308.87/291.47 WORST_CASE(Omega(n^2), ?) 308.87/291.48 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 308.87/291.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 308.87/291.48 308.87/291.48 308.87/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 308.87/291.48 308.87/291.48 (0) CpxTRS 308.87/291.48 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 308.87/291.48 (2) CpxTRS 308.87/291.48 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 308.87/291.48 (4) typed CpxTrs 308.87/291.48 (5) OrderProof [LOWER BOUND(ID), 0 ms] 308.87/291.48 (6) typed CpxTrs 308.87/291.48 (7) RewriteLemmaProof [LOWER BOUND(ID), 271 ms] 308.87/291.48 (8) BEST 308.87/291.48 (9) proven lower bound 308.87/291.48 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 308.87/291.48 (11) BOUNDS(n^1, INF) 308.87/291.48 (12) typed CpxTrs 308.87/291.48 (13) RewriteLemmaProof [LOWER BOUND(ID), 67 ms] 308.87/291.48 (14) typed CpxTrs 308.87/291.48 (15) RewriteLemmaProof [LOWER BOUND(ID), 109 ms] 308.87/291.48 (16) proven lower bound 308.87/291.48 (17) LowerBoundPropagationProof [FINISHED, 0 ms] 308.87/291.48 (18) BOUNDS(n^2, INF) 308.87/291.48 308.87/291.48 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (0) 308.87/291.48 Obligation: 308.87/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 308.87/291.48 308.87/291.48 308.87/291.48 The TRS R consists of the following rules: 308.87/291.48 308.87/291.48 isEmpty(cons(x, xs)) -> false 308.87/291.48 isEmpty(nil) -> true 308.87/291.48 isZero(0) -> true 308.87/291.48 isZero(s(x)) -> false 308.87/291.48 head(cons(x, xs)) -> x 308.87/291.48 tail(cons(x, xs)) -> xs 308.87/291.48 tail(nil) -> nil 308.87/291.48 p(s(s(x))) -> s(p(s(x))) 308.87/291.48 p(s(0)) -> 0 308.87/291.48 p(0) -> 0 308.87/291.48 inc(s(x)) -> s(inc(x)) 308.87/291.48 inc(0) -> s(0) 308.87/291.48 sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y)) 308.87/291.48 if(true, b, y, xs, ys, x) -> y 308.87/291.48 if(false, true, y, xs, ys, x) -> sumList(xs, y) 308.87/291.48 if(false, false, y, xs, ys, x) -> sumList(ys, x) 308.87/291.48 sum(xs) -> sumList(xs, 0) 308.87/291.48 308.87/291.48 S is empty. 308.87/291.48 Rewrite Strategy: FULL 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 308.87/291.48 Renamed function symbols to avoid clashes with predefined symbol. 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (2) 308.87/291.48 Obligation: 308.87/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 308.87/291.48 308.87/291.48 308.87/291.48 The TRS R consists of the following rules: 308.87/291.48 308.87/291.48 isEmpty(cons(x, xs)) -> false 308.87/291.48 isEmpty(nil) -> true 308.87/291.48 isZero(0') -> true 308.87/291.48 isZero(s(x)) -> false 308.87/291.48 head(cons(x, xs)) -> x 308.87/291.48 tail(cons(x, xs)) -> xs 308.87/291.48 tail(nil) -> nil 308.87/291.48 p(s(s(x))) -> s(p(s(x))) 308.87/291.48 p(s(0')) -> 0' 308.87/291.48 p(0') -> 0' 308.87/291.48 inc(s(x)) -> s(inc(x)) 308.87/291.48 inc(0') -> s(0') 308.87/291.48 sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y)) 308.87/291.48 if(true, b, y, xs, ys, x) -> y 308.87/291.48 if(false, true, y, xs, ys, x) -> sumList(xs, y) 308.87/291.48 if(false, false, y, xs, ys, x) -> sumList(ys, x) 308.87/291.48 sum(xs) -> sumList(xs, 0') 308.87/291.48 308.87/291.48 S is empty. 