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:
{(leq([n, m])) -> max([n, m, m])
(not leq([n, m])) -> max([n, m, n])}
(max([_an, _bn, _cn]) /\ max([_an, _bn, _dn])) -> eq_nat([_cn, _dn])
)
}

properties:
{(max([i, j, _en])) -> leq([i, _en]), (max([i, j, _fn])) -> leq([j, _fn])}


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, _en])) -> leq([i, _en]) -> 0
(max([i, j, _fn])) -> leq([j, _fn]) -> 0
}


Solving took 0.155366 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_3384, q_gen_3388, q_gen_3391},
Q_f={q_gen_3384},
Delta=
{
<s>(q_gen_3388) -> q_gen_3388
<z>() -> q_gen_3388
<s, s>(q_gen_3384) -> q_gen_3384
<z, s>(q_gen_3388) -> q_gen_3384
<z, z>() -> q_gen_3384
<s, z>(q_gen_3388) -> q_gen_3391
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385, q_gen_3389, q_gen_3390, q_gen_3395},
Q_f={q_gen_3385},
Delta=
{
<z>() -> q_gen_3390
<s, s>(q_gen_3395) -> q_gen_3395
<z, z>() -> q_gen_3395
<s, s, s>(q_gen_3385) -> q_gen_3385
<s, z, s>(q_gen_3395) -> q_gen_3385
<z, s, s>(q_gen_3395) -> q_gen_3385
<z, z, z>() -> q_gen_3385
<z, s, z>(q_gen_3390) -> q_gen_3389
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.020603 s (model generation: 0.020416,  model checking: 0.000187):

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
(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, _en])) -> leq([i, _en]) -> 1
(max([i, j, _fn])) -> leq([j, _fn]) -> 1
}
Sat witness: Found: ((not leq([n, m])) -> max([n, m, n]), {
m -> z  ;  n -> z
})

-------------------------------------------
Step 1, which took 0.009035 s (model generation: 0.008903,  model checking: 0.000132):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384},
Q_f={},
Delta=
{
<z, z>() -> q_gen_3384
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385},
Q_f={q_gen_3385},
Delta=
{
<z, z, z>() -> q_gen_3385
}

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

}
--
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, _en])) -> leq([i, _en]) -> 1
(max([i, j, _fn])) -> leq([j, _fn]) -> 1
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> z
})

-------------------------------------------
Step 2, which took 0.010778 s (model generation: 0.010650,  model checking: 0.000128):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384},
Q_f={q_gen_3384},
Delta=
{
<z, z>() -> q_gen_3384
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385},
Q_f={},
Delta=
{
<z, z, z>() -> q_gen_3385
}

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

}
--
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, _en])) -> leq([i, _en]) -> 2
(max([i, j, _fn])) -> leq([j, _fn]) -> 2
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 3, which took 0.014137 s (model generation: 0.014020,  model checking: 0.000117):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384},
Q_f={q_gen_3384},
Delta=
{
<s, s>(q_gen_3384) -> q_gen_3384
<z, z>() -> q_gen_3384
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385},
Q_f={},
Delta=
{
<z, z, z>() -> q_gen_3385
}

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

}
--
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, _en])) -> leq([i, _en]) -> 2
(max([i, j, _fn])) -> leq([j, _fn]) -> 2
}
Sat witness: Found: ((leq([n, m])) -> max([n, m, m]), {
m -> z  ;  n -> z
})

-------------------------------------------
Step 4, which took 0.010656 s (model generation: 0.010343,  model checking: 0.000313):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384},
Q_f={q_gen_3384},
Delta=
{
<s, s>(q_gen_3384) -> q_gen_3384
<z, z>() -> q_gen_3384
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385},
Q_f={q_gen_3385},
Delta=
{
<z, z, z>() -> q_gen_3385
}

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

}
--
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, _en])) -> leq([i, _en]) -> 3
(max([i, j, _fn])) -> leq([j, _fn]) -> 3
}
Sat witness: Found: ((not leq([n, m])) -> max([n, m, n]), {
m -> s(z)  ;  n -> z
})

-------------------------------------------
Step 5, which took 0.013277 s (model generation: 0.013186,  model checking: 0.000091):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384, q_gen_3388},
Q_f={q_gen_3384},
Delta=
{
<z>() -> q_gen_3388
<s, s>(q_gen_3384) -> q_gen_3384
<z, s>(q_gen_3388) -> q_gen_3384
<z, z>() -> q_gen_3384
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385, q_gen_3390},
Q_f={q_gen_3385},
Delta=
{
<z>() -> q_gen_3390
<z, s, z>(q_gen_3390) -> q_gen_3385
<z, z, z>() -> q_gen_3385
}

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

}
--
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, _en])) -> leq([i, _en]) -> 3
(max([i, j, _fn])) -> leq([j, _fn]) -> 6
}
Sat witness: Found: ((max([i, j, _fn])) -> leq([j, _fn]), {
_fn -> z  ;  i -> z  ;  j -> s(z)
})

