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, _dfa])) -> double([s(nn), s(s(_dfa))])}
(double([_efa, _ffa]) /\ double([_efa, _gfa])) -> eq_nat([_ffa, _gfa])
)
(is_zero, P:
{(double([n, _hfa]) /\ is_zero([n]) /\ not eq_nat([_hfa, 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, _hfa]) /\ is_zero([n]) /\ not eq_nat([_hfa, z])) -> BOT -> 0
(double([n, z])) -> is_zero([n]) -> 0
(double([nn, _dfa])) -> double([s(nn), s(s(_dfa))]) -> 0
}


Solving took 0.198033 seconds.
Proved
Model:
|_
{
double -> 
{{{
Q={q_gen_8404, q_gen_8406, q_gen_8407},
Q_f={q_gen_8404},
Delta=
{
<s>(q_gen_8407) -> q_gen_8407
<z>() -> q_gen_8407
<s, s>(q_gen_8404) -> q_gen_8404
<s, s>(q_gen_8406) -> q_gen_8404
<z, z>() -> q_gen_8404
<z, s>(q_gen_8407) -> q_gen_8406
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_8403},
Q_f={q_gen_8403},
Delta=
{
<z>() -> q_gen_8403
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.066617 s (model generation: 0.066424,  model checking: 0.000193):

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

})

-------------------------------------------
Step 1, which took 0.021118 s (model generation: 0.021016,  model checking: 0.000102):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_8403},
Q_f={q_gen_8403},
Delta=
{
<z>() -> q_gen_8403
}

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

})

-------------------------------------------
Step 2, which took 0.019939 s (model generation: 0.019730,  model checking: 0.000209):

Model:
|_
{
double -> 
{{{
Q={q_gen_8404},
Q_f={q_gen_8404},
Delta=
{
<z, z>() -> q_gen_8404
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_8403},
Q_f={q_gen_8403},
Delta=
{
<z>() -> q_gen_8403
}

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

-------------------------------------------
Step 3, which took 0.021093 s (model generation: 0.020890,  model checking: 0.000203):

Model:
|_
{
double -> 
{{{
Q={q_gen_8404, q_gen_8407},
Q_f={q_gen_8404},
Delta=
{
<z>() -> q_gen_8407
<s, s>(q_gen_8404) -> q_gen_8404
<z, s>(q_gen_8407) -> q_gen_8404
<z, z>() -> q_gen_8404
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_8403},
Q_f={q_gen_8403},
Delta=
{
<z>() -> q_gen_8403
}

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

-------------------------------------------
Step 4, which took 0.043317 s (model generation: 0.022842,  model checking: 0.020475):

Model:
|_
{
double -> 
{{{
Q={q_gen_8404, q_gen_8406, q_gen_8407},
Q_f={q_gen_8404},
Delta=
{
<z>() -> q_gen_8407
<s, s>(q_gen_8406) -> q_gen_8404
<z, z>() -> q_gen_8404
<z, s>(q_gen_8407) -> q_gen_8406
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  is_zero -> 
{{{
Q={q_gen_8403},
Q_f={q_gen_8403},
Delta=
{
<z>() -> q_gen_8403
}

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

Total time: 0.198033
Reason for stopping: Proved