Inference procedure has parameters:
Ice fuel: 200
Timeout: 60s
Convolution: left


Learning problem is:

env: {
nat -> {s, z}
}
definition:
{
(plus, F:
{() -> plus([n, z, n])
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)])}
(plus([_mj, _nj, _oj]) /\ plus([_mj, _nj, _pj])) -> eq_nat([_oj, _pj])
)
(le, P:
{() -> le([z, s(nn2)])
(le([nn1, nn2])) -> le([s(nn1), s(nn2)])
(le([s(nn1), s(nn2)])) -> le([nn1, nn2])
(le([s(nn1), z])) -> BOT
(le([z, z])) -> BOT}
)
}

properties:
{(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj])}


over-approximation: {plus}
under-approximation: {}

Clause system for inference is:

{
() -> le([z, s(nn2)]) -> 0
() -> plus([n, z, n]) -> 0
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 0
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 0
(le([s(nn1), z])) -> BOT -> 0
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 0
(le([z, z])) -> BOT -> 0
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 0
}


Solving took 0.293713 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2436, q_gen_2437, q_gen_2446, q_gen_2447, q_gen_2452},
Q_f={q_gen_2433, q_gen_2436},
Delta=
{
<s>(q_gen_2446) -> q_gen_2446
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, s>(q_gen_2452) -> q_gen_2452
<z, s>(q_gen_2446) -> q_gen_2452
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<s, s, s>(q_gen_2436) -> q_gen_2436
<s, z, s>(q_gen_2452) -> q_gen_2436
<z, s, s>(q_gen_2437) -> q_gen_2436
<z, s, s>(q_gen_2452) -> q_gen_2436
<z, z, s>(q_gen_2446) -> q_gen_2436
<s, s, s>(q_gen_2433) -> q_gen_2447
<s, s, s>(q_gen_2447) -> q_gen_2447
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.009396 s (model generation: 0.009195,  model checking: 0.000201):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={},
Q_f={},
Delta=
{

}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 0
() -> plus([n, z, n]) -> 3
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 1
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 1
(le([s(nn1), z])) -> BOT -> 1
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 1
(le([z, z])) -> BOT -> 1
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 1
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.011485 s (model generation: 0.011407,  model checking: 0.000078):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433},
Q_f={q_gen_2433},
Delta=
{
<z, z, z>() -> q_gen_2433
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3
() -> plus([n, z, n]) -> 3
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 1
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 1
(le([s(nn1), z])) -> BOT -> 1
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 1
(le([z, z])) -> BOT -> 1
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 1
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.009445 s (model generation: 0.009342,  model checking: 0.000103):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435},
Q_f={q_gen_2434},
Delta=
{
<z>() -> q_gen_2435
<z, s>(q_gen_2435) -> q_gen_2434
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433},
Q_f={q_gen_2433},
Delta=
{
<z, z, z>() -> q_gen_2433
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3
() -> plus([n, z, n]) -> 3
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 1
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 1
(le([s(nn1), z])) -> BOT -> 1
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 1
(le([z, z])) -> BOT -> 1
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 4
}
Sat witness: Found: ((plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]), {
_lj -> z  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 3, which took 0.011517 s (model generation: 0.011124,  model checking: 0.000393):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435},
Q_f={q_gen_2434},
Delta=
{
<z>() -> q_gen_2435
<z, s>(q_gen_2435) -> q_gen_2434
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2437},
Q_f={q_gen_2433},
Delta=
{
<z, z>() -> q_gen_2437
<z, s, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3
() -> plus([n, z, n]) -> 3
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 2
(le([s(nn1), z])) -> BOT -> 2
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 2
(le([z, z])) -> BOT -> 2
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 4
}
Sat witness: Found: ((le([nn1, nn2])) -> le([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> s(z)
})

-------------------------------------------
Step 4, which took 0.016644 s (model generation: 0.010001,  model checking: 0.006643):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435},
Q_f={q_gen_2434},
Delta=
{
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2437},
Q_f={q_gen_2433},
Delta=
{
<z, z>() -> q_gen_2437
<z, s, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3
() -> plus([n, z, n]) -> 6
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 3
(le([s(nn1), z])) -> BOT -> 3
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 3
(le([z, z])) -> BOT -> 3
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 4
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 5, which took 0.008997 s (model generation: 0.008794,  model checking: 0.000203):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435},
Q_f={q_gen_2434},
Delta=
{
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2437},
Q_f={q_gen_2433},
Delta=
{
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, s, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6
() -> plus([n, z, n]) -> 6
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 4
(le([s(nn1), z])) -> BOT -> 4
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 4
(le([z, z])) -> BOT -> 4
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 4
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 6, which took 0.009797 s (model generation: 0.009449,  model checking: 0.000348):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2437},
Q_f={q_gen_2433},
Delta=
{
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, s, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6
() -> plus([n, z, n]) -> 6
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 4
(le([s(nn1), z])) -> BOT -> 4
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 4
(le([z, z])) -> BOT -> 4
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 7
}
Sat witness: Found: ((plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]), {
_lj -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 7, which took 0.011483 s (model generation: 0.011370,  model checking: 0.000113):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2437, q_gen_2446},
Q_f={q_gen_2433},
Delta=
{
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, s, s>(q_gen_2433) -> q_gen_2433
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, s, s>(q_gen_2437) -> q_gen_2433
<z, z, s>(q_gen_2446) -> q_gen_2433
<z, z, z>() -> q_gen_2433
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6
() -> plus([n, z, n]) -> 6
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 4
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 4
(le([s(nn1), z])) -> BOT -> 4
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 7
(le([z, z])) -> BOT -> 5
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 7
}
Sat witness: Found: ((le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]), {
_qj -> s(z)  ;  m -> s(z)  ;  n -> s(z)
})

