/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 284 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 10 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 15 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 86 ms]
        (18) BOUNDS(1, INF)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(xs) -> sum2(xs, 0)
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
   ifsum(true, b, xs, y) -> y
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
   ifsum2(true, xs, y) -> sum2(tail(xs), y)
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
   isNil(nil) -> true
   isNil(cons(x, xs)) -> false
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   head(cons(x, xs)) -> x
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> c
   a -> d

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0')
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(xs) -> sum2(xs, 0')
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
   ifsum(true, b, xs, y) -> y
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
   ifsum2(true, xs, y) -> sum2(tail(xs), y)
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
   isNil(nil) -> true
   isNil(cons(x, xs)) -> false
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   head(cons(x, xs)) -> x
   head(nil) -> error
   isZero(0') -> true
   isZero(s(0')) -> false
   isZero(s(s(x))) -> isZero(s(x))
   p(0') -> s(s(0'))
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> c
   a -> d

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(xs) -> sum2(xs, 0')
sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
ifsum(true, b, xs, y) -> y
ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
ifsum2(true, xs, y) -> sum2(tail(xs), y)
ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
isNil(nil) -> true
isNil(cons(x, xs)) -> false
tail(nil) -> nil
tail(cons(x, xs)) -> xs
head(cons(x, xs)) -> x
head(nil) -> error
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
p(0') -> s(s(0'))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> c
a -> d

Types:
times :: 0':s:error -> 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
generate :: 0':s:error -> 0':s:error -> nil:cons
gen :: 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
0' :: 0':s:error
if :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
ge :: 0':s:error -> 0':s:error -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s:error -> nil:cons -> nil:cons
s :: 0':s:error -> 0':s:error
sum2 :: nil:cons -> 0':s:error -> 0':s:error
ifsum :: true:false -> true:false -> nil:cons -> 0':s:error -> 0':s:error
isNil :: nil:cons -> true:false
isZero :: 0':s:error -> true:false
head :: nil:cons -> 0':s:error
ifsum2 :: true:false -> nil:cons -> 0':s:error -> 0':s:error
tail :: nil:cons -> nil:cons
p :: 0':s:error -> 0':s:error
error :: 0':s:error
a :: c:d
c :: c:d
d :: c:d
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_c:d4_0 :: c:d
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons

----------------------------------------

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
gen, ge, sum2, isZero, p

They will be analysed ascendingly in the following order:
ge < gen
isZero < sum2
p < sum2

----------------------------------------

(6)
Obligation:
TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(xs) -> sum2(xs, 0')
sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
ifsum(true, b, xs, y) -> y
ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
ifsum2(true, xs, y) -> sum2(tail(xs), y)
ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
isNil(nil) -> true
isNil(cons(x, xs)) -> false
tail(nil) -> nil
tail(cons(x, xs)) -> xs
head(cons(x, xs)) -> x
head(nil) -> error
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
p(0') -> s(s(0'))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> c
a -> d

Types:
times :: 0':s:error -> 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
generate :: 0':s:error -> 0':s:error -> nil:cons
gen :: 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
0' :: 0':s:error
if :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
ge :: 0':s:error -> 0':s:error -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s:error -> nil:cons -> nil:cons
s :: 0':s:error -> 0':s:error
sum2 :: nil:cons -> 0':s:error -> 0':s:error
ifsum :: true:false -> true:false -> nil:cons -> 0':s:error -> 0':s:error
isNil :: nil:cons -> true:false
isZero :: 0':s:error -> true:false
head :: nil:cons -> 0':s:error
ifsum2 :: true:false -> nil:cons -> 0':s:error -> 0':s:error
tail :: nil:cons -> nil:cons
p :: 0':s:error -> 0':s:error
error :: 0':s:error
a :: c:d
c :: c:d
d :: c:d
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_c:d4_0 :: c:d
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s:error5_0(0) <=> 0'
gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
ge, gen, sum2, isZero, p

They will be analysed ascendingly in the following order:
ge < gen
isZero < sum2
p < sum2

----------------------------------------

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s:error5_0(n8_0), gen_0':s:error5_0(n8_0)) -> true, rt in Omega(1 + n8_0)

Induction Base:
ge(gen_0':s:error5_0(0), gen_0':s:error5_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s:error5_0(+(n8_0, 1)), gen_0':s:error5_0(+(n8_0, 1))) ->_R^Omega(1)
ge(gen_0':s:error5_0(n8_0), gen_0':s:error5_0(n8_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(xs) -> sum2(xs, 0')
sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
ifsum(true, b, xs, y) -> y
ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
ifsum2(true, xs, y) -> sum2(tail(xs), y)
ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
isNil(nil) -> true
isNil(cons(x, xs)) -> false
tail(nil) -> nil
tail(cons(x, xs)) -> xs
head(cons(x, xs)) -> x
head(nil) -> error
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
p(0') -> s(s(0'))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> c
a -> d

