in Omega(1 + n335_0) 310.90/291.51 310.90/291.51 Induction Base: 310.90/291.51 eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 310.90/291.51 true 310.90/291.51 310.90/291.51 Induction Step: 310.90/291.51 eq(gen_0':s4_0(+(n335_0, 1)), gen_0':s4_0(+(n335_0, 1))) ->_R^Omega(1) 310.90/291.51 eq(gen_0':s4_0(n335_0), gen_0':s4_0(n335_0)) ->_IH 310.90/291.51 true 310.90/291.51 310.90/291.51 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (14) 310.90/291.51 Obligation: 310.90/291.51 TRS: 310.90/291.51 Rules: 310.90/291.51 last(nil) -> 0' 310.90/291.51 last(cons(x, nil)) -> x 310.90/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 310.90/291.51 del(x, nil) -> nil 310.90/291.51 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 310.90/291.51 if(true, x, y, xs) -> xs 310.90/291.51 if(false, x, y, xs) -> cons(y, del(x, xs)) 310.90/291.51 eq(0', 0') -> true 310.90/291.51 eq(0', s(y)) -> false 310.90/291.51 eq(s(x), 0') -> false 310.90/291.51 eq(s(x), s(y)) -> eq(x, y) 310.90/291.51 reverse(nil) -> nil 310.90/291.51 reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) 310.90/291.51 310.90/291.51 Types: 310.90/291.51 last :: nil:cons -> 0':s 310.90/291.51 nil :: nil:cons 310.90/291.51 0' :: 0':s 310.90/291.51 cons :: 0':s -> nil:cons -> nil:cons 310.90/291.51 del :: 0':s -> nil:cons -> nil:cons 310.90/291.51 if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 310.90/291.51 eq :: 0':s -> 0':s -> true:false 310.90/291.51 true :: true:false 310.90/291.51 false :: true:false 310.90/291.51 s :: 0':s -> 0':s 310.90/291.51 reverse :: nil:cons -> nil:cons 310.90/291.51 hole_0':s1_0 :: 0':s 310.90/291.51 hole_nil:cons2_0 :: nil:cons 310.90/291.51 hole_true:false3_0 :: true:false 310.90/291.51 gen_0':s4_0 :: Nat -> 0':s 310.90/291.51 gen_nil:cons5_0 :: Nat -> nil:cons 310.90/291.51 310.90/291.51 310.90/291.51 Lemmas: 310.90/291.51 last(gen_nil:cons5_0(+(1, n7_0))) -> gen_0':s4_0(0), rt in Omega(1 + n7_0) 310.90/291.51 eq(gen_0':s4_0(n335_0), gen_0':s4_0(n335_0)) -> true, rt in Omega(1 + n335_0) 310.90/291.51 310.90/291.51 310.90/291.51 Generator Equations: 310.90/291.51 gen_0':s4_0(0) <=> 0' 310.90/291.51 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 310.90/291.51 gen_nil:cons5_0(0) <=> nil 310.90/291.51 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 310.90/291.51 310.90/291.51 310.90/291.51 The following defined symbols remain to be analysed: 310.90/291.51 del, reverse 310.90/291.51 310.90/291.51 They will be analysed ascendingly in the following order: 310.90/291.51 del < reverse 310.90/291.51 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (15) RewriteLemmaProof (LOWER BOUND(ID)) 310.90/291.51 Proved the following rewrite lemma: 310.90/291.51 reverse(gen_nil:cons5_0(n984_0)) -> gen_nil:cons5_0(n984_0), rt in Omega(1 + n984_0 + n984_0^2) 310.90/291.51 310.90/291.51 Induction Base: 310.90/291.51 reverse(gen_nil:cons5_0(0)) ->_R^Omega(1) 310.90/291.51 nil 310.90/291.51 310.90/291.51 Induction Step: 310.90/291.51 reverse(gen_nil:cons5_0(+(n984_0, 1))) ->_R^Omega(1) 310.90/291.51 cons(last(cons(0', gen_nil:cons5_0(n984_0))), reverse(del(last(cons(0', gen_nil:cons5_0(n984_0))), cons(0', gen_nil:cons5_0(n984_0))))) ->_L^Omega(1 + n984_0) 310.90/291.51 cons(gen_0':s4_0(0), reverse(del(last(cons(0', gen_nil:cons5_0(n984_0))), cons(0', gen_nil:cons5_0(n984_0))))) ->_L^Omega(1 + n984_0) 310.90/291.51 cons(gen_0':s4_0(0), reverse(del(gen_0':s4_0(0), cons(0', gen_nil:cons5_0(n984_0))))) ->_R^Omega(1) 310.90/291.51 cons(gen_0':s4_0(0), reverse(if(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_nil:cons5_0(n984_0)))) ->_L^Omega(1) 310.90/291.51 cons(gen_0':s4_0(0), reverse(if(true, gen_0':s4_0(0), 0', gen_nil:cons5_0(n984_0)))) ->_R^Omega(1) 310.90/291.51 cons(gen_0':s4_0(0), reverse(gen_nil:cons5_0(n984_0))) ->_IH 310.90/291.51 cons(gen_0':s4_0(0), gen_nil:cons5_0(c985_0)) 310.90/291.51 310.90/291.51 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (16) 310.90/291.51 Obligation: 310.90/291.51 Proved the lower bound n^2 for the following obligation: 310.90/291.51 310.90/291.51 TRS: 310.90/291.51 Rules: 310.90/291.51 last(nil) -> 0' 310.90/291.51 last(cons(x, nil)) -> x 310.90/291.51 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 310.90/291.51 del(x, nil) -> nil 310.90/291.51 del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) 310.90/291.51 if(true, x, y, xs) -> xs 310.90/291.51 if(false, x, y, xs) -> cons(y, del(x, xs)) 310.90/291.51 eq(0', 0') -> true 310.90/291.51 eq(0', s(y)) -> false 310.90/291.51 eq(s(x), 0') -> false 310.90/291.51 eq(s(x), s(y)) -> eq(x, y) 310.90/291.51 reverse(nil) -> nil 310.90/291.51 reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) 310.90/291.51 310.90/291.51 Types: 310.90/291.51 last :: nil:cons -> 0':s 310.90/291.51 nil :: nil:cons 310.90/291.51 0' :: 0':s 310.90/291.51 cons :: 0':s -> nil:cons -> nil:cons 310.90/291.51 del :: 0':s -> nil:cons -> nil:cons 310.90/291.51 if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 310.90/291.51 eq :: 0':s -> 0':s -> true:false 310.90/291.51 true :: true:false 310.90/291.51 false :: true:false 310.90/291.51 s :: 0':s -> 0':s 310.90/291.51 reverse :: nil:cons -> nil:cons 310.90/291.51 hole_0':s1_0 :: 0':s 310.90/291.51 hole_nil:cons2_0 :: nil:cons 310.90/291.51 hole_true:false3_0 :: true:false 310.90/291.51 gen_0':s4_0 :: Nat -> 0':s 310.90/291.51 gen_nil:cons5_0 :: Nat -> nil:cons 310.90/291.51 310.90/291.51 310.90/291.51 Lemmas: 310.90/291.51 last(gen_nil:cons5_0(+(1, n7_0))) -> gen_0':s4_0(0), rt in Omega(1 + n7_0) 310.90/291.51 eq(gen_0':s4_0(n335_0), gen_0':s4_0(n335_0)) -> true, rt in Omega(1 + n335_0) 310.90/291.51 310.90/291.51 310.90/291.51 Generator Equations: 310.90/291.51 gen_0':s4_0(0) <=> 0' 310.90/291.51 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 310.90/291.51 gen_nil:cons5_0(0) <=> nil 310.90/291.51 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 310.90/291.51 310.90/291.51 310.90/291.51 The following defined symbols remain to be analysed: 310.90/291.51 reverse 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (17) LowerBoundPropagationProof (FINISHED) 310.90/291.51 Propagated lower bound. 310.90/291.51 ---------------------------------------- 310.90/291.51 310.90/291.51 (18) 310.90/291.51 BOUNDS(n^2, INF) 310.90/291.55 EOF