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, _iw])) -> double([s(nn), s(s(_iw))])}
(double([_jw, _kw]) /\ double([_jw, _lw])) -> eq_nat([_kw, _lw])
)
(le, P:
{() -> le([z, s(nn2)])
(le([nn1, nn2])) -> le([s(nn1), s(nn2)])
(le([s(nn1), s(nn2)])) -> le([nn1, nn2])
(le([s(nn1), z])) -> BOT
(le([z, z])) -> BOT}
)
}

properties:
{(double([n, _mw]) /\ le([z, n])) -> le([n, _mw])}


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

Clause system for inference is:

{
() -> double([z, z]) -> 0
() -> le([z, s(nn2)]) -> 0
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 0
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 0
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 0
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 0
(le([s(nn1), z])) -> BOT -> 0
(le([z, z])) -> BOT -> 0
}


Solving took 0.126960 seconds.
Proved
Model:
|_
{
double -> 
{{{
Q={q_gen_5954, q_gen_5956, q_gen_5958, q_gen_5959},
Q_f={q_gen_5954, q_gen_5956},
Delta=
{
<s>(q_gen_5958) -> q_gen_5958
<z>() -> q_gen_5958
<z, z>() -> q_gen_5954
<s, s>(q_gen_5956) -> q_gen_5956
<z, s>(q_gen_5958) -> q_gen_5956
<s, s>(q_gen_5954) -> q_gen_5959
<s, s>(q_gen_5959) -> q_gen_5959
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953, q_gen_5960},
Q_f={q_gen_5952},
Delta=
{
<s>(q_gen_5953) -> q_gen_5953
<z>() -> q_gen_5953
<s, s>(q_gen_5952) -> q_gen_5952
<z, s>(q_gen_5953) -> q_gen_5952
<s, s>(q_gen_5960) -> q_gen_5960
<z, z>() -> q_gen_5960
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.019163 s (model generation: 0.019026,  model checking: 0.000137):

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

}

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

}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 0
() -> le([z, s(nn2)]) -> 3
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 1
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 1
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 1
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 1
(le([s(nn1), z])) -> BOT -> 1
(le([z, z])) -> BOT -> 1
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 1, which took 0.009227 s (model generation: 0.009184,  model checking: 0.000043):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953},
Q_f={q_gen_5952},
Delta=
{
<z>() -> q_gen_5953
<z, s>(q_gen_5953) -> q_gen_5952
}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> le([z, s(nn2)]) -> 3
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 1
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 1
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 1
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 1
(le([s(nn1), z])) -> BOT -> 1
(le([z, z])) -> BOT -> 1
}
Sat witness: Found: (() -> double([z, z]), {

})

-------------------------------------------
Step 2, which took 0.008595 s (model generation: 0.008461,  model checking: 0.000134):

Model:
|_
{
double -> 
{{{
Q={q_gen_5954},
Q_f={q_gen_5954},
Delta=
{
<z, z>() -> q_gen_5954
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953},
Q_f={q_gen_5952},
Delta=
{
<z>() -> q_gen_5953
<z, s>(q_gen_5953) -> q_gen_5952
}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> le([z, s(nn2)]) -> 3
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 1
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 1
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 2
(le([s(nn1), z])) -> BOT -> 2
(le([z, z])) -> BOT -> 2
}
Sat witness: Found: ((le([nn1, nn2])) -> le([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> s(z)
})

-------------------------------------------
Step 3, which took 0.009084 s (model generation: 0.008966,  model checking: 0.000118):

Model:
|_
{
double -> 
{{{
Q={q_gen_5954},
Q_f={q_gen_5954},
Delta=
{
<z, z>() -> q_gen_5954
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953},
Q_f={q_gen_5952},
Delta=
{
<z>() -> q_gen_5953
<s, s>(q_gen_5952) -> q_gen_5952
<z, s>(q_gen_5953) -> q_gen_5952
}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> le([z, s(nn2)]) -> 3
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 1
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 4
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 2
(le([s(nn1), z])) -> BOT -> 2
(le([z, z])) -> BOT -> 2
}
Sat witness: Found: ((double([nn, _iw])) -> double([s(nn), s(s(_iw))]), {
_iw -> z  ;  nn -> z
})

-------------------------------------------
Step 4, which took 0.009912 s (model generation: 0.009817,  model checking: 0.000095):

Model:
|_
{
double -> 
{{{
Q={q_gen_5954, q_gen_5958},
Q_f={q_gen_5954},
Delta=
{
<z>() -> q_gen_5958
<s, s>(q_gen_5954) -> q_gen_5954
<z, s>(q_gen_5958) -> q_gen_5954
<z, z>() -> q_gen_5954
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953},
Q_f={q_gen_5952},
Delta=
{
<z>() -> q_gen_5953
<s, s>(q_gen_5952) -> q_gen_5952
<z, s>(q_gen_5953) -> q_gen_5952
}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> le([z, s(nn2)]) -> 3
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 4
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 4
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 2
(le([s(nn1), z])) -> BOT -> 2
(le([z, z])) -> BOT -> 2
}
Sat witness: Found: ((double([n, _mw]) /\ le([z, n])) -> le([n, _mw]), {
_mw -> s(z)  ;  n -> s(z)
})

-------------------------------------------
Step 5, which took 0.009730 s (model generation: 0.009712,  model checking: 0.000018):

