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}
)
(max, F:
{() -> max([s(u), z, s(u)])
() -> max([z, y, y])
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)])}
(max([_dga, _ega, _fga]) /\ max([_dga, _ega, _gga])) -> eq_nat([_fga, _gga])
)
}

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


over-approximation: {max}
under-approximation: {}

Clause system for inference is:

{
() -> leq([z, n2]) -> 0
() -> max([s(u), z, s(u)]) -> 0
() -> max([z, y, y]) -> 0
(leq([j, i]) /\ max([i, j, _iga])) -> eq_nat([_iga, 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
(max([i, j, i])) -> leq([j, i]) -> 0
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]) -> 0
}


Solving took 0.157581 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_8603, q_gen_8606},
Q_f={q_gen_8603},
Delta=
{
<s>(q_gen_8606) -> q_gen_8606
<z>() -> q_gen_8606
<s, s>(q_gen_8603) -> q_gen_8603
<z, s>(q_gen_8606) -> q_gen_8603
<z, z>() -> q_gen_8603
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_8600, q_gen_8602},
Q_f={q_gen_8600},
Delta=
{
<s, s>(q_gen_8602) -> q_gen_8602
<z, z>() -> q_gen_8602
<s, s, s>(q_gen_8600) -> q_gen_8600
<s, z, s>(q_gen_8602) -> q_gen_8600
<z, s, s>(q_gen_8602) -> q_gen_8600
<z, z, z>() -> q_gen_8600
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.014802 s (model generation: 0.014342,  model checking: 0.000460):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
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
() -> max([s(u), z, s(u)]) -> 0
() -> max([z, y, y]) -> 3
(leq([j, i]) /\ max([i, j, _iga])) -> eq_nat([_iga, 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
(max([i, j, i])) -> leq([j, i]) -> 1
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]) -> 1
}
Sat witness: Found: (() -> max([z, y, y]), {
y -> z
})

-------------------------------------------
Step 1, which took 0.015508 s (model generation: 0.015286,  model checking: 0.000222):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_8600},
Q_f={q_gen_8600},
Delta=
{
<z, z, z>() -> q_gen_8600
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 0
() -> max([s(u), z, s(u)]) -> 3
() -> max([z, y, y]) -> 3
(leq([j, i]) /\ max([i, j, _iga])) -> eq_nat([_iga, 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
(max([i, j, i])) -> leq([j, i]) -> 1
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]) -> 1
}
Sat witness: Found: (() -> max([s(u), z, s(u)]), {
u -> z
})

-------------------------------------------
Step 2, which took 0.015692 s (model generation: 0.015587,  model checking: 0.000105):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_8600, q_gen_8602},
Q_f={q_gen_8600},
Delta=
{
<z, z>() -> q_gen_8602
<s, z, s>(q_gen_8602) -> q_gen_8600
<z, z, z>() -> q_gen_8600
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
() -> max([s(u), z, s(u)]) -> 3
() -> max([z, y, y]) -> 3
(leq([j, i]) /\ max([i, j, _iga])) -> eq_nat([_iga, 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
(max([i, j, i])) -> leq([j, i]) -> 1
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]) -> 1
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> z
})

-------------------------------------------
Step 3, which took 0.019049 s (model generation: 0.015807,  model checking: 0.003242):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8603},
Q_f={q_gen_8603},
Delta=
{
<z, z>() -> q_gen_8603
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_8600, q_gen_8602},
Q_f={q_gen_8600},
Delta=
{
<z, z>() -> q_gen_8602
<s, z, s>(q_gen_8602) -> q_gen_8600
<z, z, z>() -> q_gen_8600
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
() -> max([s(u), z, s(u)]) -> 3
() -> max([z, y, y]) -> 3
(leq([j, i]) /\ max([i, j, _iga])) -> eq_nat([_iga, 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
(max([i, j, i])) -> leq([j, i]) -> 1
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]) -> 4
}
Sat witness: Found: ((max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]), {
_cga -> z  ;  u -> z  ;  x2 -> z
})

