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


Learning problem is:

env: {
nat -> {s, z}  ;  natlist -> {cons, nil}
}
definition:
{
(length, F:
{() -> length([nil, z])
(length([ll, _yaa])) -> length([cons(x, ll), s(_yaa)])}
(length([_zaa, _aba]) /\ length([_zaa, _bba])) -> eq_nat([_aba, _bba])
)
}

properties:
{(length([l, _dba]) /\ length([cons(z, cons(z, l)), _cba])) -> eq_nat([_cba, s(s(_dba))])}


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

Clause system for inference is:

{
() -> length([nil, z]) -> 0
(length([l, _dba]) /\ length([cons(z, cons(z, l)), _cba])) -> eq_nat([_cba, s(s(_dba))]) -> 0
(length([ll, _yaa])) -> length([cons(x, ll), s(_yaa)]) -> 0
}


Solving took 0.022619 seconds.
Proved
Model:
|_
{
length -> 
{{{
Q={q_gen_5511, q_gen_5513},
Q_f={q_gen_5511},
Delta=
{
<s>(q_gen_5513) -> q_gen_5513
<z>() -> q_gen_5513
<cons, s>(q_gen_5513, q_gen_5511) -> q_gen_5511
<nil, z>() -> q_gen_5511
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.006193 s (model generation: 0.005906,  model checking: 0.000287):

Model:
|_
{
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:
{
() -> length([nil, z]) -> 3
(length([l, _dba]) /\ length([cons(z, cons(z, l)), _cba])) -> eq_nat([_cba, s(s(_dba))]) -> 1
(length([ll, _yaa])) -> length([cons(x, ll), s(_yaa)]) -> 1
}
Sat witness: Found: (() -> length([nil, z]), {

})

-------------------------------------------
Step 1, which took 0.006033 s (model generation: 0.005042,  model checking: 0.000991):

Model:
|_
{
length -> 
{{{
Q={q_gen_5511},
Q_f={q_gen_5511},
Delta=
{
<nil, z>() -> q_gen_5511
}

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

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

Teacher's answer:
New clause system:
{
() -> length([nil, z]) -> 3
(length([l, _dba]) /\ length([cons(z, cons(z, l)), _cba])) -> eq_nat([_cba, s(s(_dba))]) -> 1
(length([ll, _yaa])) -> length([cons(x, ll), s(_yaa)]) -> 4
}
Sat witness: Found: ((length([ll, _yaa])) -> length([cons(x, ll), s(_yaa)]), {
_yaa -> z  ;  ll -> nil  ;  x -> z
})

-------------------------------------------
Step 2, which took 0.004880 s (model generation: 0.004576,  model checking: 0.000304):

Model:
|_
{
length -> 
{{{
Q={q_gen_5511, q_gen_5513},
Q_f={q_gen_5511},
Delta=
{
<z>() -> q_gen_5513
<cons, s>(q_gen_5513, q_gen_5511) -> q_gen_5511
<nil, z>() -> q_gen_5511
}

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

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

Teacher's answer:
New clause system:
{
() -> length([nil, z]) -> 4
(length([l, _dba]) /\ length([cons(z, cons(z, l)), _cba])) -> eq_nat([_cba, s(s(_dba))]) -> 2
(length([ll, _yaa])) -> length([cons(x, ll), s(_yaa)]) -> 7
}
Sat witness: Found: ((length([ll, _yaa])) -> length([cons(x, ll), s(_yaa)]), {
_yaa -> z  ;  ll -> nil  ;  x -> s(z)
})

Total time: 0.022619
Reason for stopping: Proved