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


Learning problem is:

env: {
nat -> {s, z}
}
definition:
{
(leq, P:
{() -> leq([z, n2])
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)])
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2])
(leq([s(nn1), z])) -> BOT}
)
(min, F:
{() -> min([s(u), z, z])
() -> min([z, y, z])
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)])}
(min([_jm, _km, _lm]) /\ min([_jm, _km, _mm])) -> eq_nat([_lm, _mm])
)
}

properties:
{(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]), (min([i, j, i])) -> leq([i, j])}


over-approximation: {min}
under-approximation: {}

Clause system for inference is:

{
() -> leq([z, n2]) -> 0
() -> min([s(u), z, z]) -> 0
() -> min([z, y, z]) -> 0
(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]) -> 0
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 0
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 0
(leq([s(nn1), z])) -> BOT -> 0
(min([i, j, i])) -> leq([i, j]) -> 0
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]) -> 0
}


Solving took 0.102202 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_2565, q_gen_2571},
Q_f={q_gen_2565},
Delta=
{
<s>(q_gen_2571) -> q_gen_2571
<z>() -> q_gen_2571
<s, s>(q_gen_2565) -> q_gen_2565
<z, s>(q_gen_2571) -> q_gen_2565
<z, z>() -> q_gen_2565
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_2562, q_gen_2564},
Q_f={q_gen_2562},
Delta=
{
<s>(q_gen_2564) -> q_gen_2564
<z>() -> q_gen_2564
<s, s, s>(q_gen_2562) -> q_gen_2562
<s, z, z>(q_gen_2564) -> q_gen_2562
<z, s, z>(q_gen_2564) -> q_gen_2562
<z, z, z>() -> q_gen_2562
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.010482 s (model generation: 0.009908,  model checking: 0.000574):

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

}

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

}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 0
() -> min([s(u), z, z]) -> 0
() -> min([z, y, z]) -> 3
(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]) -> 1
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 1
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 1
(leq([s(nn1), z])) -> BOT -> 1
(min([i, j, i])) -> leq([i, j]) -> 1
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]) -> 1
}
Sat witness: Found: (() -> min([z, y, z]), {
y -> z
})

-------------------------------------------
Step 1, which took 0.010335 s (model generation: 0.010140,  model checking: 0.000195):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_2562},
Q_f={q_gen_2562},
Delta=
{
<z, z, z>() -> q_gen_2562
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 0
() -> min([s(u), z, z]) -> 3
() -> min([z, y, z]) -> 3
(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]) -> 1
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 1
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 1
(leq([s(nn1), z])) -> BOT -> 1
(min([i, j, i])) -> leq([i, j]) -> 1
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]) -> 1
}
Sat witness: Found: (() -> min([s(u), z, z]), {
u -> z
})

-------------------------------------------
Step 2, which took 0.010521 s (model generation: 0.010418,  model checking: 0.000103):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_2562, q_gen_2564},
Q_f={q_gen_2562},
Delta=
{
<z>() -> q_gen_2564
<s, z, z>(q_gen_2564) -> q_gen_2562
<z, z, z>() -> q_gen_2562
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
() -> min([s(u), z, z]) -> 3
() -> min([z, y, z]) -> 3
(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]) -> 1
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 1
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 1
(leq([s(nn1), z])) -> BOT -> 1
(min([i, j, i])) -> leq([i, j]) -> 1
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]) -> 1
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> z
})

-------------------------------------------
Step 3, which took 0.010758 s (model generation: 0.010434,  model checking: 0.000324):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2565},
Q_f={q_gen_2565},
Delta=
{
<z, z>() -> q_gen_2565
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_2562, q_gen_2564},
Q_f={q_gen_2562},
Delta=
{
<z>() -> q_gen_2564
<s, z, z>(q_gen_2564) -> q_gen_2562
<z, z, z>() -> q_gen_2562
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
() -> min([s(u), z, z]) -> 3
() -> min([z, y, z]) -> 3
(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]) -> 1
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 1
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 1
(leq([s(nn1), z])) -> BOT -> 1
(min([i, j, i])) -> leq([i, j]) -> 1
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]) -> 4
}
Sat witness: Found: ((min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]), {
_im -> z  ;  u -> z  ;  y1 -> z
})

