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:
{(leq([n, m])) -> max([n, m, m])
(not leq([n, m])) -> max([n, m, n])}
(max([_gp, _hp, _ip]) /\ max([_gp, _hp, _jp])) -> eq_nat([_ip, _jp])
)
}

properties:
{(max([i, j, _kp])) -> leq([i, _kp]), (max([i, j, _lp])) -> leq([j, _lp])}


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

Clause system for inference is:

{
() -> leq([z, n2]) -> 0
(leq([n, m])) -> max([n, m, m]) -> 0
(not leq([n, m])) -> max([n, m, n]) -> 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, _kp])) -> leq([i, _kp]) -> 0
(max([i, j, _lp])) -> leq([j, _lp]) -> 0
}


Solving took 0.143935 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_3059, q_gen_3063, q_gen_3066},
Q_f={q_gen_3059},
Delta=
{
<s>(q_gen_3063) -> q_gen_3063
<z>() -> q_gen_3063
<s, s>(q_gen_3059) -> q_gen_3059
<z, s>(q_gen_3063) -> q_gen_3059
<z, z>() -> q_gen_3059
<s, z>(q_gen_3063) -> q_gen_3066
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060, q_gen_3064, q_gen_3065, q_gen_3070},
Q_f={q_gen_3060},
Delta=
{
<z>() -> q_gen_3065
<s, s>(q_gen_3070) -> q_gen_3070
<z, z>() -> q_gen_3070
<s, s, s>(q_gen_3060) -> q_gen_3060
<s, z, s>(q_gen_3070) -> q_gen_3060
<z, s, s>(q_gen_3070) -> q_gen_3060
<z, z, z>() -> q_gen_3060
<z, s, z>(q_gen_3065) -> q_gen_3064
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.010204 s (model generation: 0.009818,  model checking: 0.000386):

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
(leq([n, m])) -> max([n, m, m]) -> 0
(not leq([n, m])) -> max([n, m, n]) -> 3
(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, _kp])) -> leq([i, _kp]) -> 1
(max([i, j, _lp])) -> leq([j, _lp]) -> 1
}
Sat witness: Found: ((not leq([n, m])) -> max([n, m, n]), {
m -> z  ;  n -> z
})

-------------------------------------------
Step 1, which took 0.010837 s (model generation: 0.010155,  model checking: 0.000682):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059},
Q_f={},
Delta=
{
<z, z>() -> q_gen_3059
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060},
Q_f={q_gen_3060},
Delta=
{
<z, z, z>() -> q_gen_3060
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
(leq([n, m])) -> max([n, m, m]) -> 1
(not leq([n, m])) -> max([n, m, n]) -> 3
(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, _kp])) -> leq([i, _kp]) -> 1
(max([i, j, _lp])) -> leq([j, _lp]) -> 1
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> z
})

-------------------------------------------
Step 2, which took 0.010424 s (model generation: 0.010172,  model checking: 0.000252):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059},
Q_f={q_gen_3059},
Delta=
{
<z, z>() -> q_gen_3059
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060},
Q_f={},
Delta=
{
<z, z, z>() -> q_gen_3060
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
(leq([n, m])) -> max([n, m, m]) -> 1
(not leq([n, m])) -> max([n, m, n]) -> 3
(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, _kp])) -> leq([i, _kp]) -> 2
(max([i, j, _lp])) -> leq([j, _lp]) -> 2
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 3, which took 0.010606 s (model generation: 0.010429,  model checking: 0.000177):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059},
Q_f={q_gen_3059},
Delta=
{
<s, s>(q_gen_3059) -> q_gen_3059
<z, z>() -> q_gen_3059
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060},
Q_f={},
Delta=
{
<z, z, z>() -> q_gen_3060
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
(leq([n, m])) -> max([n, m, m]) -> 4
(not leq([n, m])) -> max([n, m, n]) -> 3
(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, _kp])) -> leq([i, _kp]) -> 2
(max([i, j, _lp])) -> leq([j, _lp]) -> 2
}
Sat witness: Found: ((leq([n, m])) -> max([n, m, m]), {
m -> z  ;  n -> z
})

