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, _cy])) -> max([s(u), s(x2), s(_cy)])}
(max([_dy, _ey, _fy]) /\ max([_dy, _ey, _gy])) -> eq_nat([_fy, _gy])
)
}

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


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([i, j]) /\ max([i, j, _iy])) -> eq_nat([_iy, 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
(max([i, j, j])) -> leq([i, j]) -> 0
(max([u, x2, _cy])) -> max([s(u), s(x2), s(_cy)]) -> 0
}


Solving took 0.145412 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_6373, q_gen_6379},
Q_f={q_gen_6373},
Delta=
{
<s>(q_gen_6379) -> q_gen_6379
<z>() -> q_gen_6379
<s, s>(q_gen_6373) -> q_gen_6373
<z, s>(q_gen_6379) -> q_gen_6373
<z, z>() -> q_gen_6373
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_6370, q_gen_6372},
Q_f={q_gen_6370},
Delta=
{
<s, s>(q_gen_6372) -> q_gen_6372
<z, z>() -> q_gen_6372
<s, s, s>(q_gen_6370) -> q_gen_6370
<s, z, s>(q_gen_6372) -> q_gen_6370
<z, s, s>(q_gen_6372) -> q_gen_6370
<z, z, z>() -> q_gen_6370
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.017593 s (model generation: 0.017362,  model checking: 0.000231):

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([i, j]) /\ max([i, j, _iy])) -> eq_nat([_iy, 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
(max([i, j, j])) -> leq([i, j]) -> 1
(max([u, x2, _cy])) -> max([s(u), s(x2), s(_cy)]) -> 1
}
Sat witness: Found: (() -> max([z, y, y]), {
y -> z
})

-------------------------------------------
Step 1, which took 0.011751 s (model generation: 0.010835,  model checking: 0.000916):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_6370},
Q_f={q_gen_6370},
Delta=
{
<z, z, z>() -> q_gen_6370
}

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([i, j]) /\ max([i, j, _iy])) -> eq_nat([_iy, 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
(max([i, j, j])) -> leq([i, j]) -> 1
(max([u, x2, _cy])) -> max([s(u), s(x2), s(_cy)]) -> 1
}
Sat witness: Found: (() -> max([s(u), z, s(u)]), {
u -> z
})

-------------------------------------------
Step 2, which took 0.012170 s (model generation: 0.012100,  model checking: 0.000070):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_6370, q_gen_6372},
Q_f={q_gen_6370},
Delta=
{
<z, z>() -> q_gen_6372
<s, z, s>(q_gen_6372) -> q_gen_6370
<z, z, z>() -> q_gen_6370
}

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([i, j]) /\ max([i, j, _iy])) -> eq_nat([_iy, 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
(max([i, j, j])) -> leq([i, j]) -> 1
(max([u, x2, _cy])) -> max([s(u), s(x2), s(_cy)]) -> 1
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> z
})

-------------------------------------------
Step 3, which took 0.017770 s (model generation: 0.017603,  model checking: 0.000167):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6373},
Q_f={q_gen_6373},
Delta=
{
<z, z>() -> q_gen_6373
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_6370, q_gen_6372},
Q_f={q_gen_6370},
Delta=
{
<z, z>() -> q_gen_6372
<s, z, s>(q_gen_6372) -> q_gen_6370
<z, z, z>() -> q_gen_6370
}

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([i, j]) /\ max([i, j, _iy])) -> eq_nat([_iy, 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
(max([i, j, j])) -> leq([i, j]) -> 1
(max([u, x2, _cy])) -> max([s(u), s(x2), s(_cy)]) -> 4
}
Sat witness: Found: ((max([u, x2, _cy])) -> max([s(u), s(x2), s(_cy)]), {
_cy -> z  ;  u -> z  ;  x2 -> z
})

-------------------------------------------
Step 4, which took 0.017884 s (model generation: 0.017780,  model checking: 0.000104):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6373},
Q_f={q_gen_6373},
Delta=
{
<z, z>() -> q_gen_6373
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_6370, q_gen_6372},
Q_f={q_gen_6370},
Delta=
{
<z, z>() -> q_gen_6372
<s, s, s>(q_gen_6370) -> q_gen_6370
<s, z, s>(q_gen_6372) -> q_gen_6370
<z, z, z>() -> q_gen_6370
}

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

-------------------------------------------
Step 5, which took 0.017310 s (model generation: 0.016994,  model checking: 0.000316):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6373},
Q_f={q_gen_6373},
Delta=
{
<s, s>(q_gen_6373) -> q_gen_6373
<z, z>() -> q_gen_6373
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_6370, q_gen_6372},
Q_f={q_gen_6370},
Delta=
{
<z, z>() -> q_gen_6372
<s, s, s>(q_gen_6370) -> q_gen_6370
<s, z, s>(q_gen_6372) -> q_gen_6370
<z, z, z>() -> q_gen_6370
}

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([i, j]) /\ max([i, j, _iy])) -> eq_nat([_iy, j]) -> 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
(max([i, j, j])) -> leq([i, j]) -> 4
(max([u, x2, _cy])) -> max([s(u), s(x2), s(_cy)]) -> 4
}
Sat witness: Found: (() -> max([z, y, y]), {
y -> s(s(z))
})

-------------------------------------------
Step 6, which took 0.015648 s (model generation: 0.015328,  model checking: 0.000320):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6373},
Q_f={q_gen_6373},
Delta=
{
<s, s>(q_gen_6373) -> q_gen_6373
<z, z>() -> q_gen_6373
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_6370, q_gen_6372},
Q_f={q_gen_6370},
Delta=
{
<s, s>(q_gen_6372) -> q_gen_6372
<z, z>() -> q_gen_6372
<s, s, s>(q_gen_6370) -> q_gen_6370
<s, z, s>(q_gen_6372) -> q_gen_6370
<z, s, s>(q_gen_6372) -> q_gen_6370
<z, z, z>() -> q_gen_6370
}

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([i, j]) /\ max([i, j, _iy])) -> eq_nat([_iy, j]) -> 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
(max([i, j, j])) -> leq([i, j]) -> 4
(max([u, x2, _cy])) -> max([s(u), s(x2), s(_cy)]) -> 4
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(z)
})

-------------------------------------------
Step 7, which took 0.015218 s (model generation: 0.014665,  model checking: 0.000553):

Model:
|_
{
leq -> 
{{{
Q={q_gen_6373, q_gen_6379},
Q_f={q_gen_6373},
Delta=
{
<z>() -> q_gen_6379
<s, s>(q_gen_6373) -> q_gen_6373
<z, s>(q_gen_6379) -> q_gen_6373
<z, z>() -> q_gen_6373
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_6370, q_gen_6372},
Q_f={q_gen_6370},
Delta=
{
<s, s>(q_gen_6372) -> q_gen_6372
<z, z>() -> q_gen_6372
<s, s, s>(q_gen_6370) -> q_gen_6370
<s, z, s>(q_gen_6372) -> q_gen_6370
<z, s, s>(q_gen_6372) -> q_gen_6370
<z, z, z>() -> q_gen_6370
}

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([i, j]) /\ max([i, j, _iy])) -> eq_nat([_iy, j]) -> 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
(max([i, j, j])) -> leq([i, j]) -> 7
(max([u, x2, _cy])) -> max([s(u), s(x2), s(_cy)]) -> 5
}
Sat witness: Found: ((max([i, j, j])) -> leq([i, j]), {
i -> z  ;  j -> s(s(z))
})

Total time: 0.145412
Reason for stopping: Proved