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, _eea])) -> plus([n, s(mm), s(_eea)])}
(plus([_fea, _gea, _hea]) /\ plus([_fea, _gea, _iea])) -> eq_nat([_hea, _iea])
)
(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}
)
}

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


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

Clause system for inference is:

{
() -> leq([z, n2]) -> 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, _jea])) -> leq([n, _jea]) -> 0
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 0
}


Solving took 1.134633 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_8228, q_gen_8232},
Q_f={q_gen_8228},
Delta=
{
<s>(q_gen_8232) -> q_gen_8232
<z>() -> q_gen_8232
<s, s>(q_gen_8228) -> q_gen_8228
<z, s>(q_gen_8232) -> q_gen_8228
<z, z>() -> q_gen_8228
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227, q_gen_8230, q_gen_8241},
Q_f={q_gen_8227},
Delta=
{
<s>(q_gen_8241) -> q_gen_8241
<z>() -> q_gen_8241
<s, s>(q_gen_8230) -> q_gen_8230
<z, s>(q_gen_8241) -> q_gen_8230
<z, z>() -> q_gen_8230
<s, s, s>(q_gen_8227) -> q_gen_8227
<s, z, s>(q_gen_8230) -> q_gen_8227
<z, s, s>(q_gen_8230) -> q_gen_8227
<z, z, s>(q_gen_8241) -> q_gen_8227
<z, z, z>() -> q_gen_8227
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.101803 s (model generation: 0.099360,  model checking: 0.002443):

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, n2]) -> 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, _jea])) -> leq([n, _jea]) -> 1
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 1
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.096722 s (model generation: 0.096287,  model checking: 0.000435):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227},
Q_f={q_gen_8227},
Delta=
{
<z, z, z>() -> q_gen_8227
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 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, _jea])) -> leq([n, _jea]) -> 1
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 1
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> z
})

-------------------------------------------
Step 2, which took 0.090023 s (model generation: 0.088661,  model checking: 0.001362):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8228},
Q_f={q_gen_8228},
Delta=
{
<z, z>() -> q_gen_8228
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227},
Q_f={q_gen_8227},
Delta=
{
<z, z, z>() -> q_gen_8227
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 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, _jea])) -> leq([n, _jea]) -> 1
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 4
}
Sat witness: Found: ((plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]), {
_eea -> z  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 3, which took 0.088124 s (model generation: 0.087512,  model checking: 0.000612):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8228},
Q_f={q_gen_8228},
Delta=
{
<z, z>() -> q_gen_8228
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227, q_gen_8230},
Q_f={q_gen_8227},
Delta=
{
<z, z>() -> q_gen_8230
<z, s, s>(q_gen_8230) -> q_gen_8227
<z, z, z>() -> q_gen_8227
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 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, _jea])) -> leq([n, _jea]) -> 4
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 4
}
Sat witness: Found: ((plus([n, m, _jea])) -> leq([n, _jea]), {
_jea -> s(z)  ;  m -> s(z)  ;  n -> z
})

-------------------------------------------
Step 4, which took 0.085154 s (model generation: 0.084082,  model checking: 0.001072):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8228, q_gen_8232},
Q_f={q_gen_8228},
Delta=
{
<z>() -> q_gen_8232
<z, s>(q_gen_8232) -> q_gen_8228
<z, z>() -> q_gen_8228
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227, q_gen_8230},
Q_f={q_gen_8227},
Delta=
{
<z, z>() -> q_gen_8230
<z, s, s>(q_gen_8230) -> q_gen_8227
<z, z, z>() -> q_gen_8227
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 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, _jea])) -> leq([n, _jea]) -> 4
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 4
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 5, which took 0.097061 s (model generation: 0.091080,  model checking: 0.005981):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8228, q_gen_8232},
Q_f={q_gen_8228},
Delta=
{
<z>() -> q_gen_8232
<s, s>(q_gen_8228) -> q_gen_8228
<z, s>(q_gen_8232) -> q_gen_8228
<z, z>() -> q_gen_8228
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227, q_gen_8230},
Q_f={q_gen_8227},
Delta=
{
<z, z>() -> q_gen_8230
<z, s, s>(q_gen_8230) -> q_gen_8227
<z, z, z>() -> q_gen_8227
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 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, _jea])) -> leq([n, _jea]) -> 4
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 4
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 6, which took 0.166092 s (model generation: 0.164719,  model checking: 0.001373):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8228, q_gen_8232},
Q_f={q_gen_8228},
Delta=
{
<z>() -> q_gen_8232
<s, s>(q_gen_8228) -> q_gen_8228
<z, s>(q_gen_8232) -> q_gen_8228
<z, z>() -> q_gen_8228
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227, q_gen_8230},
Q_f={q_gen_8227},
Delta=
{
<s, s>(q_gen_8230) -> q_gen_8230
<z, z>() -> q_gen_8230
<s, z, s>(q_gen_8230) -> q_gen_8227
<z, s, s>(q_gen_8230) -> q_gen_8227
<z, z, z>() -> q_gen_8227
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6
() -> 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, _jea])) -> leq([n, _jea]) -> 4
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 4
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(s(z))
})

