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, _xj])) -> plus([n, s(mm), s(_xj)])}
(plus([_yj, _zj, _ak]) /\ plus([_yj, _zj, _bk])) -> eq_nat([_ak, _bk])
)
(leq, P:
{() -> leq([z, s(nn2)])
() -> leq([z, z])
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)])
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2])
(leq([s(nn1), z])) -> BOT}
)
}

properties:
{(plus([n, m, _ck])) -> leq([n, _ck])}


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

Clause system for inference is:

{
() -> leq([z, s(nn2)]) -> 0
() -> leq([z, z]) -> 0
() -> plus([n, z, 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
(plus([n, m, _ck])) -> leq([n, _ck]) -> 0
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 0
}


Solving took 0.109870 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_2589, q_gen_2591},
Q_f={q_gen_2589},
Delta=
{
<s>(q_gen_2591) -> q_gen_2591
<z>() -> q_gen_2591
<s, s>(q_gen_2589) -> q_gen_2589
<z, s>(q_gen_2591) -> q_gen_2589
<z, z>() -> q_gen_2589
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588, q_gen_2593, q_gen_2603},
Q_f={q_gen_2588},
Delta=
{
<s>(q_gen_2603) -> q_gen_2603
<z>() -> q_gen_2603
<s, s>(q_gen_2593) -> q_gen_2593
<z, s>(q_gen_2603) -> q_gen_2593
<z, z>() -> q_gen_2593
<s, s, s>(q_gen_2588) -> q_gen_2588
<s, z, s>(q_gen_2593) -> q_gen_2588
<z, s, s>(q_gen_2593) -> q_gen_2588
<z, z, s>(q_gen_2603) -> q_gen_2588
<z, z, z>() -> q_gen_2588
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.007640 s (model generation: 0.007431,  model checking: 0.000209):

Model:
|_
{
leq -> 
{{{
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:
{
() -> leq([z, s(nn2)]) -> 0
() -> leq([z, z]) -> 0
() -> plus([n, z, 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
(plus([n, m, _ck])) -> leq([n, _ck]) -> 1
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 1
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.008135 s (model generation: 0.008091,  model checking: 0.000044):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588},
Q_f={q_gen_2588},
Delta=
{
<z, z, z>() -> q_gen_2588
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 0
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _ck])) -> leq([n, _ck]) -> 1
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 1
}
Sat witness: Found: (() -> leq([z, z]), {

})

-------------------------------------------
Step 2, which took 0.007968 s (model generation: 0.007904,  model checking: 0.000064):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2589},
Q_f={q_gen_2589},
Delta=
{
<z, z>() -> q_gen_2589
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588},
Q_f={q_gen_2588},
Delta=
{
<z, z, z>() -> q_gen_2588
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _ck])) -> leq([n, _ck]) -> 1
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 1
}
Sat witness: Found: (() -> leq([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 3, which took 0.010700 s (model generation: 0.010569,  model checking: 0.000131):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2589, q_gen_2591},
Q_f={q_gen_2589},
Delta=
{
<z>() -> q_gen_2591
<z, s>(q_gen_2591) -> q_gen_2589
<z, z>() -> q_gen_2589
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588},
Q_f={q_gen_2588},
Delta=
{
<z, z, z>() -> q_gen_2588
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _ck])) -> leq([n, _ck]) -> 1
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 4
}
Sat witness: Found: ((plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]), {
_xj -> z  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 4, which took 0.009748 s (model generation: 0.009268,  model checking: 0.000480):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2589, q_gen_2591},
Q_f={q_gen_2589},
Delta=
{
<z>() -> q_gen_2591
<z, s>(q_gen_2591) -> q_gen_2589
<z, z>() -> q_gen_2589
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588, q_gen_2593},
Q_f={q_gen_2588},
Delta=
{
<z, z>() -> q_gen_2593
<z, s, s>(q_gen_2593) -> q_gen_2588
<z, z, z>() -> q_gen_2588
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _ck])) -> leq([n, _ck]) -> 2
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 4
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 5, which took 0.008388 s (model generation: 0.008082,  model checking: 0.000306):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2589, q_gen_2591},
Q_f={q_gen_2589},
Delta=
{
<z>() -> q_gen_2591
<s, s>(q_gen_2589) -> q_gen_2589
<z, s>(q_gen_2591) -> q_gen_2589
<z, z>() -> q_gen_2589
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588, q_gen_2593},
Q_f={q_gen_2588},
Delta=
{
<z, z>() -> q_gen_2593
<z, s, s>(q_gen_2593) -> q_gen_2588
<z, z, z>() -> q_gen_2588
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _ck])) -> leq([n, _ck]) -> 3
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 4
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 6, which took 0.010173 s (model generation: 0.010063,  model checking: 0.000110):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2589, q_gen_2591},
Q_f={q_gen_2589},
Delta=
{
<z>() -> q_gen_2591
<s, s>(q_gen_2589) -> q_gen_2589
<z, s>(q_gen_2591) -> q_gen_2589
<z, z>() -> q_gen_2589
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588, q_gen_2593},
Q_f={q_gen_2588},
Delta=
{
<s, s>(q_gen_2593) -> q_gen_2593
<z, z>() -> q_gen_2593
<s, z, s>(q_gen_2593) -> q_gen_2588
<z, s, s>(q_gen_2593) -> q_gen_2588
<z, z, z>() -> q_gen_2588
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 3
() -> leq([z, z]) -> 3
() -> plus([n, z, 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
(plus([n, m, _ck])) -> leq([n, _ck]) -> 6
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 4
}
Sat witness: Found: ((plus([n, m, _ck])) -> leq([n, _ck]), {
_ck -> s(s(z))  ;  m -> s(s(z))  ;  n -> z
})

