305.04/291.54 WORST_CASE(Omega(n^2), ?) 305.04/291.54 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 305.04/291.54 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 305.04/291.54 305.04/291.54 305.04/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 305.04/291.54 305.04/291.54 (0) CpxTRS 305.04/291.54 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 305.04/291.54 (2) CpxTRS 305.04/291.54 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 305.04/291.54 (4) typed CpxTrs 305.04/291.54 (5) OrderProof [LOWER BOUND(ID), 0 ms] 305.04/291.54 (6) typed CpxTrs 305.04/291.54 (7) RewriteLemmaProof [LOWER BOUND(ID), 212 ms] 305.04/291.54 (8) BEST 305.04/291.54 (9) proven lower bound 305.04/291.54 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 305.04/291.54 (11) BOUNDS(n^1, INF) 305.04/291.54 (12) typed CpxTrs 305.04/291.54 (13) RewriteLemmaProof [LOWER BOUND(ID), 126 ms] 305.04/291.54 (14) proven lower bound 305.04/291.54 (15) LowerBoundPropagationProof [FINISHED, 0 ms] 305.04/291.54 (16) BOUNDS(n^2, INF) 305.04/291.54 305.04/291.54 305.04/291.54 ---------------------------------------- 305.04/291.54 305.04/291.54 (0) 305.04/291.54 Obligation: 305.04/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 305.04/291.54 305.04/291.54 305.04/291.54 The TRS R consists of the following rules: 305.04/291.54 305.04/291.54 sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) 305.04/291.54 sum(cons(0, x), y) -> sum(x, y) 305.04/291.54 sum(nil, y) -> y 305.04/291.54 empty(nil) -> true 305.04/291.54 empty(cons(n, x)) -> false 305.04/291.54 tail(nil) -> nil 305.04/291.54 tail(cons(n, x)) -> x 305.04/291.54 head(cons(n, x)) -> n 305.04/291.54 weight(x) -> if(empty(x), empty(tail(x)), x) 305.04/291.54 if(true, b, x) -> weight_undefined_error 305.04/291.54 if(false, b, x) -> if2(b, x) 305.04/291.54 if2(true, x) -> head(x) 305.04/291.54 if2(false, x) -> weight(sum(x, cons(0, tail(tail(x))))) 305.04/291.54 305.04/291.54 S is empty. 305.04/291.54 Rewrite Strategy: FULL 305.04/291.54 ---------------------------------------- 305.04/291.54 305.04/291.54 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 305.04/291.54 Renamed function symbols to avoid clashes with predefined symbol. 305.04/291.54 ---------------------------------------- 305.04/291.54 305.04/291.54 (2) 305.04/291.54 Obligation: 305.04/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 305.04/291.54 305.04/291.54 305.04/291.54 The TRS R consists of the following rules: 305.04/291.54 305.04/291.54 sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) 305.04/291.54 sum(cons(0', x), y) -> sum(x, y) 305.04/291.54 sum(nil, y) -> y 305.04/291.54 empty(nil) -> true 305.04/291.54 empty(cons(n, x)) -> false 305.04/291.54 tail(nil) -> nil 305.04/291.54 tail(cons(n, x)) -> x 305.04/291.54 head(cons(n, x)) -> n 305.04/291.54 weight(x) -> if(empty(x), empty(tail(x)), x) 305.04/291.54 if(true, b, x) -> weight_undefined_error 305.04/291.54 if(false, b, x) -> if2(b, x) 305.04/291.54 if2(true, x) -> head(x) 305.04/291.54 if2(false, x) -> weight(sum(x, cons(0', tail(tail(x))))) 305.04/291.54 305.04/291.54 S is empty. 305.04/291.54 Rewrite Strategy: FULL 305.04/291.54 ---------------------------------------- 305.04/291.54 305.04/291.54 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 305.04/291.54 Infered types. 