-------------------------------------------
Step 4, which took 0.016333 s (model generation: 0.016165,  model checking: 0.000168):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8603},
Q_f={q_gen_8603},
Delta=
{
<z, z>() -> q_gen_8603
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_8600, q_gen_8602},
Q_f={q_gen_8600},
Delta=
{
<z, z>() -> q_gen_8602
<s, s, s>(q_gen_8600) -> q_gen_8600
<s, z, s>(q_gen_8602) -> q_gen_8600
<z, z, z>() -> q_gen_8600
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
() -> max([s(u), z, s(u)]) -> 3
() -> max([z, y, y]) -> 3
(leq([j, i]) /\ max([i, j, _iga])) -> eq_nat([_iga, 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
(max([i, j, i])) -> leq([j, i]) -> 4
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]) -> 4
}
Sat witness: Found: ((max([i, j, i])) -> leq([j, i]), {
i -> s(z)  ;  j -> z
})

-------------------------------------------
Step 5, which took 0.017080 s (model generation: 0.016846,  model checking: 0.000234):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8603, q_gen_8606},
Q_f={q_gen_8603},
Delta=
{
<z>() -> q_gen_8606
<z, s>(q_gen_8606) -> q_gen_8603
<z, z>() -> q_gen_8603
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_8600, q_gen_8602},
Q_f={q_gen_8600},
Delta=
{
<z, z>() -> q_gen_8602
<s, s, s>(q_gen_8600) -> q_gen_8600
<s, z, s>(q_gen_8602) -> q_gen_8600
<z, z, z>() -> q_gen_8600
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
() -> max([s(u), z, s(u)]) -> 3
() -> max([z, y, y]) -> 3
(leq([j, i]) /\ max([i, j, _iga])) -> eq_nat([_iga, i]) -> 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
(max([i, j, i])) -> leq([j, i]) -> 4
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]) -> 4
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 6, which took 0.017755 s (model generation: 0.017220,  model checking: 0.000535):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8603, q_gen_8606},
Q_f={q_gen_8603},
Delta=
{
<z>() -> q_gen_8606
<s, s>(q_gen_8603) -> q_gen_8603
<z, s>(q_gen_8606) -> q_gen_8603
<z, z>() -> q_gen_8603
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_8600, q_gen_8602},
Q_f={q_gen_8600},
Delta=
{
<z, z>() -> q_gen_8602
<s, s, s>(q_gen_8600) -> q_gen_8600
<s, z, s>(q_gen_8602) -> q_gen_8600
<z, z, z>() -> q_gen_8600
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
() -> max([s(u), z, s(u)]) -> 3
() -> max([z, y, y]) -> 6
(leq([j, i]) /\ max([i, j, _iga])) -> eq_nat([_iga, i]) -> 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
(max([i, j, i])) -> leq([j, i]) -> 4
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]) -> 4
}
Sat witness: Found: (() -> max([z, y, y]), {
y -> s(s(z))
})

-------------------------------------------
Step 7, which took 0.021463 s (model generation: 0.017883,  model checking: 0.003580):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8603, q_gen_8606},
Q_f={q_gen_8603},
Delta=
{
<z>() -> q_gen_8606
<s, s>(q_gen_8603) -> q_gen_8603
<z, s>(q_gen_8606) -> q_gen_8603
<z, z>() -> q_gen_8603
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_8600, q_gen_8602},
Q_f={q_gen_8600},
Delta=
{
<s, s>(q_gen_8602) -> q_gen_8602
<z, z>() -> q_gen_8602
<s, s, s>(q_gen_8600) -> q_gen_8600
<s, z, s>(q_gen_8602) -> q_gen_8600
<z, s, s>(q_gen_8602) -> q_gen_8600
<z, z, z>() -> q_gen_8600
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6
() -> max([s(u), z, s(u)]) -> 4
() -> max([z, y, y]) -> 6
(leq([j, i]) /\ max([i, j, _iga])) -> eq_nat([_iga, i]) -> 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
(max([i, j, i])) -> leq([j, i]) -> 4
(max([u, x2, _cga])) -> max([s(u), s(x2), s(_cga)]) -> 4
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(s(z))
})

Total time: 0.157581
Reason for stopping: Proved