309.10/291.49 WORST_CASE(Omega(n^1), ?) 309.10/291.50 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 309.10/291.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 309.10/291.50 309.10/291.50 309.10/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.10/291.50 309.10/291.50 (0) CpxTRS 309.10/291.50 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 309.10/291.50 (2) CpxTRS 309.10/291.50 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 309.10/291.50 (4) typed CpxTrs 309.10/291.50 (5) OrderProof [LOWER BOUND(ID), 0 ms] 309.10/291.50 (6) typed CpxTrs 309.10/291.50 (7) RewriteLemmaProof [LOWER BOUND(ID), 348 ms] 309.10/291.50 (8) BEST 309.10/291.50 (9) proven lower bound 309.10/291.50 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 309.10/291.50 (11) BOUNDS(n^1, INF) 309.10/291.50 (12) typed CpxTrs 309.10/291.50 (13) RewriteLemmaProof [LOWER BOUND(ID), 67 ms] 309.10/291.50 (14) typed CpxTrs 309.10/291.50 (15) RewriteLemmaProof [LOWER BOUND(ID), 15 ms] 309.10/291.50 (16) typed CpxTrs 309.10/291.50 309.10/291.50 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (0) 309.10/291.50 Obligation: 309.10/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.10/291.50 309.10/291.50 309.10/291.50 The TRS R consists of the following rules: 309.10/291.50 309.10/291.50 prod(xs) -> prodIter(xs, s(0)) 309.10/291.50 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 309.10/291.50 ifProd(true, xs, x) -> x 309.10/291.50 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 309.10/291.50 plus(0, y) -> y 309.10/291.50 plus(s(x), y) -> s(plus(x, y)) 309.10/291.50 times(x, y) -> timesIter(x, y, 0, 0) 309.10/291.50 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 309.10/291.50 ifTimes(true, x, y, z, u) -> z 309.10/291.50 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 309.10/291.50 isempty(nil) -> true 309.10/291.50 isempty(cons(x, xs)) -> false 309.10/291.50 head(nil) -> error 309.10/291.50 head(cons(x, xs)) -> x 309.10/291.50 tail(nil) -> nil 309.10/291.50 tail(cons(x, xs)) -> xs 309.10/291.50 ge(x, 0) -> true 309.10/291.50 ge(0, s(y)) -> false 309.10/291.50 ge(s(x), s(y)) -> ge(x, y) 309.10/291.50 a -> b 309.10/291.50 a -> c 309.10/291.50 309.10/291.50 S is empty. 309.10/291.50 Rewrite Strategy: FULL 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 309.10/291.50 Renamed function symbols to avoid clashes with predefined symbol. 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (2) 309.10/291.50 Obligation: 309.10/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 309.10/291.50 309.10/291.50 309.10/291.50 The TRS R consists of the following rules: 309.10/291.50 309.10/291.50 prod(xs) -> prodIter(xs, s(0')) 309.10/291.50 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 309.10/291.50 ifProd(true, xs, x) -> x 309.10/291.50 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 309.10/291.50 plus(0', y) -> y 309.10/291.50 plus(s(x), y) -> s(plus(x, y)) 309.10/291.50 times(x, y) -> timesIter(x, y, 0', 0') 309.10/291.50 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 309.10/291.50 ifTimes(true, x, y, z, u) -> z 309.10/291.50 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 309.10/291.50 isempty(nil) -> true 309.10/291.50 isempty(cons(x, xs)) -> false 309.10/291.50 head(nil) -> error 309.10/291.50 head(cons(x, xs)) -> x 309.10/291.50 tail(nil) -> nil 309.10/291.50 tail(cons(x, xs)) -> xs 309.10/291.50 ge(x, 0') -> true 309.10/291.50 ge(0', s(y)) -> false 309.10/291.50 ge(s(x), s(y)) -> ge(x, y) 309.10/291.50 a -> b 309.10/291.50 a -> c 309.10/291.50 309.10/291.50 S is empty. 