-------------------------------------------
Step 6, which took 0.010445 s (model generation: 0.010370,  model checking: 0.000075):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384, q_gen_3388},
Q_f={q_gen_3384},
Delta=
{
<z>() -> q_gen_3388
<s, s>(q_gen_3384) -> q_gen_3384
<s, z>(q_gen_3388) -> q_gen_3384
<z, s>(q_gen_3388) -> q_gen_3384
<z, z>() -> q_gen_3384
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385, q_gen_3390},
Q_f={q_gen_3385},
Delta=
{
<z>() -> q_gen_3390
<z, s, z>(q_gen_3390) -> q_gen_3385
<z, z, z>() -> q_gen_3385
}

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

}
--
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, _en])) -> leq([i, _en]) -> 4
(max([i, j, _fn])) -> leq([j, _fn]) -> 6
}
Sat witness: Found: ((leq([s(nn1), z])) -> BOT, {
nn1 -> z
})

-------------------------------------------
Step 7, which took 0.014120 s (model generation: 0.013874,  model checking: 0.000246):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384, q_gen_3388, q_gen_3391},
Q_f={q_gen_3384},
Delta=
{
<z>() -> q_gen_3388
<s, s>(q_gen_3384) -> q_gen_3384
<z, s>(q_gen_3388) -> q_gen_3384
<z, z>() -> q_gen_3384
<s, z>(q_gen_3388) -> q_gen_3391
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385, q_gen_3389, q_gen_3390},
Q_f={q_gen_3385},
Delta=
{
<z>() -> q_gen_3390
<z, z, z>() -> q_gen_3385
<z, s, z>(q_gen_3390) -> q_gen_3389
}

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

}
--
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, _en])) -> leq([i, _en]) -> 4
(max([i, j, _fn])) -> leq([j, _fn]) -> 6
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(s(z))
})

-------------------------------------------
Step 8, which took 0.012821 s (model generation: 0.012414,  model checking: 0.000407):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384, q_gen_3388, q_gen_3391},
Q_f={q_gen_3384},
Delta=
{
<s>(q_gen_3388) -> q_gen_3388
<z>() -> q_gen_3388
<s, s>(q_gen_3384) -> q_gen_3384
<z, s>(q_gen_3388) -> q_gen_3384
<z, z>() -> q_gen_3384
<s, z>(q_gen_3388) -> q_gen_3391
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385, q_gen_3389, q_gen_3390},
Q_f={q_gen_3385},
Delta=
{
<z>() -> q_gen_3390
<z, z, z>() -> q_gen_3385
<z, s, z>(q_gen_3390) -> q_gen_3389
}

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

}
--
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, _en])) -> leq([i, _en]) -> 5
(max([i, j, _fn])) -> leq([j, _fn]) -> 6
}
Sat witness: Found: ((leq([n, m])) -> max([n, m, m]), {
m -> s(z)  ;  n -> z
})

-------------------------------------------
Step 9, which took 0.012748 s (model generation: 0.011487,  model checking: 0.001261):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384, q_gen_3388, q_gen_3391},
Q_f={q_gen_3384},
Delta=
{
<s>(q_gen_3388) -> q_gen_3388
<z>() -> q_gen_3388
<s, s>(q_gen_3384) -> q_gen_3384
<z, s>(q_gen_3388) -> q_gen_3384
<z, z>() -> q_gen_3384
<s, z>(q_gen_3388) -> q_gen_3391
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385, q_gen_3389, q_gen_3390, q_gen_3395},
Q_f={q_gen_3385},
Delta=
{
<z>() -> q_gen_3390
<z, z>() -> q_gen_3395
<z, s, s>(q_gen_3395) -> q_gen_3385
<z, z, z>() -> q_gen_3385
<z, s, z>(q_gen_3390) -> q_gen_3389
}

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

}
--
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, _en])) -> leq([i, _en]) -> 6
(max([i, j, _fn])) -> leq([j, _fn]) -> 7
}
Sat witness: Found: ((not leq([n, m])) -> max([n, m, n]), {
m -> z  ;  n -> s(s(z))
})

-------------------------------------------
Step 10, which took 0.014316 s (model generation: 0.013532,  model checking: 0.000784):

Model:
|_
{
leq -> 
{{{
Q={q_gen_3384, q_gen_3388, q_gen_3391},
Q_f={q_gen_3384},
Delta=
{
<s>(q_gen_3388) -> q_gen_3388
<z>() -> q_gen_3388
<s, s>(q_gen_3384) -> q_gen_3384
<z, s>(q_gen_3388) -> q_gen_3384
<z, z>() -> q_gen_3384
<s, z>(q_gen_3388) -> q_gen_3391
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  max -> 
{{{
Q={q_gen_3385, q_gen_3389, q_gen_3390, q_gen_3395},
Q_f={q_gen_3385},
Delta=
{
<z>() -> q_gen_3390
<s, s>(q_gen_3395) -> q_gen_3395
<z, z>() -> q_gen_3395
<s, z, s>(q_gen_3395) -> q_gen_3385
<z, s, s>(q_gen_3395) -> q_gen_3385
<z, z, z>() -> q_gen_3385
<z, s, z>(q_gen_3390) -> q_gen_3389
}

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

}
--
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, _en])) -> leq([i, _en]) -> 7
(max([i, j, _fn])) -> leq([j, _fn]) -> 8
}
Sat witness: Found: ((leq([n, m])) -> max([n, m, m]), {
m -> s(z)  ;  n -> s(z)
})

Total time: 0.155366
Reason for stopping: Proved