Types:
times :: 0':s:error -> 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
generate :: 0':s:error -> 0':s:error -> nil:cons
gen :: 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
0' :: 0':s:error
if :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
ge :: 0':s:error -> 0':s:error -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s:error -> nil:cons -> nil:cons
s :: 0':s:error -> 0':s:error
sum2 :: nil:cons -> 0':s:error -> 0':s:error
ifsum :: true:false -> true:false -> nil:cons -> 0':s:error -> 0':s:error
isNil :: nil:cons -> true:false
isZero :: 0':s:error -> true:false
head :: nil:cons -> 0':s:error
ifsum2 :: true:false -> nil:cons -> 0':s:error -> 0':s:error
tail :: nil:cons -> nil:cons
p :: 0':s:error -> 0':s:error
error :: 0':s:error
a :: c:d
c :: c:d
d :: c:d
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_c:d4_0 :: c:d
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s:error5_0(0) <=> 0'
gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
ge, gen, sum2, isZero, p

They will be analysed ascendingly in the following order:
ge < gen
isZero < sum2
p < sum2

----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(xs) -> sum2(xs, 0')
sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
ifsum(true, b, xs, y) -> y
ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
ifsum2(true, xs, y) -> sum2(tail(xs), y)
ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
isNil(nil) -> true
isNil(cons(x, xs)) -> false
tail(nil) -> nil
tail(cons(x, xs)) -> xs
head(cons(x, xs)) -> x
head(nil) -> error
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
p(0') -> s(s(0'))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> c
a -> d

Types:
times :: 0':s:error -> 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
generate :: 0':s:error -> 0':s:error -> nil:cons
gen :: 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
0' :: 0':s:error
if :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
ge :: 0':s:error -> 0':s:error -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s:error -> nil:cons -> nil:cons
s :: 0':s:error -> 0':s:error
sum2 :: nil:cons -> 0':s:error -> 0':s:error
ifsum :: true:false -> true:false -> nil:cons -> 0':s:error -> 0':s:error
isNil :: nil:cons -> true:false
isZero :: 0':s:error -> true:false
head :: nil:cons -> 0':s:error
ifsum2 :: true:false -> nil:cons -> 0':s:error -> 0':s:error
tail :: nil:cons -> nil:cons
p :: 0':s:error -> 0':s:error
error :: 0':s:error
a :: c:d
c :: c:d
d :: c:d
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_c:d4_0 :: c:d
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
ge(gen_0':s:error5_0(n8_0), gen_0':s:error5_0(n8_0)) -> true, rt in Omega(1 + n8_0)


Generator Equations:
gen_0':s:error5_0(0) <=> 0'
gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
gen, sum2, isZero, p

They will be analysed ascendingly in the following order:
isZero < sum2
p < sum2

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
isZero(gen_0':s:error5_0(+(1, n469_0))) -> false, rt in Omega(1 + n469_0)

Induction Base:
isZero(gen_0':s:error5_0(+(1, 0))) ->_R^Omega(1)
false

Induction Step:
isZero(gen_0':s:error5_0(+(1, +(n469_0, 1)))) ->_R^Omega(1)
isZero(s(gen_0':s:error5_0(n469_0))) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(xs) -> sum2(xs, 0')
sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
ifsum(true, b, xs, y) -> y
ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
ifsum2(true, xs, y) -> sum2(tail(xs), y)
ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
isNil(nil) -> true
isNil(cons(x, xs)) -> false
tail(nil) -> nil
tail(cons(x, xs)) -> xs
head(cons(x, xs)) -> x
head(nil) -> error
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
p(0') -> s(s(0'))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> c
a -> d

Types:
times :: 0':s:error -> 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
generate :: 0':s:error -> 0':s:error -> nil:cons
gen :: 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
0' :: 0':s:error
if :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
ge :: 0':s:error -> 0':s:error -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s:error -> nil:cons -> nil:cons
s :: 0':s:error -> 0':s:error
sum2 :: nil:cons -> 0':s:error -> 0':s:error
ifsum :: true:false -> true:false -> nil:cons -> 0':s:error -> 0':s:error
isNil :: nil:cons -> true:false
isZero :: 0':s:error -> true:false
head :: nil:cons -> 0':s:error
ifsum2 :: true:false -> nil:cons -> 0':s:error -> 0':s:error
tail :: nil:cons -> nil:cons
p :: 0':s:error -> 0':s:error
error :: 0':s:error
a :: c:d
c :: c:d
d :: c:d
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_c:d4_0 :: c:d
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
ge(gen_0':s:error5_0(n8_0), gen_0':s:error5_0(n8_0)) -> true, rt in Omega(1 + n8_0)
isZero(gen_0':s:error5_0(+(1, n469_0))) -> false, rt in Omega(1 + n469_0)


