Inference procedure has parameters:
Ice fuel: 200
Timeout: 60s
Convolution: right


Learning problem is:

env: {
elt -> {a, b}  ;  eltlist -> {cons, nil}  ;  nat -> {s, z}
}
definition:
{
(leq, P:
{() -> leq([a, y])
() -> leq([b, b])
(leq([b, a])) -> BOT}
)
(insert, F:
{() -> insert([x, nil, cons(x, nil)])
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)])
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))])}
(insert([_z, _aa, _ba]) /\ insert([_z, _aa, _ca])) -> eq_eltlist([_ba, _ca])
)
(length, F:
{() -> length([nil, z])
(length([ll, _da])) -> length([cons(x, ll), s(_da)])}
(length([_ea, _fa]) /\ length([_ea, _ga])) -> eq_nat([_fa, _ga])
)
(leqnat, P:
{() -> leqnat([z, s(nn2)])
() -> leqnat([z, z])
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)])
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2])
(leqnat([s(nn1), z])) -> BOT}
)
}

properties:
{(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja])}


over-approximation: {insert, length}
under-approximation: {leq, leqnat}

Clause system for inference is:

{
() -> insert([x, nil, cons(x, nil)]) -> 0
() -> length([nil, z]) -> 0
() -> leq([a, y]) -> 0
() -> leq([b, b]) -> 0
() -> leqnat([z, s(nn2)]) -> 0
() -> leqnat([z, z]) -> 0
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 0
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 0
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 0
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 0
(leq([b, a])) -> BOT -> 0
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 0
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 0
(leqnat([s(nn1), z])) -> BOT -> 0
}


Solving took 0.212769 seconds.
Proved
Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<a>() -> q_gen_244
<b>() -> q_gen_244
<cons, cons>(q_gen_248, q_gen_247) -> q_gen_247
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<a, a>() -> q_gen_248
<a, b>() -> q_gen_248
<b, a>() -> q_gen_248
<b, b>() -> q_gen_248
<a, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<a, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<a>() -> q_gen_250
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239, q_gen_251},
Q_f={q_gen_239},
Delta=
{
<a, a>() -> q_gen_239
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
<b, a>() -> q_gen_251
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<s>(q_gen_238) -> q_gen_238
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.005614 s (model generation: 0.005328,  model checking: 0.000286):

Model:
|_
{
insert -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0
() -> length([nil, z]) -> 0
() -> leq([a, y]) -> 0
() -> leq([b, b]) -> 0
() -> leqnat([z, s(nn2)]) -> 0
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1
(leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Found: (() -> leqnat([z, z]), {

})

-------------------------------------------
Step 1, which took 0.004464 s (model generation: 0.004287,  model checking: 0.000177):

Model:
|_
{
insert -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236},
Q_f={q_gen_236},
Delta=
{
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0
() -> length([nil, z]) -> 0
() -> leq([a, y]) -> 0
() -> leq([b, b]) -> 0
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1
(leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Found: (() -> leqnat([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.004493 s (model generation: 0.004406,  model checking: 0.000087):

Model:
|_
{
insert -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0
() -> length([nil, z]) -> 0
() -> leq([a, y]) -> 0
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1
(leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Found: (() -> leq([b, b]), {

})

-------------------------------------------
Step 3, which took 0.005060 s (model generation: 0.004957,  model checking: 0.000103):

Model:
|_
{
insert -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239},
Q_f={q_gen_239},
Delta=
{
<b, b>() -> q_gen_239
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0
() -> length([nil, z]) -> 0
() -> leq([a, y]) -> 3
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1
(leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Found: (() -> leq([a, y]), {
y -> b
})

-------------------------------------------
Step 4, which took 0.004428 s (model generation: 0.004352,  model checking: 0.000076):

Model:
|_
{
insert -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239},
Q_f={q_gen_239},
Delta=
{
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0
() -> length([nil, z]) -> 3
() -> leq([a, y]) -> 3
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1
(leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Found: (() -> length([nil, z]), {

})

-------------------------------------------
Step 5, which took 0.006895 s (model generation: 0.005083,  model checking: 0.001812):

Model:
|_
{
insert -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241},
Q_f={q_gen_241},
Delta=
{
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239},
Q_f={q_gen_239},
Delta=
{
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3
() -> length([nil, z]) -> 3
() -> leq([a, y]) -> 3
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1
(leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Found: (() -> insert([x, nil, cons(x, nil)]), {
x -> b
})

-------------------------------------------
Step 6, which took 0.007741 s (model generation: 0.007537,  model checking: 0.000204):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<b>() -> q_gen_244
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241},
Q_f={q_gen_241},
Delta=
{
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239},
Q_f={q_gen_239},
Delta=
{
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3
() -> length([nil, z]) -> 3
() -> leq([a, y]) -> 3
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 2
(leqnat([s(nn1), z])) -> BOT -> 2
}
Sat witness: Found: ((leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 7, which took 0.010850 s (model generation: 0.010288,  model checking: 0.000562):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<b>() -> q_gen_244
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241},
Q_f={q_gen_241},
Delta=
{
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239},
Q_f={q_gen_239},
Delta=
{
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3
() -> length([nil, z]) -> 3
() -> leq([a, y]) -> 3
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 2
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 2
(leqnat([s(nn1), z])) -> BOT -> 2
}
Sat witness: Found: ((leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]), {
x -> b  ;  y -> b  ;  z -> nil
})

