/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 364 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), 36 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 53 ms]
        (16) typed CpxTrs


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

(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:

   prod(xs) -> prodIter(xs, s(0))
   prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
   ifProd(true, xs, x) -> x
   ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0, 0)
   timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
   ifTimes(true, x, y, z, u) -> z
   ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

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:

   prod(xs) -> prodIter(xs, s(0'))
   prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
   ifProd(true, xs, x) -> x
   ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0', 0')
   timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
   ifTimes(true, x, y, z, u) -> z
   ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

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

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

(4)
Obligation:
TRS:
Rules:
prod(xs) -> prodIter(xs, s(0'))
prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) -> x
ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0', 0')
timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) -> z
ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> b
a -> c

Types:
prod :: nil:cons -> 0':s:error
prodIter :: nil:cons -> 0':s:error -> 0':s:error
s :: 0':s:error -> 0':s:error
0' :: 0':s:error
ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
true :: true:false
false :: true:false
tail :: nil:cons -> nil:cons
times :: 0':s:error -> 0':s:error -> 0':s:error
head :: nil:cons -> 0':s:error
plus :: 0':s:error -> 0':s:error -> 0':s:error
timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ge :: 0':s:error -> 0':s:error -> true:false
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
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:
prodIter, plus, timesIter, ge

They will be analysed ascendingly in the following order:
plus < timesIter
ge < timesIter

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

(6)
Obligation:
TRS:
Rules:
prod(xs) -> prodIter(xs, s(0'))
prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) -> x
ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0', 0')
timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) -> z
ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> b
a -> c

Types:
prod :: nil:cons -> 0':s:error
prodIter :: nil:cons -> 0':s:error -> 0':s:error
s :: 0':s:error -> 0':s:error
0' :: 0':s:error
ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
true :: true:false
false :: true:false
tail :: nil:cons -> nil:cons
times :: 0':s:error -> 0':s:error -> 0':s:error
head :: nil:cons -> 0':s:error
plus :: 0':s:error -> 0':s:error -> 0':s:error
timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ge :: 0':s:error -> 0':s:error -> true:false
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
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:
prodIter, plus, timesIter, ge

They will be analysed ascendingly in the following order:
plus < timesIter
ge < timesIter

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
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)

Induction Base:
prodIter(gen_nil:cons6_0(0), gen_0':s:error5_0(0)) ->_R^Omega(1)
ifProd(isempty(gen_nil:cons6_0(0)), gen_nil:cons6_0(0), gen_0':s:error5_0(0)) ->_R^Omega(1)
ifProd(true, gen_nil:cons6_0(0), gen_0':s:error5_0(0)) ->_R^Omega(1)
gen_0':s:error5_0(0)

Induction Step:
prodIter(gen_nil:cons6_0(+(n8_0, 1)), gen_0':s:error5_0(0)) ->_R^Omega(1)
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)
ifProd(false, gen_nil:cons6_0(+(1, n8_0)), gen_0':s:error5_0(0)) ->_R^Omega(1)
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)
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)
prodIter(gen_nil:cons6_0(n8_0), times(gen_0':s:error5_0(0), 0')) ->_R^Omega(1)
prodIter(gen_nil:cons6_0(n8_0), timesIter(gen_0':s:error5_0(0), 0', 0', 0')) ->_R^Omega(1)
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)
prodIter(gen_nil:cons6_0(n8_0), ifTimes(true, gen_0':s:error5_0(0), 0', 0', 0')) ->_R^Omega(1)
prodIter(gen_nil:cons6_0(n8_0), 0') ->_IH
gen_0':s:error5_0(0)

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:
prod(xs) -> prodIter(xs, s(0'))
prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) -> x
ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0', 0')
timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) -> z
ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> b
a -> c