-------------------------------------------
Step 4, which took 0.011023 s (model generation: 0.010480,  model checking: 0.000543):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059},
Q_f={q_gen_3059},
Delta=
{
<s, s>(q_gen_3059) -> q_gen_3059
<z, z>() -> q_gen_3059
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060},
Q_f={q_gen_3060},
Delta=
{
<z, z, z>() -> q_gen_3060
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
(leq([n, m])) -> max([n, m, m]) -> 4
(not leq([n, m])) -> max([n, m, n]) -> 6
(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, _kp])) -> leq([i, _kp]) -> 3
(max([i, j, _lp])) -> leq([j, _lp]) -> 3
}
Sat witness: Found: ((not leq([n, m])) -> max([n, m, n]), {
m -> s(z)  ;  n -> z
})

-------------------------------------------
Step 5, which took 0.011282 s (model generation: 0.011081,  model checking: 0.000201):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059, q_gen_3063},
Q_f={q_gen_3059},
Delta=
{
<z>() -> q_gen_3063
<s, s>(q_gen_3059) -> q_gen_3059
<z, s>(q_gen_3063) -> q_gen_3059
<z, z>() -> q_gen_3059
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060, q_gen_3065},
Q_f={q_gen_3060},
Delta=
{
<z>() -> q_gen_3065
<z, s, z>(q_gen_3065) -> q_gen_3060
<z, z, z>() -> q_gen_3060
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
(leq([n, m])) -> max([n, m, m]) -> 4
(not leq([n, m])) -> max([n, m, n]) -> 6
(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, _kp])) -> leq([i, _kp]) -> 3
(max([i, j, _lp])) -> leq([j, _lp]) -> 6
}
Sat witness: Found: ((max([i, j, _lp])) -> leq([j, _lp]), {
_lp -> z  ;  i -> z  ;  j -> s(z)
})

-------------------------------------------
Step 6, which took 0.012515 s (model generation: 0.011414,  model checking: 0.001101):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059, q_gen_3063},
Q_f={q_gen_3059},
Delta=
{
<z>() -> q_gen_3063
<s, s>(q_gen_3059) -> q_gen_3059
<s, z>(q_gen_3063) -> q_gen_3059
<z, s>(q_gen_3063) -> q_gen_3059
<z, z>() -> q_gen_3059
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060, q_gen_3065},
Q_f={q_gen_3060},
Delta=
{
<z>() -> q_gen_3065
<z, s, z>(q_gen_3065) -> q_gen_3060
<z, z, z>() -> q_gen_3060
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 3
(leq([n, m])) -> max([n, m, m]) -> 4
(not leq([n, m])) -> max([n, m, n]) -> 6
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 3
(leq([s(nn1), z])) -> BOT -> 6
(max([i, j, _kp])) -> leq([i, _kp]) -> 4
(max([i, j, _lp])) -> leq([j, _lp]) -> 6
}
Sat witness: Found: ((leq([s(nn1), z])) -> BOT, {
nn1 -> z
})

-------------------------------------------
Step 7, which took 0.011785 s (model generation: 0.011416,  model checking: 0.000369):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059, q_gen_3063, q_gen_3066},
Q_f={q_gen_3059},
Delta=
{
<z>() -> q_gen_3063
<s, s>(q_gen_3059) -> q_gen_3059
<z, s>(q_gen_3063) -> q_gen_3059
<z, z>() -> q_gen_3059
<s, z>(q_gen_3063) -> q_gen_3066
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060, q_gen_3064, q_gen_3065},
Q_f={q_gen_3060},
Delta=
{
<z>() -> q_gen_3065
<z, z, z>() -> q_gen_3060
<z, s, z>(q_gen_3065) -> q_gen_3064
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6
(leq([n, m])) -> max([n, m, m]) -> 4
(not leq([n, m])) -> max([n, m, n]) -> 6
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 4
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 4
(leq([s(nn1), z])) -> BOT -> 6
(max([i, j, _kp])) -> leq([i, _kp]) -> 4
(max([i, j, _lp])) -> leq([j, _lp]) -> 6
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(s(z))
})