Model:
|_
{
double -> 
{{{
Q={q_gen_5954, q_gen_5958},
Q_f={q_gen_5954},
Delta=
{
<z>() -> q_gen_5958
<s, s>(q_gen_5954) -> q_gen_5954
<z, s>(q_gen_5958) -> q_gen_5954
<z, z>() -> q_gen_5954
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953},
Q_f={q_gen_5952},
Delta=
{
<z>() -> q_gen_5953
<s, s>(q_gen_5952) -> q_gen_5952
<z, s>(q_gen_5953) -> q_gen_5952
<z, z>() -> q_gen_5952
}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> le([z, s(nn2)]) -> 3
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 4
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 4
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 2
(le([s(nn1), z])) -> BOT -> 2
(le([z, z])) -> BOT -> 5
}
Sat witness: Found: ((le([z, z])) -> BOT, {

})

-------------------------------------------
Step 6, which took 0.008872 s (model generation: 0.008795,  model checking: 0.000077):

Model:
|_
{
double -> 
{{{
Q={q_gen_5954, q_gen_5958},
Q_f={q_gen_5954},
Delta=
{
<z>() -> q_gen_5958
<s, s>(q_gen_5954) -> q_gen_5954
<z, s>(q_gen_5958) -> q_gen_5954
<z, z>() -> q_gen_5954
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953, q_gen_5961},
Q_f={q_gen_5952},
Delta=
{
<z>() -> q_gen_5953
<s, s>(q_gen_5952) -> q_gen_5952
<s, s>(q_gen_5961) -> q_gen_5952
<z, s>(q_gen_5953) -> q_gen_5952
<z, z>() -> q_gen_5961
}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> le([z, s(nn2)]) -> 3
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 4
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 4
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 5
(le([s(nn1), z])) -> BOT -> 3
(le([z, z])) -> BOT -> 5
}
Sat witness: Found: ((le([s(nn1), s(nn2)])) -> le([nn1, nn2]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 7, which took 0.012454 s (model generation: 0.012346,  model checking: 0.000108):

Model:
|_
{
double -> 
{{{
Q={q_gen_5954, q_gen_5957, q_gen_5958},
Q_f={q_gen_5954},
Delta=
{
<z>() -> q_gen_5958
<s, s>(q_gen_5957) -> q_gen_5954
<z, z>() -> q_gen_5954
<s, s>(q_gen_5954) -> q_gen_5957
<z, s>(q_gen_5958) -> q_gen_5957
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953, q_gen_5960},
Q_f={q_gen_5952},
Delta=
{
<z>() -> q_gen_5953
<s, s>(q_gen_5952) -> q_gen_5952
<z, s>(q_gen_5953) -> q_gen_5952
<s, s>(q_gen_5960) -> q_gen_5960
<z, z>() -> q_gen_5960
}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 3
() -> le([z, s(nn2)]) -> 6
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 4
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 4
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 5
(le([s(nn1), z])) -> BOT -> 4
(le([z, z])) -> BOT -> 5
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 8, which took 0.014050 s (model generation: 0.013723,  model checking: 0.000327):

Model:
|_
{
double -> 
{{{
Q={q_gen_5954, q_gen_5957, q_gen_5958},
Q_f={q_gen_5954},
Delta=
{
<z>() -> q_gen_5958
<s, s>(q_gen_5957) -> q_gen_5954
<z, z>() -> q_gen_5954
<s, s>(q_gen_5954) -> q_gen_5957
<z, s>(q_gen_5958) -> q_gen_5957
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953, q_gen_5960},
Q_f={q_gen_5952},
Delta=
{
<s>(q_gen_5953) -> q_gen_5953
<z>() -> q_gen_5953
<s, s>(q_gen_5952) -> q_gen_5952
<z, s>(q_gen_5953) -> q_gen_5952
<s, s>(q_gen_5960) -> q_gen_5960
<z, z>() -> q_gen_5960
}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 4
() -> le([z, s(nn2)]) -> 6
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 4
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 7
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 5
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 5
(le([s(nn1), z])) -> BOT -> 5
(le([z, z])) -> BOT -> 5
}
Sat witness: Found: ((double([nn, _iw])) -> double([s(nn), s(s(_iw))]), {
_iw -> s(s(z))  ;  nn -> s(z)
})

-------------------------------------------
Step 9, which took 0.014439 s (model generation: 0.014338,  model checking: 0.000101):

Model:
|_
{
double -> 
{{{
Q={q_gen_5954, q_gen_5957, q_gen_5958, q_gen_5967},
Q_f={q_gen_5954},
Delta=
{
<z>() -> q_gen_5958
<s>(q_gen_5958) -> q_gen_5967
<s, s>(q_gen_5957) -> q_gen_5954
<z, s>(q_gen_5967) -> q_gen_5954
<z, z>() -> q_gen_5954
<s, s>(q_gen_5954) -> q_gen_5957
<z, s>(q_gen_5958) -> q_gen_5957
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  le -> 
{{{
Q={q_gen_5952, q_gen_5953, q_gen_5960},
Q_f={q_gen_5952},
Delta=
{
<s>(q_gen_5953) -> q_gen_5953
<z>() -> q_gen_5953
<s, s>(q_gen_5952) -> q_gen_5952
<z, s>(q_gen_5953) -> q_gen_5952
<s, s>(q_gen_5960) -> q_gen_5960
<z, z>() -> q_gen_5960
}

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

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

Teacher's answer:
New clause system:
{
() -> double([z, z]) -> 4
() -> le([z, s(nn2)]) -> 6
(double([n, _mw]) /\ le([z, n])) -> le([n, _mw]) -> 7
(double([nn, _iw])) -> double([s(nn), s(s(_iw))]) -> 7
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 5
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 5
(le([s(nn1), z])) -> BOT -> 5
(le([z, z])) -> BOT -> 5
}
Sat witness: Found: ((double([n, _mw]) /\ le([z, n])) -> le([n, _mw]), {
_mw -> s(s(z))  ;  n -> s(s(z))
})

Total time: 0.126960
Reason for stopping: Proved