Types:
prod :: nil:cons -> 0':s:error
prodIter :: nil:cons -> 0':s:error -> 0':s:error
s :: 0':s:error -> 0':s:error
0' :: 0':s:error
ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
true :: true:false
false :: true:false
tail :: nil:cons -> nil:cons
times :: 0':s:error -> 0':s:error -> 0':s:error
head :: nil:cons -> 0':s:error
plus :: 0':s:error -> 0':s:error -> 0':s:error
timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ge :: 0':s:error -> 0':s:error -> true:false
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
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:
prodIter, plus, timesIter, ge

They will be analysed ascendingly in the following order:
plus < timesIter
ge < timesIter

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
prod(xs) -> prodIter(xs, s(0'))
prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) -> x
ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0', 0')
timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) -> z
ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> b
a -> c

Types:
prod :: nil:cons -> 0':s:error
prodIter :: nil:cons -> 0':s:error -> 0':s:error
s :: 0':s:error -> 0':s:error
0' :: 0':s:error
ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
true :: true:false
false :: true:false
tail :: nil:cons -> nil:cons
times :: 0':s:error -> 0':s:error -> 0':s:error
head :: nil:cons -> 0':s:error
plus :: 0':s:error -> 0':s:error -> 0':s:error
timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ge :: 0':s:error -> 0':s:error -> true:false
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
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)


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:
plus, timesIter, ge

They will be analysed ascendingly in the following order:
plus < timesIter
ge < timesIter

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
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)

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

Induction Step:
plus(gen_0':s:error5_0(+(n1637_0, 1)), gen_0':s:error5_0(b)) ->_R^Omega(1)
s(plus(gen_0':s:error5_0(n1637_0), gen_0':s:error5_0(b))) ->_IH
s(gen_0':s:error5_0(+(b, c1638_0)))

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

(14)
Obligation:
TRS:
Rules:
prod(xs) -> prodIter(xs, s(0'))
prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) -> x
ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0', 0')
timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) -> z
ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> b
a -> c

Types:
prod :: nil:cons -> 0':s:error
prodIter :: nil:cons -> 0':s:error -> 0':s:error
s :: 0':s:error -> 0':s:error
0' :: 0':s:error
ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
true :: true:false
false :: true:false
tail :: nil:cons -> nil:cons
times :: 0':s:error -> 0':s:error -> 0':s:error
head :: nil:cons -> 0':s:error
plus :: 0':s:error -> 0':s:error -> 0':s:error
timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ge :: 0':s:error -> 0':s:error -> true:false
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
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)
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)


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, timesIter

They will be analysed ascendingly in the following order:
ge < timesIter

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s:error5_0(n2462_0), gen_0':s:error5_0(n2462_0)) -> true, rt in Omega(1 + n2462_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(+(n2462_0, 1)), gen_0':s:error5_0(+(n2462_0, 1))) ->_R^Omega(1)
ge(gen_0':s:error5_0(n2462_0), gen_0':s:error5_0(n2462_0)) ->_IH
true

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

(16)
Obligation:
TRS:
Rules:
prod(xs) -> prodIter(xs, s(0'))
prodIter(xs, x) -> ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) -> x
ifProd(false, xs, x) -> prodIter(tail(xs), times(x, head(xs)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0', 0')
timesIter(x, y, z, u) -> ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) -> z
ifTimes(false, x, y, z, u) -> timesIter(x, y, plus(y, z), s(u))
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
a -> b
a -> c

Types:
prod :: nil:cons -> 0':s:error
prodIter :: nil:cons -> 0':s:error -> 0':s:error
s :: 0':s:error -> 0':s:error
0' :: 0':s:error
ifProd :: true:false -> nil:cons -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
true :: true:false
false :: true:false
tail :: nil:cons -> nil:cons
times :: 0':s:error -> 0':s:error -> 0':s:error
head :: nil:cons -> 0':s:error
plus :: 0':s:error -> 0':s:error -> 0':s:error
timesIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ifTimes :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
ge :: 0':s:error -> 0':s:error -> true:false
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
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)
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)
ge(gen_0':s:error5_0(n2462_0), gen_0':s:error5_0(n2462_0)) -> true, rt in Omega(1 + n2462_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:
timesIter