308.87/291.48 Rewrite Strategy: FULL 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 308.87/291.48 Infered types. 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (4) 308.87/291.48 Obligation: 308.87/291.48 TRS: 308.87/291.48 Rules: 308.87/291.48 isEmpty(cons(x, xs)) -> false 308.87/291.48 isEmpty(nil) -> true 308.87/291.48 isZero(0') -> true 308.87/291.48 isZero(s(x)) -> false 308.87/291.48 head(cons(x, xs)) -> x 308.87/291.48 tail(cons(x, xs)) -> xs 308.87/291.48 tail(nil) -> nil 308.87/291.48 p(s(s(x))) -> s(p(s(x))) 308.87/291.48 p(s(0')) -> 0' 308.87/291.48 p(0') -> 0' 308.87/291.48 inc(s(x)) -> s(inc(x)) 308.87/291.48 inc(0') -> s(0') 308.87/291.48 sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y)) 308.87/291.48 if(true, b, y, xs, ys, x) -> y 308.87/291.48 if(false, true, y, xs, ys, x) -> sumList(xs, y) 308.87/291.48 if(false, false, y, xs, ys, x) -> sumList(ys, x) 308.87/291.48 sum(xs) -> sumList(xs, 0') 308.87/291.48 308.87/291.48 Types: 308.87/291.48 isEmpty :: cons:nil -> false:true 308.87/291.48 cons :: 0':s -> cons:nil -> cons:nil 308.87/291.48 false :: false:true 308.87/291.48 nil :: cons:nil 308.87/291.48 true :: false:true 308.87/291.48 isZero :: 0':s -> false:true 308.87/291.48 0' :: 0':s 308.87/291.48 s :: 0':s -> 0':s 308.87/291.48 head :: cons:nil -> 0':s 308.87/291.48 tail :: cons:nil -> cons:nil 308.87/291.48 p :: 0':s -> 0':s 308.87/291.48 inc :: 0':s -> 0':s 308.87/291.48 sumList :: cons:nil -> 0':s -> 0':s 308.87/291.48 if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s 308.87/291.48 sum :: cons:nil -> 0':s 308.87/291.48 hole_false:true1_0 :: false:true 308.87/291.48 hole_cons:nil2_0 :: cons:nil 308.87/291.48 hole_0':s3_0 :: 0':s 308.87/291.48 gen_cons:nil4_0 :: Nat -> cons:nil 308.87/291.48 gen_0':s5_0 :: Nat -> 0':s 308.87/291.48 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (5) OrderProof (LOWER BOUND(ID)) 308.87/291.48 Heuristically decided to analyse the following defined symbols: 308.87/291.48 p, inc, sumList 308.87/291.48 308.87/291.48 They will be analysed ascendingly in the following order: 308.87/291.48 p < sumList 308.87/291.48 inc < sumList 308.87/291.48 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (6) 308.87/291.48 Obligation: 308.87/291.48 TRS: 308.87/291.48 Rules: 308.87/291.48 isEmpty(cons(x, xs)) -> false 308.87/291.48 isEmpty(nil) -> true 308.87/291.48 isZero(0') -> true 308.87/291.48 isZero(s(x)) -> false 308.87/291.48 head(cons(x, xs)) -> x 308.87/291.48 tail(cons(x, xs)) -> xs 308.87/291.48 tail(nil) -> nil 308.87/291.48 p(s(s(x))) -> s(p(s(x))) 308.87/291.48 p(s(0')) -> 0' 308.87/291.48 p(0') -> 0' 308.87/291.48 inc(s(x)) -> s(inc(x)) 308.87/291.48 inc(0') -> s(0') 308.87/291.48 sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y)) 308.87/291.48 if(true, b, y, xs, ys, x) -> y 308.87/291.48 if(false, true, y, xs, ys, x) -> sumList(xs, y) 308.87/291.48 