-------------------------------------------
Step 8, which took 0.006500 s (model generation: 0.004778,  model checking: 0.001722):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<b>() -> q_gen_244
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<b, b>() -> q_gen_248
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241},
Q_f={q_gen_241},
Delta=
{
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239},
Q_f={q_gen_239},
Delta=
{
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3
() -> length([nil, z]) -> 3
() -> leq([a, y]) -> 3
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 2
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 2
(leqnat([s(nn1), z])) -> BOT -> 2
}
Sat witness: Found: ((length([ll, _da])) -> length([cons(x, ll), s(_da)]), {
_da -> z  ;  ll -> nil  ;  x -> b
})

-------------------------------------------
Step 9, which took 0.011714 s (model generation: 0.011087,  model checking: 0.000627):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<b>() -> q_gen_244
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<b, b>() -> q_gen_248
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239},
Q_f={q_gen_239},
Delta=
{
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3
() -> length([nil, z]) -> 3
() -> leq([a, y]) -> 3
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 2
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 2
(leqnat([s(nn1), z])) -> BOT -> 2
}
Sat witness: Found: ((insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]), {
_y -> cons(b, nil)  ;  x -> b  ;  y -> a  ;  z -> nil
})

-------------------------------------------
Step 10, which took 0.012253 s (model generation: 0.011398,  model checking: 0.000855):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<b>() -> q_gen_244
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<a, a>() -> q_gen_248
<b, b>() -> q_gen_248
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239},
Q_f={q_gen_239},
Delta=
{
<a, b>() -> q_gen_239
<b, a>() -> q_gen_239
<b, b>() -> q_gen_239
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3
() -> length([nil, z]) -> 3
() -> leq([a, y]) -> 3
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 3
() -> leqnat([z, z]) -> 3
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 2
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 5
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 3
(leqnat([s(nn1), z])) -> BOT -> 3
}
Sat witness: Found: ((leq([b, a])) -> BOT, {

})

-------------------------------------------
Step 11, which took 0.009334 s (model generation: 0.008664,  model checking: 0.000670):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<b>() -> q_gen_244
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<a, a>() -> q_gen_248
<b, b>() -> q_gen_248
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239, q_gen_251},
Q_f={q_gen_239},
Delta=
{
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
<b, a>() -> q_gen_251
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3
() -> length([nil, z]) -> 3
() -> leq([a, y]) -> 3
() -> leq([b, b]) -> 3
() -> leqnat([z, s(nn2)]) -> 6
() -> leqnat([z, z]) -> 4
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 3
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 5
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 4
(leqnat([s(nn1), z])) -> BOT -> 4
}
Sat witness: Found: (() -> leqnat([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 12, which took 0.013201 s (model generation: 0.012926,  model checking: 0.000275):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<b>() -> q_gen_244
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<a, a>() -> q_gen_248
<b, b>() -> q_gen_248
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239, q_gen_251},
Q_f={q_gen_239},
Delta=
{
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
<b, a>() -> q_gen_251
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<s>(q_gen_238) -> q_gen_238
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3
() -> length([nil, z]) -> 3
() -> leq([a, y]) -> 6
() -> leq([b, b]) -> 4
() -> leqnat([z, s(nn2)]) -> 6
() -> leqnat([z, z]) -> 4
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 5
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 4
(leqnat([s(nn1), z])) -> BOT -> 4
}
Sat witness: Found: (() -> leq([a, y]), {
y -> a
})

-------------------------------------------
Step 13, which took 0.014889 s (model generation: 0.014157,  model checking: 0.000732):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<b>() -> q_gen_244
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<a, a>() -> q_gen_248
<b, b>() -> q_gen_248
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239, q_gen_251},
Q_f={q_gen_239},
Delta=
{
<a, a>() -> q_gen_239
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
<b, a>() -> q_gen_251
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<s>(q_gen_238) -> q_gen_238
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 6
() -> length([nil, z]) -> 4
() -> leq([a, y]) -> 6
() -> leq([b, b]) -> 4
() -> leqnat([z, s(nn2)]) -> 6
() -> leqnat([z, z]) -> 4
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 5
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 4
(leqnat([s(nn1), z])) -> BOT -> 4
}
Sat witness: Found: (() -> insert([x, nil, cons(x, nil)]), {
x -> a
})

-------------------------------------------
Step 14, which took 0.015434 s (model generation: 0.013484,  model checking: 0.001950):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<a>() -> q_gen_244
<b>() -> q_gen_244
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<a, a>() -> q_gen_248
<b, b>() -> q_gen_248
<a, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239, q_gen_251},
Q_f={q_gen_239},
Delta=
{
<a, a>() -> q_gen_239
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
<b, a>() -> q_gen_251
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<s>(q_gen_238) -> q_gen_238
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 6
() -> length([nil, z]) -> 4
() -> leq([a, y]) -> 6
() -> leq([b, b]) -> 4
() -> leqnat([z, s(nn2)]) -> 6
() -> leqnat([z, z]) -> 4
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 7
(leq([b, a])) -> BOT -> 5
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 5
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 5
(leqnat([s(nn1), z])) -> BOT -> 5
}
Sat witness: Found: ((leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]), {
x -> a  ;  y -> b  ;  z -> nil
})

