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, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)])
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))])}
(insert([_xl, _yl, _am]) /\ insert([_xl, _yl, _zl])) -> eq_eltlist([_zl, _am])
)
(length, F:
{() -> length([nil, z])
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)])}
(length([_cm, _dm]) /\ length([_cm, _em])) -> eq_nat([_dm, _em])
)
}

properties:
{(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm])}


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

Clause system for inference is:

{
() -> insert([x, nil, cons(x, nil)]) -> 0
() -> length([nil, z]) -> 0
() -> leq([a, y]) -> 0
() -> leq([b, b]) -> 0
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 0
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 0
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 0
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 0
(leq([b, a])) -> BOT -> 0
}


Solving took 0.194393 seconds.
Proved
Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<a>() -> q_gen_2539
<b>() -> q_gen_2539
<cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2541
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<a, a>() -> q_gen_2542
<a, b>() -> q_gen_2542
<b, a>() -> q_gen_2542
<b, b>() -> q_gen_2542
<a, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<a, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536, q_gen_2544},
Q_f={q_gen_2536},
Delta=
{
<a>() -> q_gen_2544
<b>() -> q_gen_2544
<cons, s>(q_gen_2544, q_gen_2536) -> q_gen_2536
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534, q_gen_2545},
Q_f={q_gen_2534},
Delta=
{
<a, a>() -> q_gen_2534
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
<b, a>() -> q_gen_2545
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.010356 s (model generation: 0.009861,  model checking: 0.000495):

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
}}}

}
--
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 1
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 1
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
}
Sat witness: Found: (() -> leq([b, b]), {

})

-------------------------------------------
Step 1, which took 0.010283 s (model generation: 0.010207,  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_2534},
Q_f={q_gen_2534},
Delta=
{
<b, b>() -> q_gen_2534
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 1
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 1
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
}
Sat witness: Found: (() -> leq([a, y]), {
y -> b
})

-------------------------------------------
Step 2, which took 0.012150 s (model generation: 0.010185,  model checking: 0.001965):

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_2534},
Q_f={q_gen_2534},
Delta=
{
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 1
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 1
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
}
Sat witness: Found: (() -> length([nil, z]), {

})

-------------------------------------------
Step 3, which took 0.010671 s (model generation: 0.010411,  model checking: 0.000260):

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

}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536},
Q_f={q_gen_2536},
Delta=
{
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534},
Q_f={q_gen_2534},
Delta=
{
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 1
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 1
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1
(leq([b, a])) -> BOT -> 1
}
Sat witness: Found: (() -> insert([x, nil, cons(x, nil)]), {
x -> b
})

-------------------------------------------
Step 4, which took 0.011077 s (model generation: 0.010531,  model checking: 0.000546):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<b>() -> q_gen_2539
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536},
Q_f={q_gen_2536},
Delta=
{
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534},
Q_f={q_gen_2534},
Delta=
{
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 1
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 1
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 1
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> 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 5, which took 0.011216 s (model generation: 0.011026,  model checking: 0.000190):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<b>() -> q_gen_2539
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<b, b>() -> q_gen_2542
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536},
Q_f={q_gen_2536},
Delta=
{
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534},
Q_f={q_gen_2534},
Delta=
{
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 1
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 1
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 2
}
Sat witness: Found: ((length([ll, _bm])) -> length([cons(x, ll), s(_bm)]), {
_bm -> z  ;  ll -> nil  ;  x -> b
})

-------------------------------------------
Step 6, which took 0.011968 s (model generation: 0.011289,  model checking: 0.000679):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<b>() -> q_gen_2539
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<b, b>() -> q_gen_2542
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536, q_gen_2544},
Q_f={q_gen_2536},
Delta=
{
<b>() -> q_gen_2544
<cons, s>(q_gen_2544, q_gen_2536) -> q_gen_2536
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534},
Q_f={q_gen_2534},
Delta=
{
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 1
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 4
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 2
}
Sat witness: Found: ((insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]), {
_wl -> cons(b, nil)  ;  x -> b  ;  y -> a  ;  z -> nil
})

-------------------------------------------
Step 7, which took 0.012228 s (model generation: 0.011867,  model checking: 0.000361):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<b>() -> q_gen_2539
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<a, a>() -> q_gen_2542
<b, b>() -> q_gen_2542
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536, q_gen_2544},
Q_f={q_gen_2536},
Delta=
{
<b>() -> q_gen_2544
<cons, s>(q_gen_2544, q_gen_2536) -> q_gen_2536
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534},
Q_f={q_gen_2534},
Delta=
{
<a, b>() -> q_gen_2534
<b, a>() -> q_gen_2534
<b, b>() -> q_gen_2534
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 2
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 4
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 5
}
Sat witness: Found: ((leq([b, a])) -> BOT, {

})

-------------------------------------------
Step 8, which took 0.012165 s (model generation: 0.011823,  model checking: 0.000342):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<b>() -> q_gen_2539
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<a, a>() -> q_gen_2542
<b, b>() -> q_gen_2542
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536, q_gen_2544},
Q_f={q_gen_2536},
Delta=
{
<b>() -> q_gen_2544
<cons, s>(q_gen_2544, q_gen_2536) -> q_gen_2536
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534, q_gen_2545},
Q_f={q_gen_2534},
Delta=
{
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
<b, a>() -> q_gen_2545
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 3
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 4
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 5
}
Sat witness: Found: (() -> leq([a, y]), {
y -> a
})

