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


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, _yu])) -> min([s(u), s(y1), s(_yu)])}
(min([_zu, _av, _bv]) /\ min([_zu, _av, _cv])) -> eq_nat([_bv, _cv])
)
}

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


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([j, i]) /\ min([i, j, _ev])) -> eq_nat([_ev, j]) -> 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, j])) -> leq([j, i]) -> 0
(min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]) -> 0
}


Solving took 0.102482 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_5529, q_gen_5532},
Q_f={q_gen_5529},
Delta=
{
<s>(q_gen_5532) -> q_gen_5532
<z>() -> q_gen_5532
<s, s>(q_gen_5529) -> q_gen_5529
<z, s>(q_gen_5532) -> q_gen_5529
<z, z>() -> q_gen_5529
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_5526, q_gen_5528},
Q_f={q_gen_5526},
Delta=
{
<s>(q_gen_5528) -> q_gen_5528
<z>() -> q_gen_5528
<s, s, s>(q_gen_5526) -> q_gen_5526
<s, z, z>(q_gen_5528) -> q_gen_5526
<z, s, z>(q_gen_5528) -> q_gen_5526
<z, z, z>() -> q_gen_5526
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.008661 s (model generation: 0.008399,  model checking: 0.000262):

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

}

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

}

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

}
--
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([j, i]) /\ min([i, j, _ev])) -> eq_nat([_ev, j]) -> 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, j])) -> leq([j, i]) -> 1
(min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]) -> 1
}
Sat witness: Found: (() -> min([z, y, z]), {
y -> z
})

-------------------------------------------
Step 1, which took 0.009797 s (model generation: 0.009702,  model checking: 0.000095):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_5526},
Q_f={q_gen_5526},
Delta=
{
<z, z, z>() -> q_gen_5526
}

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

}
--
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([j, i]) /\ min([i, j, _ev])) -> eq_nat([_ev, j]) -> 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, j])) -> leq([j, i]) -> 1
(min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]) -> 1
}
Sat witness: Found: (() -> min([s(u), z, z]), {
u -> z
})

-------------------------------------------
Step 2, which took 0.009568 s (model generation: 0.009514,  model checking: 0.000054):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_5526, q_gen_5528},
Q_f={q_gen_5526},
Delta=
{
<z>() -> q_gen_5528
<s, z, z>(q_gen_5528) -> q_gen_5526
<z, z, z>() -> q_gen_5526
}

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

}
--
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([j, i]) /\ min([i, j, _ev])) -> eq_nat([_ev, j]) -> 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, j])) -> leq([j, i]) -> 1
(min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]) -> 1
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> z
})

-------------------------------------------
Step 3, which took 0.010176 s (model generation: 0.010052,  model checking: 0.000124):

Model:
|_
{
leq -> 
{{{
Q={q_gen_5529},
Q_f={q_gen_5529},
Delta=
{
<z, z>() -> q_gen_5529
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_5526, q_gen_5528},
Q_f={q_gen_5526},
Delta=
{
<z>() -> q_gen_5528
<s, z, z>(q_gen_5528) -> q_gen_5526
<z, z, z>() -> q_gen_5526
}

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

}
--
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([j, i]) /\ min([i, j, _ev])) -> eq_nat([_ev, j]) -> 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, j])) -> leq([j, i]) -> 1
(min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]) -> 4
}
Sat witness: Found: ((min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]), {
_yu -> z  ;  u -> z  ;  y1 -> z
})

-------------------------------------------
Step 4, which took 0.009798 s (model generation: 0.009710,  model checking: 0.000088):

Model:
|_
{
leq -> 
{{{
Q={q_gen_5529},
Q_f={q_gen_5529},
Delta=
{
<z, z>() -> q_gen_5529
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_5526, q_gen_5528},
Q_f={q_gen_5526},
Delta=
{
<z>() -> q_gen_5528
<s, s, s>(q_gen_5526) -> q_gen_5526
<s, z, z>(q_gen_5528) -> q_gen_5526
<z, z, z>() -> q_gen_5526
}

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

}
--
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([j, i]) /\ min([i, j, _ev])) -> eq_nat([_ev, j]) -> 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, j])) -> leq([j, i]) -> 4
(min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]) -> 4
}
Sat witness: Found: ((min([i, j, j])) -> leq([j, i]), {
i -> s(z)  ;  j -> z
})

-------------------------------------------
Step 5, which took 0.016051 s (model generation: 0.015927,  model checking: 0.000124):

Model:
|_
{
leq -> 
{{{
Q={q_gen_5529, q_gen_5532},
Q_f={q_gen_5529},
Delta=
{
<z>() -> q_gen_5532
<z, s>(q_gen_5532) -> q_gen_5529
<z, z>() -> q_gen_5529
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_5526, q_gen_5528},
Q_f={q_gen_5526},
Delta=
{
<z>() -> q_gen_5528
<s, s, s>(q_gen_5526) -> q_gen_5526
<s, z, z>(q_gen_5528) -> q_gen_5526
<z, z, z>() -> q_gen_5526
}

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

}
--
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([j, i]) /\ min([i, j, _ev])) -> eq_nat([_ev, j]) -> 1
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 2
(leq([s(nn1), z])) -> BOT -> 2
(min([i, j, j])) -> leq([j, i]) -> 4
(min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]) -> 4
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 6, which took 0.012797 s (model generation: 0.012206,  model checking: 0.000591):

Model:
|_
{
leq -> 
{{{
Q={q_gen_5529, q_gen_5532},
Q_f={q_gen_5529},
Delta=
{
<z>() -> q_gen_5532
<s, s>(q_gen_5529) -> q_gen_5529
<z, s>(q_gen_5532) -> q_gen_5529
<z, z>() -> q_gen_5529
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_5526, q_gen_5528},
Q_f={q_gen_5526},
Delta=
{
<z>() -> q_gen_5528
<s, s, s>(q_gen_5526) -> q_gen_5526
<s, z, z>(q_gen_5528) -> q_gen_5526
<z, z, z>() -> q_gen_5526
}

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

}
--
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([j, i]) /\ min([i, j, _ev])) -> eq_nat([_ev, j]) -> 2
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 3
(leq([s(nn1), z])) -> BOT -> 3
(min([i, j, j])) -> leq([j, i]) -> 4
(min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]) -> 4
}
Sat witness: Found: (() -> min([z, y, z]), {
y -> s(s(z))
})

-------------------------------------------
Step 7, which took 0.013059 s (model generation: 0.011423,  model checking: 0.001636):

Model:
|_
{
leq -> 
{{{
Q={q_gen_5529, q_gen_5532},
Q_f={q_gen_5529},
Delta=
{
<z>() -> q_gen_5532
<s, s>(q_gen_5529) -> q_gen_5529
<z, s>(q_gen_5532) -> q_gen_5529
<z, z>() -> q_gen_5529
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_5526, q_gen_5528},
Q_f={q_gen_5526},
Delta=
{
<s>(q_gen_5528) -> q_gen_5528
<z>() -> q_gen_5528
<s, s, s>(q_gen_5526) -> q_gen_5526
<s, z, z>(q_gen_5528) -> q_gen_5526
<z, s, z>(q_gen_5528) -> q_gen_5526
<z, z, z>() -> q_gen_5526
}

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

}
--
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([j, i]) /\ min([i, j, _ev])) -> eq_nat([_ev, j]) -> 3
(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, j])) -> leq([j, i]) -> 4
(min([u, y1, _yu])) -> min([s(u), s(y1), s(_yu)]) -> 4
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(s(z))
})

Total time: 0.102482
Reason for stopping: Proved