305.04/291.54 ---------------------------------------- 305.04/291.54 305.04/291.54 (4) 305.04/291.54 Obligation: 305.04/291.54 TRS: 305.04/291.54 Rules: 305.04/291.54 sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) 305.04/291.54 sum(cons(0', x), y) -> sum(x, y) 305.04/291.54 sum(nil, y) -> y 305.04/291.54 empty(nil) -> true 305.04/291.54 empty(cons(n, x)) -> false 305.04/291.54 tail(nil) -> nil 305.04/291.54 tail(cons(n, x)) -> x 305.04/291.54 head(cons(n, x)) -> n 305.04/291.54 weight(x) -> if(empty(x), empty(tail(x)), x) 305.04/291.54 if(true, b, x) -> weight_undefined_error 305.04/291.54 if(false, b, x) -> if2(b, x) 305.04/291.54 if2(true, x) -> head(x) 305.04/291.54 if2(false, x) -> weight(sum(x, cons(0', tail(tail(x))))) 305.04/291.54 305.04/291.54 Types: 305.04/291.54 sum :: cons:nil -> cons:nil -> cons:nil 305.04/291.54 cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil 305.04/291.54 s :: s:0':weight_undefined_error -> s:0':weight_undefined_error 305.04/291.54 0' :: s:0':weight_undefined_error 305.04/291.54 nil :: cons:nil 305.04/291.54 empty :: cons:nil -> true:false 305.04/291.54 true :: true:false 305.04/291.54 false :: true:false 305.04/291.54 tail :: cons:nil -> cons:nil 305.04/291.54 head :: cons:nil -> s:0':weight_undefined_error 305.04/291.54 weight :: cons:nil -> s:0':weight_undefined_error 305.04/291.54 if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.54 weight_undefined_error :: s:0':weight_undefined_error 305.04/291.54 if2 :: true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.54 hole_cons:nil1_0 :: cons:nil 305.04/291.54 hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error 305.04/291.54 hole_true:false3_0 :: true:false 305.04/291.54 gen_cons:nil4_0 :: Nat -> cons:nil 305.04/291.54 gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error 305.04/291.54 305.04/291.54 ---------------------------------------- 305.04/291.54 305.04/291.54 (5) OrderProof (LOWER BOUND(ID)) 305.04/291.54 Heuristically decided to analyse the following defined symbols: 305.04/291.54 sum, weight 305.04/291.54 305.04/291.54 They will be analysed ascendingly in the following order: 305.04/291.54 sum < weight 305.04/291.54 305.04/291.54 ---------------------------------------- 305.04/291.54 305.04/291.54 (6) 305.04/291.54 Obligation: 305.04/291.54 TRS: 305.04/291.54 Rules: 305.04/291.54 sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) 305.04/291.55 sum(cons(0', x), y) -> sum(x, y) 305.04/291.55 sum(nil, y) -> y 305.04/291.55 empty(nil) -> true 305.04/291.55 empty(cons(n, x)) -> false 305.04/291.55 tail(nil) -> nil 305.04/291.55 tail(cons(n, x)) -> x 305.04/291.55 head(cons(n, x)) -> n 305.04/291.55 weight(x) -> if(empty(x), empty(tail(x)), x) 305.04/291.55 if(true, b, x) -> weight_undefined_error 305.04/291.55 if(false, b, x) -> if2(b, x) 305.04/291.55 if2(true, x) -> head(x) 305.04/291.55 if2(false, x) -> weight(sum(x, cons(0', tail(tail(x))))) 305.04/291.55 305.04/291.55 Types: 305.04/291.55 sum :: cons:nil -> cons:nil -> cons:nil 305.04/291.55 cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil 305.04/291.55 s :: s:0':weight_undefined_error -> s:0':weight_undefined_error 305.04/291.55 0' :: s:0':weight_undefined_error 305.04/291.55 nil :: cons:nil 305.04/291.55 empty :: cons:nil -> true:false 305.04/291.55 true :: true:false 305.04/291.55 false :: true:false 305.04/291.55 tail :: cons:nil -> cons:nil 305.04/291.55 head :: cons:nil -> s:0':weight_undefined_error 305.04/291.55 weight :: cons:nil -> s:0':weight_undefined_error 305.04/291.55 if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.55 weight_undefined_error :: s:0':weight_undefined_error 305.04/291.55 if2 :: true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.55 hole_cons:nil1_0 :: cons:nil 305.04/291.55 hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error 305.04/291.55 hole_true:false3_0 :: true:false 305.04/291.55 gen_cons:nil4_0 :: Nat -> cons:nil 305.04/291.55 gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error 305.04/291.55 305.04/291.55 305.04/291.55 Generator Equations: 305.04/291.55 gen_cons:nil4_0(0) <=> nil 