/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 280 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), 40 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 29 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:

   div(x, y) -> div2(x, y, 0)
   div2(x, y, i) -> if1(le(y, 0), le(y, x), x, y, plus(i, 0), inc(i))
   if1(true, b, x, y, i, j) -> divZeroError
   if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
   if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
   if2(false, x, y, i, j) -> i
   inc(0) -> 0
   inc(s(i)) -> s(inc(i))
   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   plus(x, y) -> plusIter(x, y, 0)
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   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:

   div(x, y) -> div2(x, y, 0')
   div2(x, y, i) -> if1(le(y, 0'), le(y, x), x, y, plus(i, 0'), inc(i))
   if1(true, b, x, y, i, j) -> divZeroError
   if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
   if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
   if2(false, x, y, i, j) -> i
   inc(0') -> 0'
   inc(s(i)) -> s(inc(i))
   le(s(x), 0') -> false
   le(0', y) -> true
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0') -> x
   minus(0', y) -> 0'
   minus(s(x), s(y)) -> minus(x, y)
   plus(x, y) -> plusIter(x, y, 0')
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   a -> c
   a -> d

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

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

(4)
Obligation:
TRS:
Rules:
div(x, y) -> div2(x, y, 0')
div2(x, y, i) -> if1(le(y, 0'), le(y, x), x, y, plus(i, 0'), inc(i))
if1(true, b, x, y, i, j) -> divZeroError
if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
if2(false, x, y, i, j) -> i
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
a -> c
a -> d

Types:
div :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
div2 :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
0' :: 0':divZeroError:s
if1 :: true:false -> true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
le :: 0':divZeroError:s -> 0':divZeroError:s -> true:false
plus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
inc :: 0':divZeroError:s -> 0':divZeroError:s
true :: true:false
divZeroError :: 0':divZeroError:s
false :: true:false
if2 :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
minus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
s :: 0':divZeroError:s -> 0':divZeroError:s
plusIter :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
ifPlus :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
a :: c:d
c :: c:d
d :: c:d
hole_0':divZeroError:s1_0 :: 0':divZeroError:s
hole_true:false2_0 :: true:false
hole_c:d3_0 :: c:d
gen_0':divZeroError:s4_0 :: Nat -> 0':divZeroError:s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
div2, le, inc, minus, plusIter

They will be analysed ascendingly in the following order:
le < div2
inc < div2
minus < div2
le < plusIter

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

(6)
Obligation:
TRS:
Rules:
div(x, y) -> div2(x, y, 0')
div2(x, y, i) -> if1(le(y, 0'), le(y, x), x, y, plus(i, 0'), inc(i))
if1(true, b, x, y, i, j) -> divZeroError
if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
if2(false, x, y, i, j) -> i
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
a -> c
a -> d

Types:
div :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
div2 :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
0' :: 0':divZeroError:s
if1 :: true:false -> true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
le :: 0':divZeroError:s -> 0':divZeroError:s -> true:false
plus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
inc :: 0':divZeroError:s -> 0':divZeroError:s
true :: true:false
divZeroError :: 0':divZeroError:s
false :: true:false
if2 :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
minus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
s :: 0':divZeroError:s -> 0':divZeroError:s
plusIter :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
ifPlus :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
a :: c:d
c :: c:d
d :: c:d
hole_0':divZeroError:s1_0 :: 0':divZeroError:s
hole_true:false2_0 :: true:false
hole_c:d3_0 :: c:d
gen_0':divZeroError:s4_0 :: Nat -> 0':divZeroError:s


Generator Equations:
gen_0':divZeroError:s4_0(0) <=> 0'
gen_0':divZeroError:s4_0(+(x, 1)) <=> s(gen_0':divZeroError:s4_0(x))


The following defined symbols remain to be analysed:
le, div2, inc, minus, plusIter

They will be analysed ascendingly in the following order:
le < div2
inc < div2
minus < div2
le < plusIter

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':divZeroError:s4_0(+(1, n6_0)), gen_0':divZeroError:s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
le(gen_0':divZeroError:s4_0(+(1, 0)), gen_0':divZeroError:s4_0(0)) ->_R^Omega(1)
false