309.10/291.50 Rewrite Strategy: FULL 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 309.10/291.50 Infered types. 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (4) 309.10/291.50 Obligation: 309.10/291.50 TRS: 309.10/291.50 Rules: 309.10/291.50 prod(xs) -> prodIter(xs, s(0')) 309.10/291.50 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 309.10/291.50 ifProd(true, xs, x) -> x 309.10/291.50 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 309.10/291.50 plus(0', y) -> y 309.10/291.50 plus(s(x), y) -> s(plus(x, y)) 309.10/291.50 times(x, y) -> timesIter(x, y, 0', 0') 309.10/291.50 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 309.10/291.50 ifTimes(true, x, y, z, u) -> z 309.10/291.50 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 309.10/291.50 isempty(nil) -> true 309.10/291.50 isempty(cons(x, xs)) -> false 309.10/291.50 head(nil) -> error 309.10/291.50 head(cons(x, xs)) -> x 309.10/291.50 tail(nil) -> nil 309.10/291.50 tail(cons(x, xs)) -> xs 309.10/291.50 ge(x, 0') -> true 309.10/291.50 ge(0', s(y)) -> false 309.10/291.50 ge(s(x), s(y)) -> ge(x, y) 309.10/291.50 a -> b 309.10/291.50 a -> c 309.10/291.50 309.10/291.50 Types: 309.10/291.50 prod :: nil:cons -> 0':s:error 309.10/291.50 prodIter :: nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 s :: 0':s:error -> 0':s:error 309.10/291.50 0' :: 0':s:error 309.10/291.50 ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 isempty :: nil:cons -> true:false 309.10/291.50 true :: true:false 309.10/291.50 false :: true:false 309.10/291.50 tail :: nil:cons -> nil:cons 309.10/291.50 times :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 head :: nil:cons -> 0':s:error 309.10/291.50 plus :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ge :: 0':s:error -> 0':s:error -> true:false 309.10/291.50 nil :: nil:cons 309.10/291.50 cons :: 0':s:error -> nil:cons -> nil:cons 309.10/291.50 error :: 0':s:error 309.10/291.50 a :: b:c 309.10/291.50 b :: b:c 309.10/291.50 c :: b:c 309.10/291.50 hole_0':s:error1_0 :: 0':s:error 309.10/291.50 hole_nil:cons2_0 :: nil:cons 309.10/291.50 hole_true:false3_0 :: true:false 309.10/291.50 hole_b:c4_0 :: b:c 309.10/291.50 gen_0':s:error5_0 :: Nat -> 0':s:error 309.10/291.50 gen_nil:cons6_0 :: Nat -> nil:cons 309.10/291.50 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (5) OrderProof (LOWER BOUND(ID)) 309.10/291.50 Heuristically decided to analyse the following defined symbols: 309.10/291.50 prodIter, plus, timesIter, ge 309.10/291.50 309.10/291.50 They will be analysed ascendingly in the following order: 309.10/291.50 plus < timesIter 309.10/291.50 ge < timesIter 309.10/291.50 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (6) 309.10/291.50 Obligation: 309.10/291.50 TRS: 309.10/291.50 Rules: 309.10/291.50 prod(xs) -> prodIter(xs, s(0')) 309.10/291.50 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 309.10/291.50 ifProd(true, xs, x) -> x 309.10/291.50 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 309.10/291.50 plus(0', y) -> y 309.10/291.50 plus(s(x), y) -> s(plus(x, y)) 309.10/291.50 times(x, y) -> timesIter(x, y, 0', 0') 309.10/291.50 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 309.10/291.50 ifTimes(true, x, y, z, u) -> z 309.10/291.50 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 309.10/291.50 isempty(nil) -> true 309.10/291.50 