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, _we])) -> min([s(u), s(y1), s(_we)])}
(min([_xe, _ye, _af]) /\ min([_xe, _ye, _ze])) -> eq_nat([_ze, _af])
)
}

properties:
{(leq([j, i]) /\ min([i, j, _cf])) -> eq_nat([_cf, 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, _cf])) -> eq_nat([_cf, 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, _we])) -> min([s(u), s(y1), s(_we)]) -> 0
}


Solving took 0.111825 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_1000, q_gen_1001},
Q_f={q_gen_1000},
Delta=
{
<s>(q_gen_1001) -> q_gen_1001
<z>() -> q_gen_1001
<s, s>(q_gen_1000) -> q_gen_1000
<z, s>(q_gen_1001) -> q_gen_1000
<z, z>() -> q_gen_1000
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_1003, q_gen_1004},
Q_f={q_gen_1003},
Delta=
{
<s>(q_gen_1004) -> q_gen_1004
<z>() -> q_gen_1004
<s, s, s>(q_gen_1003) -> q_gen_1003
<s, z, z>(q_gen_1004) -> q_gen_1003
<z, s, z>(q_gen_1004) -> q_gen_1003
<z, z, z>() -> q_gen_1003
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.011623 s (model generation: 0.010357,  model checking: 0.001266):

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

-------------------------------------------
Step 1, which took 0.010845 s (model generation: 0.010631,  model checking: 0.000214):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_995},
Q_f={q_gen_995},
Delta=
{
<z, z, z>() -> q_gen_995
}

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

-------------------------------------------
Step 2, which took 0.011062 s (model generation: 0.010951,  model checking: 0.000111):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_995, q_gen_997},
Q_f={q_gen_995},
Delta=
{
<z>() -> q_gen_997
<s, z, z>(q_gen_997) -> q_gen_995
<z, z, z>() -> q_gen_995
}

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

-------------------------------------------
Step 3, which took 0.011475 s (model generation: 0.011172,  model checking: 0.000303):

Model:
|_
{
leq -> 
{{{
Q={q_gen_998},
Q_f={q_gen_998},
Delta=
{
<z, z>() -> q_gen_998
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_995, q_gen_997},
Q_f={q_gen_995},
Delta=
{
<z>() -> q_gen_997
<s, z, z>(q_gen_997) -> q_gen_995
<z, z, z>() -> q_gen_995
}

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

-------------------------------------------
Step 4, which took 0.011348 s (model generation: 0.011177,  model checking: 0.000171):

Model:
|_
{
leq -> 
{{{
Q={q_gen_998},
Q_f={q_gen_998},
Delta=
{
<z, z>() -> q_gen_998
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_995, q_gen_997},
Q_f={q_gen_995},
Delta=
{
<z>() -> q_gen_997
<s, s, s>(q_gen_995) -> q_gen_995
<s, z, z>(q_gen_997) -> q_gen_995
<z, z, z>() -> q_gen_995
}

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

-------------------------------------------
Step 5, which took 0.011489 s (model generation: 0.011252,  model checking: 0.000237):

Model:
|_
{
leq -> 
{{{
Q={q_gen_1000, q_gen_1001},
Q_f={q_gen_1000},
Delta=
{
<z>() -> q_gen_1001
<z, s>(q_gen_1001) -> q_gen_1000
<z, z>() -> q_gen_1000
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_995, q_gen_997},
Q_f={q_gen_995},
Delta=
{
<z>() -> q_gen_997
<s, s, s>(q_gen_995) -> q_gen_995
<s, z, z>(q_gen_997) -> q_gen_995
<z, z, z>() -> q_gen_995
}

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

-------------------------------------------
Step 6, which took 0.016637 s (model generation: 0.011243,  model checking: 0.005394):

Model:
|_
{
leq -> 
{{{
Q={q_gen_1000, q_gen_1001},
Q_f={q_gen_1000},
Delta=
{
<z>() -> q_gen_1001
<s, s>(q_gen_1000) -> q_gen_1000
<z, s>(q_gen_1001) -> q_gen_1000
<z, z>() -> q_gen_1000
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_995, q_gen_997},
Q_f={q_gen_995},
Delta=
{
<z>() -> q_gen_997
<s, s, s>(q_gen_995) -> q_gen_995
<s, z, z>(q_gen_997) -> q_gen_995
<z, z, z>() -> q_gen_995
}

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([j, i]) /\ min([i, j, _cf])) -> eq_nat([_cf, 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, _we])) -> min([s(u), s(y1), s(_we)]) -> 4
}
Sat witness: Found: (() -> min([z, y, z]), {
y -> s(s(z))
})

-------------------------------------------
Step 7, which took 0.012325 s (model generation: 0.011668,  model checking: 0.000657):

Model:
|_
{
leq -> 
{{{
Q={q_gen_1000, q_gen_1001},
Q_f={q_gen_1000},
Delta=
{
<z>() -> q_gen_1001
<s, s>(q_gen_1000) -> q_gen_1000
<z, s>(q_gen_1001) -> q_gen_1000
<z, z>() -> q_gen_1000
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  min -> 
{{{
Q={q_gen_1003, q_gen_1004},
Q_f={q_gen_1003},
Delta=
{
<s>(q_gen_1004) -> q_gen_1004
<z>() -> q_gen_1004
<s, s, s>(q_gen_1003) -> q_gen_1003
<s, z, z>(q_gen_1004) -> q_gen_1003
<z, s, z>(q_gen_1004) -> q_gen_1003
<z, z, z>() -> q_gen_1003
}

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

Total time: 0.111825
Reason for stopping: Proved