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, _vx])) -> min([s(u), s(y1), s(_vx)])}
(min([_wx, _xx, _yx]) /\ min([_wx, _xx, _zx])) -> eq_nat([_yx, _zx])
)
}

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


Solving took 0.170910 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_6361, q_gen_6367},
Q_f={q_gen_6361},
Delta=
{
<s>(q_gen_6367) -> q_gen_6367
<z>() -> q_gen_6367
<s, s>(q_gen_6361) -> q_gen_6361
<z, s>(q_gen_6367) -> q_gen_6361
<z, z>() -> q_gen_6361
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_6358, q_gen_6360},
Q_f={q_gen_6358},
Delta=
{
<s>(q_gen_6360) -> q_gen_6360
<z>() -> q_gen_6360
<s, s, s>(q_gen_6358) -> q_gen_6358
<s, z, z>(q_gen_6360) -> q_gen_6358
<z, s, z>(q_gen_6360) -> q_gen_6358
<z, z, z>() -> q_gen_6358
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.036720 s (model generation: 0.036107,  model checking: 0.000613):

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

-------------------------------------------
Step 1, which took 0.010715 s (model generation: 0.010572,  model checking: 0.000143):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_6358},
Q_f={q_gen_6358},
Delta=
{
<z, z, z>() -> q_gen_6358
}

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

-------------------------------------------
Step 2, which took 0.011929 s (model generation: 0.011856,  model checking: 0.000073):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_6358, q_gen_6360},
Q_f={q_gen_6358},
Delta=
{
<z>() -> q_gen_6360
<s, z, z>(q_gen_6360) -> q_gen_6358
<z, z, z>() -> q_gen_6358
}

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

-------------------------------------------
Step 3, which took 0.016013 s (model generation: 0.015853,  model checking: 0.000160):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6361},
Q_f={q_gen_6361},
Delta=
{
<z, z>() -> q_gen_6361
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_6358, q_gen_6360},
Q_f={q_gen_6358},
Delta=
{
<z>() -> q_gen_6360
<s, z, z>(q_gen_6360) -> q_gen_6358
<z, z, z>() -> q_gen_6358
}

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

-------------------------------------------
Step 4, which took 0.021080 s (model generation: 0.020916,  model checking: 0.000164):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6361},
Q_f={q_gen_6361},
Delta=
{
<z, z>() -> q_gen_6361
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_6358, q_gen_6360},
Q_f={q_gen_6358},
Delta=
{
<z>() -> q_gen_6360
<s, s, s>(q_gen_6358) -> q_gen_6358
<s, z, z>(q_gen_6360) -> q_gen_6358
<z, z, z>() -> q_gen_6358
}

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

-------------------------------------------
Step 5, which took 0.021789 s (model generation: 0.021396,  model checking: 0.000393):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6361},
Q_f={q_gen_6361},
Delta=
{
<s, s>(q_gen_6361) -> q_gen_6361
<z, z>() -> q_gen_6361
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_6358, q_gen_6360},
Q_f={q_gen_6358},
Delta=
{
<z>() -> q_gen_6360
<s, s, s>(q_gen_6358) -> q_gen_6358
<s, z, z>(q_gen_6360) -> q_gen_6358
<z, z, z>() -> q_gen_6358
}

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

-------------------------------------------
Step 6, which took 0.018956 s (model generation: 0.018605,  model checking: 0.000351):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6361},
Q_f={q_gen_6361},
Delta=
{
<s, s>(q_gen_6361) -> q_gen_6361
<z, z>() -> q_gen_6361
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_6358, q_gen_6360},
Q_f={q_gen_6358},
Delta=
{
<s>(q_gen_6360) -> q_gen_6360
<z>() -> q_gen_6360
<s, s, s>(q_gen_6358) -> q_gen_6358
<s, z, z>(q_gen_6360) -> q_gen_6358
<z, s, z>(q_gen_6360) -> q_gen_6358
<z, z, z>() -> q_gen_6358
}

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

-------------------------------------------
Step 7, which took 0.016044 s (model generation: 0.015540,  model checking: 0.000504):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6361, q_gen_6367},
Q_f={q_gen_6361},
Delta=
{
<z>() -> q_gen_6367
<s, s>(q_gen_6361) -> q_gen_6361
<z, s>(q_gen_6367) -> q_gen_6361
<z, z>() -> q_gen_6361
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  min -> 
{{{
Q={q_gen_6358, q_gen_6360},
Q_f={q_gen_6358},
Delta=
{
<s>(q_gen_6360) -> q_gen_6360
<z>() -> q_gen_6360
<s, s, s>(q_gen_6358) -> q_gen_6358
<s, z, z>(q_gen_6360) -> q_gen_6358
<z, s, z>(q_gen_6360) -> q_gen_6358
<z, z, z>() -> q_gen_6358
}

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([i, j]) /\ min([i, j, _by])) -> eq_nat([_by, 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, _vx])) -> min([s(u), s(y1), s(_vx)]) -> 5
}
Sat witness: Found: ((min([i, j, i])) -> leq([i, j]), {
i -> z  ;  j -> s(s(z))
})

Total time: 0.170910
Reason for stopping: Proved