305.04/291.55 gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x)) 305.04/291.55 gen_s:0':weight_undefined_error5_0(0) <=> 0' 305.04/291.55 gen_s:0':weight_undefined_error5_0(+(x, 1)) <=> s(gen_s:0':weight_undefined_error5_0(x)) 305.04/291.55 305.04/291.55 305.04/291.55 The following defined symbols remain to be analysed: 305.04/291.55 sum, weight 305.04/291.55 305.04/291.55 They will be analysed ascendingly in the following order: 305.04/291.55 sum < weight 305.04/291.55 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (7) RewriteLemmaProof (LOWER BOUND(ID)) 305.04/291.55 Proved the following rewrite lemma: 305.04/291.55 sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) -> gen_cons:nil4_0(b), rt in Omega(1 + n7_0) 305.04/291.55 305.04/291.55 Induction Base: 305.04/291.55 sum(gen_cons:nil4_0(0), gen_cons:nil4_0(b)) ->_R^Omega(1) 305.04/291.55 gen_cons:nil4_0(b) 305.04/291.55 305.04/291.55 Induction Step: 305.04/291.55 sum(gen_cons:nil4_0(+(n7_0, 1)), gen_cons:nil4_0(b)) ->_R^Omega(1) 305.04/291.55 sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) ->_IH 305.04/291.55 gen_cons:nil4_0(b) 305.04/291.55 305.04/291.55 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (8) 305.04/291.55 Complex Obligation (BEST) 305.04/291.55 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (9) 305.04/291.55 Obligation: 305.04/291.55 Proved the lower bound n^1 for the following obligation: 305.04/291.55 305.04/291.55 TRS: 305.04/291.55 Rules: 305.04/291.55 sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) 305.04/291.55 sum(cons(0', x), y) -> sum(x, y) 305.04/291.55 sum(nil, y) -> y 305.04/291.55 empty(nil) -> true 305.04/291.55 empty(cons(n, x)) -> false 305.04/291.55 tail(nil) -> nil 305.04/291.55 tail(cons(n, x)) -> x 305.04/291.55 head(cons(n, x)) -> n 305.04/291.55 weight(x) -> if(empty(x), empty(tail(x)), x) 305.04/291.55 if(true, b, x) -> weight_undefined_error 305.04/291.55 if(false, b, x) -> if2(b, x) 305.04/291.55 if2(true, x) -> head(x) 305.04/291.55 if2(false, x) -> weight(sum(x, cons(0', tail(tail(x))))) 305.04/291.55 305.04/291.55 Types: 305.04/291.55 sum :: cons:nil -> cons:nil -> cons:nil 305.04/291.55 cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil 305.04/291.55 s :: s:0':weight_undefined_error -> s:0':weight_undefined_error 305.04/291.55 0' :: s:0':weight_undefined_error 305.04/291.55 nil :: cons:nil 305.04/291.55 empty :: cons:nil -> true:false 305.04/291.55 true :: true:false 305.04/291.55 false :: true:false 305.04/291.55 tail :: cons:nil -> cons:nil 305.04/291.55 head :: cons:nil -> s:0':weight_undefined_error 305.04/291.55 weight :: cons:nil -> s:0':weight_undefined_error 305.04/291.55 if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.55 weight_undefined_error :: s:0':weight_undefined_error 305.04/291.55 if2 :: true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.55 hole_cons:nil1_0 :: cons:nil 305.04/291.55 hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error 305.04/291.55 hole_true:false3_0 :: true:false 305.04/291.55 gen_cons:nil4_0 :: Nat -> cons:nil 305.04/291.55 gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error 305.04/291.55 305.04/291.55 305.04/291.55 Generator Equations: 305.04/291.55 gen_cons:nil4_0(0) <=> nil 305.04/291.55 gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x)) 305.04/291.55 gen_s:0':weight_undefined_error5_0(0) <=> 0' 305.04/291.55 gen_s:0':weight_undefined_error5_0(+(x, 1)) <=> s(gen_s:0':weight_undefined_error5_0(x)) 305.04/291.55 305.04/291.55 305.04/291.55 The following defined symbols remain to be analysed: 305.04/291.55 sum, weight 305.04/291.55 305.04/291.55 They will be analysed ascendingly in the following order: 305.04/291.55 sum < weight 305.04/291.55 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (10) LowerBoundPropagationProof (FINISHED) 305.04/291.55 Propagated lower bound. 