Induction Step:
le(gen_0':divZeroError:s4_0(+(1, +(n6_0, 1))), gen_0':divZeroError:s4_0(+(n6_0, 1))) ->_R^Omega(1)
le(gen_0':divZeroError:s4_0(+(1, n6_0)), gen_0':divZeroError:s4_0(n6_0)) ->_IH
false

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:
div(x, y) -> div2(x, y, 0')
div2(x, y, i) -> if1(le(y, 0'), le(y, x), x, y, plus(i, 0'), inc(i))
if1(true, b, x, y, i, j) -> divZeroError
if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
if2(false, x, y, i, j) -> i
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
a -> c
a -> d

Types:
div :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
div2 :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
0' :: 0':divZeroError:s
if1 :: true:false -> true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
le :: 0':divZeroError:s -> 0':divZeroError:s -> true:false
plus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
inc :: 0':divZeroError:s -> 0':divZeroError:s
true :: true:false
divZeroError :: 0':divZeroError:s
false :: true:false
if2 :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
minus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
s :: 0':divZeroError:s -> 0':divZeroError:s
plusIter :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
ifPlus :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
a :: c:d
c :: c:d
d :: c:d
hole_0':divZeroError:s1_0 :: 0':divZeroError:s
hole_true:false2_0 :: true:false
hole_c:d3_0 :: c:d
gen_0':divZeroError:s4_0 :: Nat -> 0':divZeroError:s


Generator Equations:
gen_0':divZeroError:s4_0(0) <=> 0'
gen_0':divZeroError:s4_0(+(x, 1)) <=> s(gen_0':divZeroError:s4_0(x))


The following defined symbols remain to be analysed:
le, div2, inc, minus, plusIter

They will be analysed ascendingly in the following order:
le < div2
inc < div2
minus < div2
le < plusIter

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
div(x, y) -> div2(x, y, 0')
div2(x, y, i) -> if1(le(y, 0'), le(y, x), x, y, plus(i, 0'), inc(i))
if1(true, b, x, y, i, j) -> divZeroError
if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
if2(false, x, y, i, j) -> i
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
a -> c
a -> d

Types:
div :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
div2 :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
0' :: 0':divZeroError:s
if1 :: true:false -> true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
le :: 0':divZeroError:s -> 0':divZeroError:s -> true:false
plus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
inc :: 0':divZeroError:s -> 0':divZeroError:s
true :: true:false
divZeroError :: 0':divZeroError:s
false :: true:false
if2 :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
minus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
s :: 0':divZeroError:s -> 0':divZeroError:s
plusIter :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
ifPlus :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
a :: c:d
c :: c:d
d :: c:d
hole_0':divZeroError:s1_0 :: 0':divZeroError:s
hole_true:false2_0 :: true:false
hole_c:d3_0 :: c:d
gen_0':divZeroError:s4_0 :: Nat -> 0':divZeroError:s


Lemmas:
le(gen_0':divZeroError:s4_0(+(1, n6_0)), gen_0':divZeroError:s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':divZeroError:s4_0(0) <=> 0'
gen_0':divZeroError:s4_0(+(x, 1)) <=> s(gen_0':divZeroError:s4_0(x))


The following defined symbols remain to be analysed:
inc, div2, minus, plusIter

They will be analysed ascendingly in the following order:
inc < div2
minus < div2

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':divZeroError:s4_0(n313_0)) -> gen_0':divZeroError:s4_0(n313_0), rt in Omega(1 + n313_0)

Induction Base:
inc(gen_0':divZeroError:s4_0(0)) ->_R^Omega(1)
0'

Induction Step:
inc(gen_0':divZeroError:s4_0(+(n313_0, 1))) ->_R^Omega(1)
s(inc(gen_0':divZeroError:s4_0(n313_0))) ->_IH
s(gen_0':divZeroError:s4_0(c314_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:
div(x, y) -> div2(x, y, 0')
div2(x, y, i) -> if1(le(y, 0'), le(y, x), x, y, plus(i, 0'), inc(i))
if1(true, b, x, y, i, j) -> divZeroError
if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
if2(false, x, y, i, j) -> i
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
a -> c
a -> d