if(false, false, y, xs, ys, x) -> sumList(ys, x) 308.87/291.48 sum(xs) -> sumList(xs, 0') 308.87/291.48 308.87/291.48 Types: 308.87/291.48 isEmpty :: cons:nil -> false:true 308.87/291.48 cons :: 0':s -> cons:nil -> cons:nil 308.87/291.48 false :: false:true 308.87/291.48 nil :: cons:nil 308.87/291.48 true :: false:true 308.87/291.48 isZero :: 0':s -> false:true 308.87/291.48 0' :: 0':s 308.87/291.48 s :: 0':s -> 0':s 308.87/291.48 head :: cons:nil -> 0':s 308.87/291.48 tail :: cons:nil -> cons:nil 308.87/291.48 p :: 0':s -> 0':s 308.87/291.48 inc :: 0':s -> 0':s 308.87/291.48 sumList :: cons:nil -> 0':s -> 0':s 308.87/291.48 if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s 308.87/291.48 sum :: cons:nil -> 0':s 308.87/291.48 hole_false:true1_0 :: false:true 308.87/291.48 hole_cons:nil2_0 :: cons:nil 308.87/291.48 hole_0':s3_0 :: 0':s 308.87/291.48 gen_cons:nil4_0 :: Nat -> cons:nil 308.87/291.48 gen_0':s5_0 :: Nat -> 0':s 308.87/291.48 308.87/291.48 308.87/291.48 Generator Equations: 308.87/291.48 gen_cons:nil4_0(0) <=> nil 308.87/291.48 gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x)) 308.87/291.48 gen_0':s5_0(0) <=> 0' 308.87/291.48 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 308.87/291.48 308.87/291.48 308.87/291.48 The following defined symbols remain to be analysed: 308.87/291.48 p, inc, sumList 308.87/291.48 308.87/291.48 They will be analysed ascendingly in the following order: 308.87/291.48 p < sumList 308.87/291.48 inc < sumList 308.87/291.48 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (7) RewriteLemmaProof (LOWER BOUND(ID)) 308.87/291.48 Proved the following rewrite lemma: 308.87/291.48 p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) 308.87/291.48 308.87/291.48 Induction Base: 308.87/291.48 p(gen_0':s5_0(+(1, 0))) ->_R^Omega(1) 308.87/291.48 0' 308.87/291.48 308.87/291.48 Induction Step: 308.87/291.48 p(gen_0':s5_0(+(1, +(n7_0, 1)))) ->_R^Omega(1) 308.87/291.48 s(p(s(gen_0':s5_0(n7_0)))) ->_IH 308.87/291.48 s(gen_0':s5_0(c8_0)) 308.87/291.48 308.87/291.48 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (8) 308.87/291.48 Complex Obligation (BEST) 308.87/291.48 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (9) 308.87/291.48 Obligation: 308.87/291.48 Proved the lower bound n^1 for the following obligation: 308.87/291.48 308.87/291.48 TRS: 308.87/291.48 Rules: 308.87/291.48 isEmpty(cons(x, xs)) -> false 308.87/291.48 isEmpty(nil) -> true 308.87/291.48 isZero(0') -> true 308.87/291.48 isZero(s(x)) -> false 308.87/291.48 head(cons(x, xs)) -> x 308.87/291.48 tail(cons(x, xs)) -> xs 308.87/291.48 tail(nil) -> nil 308.87/291.48 p(s(s(x))) -> s(p(s(x))) 308.87/291.48 p(s(0')) -> 0' 308.87/291.48 p(0') -> 0' 308.87/291.48 inc(s(x)) -> s(inc(x)) 308.87/291.48 inc(0') -> s(0') 308.87/291.48 sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y)) 308.87/291.48 if(true, b, y, xs, ys, x) -> y 308.87/291.48 if(false, true, y, xs, ys, x) -> sumList(xs, y) 308.87/291.48 if(false, false, y, xs, ys, x) -> sumList(ys, x) 308.87/291.48 sum(xs) -> sumList(xs, 0') 308.87/291.48 308.87/291.48 Types: 