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


Learning problem is:

env: {
nat -> {s, z}
}
definition:
{
(double, F:
{() -> double([z, z])
(double([nn, _dca])) -> double([s(nn), s(s(_dca))])}
(double([_eca, _fca]) /\ double([_eca, _gca])) -> eq_nat([_fca, _gca])
)
(is_zero, P:
{(double([n, _hca]) /\ is_zero([n])) -> eq_nat([_hca, 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, _hca]) /\ is_zero([n])) -> eq_nat([_hca, z]) -> 0
(double([n, z])) -> is_zero([n]) -> 0
(double([nn, _dca])) -> double([s(nn), s(s(_dca))]) -> 0
}


Solving took 0.065655 seconds.
Proved
Model:
|_
{
double -> 
{{{
Q={q_gen_5625, q_gen_5627, q_gen_5628},
Q_f={q_gen_5625},
Delta=
{
<s>(q_gen_5628) -> q_gen_5628
<z>() -> q_gen_5628
<s, s>(q_gen_5625) -> q_gen_5625
<s, s>(q_gen_5627) -> q_gen_5625
<z, z>() -> q_gen_5625
<z, s>(q_gen_5628) -> q_gen_5627
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_5624},
Q_f={q_gen_5624},
Delta=
{
<z>() -> q_gen_5624
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.010115 s (model generation: 0.009836,  model checking: 0.000279):

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

}

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

}

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

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

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

})

-------------------------------------------
Step 1, which took 0.010179 s (model generation: 0.010099,  model checking: 0.000080):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_5624},
Q_f={q_gen_5624},
Delta=
{
<z>() -> q_gen_5624
}

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

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

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

})

-------------------------------------------
Step 2, which took 0.010305 s (model generation: 0.010168,  model checking: 0.000137):

Model:
|_
{
double -> 
{{{
Q={q_gen_5625},
Q_f={q_gen_5625},
Delta=
{
<z, z>() -> q_gen_5625
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_5624},
Q_f={q_gen_5624},
Delta=
{
<z>() -> q_gen_5624
}

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

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

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

-------------------------------------------
Step 3, which took 0.010848 s (model generation: 0.010689,  model checking: 0.000159):

Model:
|_
{
double -> 
{{{
Q={q_gen_5625, q_gen_5628},
Q_f={q_gen_5625},
Delta=
{
<z>() -> q_gen_5628
<s, s>(q_gen_5625) -> q_gen_5625
<z, s>(q_gen_5628) -> q_gen_5625
<z, z>() -> q_gen_5625
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_5624},
Q_f={q_gen_5624},
Delta=
{
<z>() -> q_gen_5624
}

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

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

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

-------------------------------------------
Step 4, which took 0.011309 s (model generation: 0.010845,  model checking: 0.000464):

Model:
|_
{
double -> 
{{{
Q={q_gen_5625, q_gen_5627, q_gen_5628},
Q_f={q_gen_5625},
Delta=
{
<z>() -> q_gen_5628
<s, s>(q_gen_5627) -> q_gen_5625
<z, z>() -> q_gen_5625
<z, s>(q_gen_5628) -> q_gen_5627
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_5624},
Q_f={q_gen_5624},
Delta=
{
<z>() -> q_gen_5624
}

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

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

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

Total time: 0.065655
Reason for stopping: Proved