-------------------------------------------
Step 15, which took 0.014224 s (model generation: 0.013911,  model checking: 0.000313):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<a>() -> q_gen_244
<b>() -> q_gen_244
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<a, a>() -> q_gen_248
<b, a>() -> q_gen_248
<b, b>() -> q_gen_248
<a, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<a, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239, q_gen_251},
Q_f={q_gen_239},
Delta=
{
<a, a>() -> q_gen_239
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
<b, a>() -> q_gen_251
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<s>(q_gen_238) -> q_gen_238
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 6
() -> length([nil, z]) -> 4
() -> leq([a, y]) -> 6
() -> leq([b, b]) -> 4
() -> leqnat([z, s(nn2)]) -> 6
() -> leqnat([z, z]) -> 4
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 7
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 7
(leq([b, a])) -> BOT -> 5
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 5
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 5
(leqnat([s(nn1), z])) -> BOT -> 5
}
Sat witness: Found: ((length([ll, _da])) -> length([cons(x, ll), s(_da)]), {
_da -> z  ;  ll -> nil  ;  x -> a
})

-------------------------------------------
Step 16, which took 0.015893 s (model generation: 0.014768,  model checking: 0.001125):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<a>() -> q_gen_244
<b>() -> q_gen_244
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<a, a>() -> q_gen_248
<b, a>() -> q_gen_248
<b, b>() -> q_gen_248
<a, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<a, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<a>() -> q_gen_250
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239, q_gen_251},
Q_f={q_gen_239},
Delta=
{
<a, a>() -> q_gen_239
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
<b, a>() -> q_gen_251
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<s>(q_gen_238) -> q_gen_238
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 6
() -> length([nil, z]) -> 4
() -> leq([a, y]) -> 6
() -> leq([b, b]) -> 4
() -> leqnat([z, s(nn2)]) -> 6
() -> leqnat([z, z]) -> 4
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 7
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 7
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 7
(leq([b, a])) -> BOT -> 5
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 5
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 5
(leqnat([s(nn1), z])) -> BOT -> 5
}
Sat witness: Found: ((insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]), {
_y -> cons(a, cons(a, nil))  ;  x -> b  ;  y -> a  ;  z -> cons(a, nil)
})

-------------------------------------------
Step 17, which took 0.026038 s (model generation: 0.015953,  model checking: 0.010085):

Model:
|_
{
insert -> 
{{{
Q={q_gen_242, q_gen_243, q_gen_244, q_gen_247, q_gen_248},
Q_f={q_gen_242},
Delta=
{
<nil>() -> q_gen_243
<a>() -> q_gen_244
<b>() -> q_gen_244
<cons, cons>(q_gen_248, q_gen_247) -> q_gen_247
<nil, cons>(q_gen_244, q_gen_243) -> q_gen_247
<a, a>() -> q_gen_248
<b, a>() -> q_gen_248
<b, b>() -> q_gen_248
<a, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<a, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
<b, cons, cons>(q_gen_248, q_gen_247) -> q_gen_242
<b, nil, cons>(q_gen_244, q_gen_243) -> q_gen_242
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_241, q_gen_250},
Q_f={q_gen_241},
Delta=
{
<a>() -> q_gen_250
<b>() -> q_gen_250
<cons, s>(q_gen_250, q_gen_241) -> q_gen_241
<nil, z>() -> q_gen_241
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_239, q_gen_251},
Q_f={q_gen_239},
Delta=
{
<a, a>() -> q_gen_239
<a, b>() -> q_gen_239
<b, b>() -> q_gen_239
<b, a>() -> q_gen_251
}

Datatype: <elt, elt>
Convolution form: right
}}}
  ;  leqnat -> 
{{{
Q={q_gen_236, q_gen_238},
Q_f={q_gen_236},
Delta=
{
<s>(q_gen_238) -> q_gen_238
<z>() -> q_gen_238
<s, s>(q_gen_236) -> q_gen_236
<z, s>(q_gen_238) -> q_gen_236
<z, z>() -> q_gen_236
}

Datatype: <nat, nat>
Convolution form: right
}}}

}
--
Equality automata are defined for: {eq_elt, eq_eltlist, eq_nat}
_|

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 7
() -> length([nil, z]) -> 5
() -> leq([a, y]) -> 7
() -> leq([b, b]) -> 5
() -> leqnat([z, s(nn2)]) -> 7
() -> leqnat([z, z]) -> 5
(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 5
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 7
(length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 7
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 10
(leq([b, a])) -> BOT -> 6
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 6
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 6
(leqnat([s(nn1), z])) -> BOT -> 6
}
Sat witness: Found: ((leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]), {
x -> b  ;  y -> b  ;  z -> cons(a, nil)
})

Total time: 0.212769
Reason for stopping: Proved