-------------------------------------------
Step 4, which took 0.010803 s (model generation: 0.010644,  model checking: 0.000159):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2565},
Q_f={q_gen_2565},
Delta=
{
<z, z>() -> q_gen_2565
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_2562, q_gen_2564},
Q_f={q_gen_2562},
Delta=
{
<z>() -> q_gen_2564
<s, s, s>(q_gen_2562) -> q_gen_2562
<s, z, z>(q_gen_2564) -> q_gen_2562
<z, z, z>() -> q_gen_2562
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
() -> min([s(u), z, z]) -> 3
() -> min([z, y, z]) -> 3
(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]) -> 1
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 1
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 1
(leq([s(nn1), z])) -> BOT -> 1
(min([i, j, i])) -> leq([i, j]) -> 4
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]) -> 4
}
Sat witness: Found: ((min([i, j, i])) -> leq([i, j]), {
i -> s(z)  ;  j -> s(z)
})

-------------------------------------------
Step 5, which took 0.011799 s (model generation: 0.010735,  model checking: 0.001064):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2565},
Q_f={q_gen_2565},
Delta=
{
<s, s>(q_gen_2565) -> q_gen_2565
<z, z>() -> q_gen_2565
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_2562, q_gen_2564},
Q_f={q_gen_2562},
Delta=
{
<z>() -> q_gen_2564
<s, s, s>(q_gen_2562) -> q_gen_2562
<s, z, z>(q_gen_2564) -> q_gen_2562
<z, z, z>() -> q_gen_2562
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
() -> min([s(u), z, z]) -> 3
() -> min([z, y, z]) -> 6
(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]) -> 2
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 2
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 2
(leq([s(nn1), z])) -> BOT -> 2
(min([i, j, i])) -> leq([i, j]) -> 4
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]) -> 4
}
Sat witness: Found: (() -> min([z, y, z]), {
y -> s(s(z))
})

-------------------------------------------
Step 6, which took 0.011757 s (model generation: 0.011232,  model checking: 0.000525):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2565},
Q_f={q_gen_2565},
Delta=
{
<s, s>(q_gen_2565) -> q_gen_2565
<z, z>() -> q_gen_2565
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_2562, q_gen_2564},
Q_f={q_gen_2562},
Delta=
{
<s>(q_gen_2564) -> q_gen_2564
<z>() -> q_gen_2564
<s, s, s>(q_gen_2562) -> q_gen_2562
<s, z, z>(q_gen_2564) -> q_gen_2562
<z, s, z>(q_gen_2564) -> q_gen_2562
<z, z, z>() -> q_gen_2562
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6
() -> min([s(u), z, z]) -> 4
() -> min([z, y, z]) -> 6
(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]) -> 3
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 3
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 3
(leq([s(nn1), z])) -> BOT -> 3
(min([i, j, i])) -> leq([i, j]) -> 4
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]) -> 4
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(z)
})

-------------------------------------------
Step 7, which took 0.012580 s (model generation: 0.011561,  model checking: 0.001019):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2565, q_gen_2571},
Q_f={q_gen_2565},
Delta=
{
<z>() -> q_gen_2571
<s, s>(q_gen_2565) -> q_gen_2565
<z, s>(q_gen_2571) -> q_gen_2565
<z, z>() -> q_gen_2565
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_2562, q_gen_2564},
Q_f={q_gen_2562},
Delta=
{
<s>(q_gen_2564) -> q_gen_2564
<z>() -> q_gen_2564
<s, s, s>(q_gen_2562) -> q_gen_2562
<s, z, z>(q_gen_2564) -> q_gen_2562
<z, s, z>(q_gen_2564) -> q_gen_2562
<z, z, z>() -> q_gen_2562
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6
() -> min([s(u), z, z]) -> 4
() -> min([z, y, z]) -> 6
(leq([i, j]) /\ min([i, j, _om])) -> eq_nat([_om, i]) -> 4
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 4
(leq([s(nn1), z])) -> BOT -> 4
(min([i, j, i])) -> leq([i, j]) -> 7
(min([u, y1, _im])) -> min([s(u), s(y1), s(_im)]) -> 5
}
Sat witness: Found: ((min([i, j, i])) -> leq([i, j]), {
i -> z  ;  j -> s(s(z))
})

Total time: 0.102202
Reason for stopping: Proved