isempty(cons(x, xs)) -> false 309.10/291.50 head(nil) -> error 309.10/291.50 head(cons(x, xs)) -> x 309.10/291.50 tail(nil) -> nil 309.10/291.50 tail(cons(x, xs)) -> xs 309.10/291.50 ge(x, 0') -> true 309.10/291.50 ge(0', s(y)) -> false 309.10/291.50 ge(s(x), s(y)) -> ge(x, y) 309.10/291.50 a -> b 309.10/291.50 a -> c 309.10/291.50 309.10/291.50 Types: 309.10/291.50 prod :: nil:cons -> 0':s:error 309.10/291.50 prodIter :: nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 s :: 0':s:error -> 0':s:error 309.10/291.50 0' :: 0':s:error 309.10/291.50 ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 isempty :: nil:cons -> true:false 309.10/291.50 true :: true:false 309.10/291.50 false :: true:false 309.10/291.50 tail :: nil:cons -> nil:cons 309.10/291.50 times :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 head :: nil:cons -> 0':s:error 309.10/291.50 plus :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ge :: 0':s:error -> 0':s:error -> true:false 309.10/291.50 nil :: nil:cons 309.10/291.50 cons :: 0':s:error -> nil:cons -> nil:cons 309.10/291.50 error :: 0':s:error 309.10/291.50 a :: b:c 309.10/291.50 b :: b:c 309.10/291.50 c :: b:c 309.10/291.50 hole_0':s:error1_0 :: 0':s:error 309.10/291.50 hole_nil:cons2_0 :: nil:cons 309.10/291.50 hole_true:false3_0 :: true:false 309.10/291.50 hole_b:c4_0 :: b:c 309.10/291.50 gen_0':s:error5_0 :: Nat -> 0':s:error 309.10/291.50 gen_nil:cons6_0 :: Nat -> nil:cons 309.10/291.50 309.10/291.50 309.10/291.50 Generator Equations: 309.10/291.50 gen_0':s:error5_0(0) <=> 0' 309.10/291.50 gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x)) 309.10/291.50 gen_nil:cons6_0(0) <=> nil 309.10/291.50 gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x)) 309.10/291.50 309.10/291.50 309.10/291.50 The following defined symbols remain to be analysed: 309.10/291.50 prodIter, plus, timesIter, ge 309.10/291.50 309.10/291.50 They will be analysed ascendingly in the following order: 309.10/291.50 plus < timesIter 309.10/291.50 ge < timesIter 309.10/291.50 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (7) RewriteLemmaProof (LOWER BOUND(ID)) 309.10/291.50 Proved the following rewrite lemma: 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) -> gen_0':s:error5_0(0), rt in Omega(1 + n8_0) 309.10/291.50 309.10/291.50 Induction Base: 309.10/291.50 prodIter(gen_nil:cons6_0(0), gen_0':s:error5_0(0)) ->_R^Omega(1) 309.10/291.50 ifProd(isempty(gen_nil:cons6_0(0)), gen_nil:cons6_0(0), gen_0':s:error5_0(0)) ->_R^Omega(1) 309.10/291.50 ifProd(true, gen_nil:cons6_0(0), gen_0':s:error5_0(0)) ->_R^Omega(1) 309.10/291.50 gen_0':s:error5_0(0) 309.10/291.50 309.10/291.50 Induction Step: 309.10/291.50 prodIter(gen_nil:cons6_0(+(n8_0, 1)), gen_0':s:error5_0(0)) ->_R^Omega(1) 309.10/291.50 ifProd(isempty(gen_nil:cons6_0(+(n8_0, 1))), gen_nil:cons6_0(+(n8_0, 1)), gen_0':s:error5_0(0)) ->_R^Omega(1) 309.10/291.50 ifProd(false, gen_nil:cons6_0(+(1, n8_0)), gen_0':s:error5_0(0)) ->_R^Omega(1) 309.10/291.50 prodIter(tail(gen_nil:cons6_0(+(1, n8_0))), times(gen_0':s:error5_0(0), head(gen_nil:cons6_0(+(1, n8_0))))) ->_R^Omega(1) 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), times(gen_0':s:error5_0(0), head(gen_nil:cons6_0(+(1, n8_0))))) ->_R^Omega(1) 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), times(gen_0':s:error5_0(0), 0')) ->_R^Omega(1) 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), timesIter(gen_0':s:error5_0(0), 0', 0', 0')) ->_R^Omega(1) 