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


Learning problem is:

env: {
nat -> {s, z}  ;  natlist -> {cons, nil}
}
definition:
{
(le, P:
{() -> le([z, s(nn2)])
(le([nn1, nn2])) -> le([s(nn1), s(nn2)])
(le([s(nn1), s(nn2)])) -> le([nn1, nn2])
(le([s(nn1), z])) -> BOT
(le([z, z])) -> BOT}
)
(length, F:
{() -> length([nil, z])
(length([ll, _vj])) -> length([cons(x, ll), s(_vj)])}
(length([_wj, _xj]) /\ length([_wj, _yj])) -> eq_nat([_xj, _yj])
)
}

properties:
{(length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak])}


over-approximation: {length}
under-approximation: {le}

Clause system for inference is:

{
() -> le([z, s(nn2)]) -> 0
() -> length([nil, z]) -> 0
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 0
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 0
(le([s(nn1), z])) -> BOT -> 0
(le([z, z])) -> BOT -> 0
(length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 0
(length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 0
}


Solving took 0.033818 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_2173, q_gen_2174},
Q_f={q_gen_2173},
Delta=
{
<s>(q_gen_2174) -> q_gen_2174
<z>() -> q_gen_2174
<s, s>(q_gen_2173) -> q_gen_2173
<z, s>(q_gen_2174) -> q_gen_2173
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2172, q_gen_2176},
Q_f={q_gen_2172},
Delta=
{
<s>(q_gen_2176) -> q_gen_2176
<z>() -> q_gen_2176
<cons, s>(q_gen_2176, q_gen_2172) -> q_gen_2172
<nil, z>() -> q_gen_2172
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.004280 s (model generation: 0.004171,  model checking: 0.000109):

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

}

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

}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 0
() -> length([nil, z]) -> 3
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 1
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 1
(le([s(nn1), z])) -> BOT -> 1
(le([z, z])) -> BOT -> 1
(length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 1
(length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 1
}
Sat witness: Found: (() -> length([nil, z]), {

})

-------------------------------------------
Step 1, which took 0.004638 s (model generation: 0.004508,  model checking: 0.000130):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2172},
Q_f={q_gen_2172},
Delta=
{
<nil, z>() -> q_gen_2172
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3
() -> length([nil, z]) -> 3
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 1
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 1
(le([s(nn1), z])) -> BOT -> 1
(le([z, z])) -> BOT -> 1
(length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 1
(length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 1
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.005212 s (model generation: 0.004866,  model checking: 0.000346):

Model:
|_
{
le -> 
{{{
Q={q_gen_2173, q_gen_2174},
Q_f={q_gen_2173},
Delta=
{
<z>() -> q_gen_2174
<z, s>(q_gen_2174) -> q_gen_2173
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2172},
Q_f={q_gen_2172},
Delta=
{
<nil, z>() -> q_gen_2172
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3
() -> length([nil, z]) -> 3
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 1
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 1
(le([s(nn1), z])) -> BOT -> 1
(le([z, z])) -> BOT -> 1
(length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 1
(length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 4
}
Sat witness: Found: ((length([ll, _vj])) -> length([cons(x, ll), s(_vj)]), {
_vj -> z  ;  ll -> nil  ;  x -> z
})

-------------------------------------------
Step 3, which took 0.004242 s (model generation: 0.003943,  model checking: 0.000299):

Model:
|_
{
le -> 
{{{
Q={q_gen_2173, q_gen_2174},
Q_f={q_gen_2173},
Delta=
{
<z>() -> q_gen_2174
<z, s>(q_gen_2174) -> q_gen_2173
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2172, q_gen_2176},
Q_f={q_gen_2172},
Delta=
{
<z>() -> q_gen_2176
<cons, s>(q_gen_2176, q_gen_2172) -> q_gen_2172
<nil, z>() -> q_gen_2172
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3
() -> length([nil, z]) -> 3
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 1
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 1
(le([s(nn1), z])) -> BOT -> 1
(le([z, z])) -> BOT -> 1
(length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 4
(length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 4
}
Sat witness: Found: ((length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]), {
_ak -> s(s(z))  ;  _zj -> s(z)  ;  l -> cons(z, nil)  ;  x -> z
})

-------------------------------------------
Step 4, which took 0.004466 s (model generation: 0.004214,  model checking: 0.000252):

Model:
|_
{
le -> 
{{{
Q={q_gen_2173, q_gen_2174},
Q_f={q_gen_2173},
Delta=
{
<z>() -> q_gen_2174
<s, s>(q_gen_2173) -> q_gen_2173
<z, s>(q_gen_2174) -> q_gen_2173
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2172, q_gen_2176},
Q_f={q_gen_2172},
Delta=
{
<z>() -> q_gen_2176
<cons, s>(q_gen_2176, q_gen_2172) -> q_gen_2172
<nil, z>() -> q_gen_2172
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6
() -> length([nil, z]) -> 4
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 2
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 2
(le([s(nn1), z])) -> BOT -> 2
(le([z, z])) -> BOT -> 2
(length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 4
(length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 4
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 5, which took 0.005449 s (model generation: 0.004967,  model checking: 0.000482):

Model:
|_
{
le -> 
{{{
Q={q_gen_2173, q_gen_2174},
Q_f={q_gen_2173},
Delta=
{
<s>(q_gen_2174) -> q_gen_2174
<z>() -> q_gen_2174
<s, s>(q_gen_2173) -> q_gen_2173
<z, s>(q_gen_2174) -> q_gen_2173
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_2172, q_gen_2176},
Q_f={q_gen_2172},
Delta=
{
<z>() -> q_gen_2176
<cons, s>(q_gen_2176, q_gen_2172) -> q_gen_2172
<nil, z>() -> q_gen_2172
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6
() -> length([nil, z]) -> 4
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 3
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 3
(le([s(nn1), z])) -> BOT -> 3
(le([z, z])) -> BOT -> 3
(length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 4
(length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 7
}
Sat witness: Found: ((length([ll, _vj])) -> length([cons(x, ll), s(_vj)]), {
_vj -> z  ;  ll -> nil  ;  x -> s(z)
})

Total time: 0.033818
Reason for stopping: Proved