308.87/291.48 isEmpty :: cons:nil -> false:true 308.87/291.48 cons :: 0':s -> cons:nil -> cons:nil 308.87/291.48 false :: false:true 308.87/291.48 nil :: cons:nil 308.87/291.48 true :: false:true 308.87/291.48 isZero :: 0':s -> false:true 308.87/291.48 0' :: 0':s 308.87/291.48 s :: 0':s -> 0':s 308.87/291.48 head :: cons:nil -> 0':s 308.87/291.48 tail :: cons:nil -> cons:nil 308.87/291.48 p :: 0':s -> 0':s 308.87/291.48 inc :: 0':s -> 0':s 308.87/291.48 sumList :: cons:nil -> 0':s -> 0':s 308.87/291.48 if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s 308.87/291.48 sum :: cons:nil -> 0':s 308.87/291.48 hole_false:true1_0 :: false:true 308.87/291.48 hole_cons:nil2_0 :: cons:nil 308.87/291.48 hole_0':s3_0 :: 0':s 308.87/291.48 gen_cons:nil4_0 :: Nat -> cons:nil 308.87/291.48 gen_0':s5_0 :: Nat -> 0':s 308.87/291.48 308.87/291.48 308.87/291.48 Generator Equations: 308.87/291.48 gen_cons:nil4_0(0) <=> nil 308.87/291.48 gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x)) 308.87/291.48 gen_0':s5_0(0) <=> 0' 308.87/291.48 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 308.87/291.48 308.87/291.48 308.87/291.48 The following defined symbols remain to be analysed: 308.87/291.48 p, inc, sumList 308.87/291.48 308.87/291.48 They will be analysed ascendingly in the following order: 308.87/291.48 p < sumList 308.87/291.48 inc < sumList 308.87/291.48 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (10) LowerBoundPropagationProof (FINISHED) 308.87/291.48 Propagated lower bound. 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (11) 308.87/291.48 BOUNDS(n^1, INF) 308.87/291.48 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (12) 308.87/291.48 Obligation: 308.87/291.48 TRS: 308.87/291.48 Rules: 308.87/291.48 isEmpty(cons(x, xs)) -> false 308.87/291.48 isEmpty(nil) -> true 308.87/291.48 isZero(0') -> true 308.87/291.48 isZero(s(x)) -> false 308.87/291.48 head(cons(x, xs)) -> x 308.87/291.48 tail(cons(x, xs)) -> xs 308.87/291.48 tail(nil) -> nil 308.87/291.48 p(s(s(x))) -> s(p(s(x))) 308.87/291.48 p(s(0')) -> 0' 308.87/291.48 p(0') -> 0' 308.87/291.48 inc(s(x)) -> s(inc(x)) 308.87/291.48 inc(0') -> s(0') 308.87/291.48 sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y)) 308.87/291.48 if(true, b, y, xs, ys, x) -> y 308.87/291.48 if(false, true, y, xs, ys, x) -> sumList(xs, y) 308.87/291.48 if(false, false, y, xs, ys, x) -> sumList(ys, x) 308.87/291.48 sum(xs) -> sumList(xs, 0') 308.87/291.48 308.87/291.48 Types: 308.87/291.48 isEmpty :: cons:nil -> false:true 308.87/291.48 cons :: 0':s -> cons:nil -> cons:nil 308.87/291.48 false :: false:true 308.87/291.48 nil :: cons:nil 308.87/291.48 true :: false:true 308.87/291.48 isZero :: 0':s -> false:true 308.87/291.48 0' :: 0':s 308.87/291.48 s :: 0':s -> 0':s 308.87/291.48 head :: cons:nil -> 0':s 308.87/291.48 tail :: cons:nil -> cons:nil 308.87/291.48 p :: 0':s -> 0':s 308.87/291.48 inc :: 0':s -> 0':s 308.87/291.48 sumList :: cons:nil -> 0':s -> 0':s 308.87/291.48 if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s 308.87/291.48 sum :: cons:nil -> 0':s 308.87/291.48 