Generator Equations:
gen_0':s:error5_0(0) <=> 0'
gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
p, sum2

They will be analysed ascendingly in the following order:
p < sum2

----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_0':s:error5_0(+(1, n641_0))) -> gen_0':s:error5_0(n641_0), rt in Omega(1 + n641_0)

Induction Base:
p(gen_0':s:error5_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
p(gen_0':s:error5_0(+(1, +(n641_0, 1)))) ->_R^Omega(1)
s(p(s(gen_0':s:error5_0(n641_0)))) ->_IH
s(gen_0':s:error5_0(c642_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Obligation:
TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(xs) -> sum2(xs, 0')
sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
ifsum(true, b, xs, y) -> y
ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
ifsum2(true, xs, y) -> sum2(tail(xs), y)
ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
isNil(nil) -> true
isNil(cons(x, xs)) -> false
tail(nil) -> nil
tail(cons(x, xs)) -> xs
head(cons(x, xs)) -> x
head(nil) -> error
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
p(0') -> s(s(0'))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> c
a -> d

Types:
times :: 0':s:error -> 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
generate :: 0':s:error -> 0':s:error -> nil:cons
gen :: 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
0' :: 0':s:error
if :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> nil:cons
ge :: 0':s:error -> 0':s:error -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s:error -> nil:cons -> nil:cons
s :: 0':s:error -> 0':s:error
sum2 :: nil:cons -> 0':s:error -> 0':s:error
ifsum :: true:false -> true:false -> nil:cons -> 0':s:error -> 0':s:error
isNil :: nil:cons -> true:false
isZero :: 0':s:error -> true:false
head :: nil:cons -> 0':s:error
ifsum2 :: true:false -> nil:cons -> 0':s:error -> 0':s:error
tail :: nil:cons -> nil:cons
p :: 0':s:error -> 0':s:error
error :: 0':s:error
a :: c:d
c :: c:d
d :: c:d
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_c:d4_0 :: c:d
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
ge(gen_0':s:error5_0(n8_0), gen_0':s:error5_0(n8_0)) -> true, rt in Omega(1 + n8_0)
isZero(gen_0':s:error5_0(+(1, n469_0))) -> false, rt in Omega(1 + n469_0)
p(gen_0':s:error5_0(+(1, n641_0))) -> gen_0':s:error5_0(n641_0), rt in Omega(1 + n641_0)


Generator Equations:
gen_0':s:error5_0(0) <=> 0'
gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
sum2
----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum2(gen_nil:cons6_0(n950_0), gen_0':s:error5_0(b)) -> gen_0':s:error5_0(b), rt in Omega(1 + n950_0)

Induction Base:
sum2(gen_nil:cons6_0(0), gen_0':s:error5_0(b)) ->_R^Omega(1)
ifsum(isNil(gen_nil:cons6_0(0)), isZero(head(gen_nil:cons6_0(0))), gen_nil:cons6_0(0), gen_0':s:error5_0(b)) ->_R^Omega(1)
ifsum(true, isZero(head(gen_nil:cons6_0(0))), gen_nil:cons6_0(0), gen_0':s:error5_0(b)) ->_R^Omega(1)
ifsum(true, isZero(error), gen_nil:cons6_0(0), gen_0':s:error5_0(b)) ->_R^Omega(1)
gen_0':s:error5_0(b)

Induction Step:
sum2(gen_nil:cons6_0(+(n950_0, 1)), gen_0':s:error5_0(b)) ->_R^Omega(1)
ifsum(isNil(gen_nil:cons6_0(+(n950_0, 1))), isZero(head(gen_nil:cons6_0(+(n950_0, 1)))), gen_nil:cons6_0(+(n950_0, 1)), gen_0':s:error5_0(b)) ->_R^Omega(1)
ifsum(false, isZero(head(gen_nil:cons6_0(+(1, n950_0)))), gen_nil:cons6_0(+(1, n950_0)), gen_0':s:error5_0(b)) ->_R^Omega(1)
ifsum(false, isZero(0'), gen_nil:cons6_0(+(1, n950_0)), gen_0':s:error5_0(b)) ->_R^Omega(1)
ifsum(false, true, gen_nil:cons6_0(+(1, n950_0)), gen_0':s:error5_0(b)) ->_R^Omega(1)
ifsum2(true, gen_nil:cons6_0(+(1, n950_0)), gen_0':s:error5_0(b)) ->_R^Omega(1)
sum2(tail(gen_nil:cons6_0(+(1, n950_0))), gen_0':s:error5_0(b)) ->_R^Omega(1)
sum2(gen_nil:cons6_0(n950_0), gen_0':s:error5_0(b)) ->_IH
gen_0':s:error5_0(b)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(18)
BOUNDS(1, INF)