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), ifTimes(ge(0', gen_0':s:error5_0(0)), gen_0':s:error5_0(0), 0', 0', 0')) ->_R^Omega(1) 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), ifTimes(true, gen_0':s:error5_0(0), 0', 0', 0')) ->_R^Omega(1) 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), 0') ->_IH 309.10/291.50 gen_0':s:error5_0(0) 309.10/291.50 309.10/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (8) 309.10/291.50 Complex Obligation (BEST) 309.10/291.50 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (9) 309.10/291.50 Obligation: 309.10/291.50 Proved the lower bound n^1 for the following obligation: 309.10/291.50 309.10/291.50 TRS: 309.10/291.50 Rules: 309.10/291.50 prod(xs) -> prodIter(xs, s(0')) 309.10/291.50 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 309.10/291.50 ifProd(true, xs, x) -> x 309.10/291.50 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 309.10/291.50 plus(0', y) -> y 309.10/291.50 plus(s(x), y) -> s(plus(x, y)) 309.10/291.50 times(x, y) -> timesIter(x, y, 0', 0') 309.10/291.50 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 309.10/291.50 ifTimes(true, x, y, z, u) -> z 309.10/291.50 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 309.10/291.50 isempty(nil) -> true 309.10/291.50 isempty(cons(x, xs)) -> false 309.10/291.50 head(nil) -> error 309.10/291.50 head(cons(x, xs)) -> x 309.10/291.50 tail(nil) -> nil 309.10/291.50 tail(cons(x, xs)) -> xs 309.10/291.50 ge(x, 0') -> true 309.10/291.50 ge(0', s(y)) -> false 309.10/291.50 ge(s(x), s(y)) -> ge(x, y) 309.10/291.50 a -> b 309.10/291.50 a -> c 309.10/291.50 309.10/291.50 Types: 309.10/291.50 prod :: nil:cons -> 0':s:error 309.10/291.50 prodIter :: nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 s :: 0':s:error -> 0':s:error 309.10/291.50 0' :: 0':s:error 309.10/291.50 ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 isempty :: nil:cons -> true:false 309.10/291.50 true :: true:false 309.10/291.50 false :: true:false 309.10/291.50 tail :: nil:cons -> nil:cons 309.10/291.50 times :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 head :: nil:cons -> 0':s:error 309.10/291.50 plus :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ge :: 0':s:error -> 0':s:error -> true:false 309.10/291.50 nil :: nil:cons 309.10/291.50 cons :: 0':s:error -> nil:cons -> nil:cons 309.10/291.50 error :: 0':s:error 309.10/291.50 a :: b:c 309.10/291.50 b :: b:c 309.10/291.50 c :: b:c 309.10/291.50 hole_0':s:error1_0 :: 0':s:error 309.10/291.50 hole_nil:cons2_0 :: nil:cons 309.10/291.50 hole_true:false3_0 :: true:false 309.10/291.50 hole_b:c4_0 :: b:c 309.10/291.50 gen_0':s:error5_0 :: Nat -> 0':s:error 309.10/291.50 gen_nil:cons6_0 :: Nat -> nil:cons 309.10/291.50 309.10/291.50 309.10/291.50 Generator Equations: 309.10/291.50 gen_0':s:error5_0(0) <=> 0' 309.10/291.50 gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x)) 309.10/291.50 gen_nil:cons6_0(0) <=> nil 309.10/291.50 gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x)) 309.10/291.50 309.10/291.50 309.10/291.50 The following defined symbols remain to be analysed: 309.10/291.50 prodIter, plus, timesIter, ge 309.10/291.50 309.10/291.50 They will be analysed ascendingly in the following order: 309.10/291.50 plus < timesIter 309.10/291.50 ge < timesIter 309.10/291.50 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (10) LowerBoundPropagationProof (FINISHED) 309.10/291.50 Propagated lower bound. 