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


Learning problem is:

env: {
nat -> {s, z}
}
definition:
{
(plus, F:
{() -> plus([n, z, n])
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)])}
(plus([_qj, _rj, _sj]) /\ plus([_qj, _rj, _tj])) -> eq_nat([_sj, _tj])
)
(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, _uj])) -> leq([n, _uj])}


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, _uj])) -> leq([n, _uj]) -> 0
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 0
}


Solving took 0.059919 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_2152, q_gen_2156},
Q_f={q_gen_2152},
Delta=
{
<s>(q_gen_2156) -> q_gen_2156
<z>() -> q_gen_2156
<s, s>(q_gen_2152) -> q_gen_2152
<z, s>(q_gen_2156) -> q_gen_2152
<z, z>() -> q_gen_2152
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151, q_gen_2154, q_gen_2165},
Q_f={q_gen_2151},
Delta=
{
<s>(q_gen_2165) -> q_gen_2165
<z>() -> q_gen_2165
<s, s>(q_gen_2154) -> q_gen_2154
<z, s>(q_gen_2165) -> q_gen_2154
<z, z>() -> q_gen_2154
<s, s, s>(q_gen_2151) -> q_gen_2151
<s, z, s>(q_gen_2154) -> q_gen_2151
<z, s, s>(q_gen_2154) -> q_gen_2151
<z, z, s>(q_gen_2165) -> q_gen_2151
<z, z, z>() -> q_gen_2151
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.005201 s (model generation: 0.004988,  model checking: 0.000213):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
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
() -> 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, _uj])) -> leq([n, _uj]) -> 1
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 1
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.005214 s (model generation: 0.005134,  model checking: 0.000080):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151},
Q_f={q_gen_2151},
Delta=
{
<z, z, z>() -> q_gen_2151
}

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

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

-------------------------------------------
Step 2, which took 0.005132 s (model generation: 0.004604,  model checking: 0.000528):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2152},
Q_f={q_gen_2152},
Delta=
{
<z, z>() -> q_gen_2152
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151},
Q_f={q_gen_2151},
Delta=
{
<z, z, z>() -> q_gen_2151
}

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

}
--
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, _uj])) -> leq([n, _uj]) -> 1
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 4
}
Sat witness: Found: ((plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]), {
_pj -> z  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 3, which took 0.005633 s (model generation: 0.005517,  model checking: 0.000116):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2152},
Q_f={q_gen_2152},
Delta=
{
<z, z>() -> q_gen_2152
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151, q_gen_2154},
Q_f={q_gen_2151},
Delta=
{
<z, z>() -> q_gen_2154
<z, s, s>(q_gen_2154) -> q_gen_2151
<z, z, z>() -> q_gen_2151
}

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

}
--
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, _uj])) -> leq([n, _uj]) -> 4
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 4
}
Sat witness: Found: ((plus([n, m, _uj])) -> leq([n, _uj]), {
_uj -> s(z)  ;  m -> s(z)  ;  n -> z
})

-------------------------------------------
Step 4, which took 0.004875 s (model generation: 0.004773,  model checking: 0.000102):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2152, q_gen_2156},
Q_f={q_gen_2152},
Delta=
{
<z>() -> q_gen_2156
<z, s>(q_gen_2156) -> q_gen_2152
<z, z>() -> q_gen_2152
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151, q_gen_2154},
Q_f={q_gen_2151},
Delta=
{
<z, z>() -> q_gen_2154
<z, s, s>(q_gen_2154) -> q_gen_2151
<z, z, z>() -> q_gen_2151
}

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

}
--
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, _uj])) -> leq([n, _uj]) -> 4
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 4
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 5, which took 0.004528 s (model generation: 0.004292,  model checking: 0.000236):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2152, q_gen_2156},
Q_f={q_gen_2152},
Delta=
{
<z>() -> q_gen_2156
<s, s>(q_gen_2152) -> q_gen_2152
<z, s>(q_gen_2156) -> q_gen_2152
<z, z>() -> q_gen_2152
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151, q_gen_2154},
Q_f={q_gen_2151},
Delta=
{
<z, z>() -> q_gen_2154
<z, s, s>(q_gen_2154) -> q_gen_2151
<z, z, z>() -> q_gen_2151
}