hole_false:true1_0 :: false:true 308.87/291.48 hole_cons:nil2_0 :: cons:nil 308.87/291.48 hole_0':s3_0 :: 0':s 308.87/291.48 gen_cons:nil4_0 :: Nat -> cons:nil 308.87/291.48 gen_0':s5_0 :: Nat -> 0':s 308.87/291.48 308.87/291.48 308.87/291.48 Lemmas: 308.87/291.48 p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) 308.87/291.48 308.87/291.48 308.87/291.48 Generator Equations: 308.87/291.48 gen_cons:nil4_0(0) <=> nil 308.87/291.48 gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x)) 308.87/291.48 gen_0':s5_0(0) <=> 0' 308.87/291.48 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 308.87/291.48 308.87/291.48 308.87/291.48 The following defined symbols remain to be analysed: 308.87/291.48 inc, sumList 308.87/291.48 308.87/291.48 They will be analysed ascendingly in the following order: 308.87/291.48 inc < sumList 308.87/291.48 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (13) RewriteLemmaProof (LOWER BOUND(ID)) 308.87/291.48 Proved the following rewrite lemma: 308.87/291.48 inc(gen_0':s5_0(n264_0)) -> gen_0':s5_0(+(1, n264_0)), rt in Omega(1 + n264_0) 308.87/291.48 308.87/291.48 Induction Base: 308.87/291.48 inc(gen_0':s5_0(0)) ->_R^Omega(1) 308.87/291.48 s(0') 308.87/291.48 308.87/291.48 Induction Step: 308.87/291.48 inc(gen_0':s5_0(+(n264_0, 1))) ->_R^Omega(1) 308.87/291.48 s(inc(gen_0':s5_0(n264_0))) ->_IH 308.87/291.48 s(gen_0':s5_0(+(1, c265_0))) 308.87/291.48 308.87/291.48 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (14) 308.87/291.48 Obligation: 308.87/291.48 TRS: 308.87/291.48 Rules: 308.87/291.48 isEmpty(cons(x, xs)) -> false 308.87/291.48 isEmpty(nil) -> true 308.87/291.48 isZero(0') -> true 308.87/291.48 isZero(s(x)) -> false 308.87/291.48 head(cons(x, xs)) -> x 308.87/291.48 tail(cons(x, xs)) -> xs 308.87/291.48 tail(nil) -> nil 308.87/291.48 p(s(s(x))) -> s(p(s(x))) 308.87/291.48 p(s(0')) -> 0' 308.87/291.48 p(0') -> 0' 308.87/291.48 inc(s(x)) -> s(inc(x)) 308.87/291.48 inc(0') -> s(0') 308.87/291.48 sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y)) 308.87/291.48 if(true, b, y, xs, ys, x) -> y 308.87/291.48 if(false, true, y, xs, ys, x) -> sumList(xs, y) 308.87/291.48 if(false, false, y, xs, ys, x) -> sumList(ys, x) 308.87/291.48 sum(xs) -> sumList(xs, 0') 308.87/291.48 308.87/291.48 Types: 308.87/291.48 isEmpty :: cons:nil -> false:true 308.87/291.48 cons :: 0':s -> cons:nil -> cons:nil 308.87/291.48 false :: false:true 308.87/291.48 nil :: cons:nil 308.87/291.48 true :: false:true 308.87/291.48 isZero :: 0':s -> false:true 308.87/291.48 0' :: 0':s 308.87/291.48 s :: 0':s -> 0':s 308.87/291.48 head :: cons:nil -> 0':s 308.87/291.48 tail :: cons:nil -> cons:nil 308.87/291.48 p :: 0':s -> 0':s 308.87/291.48 inc :: 0':s -> 0':s 308.87/291.48 sumList :: cons:nil -> 0':s -> 0':s 308.87/291.48 if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s 308.87/291.48 sum :: cons:nil -> 0':s 308.87/291.48 hole_false:true1_0 :: false:true 308.87/291.48 hole_cons:nil2_0 :: cons:nil 308.87/291.48 hole_0':s3_0 :: 0':s 308.87/291.48 gen_cons:nil4_0 :: Nat -> cons:nil 308.87/291.48 gen_0':s5_0 :: Nat -> 0':s 308.87/291.48 308.87/291.48 308.87/291.48 Lemmas: 308.87/291.48 p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) 308.87/291.48 inc(gen_0':s5_0(n264_0)) -> gen_0':s5_0(+(1, n264_0)), rt in Omega(1 + n264_0) 308.87/291.48 308.87/291.48 308.87/291.48 Generator Equations: 308.87/291.48 gen_cons:nil4_0(0) <=> nil 308.87/291.48 gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x)) 308.87/291.48 gen_0':s5_0(0) <=> 0' 308.87/291.48 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 308.87/291.48 308.87/291.48 308.87/291.48 The following defined symbols remain to be analysed: 308.87/291.48 sumList 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (15) RewriteLemmaProof (LOWER BOUND(ID)) 308.87/291.48 Proved the following rewrite lemma: 308.87/291.48 sumList(gen_cons:nil4_0(n531_0), gen_0':s5_0(b)) -> gen_0':s5_0(b), rt in Omega(1 + b + b*n531_0 + n531_0) 308.87/291.48 308.87/291.48 Induction Base: 308.87/291.48 sumList(gen_cons:nil4_0(0), gen_0':s5_0(b)) ->_R^Omega(1) 308.87/291.48 if(isEmpty(gen_cons:nil4_0(0)), isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), tail(gen_cons:nil4_0(0)), cons(p(head(gen_cons:nil4_0(0))), tail(gen_cons:nil4_0(0))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(true, isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), tail(gen_cons:nil4_0(0)), cons(p(head(gen_cons:nil4_0(0))), tail(gen_cons:nil4_0(0))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(true, isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), nil, cons(p(head(gen_cons:nil4_0(0))), tail(gen_cons:nil4_0(0))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(true, isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), nil, cons(p(head(gen_cons:nil4_0(0))), nil), inc(gen_0':s5_0(b))) ->_L^Omega(1 + b) 308.87/291.48 if(true, isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), nil, cons(p(head(gen_cons:nil4_0(0))), nil), gen_0':s5_0(+(1, b))) ->_R^Omega(1) 308.87/291.48 gen_0':s5_0(b) 308.87/291.48 308.87/291.48 Induction Step: 308.87/291.48 sumList(gen_cons:nil4_0(+(n531_0, 1)), gen_0':s5_0(b)) ->_R^Omega(1) 308.87/291.48 if(isEmpty(gen_cons:nil4_0(+(n531_0, 1))), isZero(head(gen_cons:nil4_0(+(n531_0, 1)))), gen_0':s5_0(b), tail(gen_cons:nil4_0(+(n531_0, 1))), cons(p(head(gen_cons:nil4_0(+(n531_0, 1)))), tail(gen_cons:nil4_0(+(n531_0, 1)))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(false, isZero(head(gen_cons:nil4_0(+(1, n531_0)))), gen_0':s5_0(b), tail(gen_cons:nil4_0(+(1, n531_0))), cons(p(head(gen_cons:nil4_0(+(1, n531_0)))), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(false, isZero(0'), gen_0':s5_0(b), tail(gen_cons:nil4_0(+(1, n531_0))), cons(p(head(gen_cons:nil4_0(+(1, n531_0)))), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(false, true, gen_0':s5_0(b), tail(gen_cons:nil4_0(+(1, n531_0))), cons(p(head(gen_cons:nil4_0(+(1, n531_0)))), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(p(head(gen_cons:nil4_0(+(1, n531_0)))), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(p(0'), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(0', tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1) 