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


Learning problem is:

env: {
nat -> {s, z}
}
definition:
{
(double, F:
{() -> double([z, z])
(double([nn, _saa])) -> double([s(nn), s(s(_saa))])}
(double([_taa, _uaa]) /\ double([_taa, _vaa])) -> eq_nat([_uaa, _vaa])
)
(is_zero, P:
{(double([n, _waa]) /\ is_zero([n])) -> eq_nat([_waa, z])
(double([n, z])) -> is_zero([n])}
)
}

properties:
{() -> is_zero([z])}


over-approximation: {double}
under-approximation: {is_zero}

Clause system for inference is:

{
() -> double([z, z]) -> 0
() -> is_zero([z]) -> 0
(double([n, _waa]) /\ is_zero([n])) -> eq_nat([_waa, z]) -> 0
(double([n, z])) -> is_zero([n]) -> 0
(double([nn, _saa])) -> double([s(nn), s(s(_saa))]) -> 0
}


Solving took 0.073851 seconds.
Proved
Model:
|_
{
double -> 
{{{
Q={q_gen_7148, q_gen_7150, q_gen_7151},
Q_f={q_gen_7148},
Delta=
{
<s>(q_gen_7151) -> q_gen_7151
<z>() -> q_gen_7151
<s, s>(q_gen_7148) -> q_gen_7148
<s, s>(q_gen_7150) -> q_gen_7148
<z, z>() -> q_gen_7148
<z, s>(q_gen_7151) -> q_gen_7150
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_7147},
Q_f={q_gen_7147},
Delta=
{
<z>() -> q_gen_7147
}

Datatype: <nat>
Convolution form: left
}}}

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.030528 s (model generation: 0.030431,  model checking: 0.000097):

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

}

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

}

Datatype: <nat>
Convolution form: left
}}}

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 0
() -> is_zero([z]) -> 3
(double([n, _waa]) /\ is_zero([n])) -> eq_nat([_waa, z]) -> 1
(double([n, z])) -> is_zero([n]) -> 1
(double([nn, _saa])) -> double([s(nn), s(s(_saa))]) -> 1
}
Sat witness: Found: (() -> is_zero([z]), {

})

-------------------------------------------
Step 1, which took 0.008516 s (model generation: 0.008470,  model checking: 0.000046):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_7147},
Q_f={q_gen_7147},
Delta=
{
<z>() -> q_gen_7147
}

Datatype: <nat>
Convolution form: left
}}}

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> is_zero([z]) -> 3
(double([n, _waa]) /\ is_zero([n])) -> eq_nat([_waa, z]) -> 1
(double([n, z])) -> is_zero([n]) -> 1
(double([nn, _saa])) -> double([s(nn), s(s(_saa))]) -> 1
}
Sat witness: Found: (() -> double([z, z]), {

})

-------------------------------------------
Step 2, which took 0.008184 s (model generation: 0.008075,  model checking: 0.000109):

Model:
|_
{
double -> 
{{{
Q={q_gen_7148},
Q_f={q_gen_7148},
Delta=
{
<z, z>() -> q_gen_7148
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_7147},
Q_f={q_gen_7147},
Delta=
{
<z>() -> q_gen_7147
}

Datatype: <nat>
Convolution form: left
}}}

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> is_zero([z]) -> 3
(double([n, _waa]) /\ is_zero([n])) -> eq_nat([_waa, z]) -> 1
(double([n, z])) -> is_zero([n]) -> 1
(double([nn, _saa])) -> double([s(nn), s(s(_saa))]) -> 4
}
Sat witness: Found: ((double([nn, _saa])) -> double([s(nn), s(s(_saa))]), {
_saa -> z  ;  nn -> z
})

-------------------------------------------
Step 3, which took 0.007796 s (model generation: 0.007708,  model checking: 0.000088):

Model:
|_
{
double -> 
{{{
Q={q_gen_7148, q_gen_7151},
Q_f={q_gen_7148},
Delta=
{
<z>() -> q_gen_7151
<s, s>(q_gen_7148) -> q_gen_7148
<z, s>(q_gen_7151) -> q_gen_7148
<z, z>() -> q_gen_7148
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_7147},
Q_f={q_gen_7147},
Delta=
{
<z>() -> q_gen_7147
}

Datatype: <nat>
Convolution form: left
}}}

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> is_zero([z]) -> 3
(double([n, _waa]) /\ is_zero([n])) -> eq_nat([_waa, z]) -> 4
(double([n, z])) -> is_zero([n]) -> 2
(double([nn, _saa])) -> double([s(nn), s(s(_saa))]) -> 4
}
Sat witness: Found: ((double([n, _waa]) /\ is_zero([n])) -> eq_nat([_waa, z]), {
_waa -> s(z)  ;  n -> z
})

-------------------------------------------
Step 4, which took 0.007768 s (model generation: 0.007567,  model checking: 0.000201):

Model:
|_
{
double -> 
{{{
Q={q_gen_7148, q_gen_7150, q_gen_7151},
Q_f={q_gen_7148},
Delta=
{
<z>() -> q_gen_7151
<s, s>(q_gen_7150) -> q_gen_7148
<z, z>() -> q_gen_7148
<z, s>(q_gen_7151) -> q_gen_7150
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_7147},
Q_f={q_gen_7147},
Delta=
{
<z>() -> q_gen_7147
}

Datatype: <nat>
Convolution form: left
}}}

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 4
() -> is_zero([z]) -> 4
(double([n, _waa]) /\ is_zero([n])) -> eq_nat([_waa, z]) -> 4
(double([n, z])) -> is_zero([n]) -> 3
(double([nn, _saa])) -> double([s(nn), s(s(_saa))]) -> 7
}
Sat witness: Found: ((double([nn, _saa])) -> double([s(nn), s(s(_saa))]), {
_saa -> s(s(z))  ;  nn -> s(z)
})

Total time: 0.073851
Reason for stopping: Proved