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


Learning problem is:

env: {
nat -> {s, z}
}
definition:
{
(min, F:
{() -> min([s(u), z, z])
() -> min([z, y, z])
(min([u, y1, _to])) -> min([s(u), s(y1), s(_to)])}
(min([_uo, _vo, _wo]) /\ min([_uo, _vo, _xo])) -> eq_nat([_wo, _xo])
)
}

properties:
{(min([a, b, _yo]) /\ min([b, a, _zo])) -> eq_nat([_yo, _zo])}


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

Clause system for inference is:

{
() -> min([s(u), z, z]) -> 0
() -> min([z, y, z]) -> 0
(min([a, b, _yo]) /\ min([b, a, _zo])) -> eq_nat([_yo, _zo]) -> 0
(min([u, y1, _to])) -> min([s(u), s(y1), s(_to)]) -> 0
}


Solving took 0.047461 seconds.
Proved
Model:
|_
{
min -> 
{{{
Q={q_gen_3031, q_gen_3033},
Q_f={q_gen_3031},
Delta=
{
<s>(q_gen_3033) -> q_gen_3033
<z>() -> q_gen_3033
<s, s, s>(q_gen_3031) -> q_gen_3031
<s, z, z>(q_gen_3033) -> q_gen_3031
<z, s, z>(q_gen_3033) -> q_gen_3031
<z, z, z>() -> q_gen_3031
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.005655 s (model generation: 0.005250,  model checking: 0.000405):

Model:
|_
{
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:
{
() -> min([s(u), z, z]) -> 0
() -> min([z, y, z]) -> 3
(min([a, b, _yo]) /\ min([b, a, _zo])) -> eq_nat([_yo, _zo]) -> 1
(min([u, y1, _to])) -> min([s(u), s(y1), s(_to)]) -> 1
}
Sat witness: Found: (() -> min([z, y, z]), {
y -> z
})

-------------------------------------------
Step 1, which took 0.008024 s (model generation: 0.007841,  model checking: 0.000183):

Model:
|_
{
min -> 
{{{
Q={q_gen_3031},
Q_f={q_gen_3031},
Delta=
{
<z, z, z>() -> q_gen_3031
}

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

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

Teacher's answer:
New clause system:
{
() -> min([s(u), z, z]) -> 3
() -> min([z, y, z]) -> 3
(min([a, b, _yo]) /\ min([b, a, _zo])) -> eq_nat([_yo, _zo]) -> 1
(min([u, y1, _to])) -> min([s(u), s(y1), s(_to)]) -> 1
}
Sat witness: Found: (() -> min([s(u), z, z]), {
u -> z
})

-------------------------------------------
Step 2, which took 0.010567 s (model generation: 0.010285,  model checking: 0.000282):

Model:
|_
{
min -> 
{{{
Q={q_gen_3031, q_gen_3033},
Q_f={q_gen_3031},
Delta=
{
<z>() -> q_gen_3033
<s, z, z>(q_gen_3033) -> q_gen_3031
<z, z, z>() -> q_gen_3031
}

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

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

Teacher's answer:
New clause system:
{
() -> min([s(u), z, z]) -> 3
() -> min([z, y, z]) -> 3
(min([a, b, _yo]) /\ min([b, a, _zo])) -> eq_nat([_yo, _zo]) -> 1
(min([u, y1, _to])) -> min([s(u), s(y1), s(_to)]) -> 4
}
Sat witness: Found: ((min([u, y1, _to])) -> min([s(u), s(y1), s(_to)]), {
_to -> z  ;  u -> z  ;  y1 -> z
})

-------------------------------------------
Step 3, which took 0.011163 s (model generation: 0.010721,  model checking: 0.000442):

Model:
|_
{
min -> 
{{{
Q={q_gen_3031, q_gen_3033},
Q_f={q_gen_3031},
Delta=
{
<z>() -> q_gen_3033
<s, s, s>(q_gen_3031) -> q_gen_3031
<s, z, z>(q_gen_3033) -> q_gen_3031
<z, z, z>() -> q_gen_3031
}

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

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

Teacher's answer:
New clause system:
{
() -> min([s(u), z, z]) -> 3
() -> min([z, y, z]) -> 6
(min([a, b, _yo]) /\ min([b, a, _zo])) -> eq_nat([_yo, _zo]) -> 2
(min([u, y1, _to])) -> min([s(u), s(y1), s(_to)]) -> 4
}
Sat witness: Found: (() -> min([z, y, z]), {
y -> s(s(z))
})

Total time: 0.047461
Reason for stopping: Proved