Types:
div :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
div2 :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
0' :: 0':divZeroError:s
if1 :: true:false -> true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
le :: 0':divZeroError:s -> 0':divZeroError:s -> true:false
plus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
inc :: 0':divZeroError:s -> 0':divZeroError:s
true :: true:false
divZeroError :: 0':divZeroError:s
false :: true:false
if2 :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
minus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
s :: 0':divZeroError:s -> 0':divZeroError:s
plusIter :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
ifPlus :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
a :: c:d
c :: c:d
d :: c:d
hole_0':divZeroError:s1_0 :: 0':divZeroError:s
hole_true:false2_0 :: true:false
hole_c:d3_0 :: c:d
gen_0':divZeroError:s4_0 :: Nat -> 0':divZeroError:s


Lemmas:
le(gen_0':divZeroError:s4_0(+(1, n6_0)), gen_0':divZeroError:s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
inc(gen_0':divZeroError:s4_0(n313_0)) -> gen_0':divZeroError:s4_0(n313_0), rt in Omega(1 + n313_0)


Generator Equations:
gen_0':divZeroError:s4_0(0) <=> 0'
gen_0':divZeroError:s4_0(+(x, 1)) <=> s(gen_0':divZeroError:s4_0(x))


The following defined symbols remain to be analysed:
minus, div2, plusIter

They will be analysed ascendingly in the following order:
minus < div2

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':divZeroError:s4_0(n543_0), gen_0':divZeroError:s4_0(n543_0)) -> gen_0':divZeroError:s4_0(0), rt in Omega(1 + n543_0)

Induction Base:
minus(gen_0':divZeroError:s4_0(0), gen_0':divZeroError:s4_0(0)) ->_R^Omega(1)
gen_0':divZeroError:s4_0(0)

Induction Step:
minus(gen_0':divZeroError:s4_0(+(n543_0, 1)), gen_0':divZeroError:s4_0(+(n543_0, 1))) ->_R^Omega(1)
minus(gen_0':divZeroError:s4_0(n543_0), gen_0':divZeroError:s4_0(n543_0)) ->_IH
gen_0':divZeroError:s4_0(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:
div(x, y) -> div2(x, y, 0')
div2(x, y, i) -> if1(le(y, 0'), le(y, x), x, y, plus(i, 0'), inc(i))
if1(true, b, x, y, i, j) -> divZeroError
if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
if2(false, x, y, i, j) -> i
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
a -> c
a -> d

Types:
div :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
div2 :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
0' :: 0':divZeroError:s
if1 :: true:false -> true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
le :: 0':divZeroError:s -> 0':divZeroError:s -> true:false
plus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
inc :: 0':divZeroError:s -> 0':divZeroError:s
true :: true:false
divZeroError :: 0':divZeroError:s
false :: true:false
if2 :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
minus :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
s :: 0':divZeroError:s -> 0':divZeroError:s
plusIter :: 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
ifPlus :: true:false -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s -> 0':divZeroError:s
a :: c:d
c :: c:d
d :: c:d
hole_0':divZeroError:s1_0 :: 0':divZeroError:s
hole_true:false2_0 :: true:false
hole_c:d3_0 :: c:d
gen_0':divZeroError:s4_0 :: Nat -> 0':divZeroError:s


Lemmas:
le(gen_0':divZeroError:s4_0(+(1, n6_0)), gen_0':divZeroError:s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
inc(gen_0':divZeroError:s4_0(n313_0)) -> gen_0':divZeroError:s4_0(n313_0), rt in Omega(1 + n313_0)
minus(gen_0':divZeroError:s4_0(n543_0), gen_0':divZeroError:s4_0(n543_0)) -> gen_0':divZeroError:s4_0(0), rt in Omega(1 + n543_0)


Generator Equations:
gen_0':divZeroError:s4_0(0) <=> 0'
gen_0':divZeroError:s4_0(+(x, 1)) <=> s(gen_0':divZeroError:s4_0(x))


The following defined symbols remain to be analysed:
div2, plusIter