-------------------------------------------
Step 8, which took 0.012523 s (model generation: 0.011755,  model checking: 0.000768):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059, q_gen_3063, q_gen_3066},
Q_f={q_gen_3059},
Delta=
{
<s>(q_gen_3063) -> q_gen_3063
<z>() -> q_gen_3063
<s, s>(q_gen_3059) -> q_gen_3059
<z, s>(q_gen_3063) -> q_gen_3059
<z, z>() -> q_gen_3059
<s, z>(q_gen_3063) -> q_gen_3066
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060, q_gen_3064, q_gen_3065},
Q_f={q_gen_3060},
Delta=
{
<z>() -> q_gen_3065
<z, z, z>() -> q_gen_3060
<z, s, z>(q_gen_3065) -> q_gen_3064
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6
(leq([n, m])) -> max([n, m, m]) -> 7
(not leq([n, m])) -> max([n, m, n]) -> 6
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 5
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 5
(leq([s(nn1), z])) -> BOT -> 6
(max([i, j, _kp])) -> leq([i, _kp]) -> 5
(max([i, j, _lp])) -> leq([j, _lp]) -> 6
}
Sat witness: Found: ((leq([n, m])) -> max([n, m, m]), {
m -> s(z)  ;  n -> z
})

-------------------------------------------
Step 9, which took 0.013510 s (model generation: 0.012566,  model checking: 0.000944):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059, q_gen_3063, q_gen_3066},
Q_f={q_gen_3059},
Delta=
{
<s>(q_gen_3063) -> q_gen_3063
<z>() -> q_gen_3063
<s, s>(q_gen_3059) -> q_gen_3059
<z, s>(q_gen_3063) -> q_gen_3059
<z, z>() -> q_gen_3059
<s, z>(q_gen_3063) -> q_gen_3066
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060, q_gen_3064, q_gen_3065, q_gen_3070},
Q_f={q_gen_3060},
Delta=
{
<z>() -> q_gen_3065
<z, z>() -> q_gen_3070
<z, s, s>(q_gen_3070) -> q_gen_3060
<z, z, z>() -> q_gen_3060
<z, s, z>(q_gen_3065) -> q_gen_3064
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6
(leq([n, m])) -> max([n, m, m]) -> 7
(not leq([n, m])) -> max([n, m, n]) -> 9
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 6
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 6
(leq([s(nn1), z])) -> BOT -> 7
(max([i, j, _kp])) -> leq([i, _kp]) -> 6
(max([i, j, _lp])) -> leq([j, _lp]) -> 7
}
Sat witness: Found: ((not leq([n, m])) -> max([n, m, n]), {
m -> z  ;  n -> s(s(z))
})

-------------------------------------------
Step 10, which took 0.014201 s (model generation: 0.012854,  model checking: 0.001347):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3059, q_gen_3063, q_gen_3066},
Q_f={q_gen_3059},
Delta=
{
<s>(q_gen_3063) -> q_gen_3063
<z>() -> q_gen_3063
<s, s>(q_gen_3059) -> q_gen_3059
<z, s>(q_gen_3063) -> q_gen_3059
<z, z>() -> q_gen_3059
<s, z>(q_gen_3063) -> q_gen_3066
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  max -> 
{{{
Q={q_gen_3060, q_gen_3064, q_gen_3065, q_gen_3070},
Q_f={q_gen_3060},
Delta=
{
<z>() -> q_gen_3065
<s, s>(q_gen_3070) -> q_gen_3070
<z, z>() -> q_gen_3070
<s, z, s>(q_gen_3070) -> q_gen_3060
<z, s, s>(q_gen_3070) -> q_gen_3060
<z, z, z>() -> q_gen_3060
<z, s, z>(q_gen_3065) -> q_gen_3064
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 7
(leq([n, m])) -> max([n, m, m]) -> 10
(not leq([n, m])) -> max([n, m, n]) -> 9
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 7
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 7
(leq([s(nn1), z])) -> BOT -> 8
(max([i, j, _kp])) -> leq([i, _kp]) -> 7
(max([i, j, _lp])) -> leq([j, _lp]) -> 8
}
Sat witness: Found: ((leq([n, m])) -> max([n, m, m]), {
m -> s(z)  ;  n -> s(z)
})

Total time: 0.143935
Reason for stopping: Proved