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (11) 305.04/291.55 BOUNDS(n^1, INF) 305.04/291.55 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (12) 305.04/291.55 Obligation: 305.04/291.55 TRS: 305.04/291.55 Rules: 305.04/291.55 sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) 305.04/291.55 sum(cons(0', x), y) -> sum(x, y) 305.04/291.55 sum(nil, y) -> y 305.04/291.55 empty(nil) -> true 305.04/291.55 empty(cons(n, x)) -> false 305.04/291.55 tail(nil) -> nil 305.04/291.55 tail(cons(n, x)) -> x 305.04/291.55 head(cons(n, x)) -> n 305.04/291.55 weight(x) -> if(empty(x), empty(tail(x)), x) 305.04/291.55 if(true, b, x) -> weight_undefined_error 305.04/291.55 if(false, b, x) -> if2(b, x) 305.04/291.55 if2(true, x) -> head(x) 305.04/291.55 if2(false, x) -> weight(sum(x, cons(0', tail(tail(x))))) 305.04/291.55 305.04/291.55 Types: 305.04/291.55 sum :: cons:nil -> cons:nil -> cons:nil 305.04/291.55 cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil 305.04/291.55 s :: s:0':weight_undefined_error -> s:0':weight_undefined_error 305.04/291.55 0' :: s:0':weight_undefined_error 305.04/291.55 nil :: cons:nil 305.04/291.55 empty :: cons:nil -> true:false 305.04/291.55 true :: true:false 305.04/291.55 false :: true:false 305.04/291.55 tail :: cons:nil -> cons:nil 305.04/291.55 head :: cons:nil -> s:0':weight_undefined_error 305.04/291.55 weight :: cons:nil -> s:0':weight_undefined_error 305.04/291.55 if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.55 weight_undefined_error :: s:0':weight_undefined_error 305.04/291.55 if2 :: true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.55 hole_cons:nil1_0 :: cons:nil 305.04/291.55 hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error 305.04/291.55 hole_true:false3_0 :: true:false 305.04/291.55 gen_cons:nil4_0 :: Nat -> cons:nil 305.04/291.55 gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error 305.04/291.55 305.04/291.55 305.04/291.55 Lemmas: 305.04/291.55 sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) -> gen_cons:nil4_0(b), rt in Omega(1 + n7_0) 305.04/291.55 305.04/291.55 305.04/291.55 Generator Equations: 305.04/291.55 gen_cons:nil4_0(0) <=> nil 305.04/291.55 gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x)) 305.04/291.55 gen_s:0':weight_undefined_error5_0(0) <=> 0' 305.04/291.55 gen_s:0':weight_undefined_error5_0(+(x, 1)) <=> s(gen_s:0':weight_undefined_error5_0(x)) 305.04/291.55 305.04/291.55 305.04/291.55 The following defined symbols remain to be analysed: 305.04/291.55 weight 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (13) RewriteLemmaProof (LOWER BOUND(ID)) 305.04/291.55 Proved the following rewrite lemma: 305.04/291.55 weight(gen_cons:nil4_0(+(1, n640_0))) -> gen_s:0':weight_undefined_error5_0(0), rt in Omega(1 + n640_0 + n640_0^2) 305.04/291.55 305.04/291.55 Induction Base: 305.04/291.55 weight(gen_cons:nil4_0(+(1, 0))) ->_R^Omega(1) 305.04/291.55 if(empty(gen_cons:nil4_0(+(1, 0))), empty(tail(gen_cons:nil4_0(+(1, 0)))), gen_cons:nil4_0(+(1, 0))) ->_R^Omega(1) 305.04/291.55 if(false, empty(tail(gen_cons:nil4_0(1))), gen_cons:nil4_0(1)) ->_R^Omega(1) 305.04/291.55 if(false, empty(gen_cons:nil4_0(0)), gen_cons:nil4_0(1)) ->_R^Omega(1) 305.04/291.55 if(false, true, gen_cons:nil4_0(1)) ->_R^Omega(1) 305.04/291.55 if2(true, gen_cons:nil4_0(1)) ->_R^Omega(1) 305.04/291.55 head(gen_cons:nil4_0(1)) ->_R^Omega(1) 305.04/291.55 0' 305.04/291.55 305.04/291.55 Induction Step: 305.04/291.55 weight(gen_cons:nil4_0(+(1, +(n640_0, 1)))) ->_R^Omega(1) 305.04/291.55 if(empty(gen_cons:nil4_0(+(1, +(n640_0, 