-------------------------------------------
Step 8, which took 0.010970 s (model generation: 0.010840,  model checking: 0.000130):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<z, z>() -> q_gen_2434
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2437, q_gen_2446},
Q_f={q_gen_2433},
Delta=
{
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, s, s>(q_gen_2433) -> q_gen_2433
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, s, s>(q_gen_2437) -> q_gen_2433
<z, z, s>(q_gen_2446) -> q_gen_2433
<z, z, z>() -> q_gen_2433
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6
() -> plus([n, z, n]) -> 6
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 5
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 5
(le([s(nn1), z])) -> BOT -> 5
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 7
(le([z, z])) -> BOT -> 8
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 7
}
Sat witness: Found: ((le([z, z])) -> BOT, {

})

-------------------------------------------
Step 9, which took 0.011002 s (model generation: 0.010858,  model checking: 0.000144):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2449},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<s, s>(q_gen_2449) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<z, z>() -> q_gen_2449
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2437, q_gen_2446},
Q_f={q_gen_2433},
Delta=
{
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, s, s>(q_gen_2433) -> q_gen_2433
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, s, s>(q_gen_2437) -> q_gen_2433
<z, z, s>(q_gen_2446) -> q_gen_2433
<z, z, z>() -> q_gen_2433
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6
() -> plus([n, z, n]) -> 6
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 5
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 8
(le([s(nn1), z])) -> BOT -> 6
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 7
(le([z, z])) -> BOT -> 8
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 7
}
Sat witness: Found: ((le([s(nn1), s(nn2)])) -> le([nn1, nn2]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 10, which took 0.011507 s (model generation: 0.011017,  model checking: 0.000490):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2437, q_gen_2445, q_gen_2446},
Q_f={q_gen_2433},
Delta=
{
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, s, s>(q_gen_2445) -> q_gen_2433
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, s, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<s, s, s>(q_gen_2433) -> q_gen_2445
<z, z, s>(q_gen_2446) -> q_gen_2445
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 7
() -> plus([n, z, n]) -> 7
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 6
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 8
(le([s(nn1), z])) -> BOT -> 7
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 7
(le([z, z])) -> BOT -> 8
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]), {
_lj -> s(s(z))  ;  mm -> s(z)  ;  n -> s(z)
})

-------------------------------------------
Step 11, which took 0.011468 s (model generation: 0.011175,  model checking: 0.000293):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2436, q_gen_2437, q_gen_2446, q_gen_2447},
Q_f={q_gen_2433, q_gen_2436},
Delta=
{
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, s>(q_gen_2446) -> q_gen_2437
<z, z>() -> q_gen_2437
<z, z, z>() -> q_gen_2433
<s, s, s>(q_gen_2436) -> q_gen_2436
<s, z, s>(q_gen_2437) -> q_gen_2436
<z, s, s>(q_gen_2437) -> q_gen_2436
<z, z, s>(q_gen_2446) -> q_gen_2436
<s, s, s>(q_gen_2433) -> q_gen_2447
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 7
() -> plus([n, z, n]) -> 7
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 7
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 8
(le([s(nn1), z])) -> BOT -> 7
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 10
(le([z, z])) -> BOT -> 8
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 10
}
Sat witness: Found: ((le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]), {
_qj -> s(s(z))  ;  m -> s(z)  ;  n -> s(s(z))
})

-------------------------------------------
Step 12, which took 0.015698 s (model generation: 0.015111,  model checking: 0.000587):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2436, q_gen_2437, q_gen_2446, q_gen_2447},
Q_f={q_gen_2433, q_gen_2436},
Delta=
{
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, s>(q_gen_2446) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, s, s>(q_gen_2436) -> q_gen_2433
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<z, s, s>(q_gen_2437) -> q_gen_2436
<z, z, s>(q_gen_2446) -> q_gen_2436
<s, s, s>(q_gen_2433) -> q_gen_2447
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 8
() -> plus([n, z, n]) -> 8
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 8
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 9
(le([s(nn1), z])) -> BOT -> 8
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 10
(le([z, z])) -> BOT -> 9
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 13
}
Sat witness: Found: ((plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]), {
_lj -> s(s(z))  ;  mm -> z  ;  n -> s(s(z))
})