-------------------------------------------
Step 7, which took 0.009308 s (model generation: 0.008899,  model checking: 0.000409):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2589, q_gen_2591},
Q_f={q_gen_2589},
Delta=
{
<s>(q_gen_2591) -> q_gen_2591
<z>() -> q_gen_2591
<s, s>(q_gen_2589) -> q_gen_2589
<z, s>(q_gen_2591) -> q_gen_2589
<z, z>() -> q_gen_2589
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588, q_gen_2593},
Q_f={q_gen_2588},
Delta=
{
<s, s>(q_gen_2593) -> q_gen_2593
<z, z>() -> q_gen_2593
<s, z, s>(q_gen_2593) -> q_gen_2588
<z, s, s>(q_gen_2593) -> q_gen_2588
<z, z, z>() -> q_gen_2588
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 4
() -> leq([z, z]) -> 4
() -> plus([n, z, 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 -> 4
(plus([n, m, _ck])) -> leq([n, _ck]) -> 6
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 7
}
Sat witness: Found: ((plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]), {
_xj -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 8, which took 0.013211 s (model generation: 0.012662,  model checking: 0.000549):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2589, q_gen_2591},
Q_f={q_gen_2589},
Delta=
{
<s>(q_gen_2591) -> q_gen_2591
<z>() -> q_gen_2591
<s, s>(q_gen_2589) -> q_gen_2589
<z, s>(q_gen_2591) -> q_gen_2589
<z, z>() -> q_gen_2589
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588, q_gen_2593, q_gen_2603},
Q_f={q_gen_2588},
Delta=
{
<z>() -> q_gen_2603
<s, s>(q_gen_2593) -> q_gen_2593
<z, z>() -> q_gen_2593
<s, s, s>(q_gen_2588) -> q_gen_2588
<s, z, s>(q_gen_2593) -> q_gen_2588
<z, s, s>(q_gen_2593) -> q_gen_2588
<z, z, s>(q_gen_2603) -> q_gen_2588
<z, z, z>() -> q_gen_2588
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 5
() -> leq([z, z]) -> 5
() -> plus([n, z, n]) -> 7
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 5
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 5
(leq([s(nn1), z])) -> BOT -> 5
(plus([n, m, _ck])) -> leq([n, _ck]) -> 7
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]), {
_xj -> s(z)  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 9, which took 0.012380 s (model generation: 0.011846,  model checking: 0.000534):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2589, q_gen_2591},
Q_f={q_gen_2589},
Delta=
{
<s>(q_gen_2591) -> q_gen_2591
<z>() -> q_gen_2591
<s, s>(q_gen_2589) -> q_gen_2589
<z, s>(q_gen_2591) -> q_gen_2589
<z, z>() -> q_gen_2589
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2588, q_gen_2593, q_gen_2603},
Q_f={q_gen_2588},
Delta=
{
<z>() -> q_gen_2603
<s, s>(q_gen_2593) -> q_gen_2593
<z, s>(q_gen_2603) -> q_gen_2593
<z, z>() -> q_gen_2593
<s, s, s>(q_gen_2588) -> q_gen_2588
<s, z, s>(q_gen_2593) -> q_gen_2588
<z, s, s>(q_gen_2593) -> q_gen_2588
<z, z, s>(q_gen_2603) -> q_gen_2588
<z, z, z>() -> q_gen_2588
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, s(nn2)]) -> 6
() -> leq([z, z]) -> 6
() -> plus([n, z, n]) -> 8
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 6
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 6
(leq([s(nn1), z])) -> BOT -> 6
(plus([n, m, _ck])) -> leq([n, _ck]) -> 8
(plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]) -> 13
}
Sat witness: Found: ((plus([n, mm, _xj])) -> plus([n, s(mm), s(_xj)]), {
_xj -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

Total time: 0.109870
Reason for stopping: Proved