308.87/291.48 if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(0', gen_cons:nil4_0(n531_0)), inc(gen_0':s5_0(b))) ->_L^Omega(1 + b) 308.87/291.48 if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(0', gen_cons:nil4_0(n531_0)), gen_0':s5_0(+(1, b))) ->_R^Omega(1) 308.87/291.48 sumList(gen_cons:nil4_0(n531_0), gen_0':s5_0(b)) ->_IH 308.87/291.48 gen_0':s5_0(b) 308.87/291.48 308.87/291.48 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (16) 308.87/291.48 Obligation: 308.87/291.48 Proved the lower bound n^2 for the following obligation: 308.87/291.48 308.87/291.48 TRS: 308.87/291.48 Rules: 308.87/291.48 isEmpty(cons(x, xs)) -> false 308.87/291.48 isEmpty(nil) -> true 308.87/291.48 isZero(0') -> true 308.87/291.48 isZero(s(x)) -> false 308.87/291.48 head(cons(x, xs)) -> x 308.87/291.48 tail(cons(x, xs)) -> xs 308.87/291.48 tail(nil) -> nil 308.87/291.48 p(s(s(x))) -> s(p(s(x))) 308.87/291.48 p(s(0')) -> 0' 308.87/291.48 p(0') -> 0' 308.87/291.48 inc(s(x)) -> s(inc(x)) 308.87/291.48 inc(0') -> s(0') 308.87/291.48 sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y)) 308.87/291.48 if(true, b, y, xs, ys, x) -> y 308.87/291.48 if(false, true, y, xs, ys, x) -> sumList(xs, y) 308.87/291.48 if(false, false, y, xs, ys, x) -> sumList(ys, x) 308.87/291.48 sum(xs) -> sumList(xs, 0') 308.87/291.48 308.87/291.48 Types: 308.87/291.48 isEmpty :: cons:nil -> false:true 308.87/291.48 cons :: 0':s -> cons:nil -> cons:nil 308.87/291.48 false :: false:true 308.87/291.48 nil :: cons:nil 308.87/291.48 true :: false:true 308.87/291.48 isZero :: 0':s -> false:true 308.87/291.48 0' :: 0':s 308.87/291.48 s :: 0':s -> 0':s 308.87/291.48 head :: cons:nil -> 0':s 308.87/291.48 tail :: cons:nil -> cons:nil 308.87/291.48 p :: 0':s -> 0':s 308.87/291.48 inc :: 0':s -> 0':s 308.87/291.48 sumList :: cons:nil -> 0':s -> 0':s 308.87/291.48 if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s 308.87/291.48 sum :: cons:nil -> 0':s 308.87/291.48 hole_false:true1_0 :: false:true 308.87/291.48 hole_cons:nil2_0 :: cons:nil 308.87/291.48 hole_0':s3_0 :: 0':s 308.87/291.48 gen_cons:nil4_0 :: Nat -> cons:nil 308.87/291.48 gen_0':s5_0 :: Nat -> 0':s 308.87/291.48 308.87/291.48 308.87/291.48 Lemmas: 308.87/291.48 p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) 308.87/291.48 inc(gen_0':s5_0(n264_0)) -> gen_0':s5_0(+(1, n264_0)), rt in Omega(1 + n264_0) 308.87/291.48 308.87/291.48 308.87/291.48 Generator Equations: 308.87/291.48 gen_cons:nil4_0(0) <=> nil 308.87/291.48 gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x)) 308.87/291.48 gen_0':s5_0(0) <=> 0' 308.87/291.48 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 308.87/291.48 308.87/291.48 308.87/291.48 The following defined symbols remain to be analysed: 308.87/291.48 sumList 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (17) LowerBoundPropagationProof (FINISHED) 308.87/291.48 Propagated lower bound. 308.87/291.48 ---------------------------------------- 308.87/291.48 308.87/291.48 (18) 308.87/291.48 BOUNDS(n^2, INF) 308.87/291.52 EOF