-------------------------------------------
Step 13, which took 0.015470 s (model generation: 0.015067,  model checking: 0.000403):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2437, q_gen_2445, q_gen_2446, q_gen_2452},
Q_f={q_gen_2433},
Delta=
{
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<z, s>(q_gen_2446) -> q_gen_2452
<s, s, s>(q_gen_2445) -> q_gen_2433
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, s, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<s, s, s>(q_gen_2433) -> q_gen_2445
<s, z, s>(q_gen_2452) -> q_gen_2445
<z, s, s>(q_gen_2452) -> q_gen_2445
<z, z, s>(q_gen_2446) -> q_gen_2445
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 9
() -> plus([n, z, n]) -> 9
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 9
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 10
(le([s(nn1), z])) -> BOT -> 9
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 13
(le([z, z])) -> BOT -> 10
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 13
}
Sat witness: Found: ((le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]), {
_qj -> s(s(z))  ;  m -> s(s(z))  ;  n -> s(s(z))
})

-------------------------------------------
Step 14, which took 0.015799 s (model generation: 0.015419,  model checking: 0.000380):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448, q_gen_2449},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2449) -> q_gen_2448
<z, z>() -> q_gen_2449
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2436, q_gen_2437, q_gen_2446, q_gen_2447},
Q_f={q_gen_2433, q_gen_2436},
Delta=
{
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, s>(q_gen_2446) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, s, s>(q_gen_2447) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<s, s, s>(q_gen_2436) -> q_gen_2436
<s, z, s>(q_gen_2437) -> q_gen_2436
<z, s, s>(q_gen_2437) -> q_gen_2436
<z, z, s>(q_gen_2446) -> q_gen_2436
<s, s, s>(q_gen_2433) -> q_gen_2447
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 10
() -> plus([n, z, n]) -> 10
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 10
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13
(le([s(nn1), z])) -> BOT -> 10
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 13
(le([z, z])) -> BOT -> 11
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 13
}
Sat witness: Found: ((le([s(nn1), s(nn2)])) -> le([nn1, nn2]), {
nn1 -> s(z)  ;  nn2 -> s(z)
})

-------------------------------------------
Step 15, which took 0.015904 s (model generation: 0.015680,  model checking: 0.000224):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2436, q_gen_2437, q_gen_2440, q_gen_2446, q_gen_2447},
Q_f={q_gen_2433, q_gen_2436},
Delta=
{
<z>() -> q_gen_2446
<z, z>() -> q_gen_2437
<s, s>(q_gen_2437) -> q_gen_2440
<z, s>(q_gen_2446) -> q_gen_2440
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<s, s, s>(q_gen_2436) -> q_gen_2436
<s, z, s>(q_gen_2440) -> q_gen_2436
<z, s, s>(q_gen_2437) -> q_gen_2436
<z, s, s>(q_gen_2440) -> q_gen_2436
<z, z, s>(q_gen_2446) -> q_gen_2436
<s, s, s>(q_gen_2433) -> q_gen_2447
<s, s, s>(q_gen_2447) -> q_gen_2447
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 10
() -> plus([n, z, n]) -> 13
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 11
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13
(le([s(nn1), z])) -> BOT -> 11
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 13
(le([z, z])) -> BOT -> 11
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 13
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(s(z)))
})