1)))), empty(tail(gen_cons:nil4_0(+(1, +(n640_0, 1))))), gen_cons:nil4_0(+(1, +(n640_0, 1)))) ->_R^Omega(1) 305.04/291.55 if(false, empty(tail(gen_cons:nil4_0(+(2, n640_0)))), gen_cons:nil4_0(+(2, n640_0))) ->_R^Omega(1) 305.04/291.55 if(false, empty(gen_cons:nil4_0(+(1, n640_0))), gen_cons:nil4_0(+(2, n640_0))) ->_R^Omega(1) 305.04/291.55 if(false, false, gen_cons:nil4_0(+(2, n640_0))) ->_R^Omega(1) 305.04/291.55 if2(false, gen_cons:nil4_0(+(2, n640_0))) ->_R^Omega(1) 305.04/291.55 weight(sum(gen_cons:nil4_0(+(2, n640_0)), cons(0', tail(tail(gen_cons:nil4_0(+(2, n640_0))))))) ->_R^Omega(1) 305.04/291.55 weight(sum(gen_cons:nil4_0(+(2, n640_0)), cons(0', tail(gen_cons:nil4_0(+(1, n640_0)))))) ->_R^Omega(1) 305.04/291.55 weight(sum(gen_cons:nil4_0(+(2, n640_0)), cons(0', gen_cons:nil4_0(n640_0)))) ->_L^Omega(3 + n640_0) 305.04/291.55 weight(gen_cons:nil4_0(+(n640_0, 1))) ->_IH 305.04/291.55 gen_s:0':weight_undefined_error5_0(0) 305.04/291.55 305.04/291.55 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (14) 305.04/291.55 Obligation: 305.04/291.55 Proved the lower bound n^2 for the following obligation: 305.04/291.55 305.04/291.55 TRS: 305.04/291.55 Rules: 305.04/291.55 sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) 305.04/291.55 sum(cons(0', x), y) -> sum(x, y) 305.04/291.55 sum(nil, y) -> y 305.04/291.55 empty(nil) -> true 305.04/291.55 empty(cons(n, x)) -> false 305.04/291.55 tail(nil) -> nil 305.04/291.55 tail(cons(n, x)) -> x 305.04/291.55 head(cons(n, x)) -> n 305.04/291.55 weight(x) -> if(empty(x), empty(tail(x)), x) 305.04/291.55 if(true, b, x) -> weight_undefined_error 305.04/291.55 if(false, b, x) -> if2(b, x) 305.04/291.55 if2(true, x) -> head(x) 305.04/291.55 if2(false, x) -> weight(sum(x, cons(0', tail(tail(x))))) 305.04/291.55 305.04/291.55 Types: 305.04/291.55 sum :: cons:nil -> cons:nil -> cons:nil 305.04/291.55 cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil 305.04/291.55 s :: s:0':weight_undefined_error -> s:0':weight_undefined_error 305.04/291.55 0' :: s:0':weight_undefined_error 305.04/291.55 nil :: cons:nil 305.04/291.55 empty :: cons:nil -> true:false 305.04/291.55 true :: true:false 305.04/291.55 false :: true:false 305.04/291.55 tail :: cons:nil -> cons:nil 305.04/291.55 head :: cons:nil -> s:0':weight_undefined_error 305.04/291.55 weight :: cons:nil -> s:0':weight_undefined_error 305.04/291.55 if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.55 weight_undefined_error :: s:0':weight_undefined_error 305.04/291.55 if2 :: true:false -> cons:nil -> s:0':weight_undefined_error 305.04/291.55 hole_cons:nil1_0 :: cons:nil 305.04/291.55 hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error 305.04/291.55 hole_true:false3_0 :: true:false 305.04/291.55 gen_cons:nil4_0 :: Nat -> cons:nil 305.04/291.55 gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error 305.04/291.55 305.04/291.55 305.04/291.55 Lemmas: 305.04/291.55 sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) -> gen_cons:nil4_0(b), rt in Omega(1 + n7_0) 305.04/291.55 305.04/291.55 305.04/291.55 Generator Equations: 305.04/291.55 gen_cons:nil4_0(0) <=> nil 305.04/291.55 gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x)) 305.04/291.55 gen_s:0':weight_undefined_error5_0(0) <=> 0' 305.04/291.55 gen_s:0':weight_undefined_error5_0(+(x, 1)) <=> s(gen_s:0':weight_undefined_error5_0(x)) 305.04/291.55 305.04/291.55 305.04/291.55 The following defined symbols remain to be analysed: 305.04/291.55 weight 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (15) LowerBoundPropagationProof (FINISHED) 305.04/291.55 Propagated lower bound. 305.04/291.55 ---------------------------------------- 305.04/291.55 305.04/291.55 (16) 305.04/291.55 BOUNDS(n^2, INF) 305.04/291.58 EOF