-------------------------------------------
Step 7, which took 0.087114 s (model generation: 0.085406,  model checking: 0.001708):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8228, q_gen_8232},
Q_f={q_gen_8228},
Delta=
{
<s>(q_gen_8232) -> q_gen_8232
<z>() -> q_gen_8232
<s, s>(q_gen_8228) -> q_gen_8228
<z, s>(q_gen_8232) -> q_gen_8228
<z, z>() -> q_gen_8228
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227, q_gen_8230},
Q_f={q_gen_8227},
Delta=
{
<s, s>(q_gen_8230) -> q_gen_8230
<z, z>() -> q_gen_8230
<s, z, s>(q_gen_8230) -> q_gen_8227
<z, s, s>(q_gen_8230) -> q_gen_8227
<z, z, z>() -> q_gen_8227
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 6
() -> 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, _jea])) -> leq([n, _jea]) -> 4
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 7
}
Sat witness: Found: ((plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]), {
_eea -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 8, which took 0.098382 s (model generation: 0.091977,  model checking: 0.006405):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8228, q_gen_8232},
Q_f={q_gen_8228},
Delta=
{
<s>(q_gen_8232) -> q_gen_8232
<z>() -> q_gen_8232
<s, s>(q_gen_8228) -> q_gen_8228
<z, s>(q_gen_8232) -> q_gen_8228
<z, z>() -> q_gen_8228
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227, q_gen_8230, q_gen_8241},
Q_f={q_gen_8227},
Delta=
{
<z>() -> q_gen_8241
<s, s>(q_gen_8230) -> q_gen_8230
<z, z>() -> q_gen_8230
<s, s, s>(q_gen_8227) -> q_gen_8227
<s, z, s>(q_gen_8230) -> q_gen_8227
<z, s, s>(q_gen_8230) -> q_gen_8227
<z, z, s>(q_gen_8241) -> q_gen_8227
<z, z, z>() -> q_gen_8227
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 7
() -> 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, _jea])) -> leq([n, _jea]) -> 5
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]), {
_eea -> s(z)  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 9, which took 0.100523 s (model generation: 0.095204,  model checking: 0.005319):

Model:
|_
{
leq -> 
{{{
Q={q_gen_8228, q_gen_8232},
Q_f={q_gen_8228},
Delta=
{
<s>(q_gen_8232) -> q_gen_8232
<z>() -> q_gen_8232
<s, s>(q_gen_8228) -> q_gen_8228
<z, s>(q_gen_8232) -> q_gen_8228
<z, z>() -> q_gen_8228
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8227, q_gen_8230, q_gen_8241},
Q_f={q_gen_8227},
Delta=
{
<z>() -> q_gen_8241
<s, s>(q_gen_8230) -> q_gen_8230
<z, s>(q_gen_8241) -> q_gen_8230
<z, z>() -> q_gen_8230
<s, s, s>(q_gen_8227) -> q_gen_8227
<s, z, s>(q_gen_8230) -> q_gen_8227
<z, s, s>(q_gen_8230) -> q_gen_8227
<z, z, s>(q_gen_8241) -> q_gen_8227
<z, z, z>() -> q_gen_8227
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 8
() -> 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, _jea])) -> leq([n, _jea]) -> 6
(plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]) -> 13
}
Sat witness: Found: ((plus([n, mm, _eea])) -> plus([n, s(mm), s(_eea)]), {
_eea -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

Total time: 1.134633
Reason for stopping: Proved