-------------------------------------------
Step 16, which took 0.015763 s (model generation: 0.015305,  model checking: 0.000458):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2436, q_gen_2437, q_gen_2440, q_gen_2446, q_gen_2447},
Q_f={q_gen_2433, q_gen_2436},
Delta=
{
<z>() -> q_gen_2446
<z, z>() -> q_gen_2437
<s, s>(q_gen_2437) -> q_gen_2440
<s, s>(q_gen_2440) -> q_gen_2440
<z, s>(q_gen_2446) -> q_gen_2440
<s, s, s>(q_gen_2436) -> q_gen_2433
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<s, z, s>(q_gen_2440) -> q_gen_2436
<z, s, s>(q_gen_2437) -> q_gen_2436
<z, s, s>(q_gen_2440) -> q_gen_2436
<z, z, s>(q_gen_2446) -> q_gen_2436
<s, s, s>(q_gen_2433) -> q_gen_2447
<s, s, s>(q_gen_2447) -> q_gen_2447
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 11
() -> plus([n, z, n]) -> 13
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 12
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13
(le([s(nn1), z])) -> BOT -> 12
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 13
(le([z, z])) -> BOT -> 12
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 16
}
Sat witness: Found: ((plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]), {
_lj -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 17, which took 0.016452 s (model generation: 0.016232,  model checking: 0.000220):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2436, q_gen_2437, q_gen_2440, q_gen_2446, q_gen_2447},
Q_f={q_gen_2433, q_gen_2436},
Delta=
{
<s>(q_gen_2446) -> q_gen_2446
<z>() -> q_gen_2446
<s, s>(q_gen_2440) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, s>(q_gen_2437) -> q_gen_2440
<z, s>(q_gen_2446) -> q_gen_2440
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<s, s, s>(q_gen_2436) -> q_gen_2436
<s, z, s>(q_gen_2440) -> q_gen_2436
<z, s, s>(q_gen_2437) -> q_gen_2436
<z, s, s>(q_gen_2440) -> q_gen_2436
<z, z, s>(q_gen_2446) -> q_gen_2436
<s, s, s>(q_gen_2433) -> q_gen_2447
<s, s, s>(q_gen_2447) -> q_gen_2447
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 12
() -> plus([n, z, n]) -> 13
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 13
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13
(le([s(nn1), z])) -> BOT -> 13
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 16
(le([z, z])) -> BOT -> 13
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 16
}
Sat witness: Found: ((le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]), {
_qj -> s(s(s(z)))  ;  m -> s(z)  ;  n -> s(s(s(z)))
})

-------------------------------------------
Step 18, which took 0.021225 s (model generation: 0.020640,  model checking: 0.000585):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2436, q_gen_2437, q_gen_2446, q_gen_2447, q_gen_2452},
Q_f={q_gen_2433, q_gen_2436},
Delta=
{
<s>(q_gen_2446) -> q_gen_2446
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<z, s>(q_gen_2446) -> q_gen_2452
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<s, s, s>(q_gen_2436) -> q_gen_2436
<s, z, s>(q_gen_2452) -> q_gen_2436
<z, s, s>(q_gen_2437) -> q_gen_2436
<z, s, s>(q_gen_2452) -> q_gen_2436
<z, z, s>(q_gen_2446) -> q_gen_2436
<s, s, s>(q_gen_2433) -> q_gen_2447
<s, s, s>(q_gen_2447) -> q_gen_2447
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 13
() -> plus([n, z, n]) -> 14
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 14
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 14
(le([s(nn1), z])) -> BOT -> 14
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 16
(le([z, z])) -> BOT -> 14
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 19
}
Sat witness: Found: ((plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]), {
_lj -> s(s(z))  ;  mm -> s(z)  ;  n -> z
})

-------------------------------------------
Step 19, which took 0.020547 s (model generation: 0.019660,  model checking: 0.000887):

Model:
|_
{
le -> 
{{{
Q={q_gen_2434, q_gen_2435, q_gen_2448},
Q_f={q_gen_2434},
Delta=
{
<s>(q_gen_2435) -> q_gen_2435
<z>() -> q_gen_2435
<s, s>(q_gen_2434) -> q_gen_2434
<z, s>(q_gen_2435) -> q_gen_2434
<s, s>(q_gen_2448) -> q_gen_2448
<z, z>() -> q_gen_2448
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2433, q_gen_2436, q_gen_2437, q_gen_2446, q_gen_2447, q_gen_2452},
Q_f={q_gen_2433, q_gen_2436},
Delta=
{
<s>(q_gen_2446) -> q_gen_2446
<z>() -> q_gen_2446
<s, s>(q_gen_2437) -> q_gen_2437
<z, z>() -> q_gen_2437
<s, s>(q_gen_2452) -> q_gen_2452
<z, s>(q_gen_2446) -> q_gen_2452
<s, s, s>(q_gen_2436) -> q_gen_2433
<s, z, s>(q_gen_2437) -> q_gen_2433
<z, z, z>() -> q_gen_2433
<s, z, s>(q_gen_2452) -> q_gen_2436
<z, s, s>(q_gen_2437) -> q_gen_2436
<z, s, s>(q_gen_2452) -> q_gen_2436
<z, z, s>(q_gen_2446) -> q_gen_2436
<s, s, s>(q_gen_2433) -> q_gen_2447
<s, s, s>(q_gen_2447) -> q_gen_2447
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 14
() -> plus([n, z, n]) -> 15
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 15
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 15
(le([s(nn1), z])) -> BOT -> 15
(le([z, m]) /\ plus([n, m, _qj])) -> le([n, _qj]) -> 17
(le([z, z])) -> BOT -> 15
(plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]) -> 22
}
Sat witness: Found: ((plus([n, mm, _lj])) -> plus([n, s(mm), s(_lj)]), {
_lj -> s(s(s(z)))  ;  mm -> s(z)  ;  n -> s(s(z))
})

Total time: 0.293713
Reason for stopping: Proved