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (11) 309.10/291.50 BOUNDS(n^1, INF) 309.10/291.50 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (12) 309.10/291.50 Obligation: 309.10/291.50 TRS: 309.10/291.50 Rules: 309.10/291.50 prod(xs) -> prodIter(xs, s(0')) 309.10/291.50 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 309.10/291.50 ifProd(true, xs, x) -> x 309.10/291.50 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 309.10/291.50 plus(0', y) -> y 309.10/291.50 plus(s(x), y) -> s(plus(x, y)) 309.10/291.50 times(x, y) -> timesIter(x, y, 0', 0') 309.10/291.50 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 309.10/291.50 ifTimes(true, x, y, z, u) -> z 309.10/291.50 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 309.10/291.50 isempty(nil) -> true 309.10/291.50 isempty(cons(x, xs)) -> false 309.10/291.50 head(nil) -> error 309.10/291.50 head(cons(x, xs)) -> x 309.10/291.50 tail(nil) -> nil 309.10/291.50 tail(cons(x, xs)) -> xs 309.10/291.50 ge(x, 0') -> true 309.10/291.50 ge(0', s(y)) -> false 309.10/291.50 ge(s(x), s(y)) -> ge(x, y) 309.10/291.50 a -> b 309.10/291.50 a -> c 309.10/291.50 309.10/291.50 Types: 309.10/291.50 prod :: nil:cons -> 0':s:error 309.10/291.50 prodIter :: nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 s :: 0':s:error -> 0':s:error 309.10/291.50 0' :: 0':s:error 309.10/291.50 ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 isempty :: nil:cons -> true:false 309.10/291.50 true :: true:false 309.10/291.50 false :: true:false 309.10/291.50 tail :: nil:cons -> nil:cons 309.10/291.50 times :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 head :: nil:cons -> 0':s:error 309.10/291.50 plus :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ge :: 0':s:error -> 0':s:error -> true:false 309.10/291.50 nil :: nil:cons 309.10/291.50 cons :: 0':s:error -> nil:cons -> nil:cons 309.10/291.50 error :: 0':s:error 309.10/291.50 a :: b:c 309.10/291.50 b :: b:c 309.10/291.50 c :: b:c 309.10/291.50 hole_0':s:error1_0 :: 0':s:error 309.10/291.50 hole_nil:cons2_0 :: nil:cons 309.10/291.50 hole_true:false3_0 :: true:false 309.10/291.50 hole_b:c4_0 :: b:c 309.10/291.50 gen_0':s:error5_0 :: Nat -> 0':s:error 309.10/291.50 gen_nil:cons6_0 :: Nat -> nil:cons 309.10/291.50 309.10/291.50 309.10/291.50 Lemmas: 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) -> gen_0':s:error5_0(0), rt in Omega(1 + n8_0) 309.10/291.50 309.10/291.50 309.10/291.50 Generator Equations: 309.10/291.50 gen_0':s:error5_0(0) <=> 0' 309.10/291.50 gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x)) 309.10/291.50 gen_nil:cons6_0(0) <=> nil 309.10/291.50 gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x)) 309.10/291.50 309.10/291.50 309.10/291.50 The following defined symbols remain to be analysed: 309.10/291.50 plus, timesIter, ge 309.10/291.50 309.10/291.50 They will be analysed ascendingly in the following order: 309.10/291.50 plus < timesIter 309.10/291.50 ge < timesIter 309.10/291.50 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (13) RewriteLemmaProof (LOWER BOUND(ID)) 309.10/291.50 Proved the following rewrite lemma: 309.10/291.50 plus(gen_0':s:error5_0(n1637_0), gen_0':s:error5_0(b)) -> gen_0':s:error5_0(+(n1637_0, b)), rt in Omega(1 + n1637_0) 309.10/291.50 309.10/291.50 Induction Base: 309.10/291.50 plus(gen_0':s:error5_0(0), gen_0':s:error5_0(b)) ->_R^Omega(1) 309.10/291.50 gen_0':s:error5_0(b) 309.10/291.50 309.10/291.50 Induction Step: 309.10/291.50 plus(gen_0':s:error5_0(+(n1637_0, 