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

}
--
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, _uj])) -> leq([n, _uj]) -> 4
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 4
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 6, which took 0.004947 s (model generation: 0.004843,  model checking: 0.000104):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2152, q_gen_2156},
Q_f={q_gen_2152},
Delta=
{
<z>() -> q_gen_2156
<s, s>(q_gen_2152) -> q_gen_2152
<z, s>(q_gen_2156) -> q_gen_2152
<z, z>() -> q_gen_2152
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151, q_gen_2154},
Q_f={q_gen_2151},
Delta=
{
<s, s>(q_gen_2154) -> q_gen_2154
<z, z>() -> q_gen_2154
<s, z, s>(q_gen_2154) -> q_gen_2151
<z, s, s>(q_gen_2154) -> q_gen_2151
<z, z, z>() -> q_gen_2151
}

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

}
--
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, _uj])) -> leq([n, _uj]) -> 4
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 4
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(s(z))
})

-------------------------------------------
Step 7, which took 0.005850 s (model generation: 0.005726,  model checking: 0.000124):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2152, q_gen_2156},
Q_f={q_gen_2152},
Delta=
{
<s>(q_gen_2156) -> q_gen_2156
<z>() -> q_gen_2156
<s, s>(q_gen_2152) -> q_gen_2152
<z, s>(q_gen_2156) -> q_gen_2152
<z, z>() -> q_gen_2152
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151, q_gen_2154},
Q_f={q_gen_2151},
Delta=
{
<s, s>(q_gen_2154) -> q_gen_2154
<z, z>() -> q_gen_2154
<s, z, s>(q_gen_2154) -> q_gen_2151
<z, s, s>(q_gen_2154) -> q_gen_2151
<z, z, z>() -> q_gen_2151
}

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

}
--
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, _uj])) -> leq([n, _uj]) -> 4
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 7
}
Sat witness: Found: ((plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]), {
_pj -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 8, which took 0.004952 s (model generation: 0.004592,  model checking: 0.000360):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2152, q_gen_2156},
Q_f={q_gen_2152},
Delta=
{
<s>(q_gen_2156) -> q_gen_2156
<z>() -> q_gen_2156
<s, s>(q_gen_2152) -> q_gen_2152
<z, s>(q_gen_2156) -> q_gen_2152
<z, z>() -> q_gen_2152
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151, q_gen_2154, q_gen_2165},
Q_f={q_gen_2151},
Delta=
{
<z>() -> q_gen_2165
<s, s>(q_gen_2154) -> q_gen_2154
<z, z>() -> q_gen_2154
<s, s, s>(q_gen_2151) -> q_gen_2151
<s, z, s>(q_gen_2154) -> q_gen_2151
<z, s, s>(q_gen_2154) -> q_gen_2151
<z, z, s>(q_gen_2165) -> q_gen_2151
<z, z, z>() -> q_gen_2151
}

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

}
--
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, _uj])) -> leq([n, _uj]) -> 5
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]), {
_pj -> s(z)  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 9, which took 0.006283 s (model generation: 0.005793,  model checking: 0.000490):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2152, q_gen_2156},
Q_f={q_gen_2152},
Delta=
{
<s>(q_gen_2156) -> q_gen_2156
<z>() -> q_gen_2156
<s, s>(q_gen_2152) -> q_gen_2152
<z, s>(q_gen_2156) -> q_gen_2152
<z, z>() -> q_gen_2152
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_2151, q_gen_2154, q_gen_2165},
Q_f={q_gen_2151},
Delta=
{
<z>() -> q_gen_2165
<s, s>(q_gen_2154) -> q_gen_2154
<z, s>(q_gen_2165) -> q_gen_2154
<z, z>() -> q_gen_2154
<s, s, s>(q_gen_2151) -> q_gen_2151
<s, z, s>(q_gen_2154) -> q_gen_2151
<z, s, s>(q_gen_2154) -> q_gen_2151
<z, z, s>(q_gen_2165) -> q_gen_2151
<z, z, z>() -> q_gen_2151
}

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

}
--
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, _uj])) -> leq([n, _uj]) -> 6
(plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]) -> 13
}
Sat witness: Found: ((plus([n, mm, _pj])) -> plus([n, s(mm), s(_pj)]), {
_pj -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

Total time: 0.059919
Reason for stopping: Proved