-------------------------------------------
Step 9, which took 0.013380 s (model generation: 0.012514,  model checking: 0.000866):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<b>() -> q_gen_2539
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<a, a>() -> q_gen_2542
<b, b>() -> q_gen_2542
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536, q_gen_2544},
Q_f={q_gen_2536},
Delta=
{
<b>() -> q_gen_2544
<cons, s>(q_gen_2544, q_gen_2536) -> q_gen_2536
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534, q_gen_2545},
Q_f={q_gen_2534},
Delta=
{
<a, a>() -> q_gen_2534
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
<b, a>() -> q_gen_2545
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 4
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 4
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4
(leq([b, a])) -> BOT -> 5
}
Sat witness: Found: (() -> insert([x, nil, cons(x, nil)]), {
x -> a
})

-------------------------------------------
Step 10, which took 0.013736 s (model generation: 0.012695,  model checking: 0.001041):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<a>() -> q_gen_2539
<b>() -> q_gen_2539
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<a, a>() -> q_gen_2542
<b, b>() -> q_gen_2542
<a, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536, q_gen_2544},
Q_f={q_gen_2536},
Delta=
{
<b>() -> q_gen_2544
<cons, s>(q_gen_2544, q_gen_2536) -> q_gen_2536
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534, q_gen_2545},
Q_f={q_gen_2534},
Delta=
{
<a, a>() -> q_gen_2534
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
<b, a>() -> q_gen_2545
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 4
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 4
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 4
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 7
(leq([b, a])) -> 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 11, which took 0.013870 s (model generation: 0.013532,  model checking: 0.000338):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<a>() -> q_gen_2539
<b>() -> q_gen_2539
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<a, a>() -> q_gen_2542
<b, a>() -> q_gen_2542
<b, b>() -> q_gen_2542
<a, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<a, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536, q_gen_2544},
Q_f={q_gen_2536},
Delta=
{
<b>() -> q_gen_2544
<cons, s>(q_gen_2544, q_gen_2536) -> q_gen_2536
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534, q_gen_2545},
Q_f={q_gen_2534},
Delta=
{
<a, a>() -> q_gen_2534
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
<b, a>() -> q_gen_2545
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 4
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 4
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 7
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 7
(leq([b, a])) -> BOT -> 5
}
Sat witness: Found: ((length([ll, _bm])) -> length([cons(x, ll), s(_bm)]), {
_bm -> z  ;  ll -> nil  ;  x -> a
})

-------------------------------------------
Step 12, which took 0.014943 s (model generation: 0.013756,  model checking: 0.001187):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<a>() -> q_gen_2539
<b>() -> q_gen_2539
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<a, a>() -> q_gen_2542
<b, a>() -> q_gen_2542
<b, b>() -> q_gen_2542
<a, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<a, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536, q_gen_2544},
Q_f={q_gen_2536},
Delta=
{
<a>() -> q_gen_2544
<b>() -> q_gen_2544
<cons, s>(q_gen_2544, q_gen_2536) -> q_gen_2536
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534, q_gen_2545},
Q_f={q_gen_2534},
Delta=
{
<a, a>() -> q_gen_2534
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
<b, a>() -> q_gen_2545
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 4
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 7
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 7
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 7
(leq([b, a])) -> BOT -> 5
}
Sat witness: Found: ((insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]), {
_wl -> cons(a, cons(a, nil))  ;  x -> b  ;  y -> a  ;  z -> cons(a, nil)
})

-------------------------------------------
Step 13, which took 0.017201 s (model generation: 0.015295,  model checking: 0.001906):

Model:
|_
{
insert -> 
{{{
Q={q_gen_2537, q_gen_2538, q_gen_2539, q_gen_2541, q_gen_2542},
Q_f={q_gen_2537},
Delta=
{
<nil>() -> q_gen_2538
<a>() -> q_gen_2539
<b>() -> q_gen_2539
<cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2541
<nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2541
<a, a>() -> q_gen_2542
<b, a>() -> q_gen_2542
<b, b>() -> q_gen_2542
<a, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<a, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
<b, cons, cons>(q_gen_2542, q_gen_2541) -> q_gen_2537
<b, nil, cons>(q_gen_2539, q_gen_2538) -> q_gen_2537
}

Datatype: <elt, eltlist, eltlist>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2536, q_gen_2544},
Q_f={q_gen_2536},
Delta=
{
<a>() -> q_gen_2544
<b>() -> q_gen_2544
<cons, s>(q_gen_2544, q_gen_2536) -> q_gen_2536
<nil, z>() -> q_gen_2536
}

Datatype: <eltlist, nat>
Convolution form: right
}}}
  ;  leq -> 
{{{
Q={q_gen_2534, q_gen_2545},
Q_f={q_gen_2534},
Delta=
{
<a, a>() -> q_gen_2534
<a, b>() -> q_gen_2534
<b, b>() -> q_gen_2534
<b, a>() -> q_gen_2545
}

Datatype: <elt, elt>
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
(insert([x, l, _gm]) /\ length([_gm, _hm]) /\ length([l, _fm])) -> eq_nat([s(_fm), _hm]) -> 5
(insert([x, z, _wl]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _wl)]) -> 7
(length([ll, _bm])) -> length([cons(x, ll), s(_bm)]) -> 7
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 10
(leq([b, a])) -> 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: 0.194393
Reason for stopping: Proved