1)), gen_0':s:error5_0(b)) ->_R^Omega(1) 309.10/291.50 s(plus(gen_0':s:error5_0(n1637_0), gen_0':s:error5_0(b))) ->_IH 309.10/291.50 s(gen_0':s:error5_0(+(b, c1638_0))) 309.10/291.50 309.10/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (14) 309.10/291.50 Obligation: 309.10/291.50 TRS: 309.10/291.50 Rules: 309.10/291.50 prod(xs) -> prodIter(xs, s(0')) 309.10/291.50 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 309.10/291.50 ifProd(true, xs, x) -> x 309.10/291.50 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 309.10/291.50 plus(0', y) -> y 309.10/291.50 plus(s(x), y) -> s(plus(x, y)) 309.10/291.50 times(x, y) -> timesIter(x, y, 0', 0') 309.10/291.50 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 309.10/291.50 ifTimes(true, x, y, z, u) -> z 309.10/291.50 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 309.10/291.50 isempty(nil) -> true 309.10/291.50 isempty(cons(x, xs)) -> false 309.10/291.50 head(nil) -> error 309.10/291.50 head(cons(x, xs)) -> x 309.10/291.50 tail(nil) -> nil 309.10/291.50 tail(cons(x, xs)) -> xs 309.10/291.50 ge(x, 0') -> true 309.10/291.50 ge(0', s(y)) -> false 309.10/291.50 ge(s(x), s(y)) -> ge(x, y) 309.10/291.50 a -> b 309.10/291.50 a -> c 309.10/291.50 309.10/291.50 Types: 309.10/291.50 prod :: nil:cons -> 0':s:error 309.10/291.50 prodIter :: nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 s :: 0':s:error -> 0':s:error 309.10/291.50 0' :: 0':s:error 309.10/291.50 ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 isempty :: nil:cons -> true:false 309.10/291.50 true :: true:false 309.10/291.50 false :: true:false 309.10/291.50 tail :: nil:cons -> nil:cons 309.10/291.50 times :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 head :: nil:cons -> 0':s:error 309.10/291.50 plus :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ge :: 0':s:error -> 0':s:error -> true:false 309.10/291.50 nil :: nil:cons 309.10/291.50 cons :: 0':s:error -> nil:cons -> nil:cons 309.10/291.50 error :: 0':s:error 309.10/291.50 a :: b:c 309.10/291.50 b :: b:c 309.10/291.50 c :: b:c 309.10/291.50 hole_0':s:error1_0 :: 0':s:error 309.10/291.50 hole_nil:cons2_0 :: nil:cons 309.10/291.50 hole_true:false3_0 :: true:false 309.10/291.50 hole_b:c4_0 :: b:c 309.10/291.50 gen_0':s:error5_0 :: Nat -> 0':s:error 309.10/291.50 gen_nil:cons6_0 :: Nat -> nil:cons 309.10/291.50 309.10/291.50 309.10/291.50 Lemmas: 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) -> gen_0':s:error5_0(0), rt in Omega(1 + n8_0) 309.10/291.50 plus(gen_0':s:error5_0(n1637_0), gen_0':s:error5_0(b)) -> gen_0':s:error5_0(+(n1637_0, b)), rt in Omega(1 + n1637_0) 309.10/291.50 309.10/291.50 309.10/291.50 Generator Equations: 309.10/291.50 gen_0':s:error5_0(0) <=> 0' 309.10/291.50 gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x)) 309.10/291.50 gen_nil:cons6_0(0) <=> nil 309.10/291.50 gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x)) 309.10/291.50 309.10/291.50 309.10/291.50 The following defined symbols remain to be analysed: 309.10/291.50 ge, timesIter 309.10/291.50 309.10/291.50 They will be analysed ascendingly in the following order: 309.10/291.50 ge < timesIter 309.10/291.50 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (15) RewriteLemmaProof (LOWER BOUND(ID)) 309.10/291.50 Proved the following rewrite lemma: 309.10/291.50 ge(gen_0':s:error5_0(n2462_0), gen_0':s:error5_0(n2462_0)) -> true, rt in Omega(1 + n2462_0) 309.10/291.50 309.10/291.50 Induction Base: 