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}
)
(max, F:
{() -> max([s(u), z, s(u)])
() -> max([z, y, y])
(max([u, x2, _xfa])) -> max([s(u), s(x2), s(_xfa)])}
(max([_yfa, _zfa, _aga]) /\ max([_yfa, _zfa, _bga])) -> eq_nat([_aga, _bga])
)
}

properties:
{(leq([j, i]) /\ max([i, j, _dga])) -> eq_nat([_dga, 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, _dga])) -> eq_nat([_dga, 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, _xfa])) -> max([s(u), s(x2), s(_xfa)]) -> 0
}


Solving took 0.058954 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_6311, q_gen_6314},
Q_f={q_gen_6311},
Delta=
{
<s>(q_gen_6314) -> q_gen_6314
<z>() -> q_gen_6314
<s, s>(q_gen_6311) -> q_gen_6311
<z, s>(q_gen_6314) -> q_gen_6311
<z, z>() -> q_gen_6311
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_6308, q_gen_6310},
Q_f={q_gen_6308},
Delta=
{
<s, s>(q_gen_6310) -> q_gen_6310
<z, z>() -> q_gen_6310
<s, s, s>(q_gen_6308) -> q_gen_6308
<s, z, s>(q_gen_6310) -> q_gen_6308
<z, s, s>(q_gen_6310) -> q_gen_6308
<z, z, z>() -> q_gen_6308
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.005925 s (model generation: 0.005090,  model checking: 0.000835):

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

}

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

-------------------------------------------
Step 1, which took 0.004595 s (model generation: 0.004475,  model checking: 0.000120):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_6308},
Q_f={q_gen_6308},
Delta=
{
<z, z, z>() -> q_gen_6308
}

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

}
--
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, _dga])) -> eq_nat([_dga, 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, _xfa])) -> max([s(u), s(x2), s(_xfa)]) -> 1
}
Sat witness: Found: (() -> max([s(u), z, s(u)]), {
u -> z
})

-------------------------------------------
Step 2, which took 0.004557 s (model generation: 0.004420,  model checking: 0.000137):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_6308, q_gen_6310},
Q_f={q_gen_6308},
Delta=
{
<z, z>() -> q_gen_6310
<s, z, s>(q_gen_6310) -> q_gen_6308
<z, z, z>() -> q_gen_6308
}

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

}
--
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, _dga])) -> eq_nat([_dga, 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, _xfa])) -> max([s(u), s(x2), s(_xfa)]) -> 1
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> z
})

-------------------------------------------
Step 3, which took 0.004716 s (model generation: 0.004598,  model checking: 0.000118):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6311},
Q_f={q_gen_6311},
Delta=
{
<z, z>() -> q_gen_6311
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_6308, q_gen_6310},
Q_f={q_gen_6308},
Delta=
{
<z, z>() -> q_gen_6310
<s, z, s>(q_gen_6310) -> q_gen_6308
<z, z, z>() -> q_gen_6308
}

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

}
--
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, _dga])) -> eq_nat([_dga, 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, _xfa])) -> max([s(u), s(x2), s(_xfa)]) -> 4
}
Sat witness: Found: ((max([u, x2, _xfa])) -> max([s(u), s(x2), s(_xfa)]), {
_xfa -> z  ;  u -> z  ;  x2 -> z
})

-------------------------------------------
Step 4, which took 0.005128 s (model generation: 0.005047,  model checking: 0.000081):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6311},
Q_f={q_gen_6311},
Delta=
{
<z, z>() -> q_gen_6311
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_6308, q_gen_6310},
Q_f={q_gen_6308},
Delta=
{
<z, z>() -> q_gen_6310
<s, s, s>(q_gen_6308) -> q_gen_6308
<s, z, s>(q_gen_6310) -> q_gen_6308
<z, z, z>() -> q_gen_6308
}

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

}
--
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, _dga])) -> eq_nat([_dga, 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, _xfa])) -> max([s(u), s(x2), s(_xfa)]) -> 4
}
Sat witness: Found: ((max([i, j, i])) -> leq([j, i]), {
i -> s(z)  ;  j -> z
})

-------------------------------------------
Step 5, which took 0.005665 s (model generation: 0.005486,  model checking: 0.000179):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6311, q_gen_6314},
Q_f={q_gen_6311},
Delta=
{
<z>() -> q_gen_6314
<z, s>(q_gen_6314) -> q_gen_6311
<z, z>() -> q_gen_6311
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_6308, q_gen_6310},
Q_f={q_gen_6308},
Delta=
{
<z, z>() -> q_gen_6310
<s, s, s>(q_gen_6308) -> q_gen_6308
<s, z, s>(q_gen_6310) -> q_gen_6308
<z, z, z>() -> q_gen_6308
}

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

}
--
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, _dga])) -> eq_nat([_dga, 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, _xfa])) -> max([s(u), s(x2), s(_xfa)]) -> 4
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 6, which took 0.010668 s (model generation: 0.010457,  model checking: 0.000211):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6311, q_gen_6314},
Q_f={q_gen_6311},
Delta=
{
<z>() -> q_gen_6314
<s, s>(q_gen_6311) -> q_gen_6311
<z, s>(q_gen_6314) -> q_gen_6311
<z, z>() -> q_gen_6311
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_6308, q_gen_6310},
Q_f={q_gen_6308},
Delta=
{
<z, z>() -> q_gen_6310
<s, s, s>(q_gen_6308) -> q_gen_6308
<s, z, s>(q_gen_6310) -> q_gen_6308
<z, z, z>() -> q_gen_6308
}

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

}
--
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, _dga])) -> eq_nat([_dga, 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, _xfa])) -> max([s(u), s(x2), s(_xfa)]) -> 4
}
Sat witness: Found: (() -> max([z, y, y]), {
y -> s(s(z))
})

-------------------------------------------
Step 7, which took 0.005513 s (model generation: 0.004895,  model checking: 0.000618):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6311, q_gen_6314},
Q_f={q_gen_6311},
Delta=
{
<z>() -> q_gen_6314
<s, s>(q_gen_6311) -> q_gen_6311
<z, s>(q_gen_6314) -> q_gen_6311
<z, z>() -> q_gen_6311
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_6308, q_gen_6310},
Q_f={q_gen_6308},
Delta=
{
<s, s>(q_gen_6310) -> q_gen_6310
<z, z>() -> q_gen_6310
<s, s, s>(q_gen_6308) -> q_gen_6308
<s, z, s>(q_gen_6310) -> q_gen_6308
<z, s, s>(q_gen_6310) -> q_gen_6308
<z, z, z>() -> q_gen_6308
}

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

}
--
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, _dga])) -> eq_nat([_dga, 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, _xfa])) -> max([s(u), s(x2), s(_xfa)]) -> 4
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(s(z))
})

Total time: 0.058954
Reason for stopping: Proved