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, _rl])) -> double([s(nn), s(s(_rl))])}
(double([_sl, _tl]) /\ double([_sl, _ul])) -> eq_nat([_tl, _ul])
)
(is_zero, P:
{(double([n, _vl]) /\ is_zero([n]) /\ not eq_nat([_vl, z])) -> BOT
(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, _vl]) /\ is_zero([n]) /\ not eq_nat([_vl, z])) -> BOT -> 0
(double([n, z])) -> is_zero([n]) -> 0
(double([nn, _rl])) -> double([s(nn), s(s(_rl))]) -> 0
}


Solving took 0.067268 seconds.
Proved
Model:
|_
{
double -> 
{{{
Q={q_gen_2526, q_gen_2528, q_gen_2529},
Q_f={q_gen_2526},
Delta=
{
<s>(q_gen_2529) -> q_gen_2529
<z>() -> q_gen_2529
<s, s>(q_gen_2526) -> q_gen_2526
<s, s>(q_gen_2528) -> q_gen_2526
<z, z>() -> q_gen_2526
<z, s>(q_gen_2529) -> q_gen_2528
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_2525},
Q_f={q_gen_2525},
Delta=
{
<z>() -> q_gen_2525
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.010004 s (model generation: 0.009823,  model checking: 0.000181):

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, _vl]) /\ is_zero([n]) /\ not eq_nat([_vl, z])) -> BOT -> 1
(double([n, z])) -> is_zero([n]) -> 1
(double([nn, _rl])) -> double([s(nn), s(s(_rl))]) -> 1
}
Sat witness: Found: (() -> is_zero([z]), {

})

-------------------------------------------
Step 1, which took 0.010049 s (model generation: 0.009969,  model checking: 0.000080):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_2525},
Q_f={q_gen_2525},
Delta=
{
<z>() -> q_gen_2525
}

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, _vl]) /\ is_zero([n]) /\ not eq_nat([_vl, z])) -> BOT -> 1
(double([n, z])) -> is_zero([n]) -> 1
(double([nn, _rl])) -> double([s(nn), s(s(_rl))]) -> 1
}
Sat witness: Found: (() -> double([z, z]), {

})

-------------------------------------------
Step 2, which took 0.010354 s (model generation: 0.010215,  model checking: 0.000139):

Model:
|_
{
double -> 
{{{
Q={q_gen_2526},
Q_f={q_gen_2526},
Delta=
{
<z, z>() -> q_gen_2526
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_2525},
Q_f={q_gen_2525},
Delta=
{
<z>() -> q_gen_2525
}

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, _vl]) /\ is_zero([n]) /\ not eq_nat([_vl, z])) -> BOT -> 1
(double([n, z])) -> is_zero([n]) -> 1
(double([nn, _rl])) -> double([s(nn), s(s(_rl))]) -> 4
}
Sat witness: Found: ((double([nn, _rl])) -> double([s(nn), s(s(_rl))]), {
_rl -> z  ;  nn -> z
})

-------------------------------------------
Step 3, which took 0.010788 s (model generation: 0.010631,  model checking: 0.000157):

Model:
|_
{
double -> 
{{{
Q={q_gen_2526, q_gen_2529},
Q_f={q_gen_2526},
Delta=
{
<z>() -> q_gen_2529
<s, s>(q_gen_2526) -> q_gen_2526
<z, s>(q_gen_2529) -> q_gen_2526
<z, z>() -> q_gen_2526
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_2525},
Q_f={q_gen_2525},
Delta=
{
<z>() -> q_gen_2525
}

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, _vl]) /\ is_zero([n]) /\ not eq_nat([_vl, z])) -> BOT -> 4
(double([n, z])) -> is_zero([n]) -> 2
(double([nn, _rl])) -> double([s(nn), s(s(_rl))]) -> 4
}
Sat witness: Found: ((double([n, _vl]) /\ is_zero([n]) /\ not eq_nat([_vl, z])) -> BOT, {
_vl -> s(z)  ;  n -> z
})

-------------------------------------------
Step 4, which took 0.013540 s (model generation: 0.010662,  model checking: 0.002878):

Model:
|_
{
double -> 
{{{
Q={q_gen_2526, q_gen_2528, q_gen_2529},
Q_f={q_gen_2526},
Delta=
{
<z>() -> q_gen_2529
<s, s>(q_gen_2528) -> q_gen_2526
<z, z>() -> q_gen_2526
<z, s>(q_gen_2529) -> q_gen_2528
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  is_zero -> 
{{{
Q={q_gen_2525},
Q_f={q_gen_2525},
Delta=
{
<z>() -> q_gen_2525
}

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, _vl]) /\ is_zero([n]) /\ not eq_nat([_vl, z])) -> BOT -> 4
(double([n, z])) -> is_zero([n]) -> 3
(double([nn, _rl])) -> double([s(nn), s(s(_rl))]) -> 7
}
Sat witness: Found: ((double([nn, _rl])) -> double([s(nn), s(s(_rl))]), {
_rl -> s(s(z))  ;  nn -> s(z)
})

Total time: 0.067268
Reason for stopping: Proved