309.10/291.50 ge(gen_0':s:error5_0(0), gen_0':s:error5_0(0)) ->_R^Omega(1) 309.10/291.50 true 309.10/291.50 309.10/291.50 Induction Step: 309.10/291.50 ge(gen_0':s:error5_0(+(n2462_0, 1)), gen_0':s:error5_0(+(n2462_0, 1))) ->_R^Omega(1) 309.10/291.50 ge(gen_0':s:error5_0(n2462_0), gen_0':s:error5_0(n2462_0)) ->_IH 309.10/291.50 true 309.10/291.50 309.10/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 309.10/291.50 ---------------------------------------- 309.10/291.50 309.10/291.50 (16) 309.10/291.50 Obligation: 309.10/291.50 TRS: 309.10/291.50 Rules: 309.10/291.50 prod(xs) -> prodIter(xs, s(0')) 309.10/291.50 prodIter(xs, x) -> ifProd(isempty(xs), xs, x) 309.10/291.50 ifProd(true, xs, x) -> x 309.10/291.50 ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs))) 309.10/291.50 plus(0', y) -> y 309.10/291.50 plus(s(x), y) -> s(plus(x, y)) 309.10/291.50 times(x, y) -> timesIter(x, y, 0', 0') 309.10/291.50 timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u) 309.10/291.50 ifTimes(true, x, y, z, u) -> z 309.10/291.50 ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u)) 309.10/291.50 isempty(nil) -> true 309.10/291.50 isempty(cons(x, xs)) -> false 309.10/291.50 head(nil) -> error 309.10/291.50 head(cons(x, xs)) -> x 309.10/291.50 tail(nil) -> nil 309.10/291.50 tail(cons(x, xs)) -> xs 309.10/291.50 ge(x, 0') -> true 309.10/291.50 ge(0', s(y)) -> false 309.10/291.50 ge(s(x), s(y)) -> ge(x, y) 309.10/291.50 a -> b 309.10/291.50 a -> c 309.10/291.50 309.10/291.50 Types: 309.10/291.50 prod :: nil:cons -> 0':s:error 309.10/291.50 prodIter :: nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 s :: 0':s:error -> 0':s:error 309.10/291.50 0' :: 0':s:error 309.10/291.50 ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error 309.10/291.50 isempty :: nil:cons -> true:false 309.10/291.50 true :: true:false 309.10/291.50 false :: true:false 309.10/291.50 tail :: nil:cons -> nil:cons 309.10/291.50 times :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 head :: nil:cons -> 0':s:error 309.10/291.50 plus :: 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error 309.10/291.50 ge :: 0':s:error -> 0':s:error -> true:false 309.10/291.50 nil :: nil:cons 309.10/291.50 cons :: 0':s:error -> nil:cons -> nil:cons 309.10/291.50 error :: 0':s:error 309.10/291.50 a :: b:c 309.10/291.50 b :: b:c 309.10/291.50 c :: b:c 309.10/291.50 hole_0':s:error1_0 :: 0':s:error 309.10/291.50 hole_nil:cons2_0 :: nil:cons 309.10/291.50 hole_true:false3_0 :: true:false 309.10/291.50 hole_b:c4_0 :: b:c 309.10/291.50 gen_0':s:error5_0 :: Nat -> 0':s:error 309.10/291.50 gen_nil:cons6_0 :: Nat -> nil:cons 309.10/291.50 309.10/291.50 309.10/291.50 Lemmas: 309.10/291.50 prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) -> gen_0':s:error5_0(0), rt in Omega(1 + n8_0) 309.10/291.50 plus(gen_0':s:error5_0(n1637_0), gen_0':s:error5_0(b)) -> gen_0':s:error5_0(+(n1637_0, b)), rt in Omega(1 + n1637_0) 309.10/291.50 ge(gen_0':s:error5_0(n2462_0), gen_0':s:error5_0(n2462_0)) -> true, rt in Omega(1 + n2462_0) 309.10/291.50 309.10/291.50 309.10/291.50 Generator Equations: 309.10/291.50 gen_0':s:error5_0(0) <=> 0' 309.10/291.50 gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x)) 309.10/291.50 gen_nil:cons6_0(0) <=> nil 309.10/291.50 gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x)) 309.10/291.50 309.10/291.50 309.10/291.50 The following defined symbols remain to be analysed: 309.10/291.50 timesIter 309.10/291.54 EOF