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, _mh])) -> plus([n, s(mm), s(_mh)])}
(plus([_nh, _oh, _ph]) /\ plus([_nh, _oh, _qh])) -> eq_nat([_ph, _qh])
)
(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:
{(leq([i, z])) -> eq_nat([i, z])}


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

Clause system for inference is:

{
() -> leq([z, n2]) -> 0
() -> plus([n, z, n]) -> 0
(leq([i, z])) -> eq_nat([i, z]) -> 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, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 0
}


Solving took 0.105596 seconds.
Proved
Model:
|_
{
leq -> 
{{{
Q={q_gen_2050, q_gen_2057},
Q_f={q_gen_2050},
Delta=
{
<s>(q_gen_2057) -> q_gen_2057
<z>() -> q_gen_2057
<s, s>(q_gen_2050) -> q_gen_2050
<z, s>(q_gen_2057) -> q_gen_2050
<z, z>() -> q_gen_2050
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049, q_gen_2052, q_gen_2061},
Q_f={q_gen_2049},
Delta=
{
<s>(q_gen_2061) -> q_gen_2061
<z>() -> q_gen_2061
<s, s>(q_gen_2052) -> q_gen_2052
<z, s>(q_gen_2061) -> q_gen_2052
<z, z>() -> q_gen_2052
<s, s, s>(q_gen_2049) -> q_gen_2049
<s, z, s>(q_gen_2052) -> q_gen_2049
<z, s, s>(q_gen_2052) -> q_gen_2049
<z, z, s>(q_gen_2061) -> q_gen_2049
<z, z, z>() -> q_gen_2049
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.008927 s (model generation: 0.008468,  model checking: 0.000459):

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([i, z])) -> eq_nat([i, z]) -> 1
(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, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 1
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.007721 s (model generation: 0.007666,  model checking: 0.000055):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049},
Q_f={q_gen_2049},
Delta=
{
<z, z, z>() -> q_gen_2049
}

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

-------------------------------------------
Step 2, which took 0.007241 s (model generation: 0.007145,  model checking: 0.000096):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2050},
Q_f={q_gen_2050},
Delta=
{
<z, z>() -> q_gen_2050
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049},
Q_f={q_gen_2049},
Delta=
{
<z, z, z>() -> q_gen_2049
}

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([i, z])) -> eq_nat([i, z]) -> 1
(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, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 4
}
Sat witness: Found: ((plus([n, mm, _mh])) -> plus([n, s(mm), s(_mh)]), {
_mh -> z  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 3, which took 0.008124 s (model generation: 0.007998,  model checking: 0.000126):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2050},
Q_f={q_gen_2050},
Delta=
{
<z, z>() -> q_gen_2050
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049, q_gen_2052},
Q_f={q_gen_2049},
Delta=
{
<z, z>() -> q_gen_2052
<z, s, s>(q_gen_2052) -> q_gen_2049
<z, z, z>() -> q_gen_2049
}

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([i, z])) -> eq_nat([i, z]) -> 1
(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, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 4
}
Sat witness: Found: ((leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 4, which took 0.011547 s (model generation: 0.009195,  model checking: 0.002352):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2050},
Q_f={q_gen_2050},
Delta=
{
<s, s>(q_gen_2050) -> q_gen_2050
<z, z>() -> q_gen_2050
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049, q_gen_2052},
Q_f={q_gen_2049},
Delta=
{
<z, z>() -> q_gen_2052
<z, s, s>(q_gen_2052) -> q_gen_2049
<z, z, z>() -> q_gen_2049
}

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([i, z])) -> eq_nat([i, z]) -> 2
(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, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 4
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 5, which took 0.008181 s (model generation: 0.008044,  model checking: 0.000137):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2050},
Q_f={q_gen_2050},
Delta=
{
<s, s>(q_gen_2050) -> q_gen_2050
<z, z>() -> q_gen_2050
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049, q_gen_2052},
Q_f={q_gen_2049},
Delta=
{
<s, s>(q_gen_2052) -> q_gen_2052
<z, z>() -> q_gen_2052
<s, z, s>(q_gen_2052) -> q_gen_2049
<z, s, s>(q_gen_2052) -> q_gen_2049
<z, z, z>() -> q_gen_2049
}

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([i, z])) -> eq_nat([i, z]) -> 3
(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, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 4
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(z)
})

-------------------------------------------
Step 6, which took 0.009526 s (model generation: 0.009349,  model checking: 0.000177):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2050, q_gen_2057},
Q_f={q_gen_2050},
Delta=
{
<z>() -> q_gen_2057
<s, s>(q_gen_2050) -> q_gen_2050
<z, s>(q_gen_2057) -> q_gen_2050
<z, z>() -> q_gen_2050
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049, q_gen_2052},
Q_f={q_gen_2049},
Delta=
{
<s, s>(q_gen_2052) -> q_gen_2052
<z, z>() -> q_gen_2052
<s, z, s>(q_gen_2052) -> q_gen_2049
<z, s, s>(q_gen_2052) -> q_gen_2049
<z, z, z>() -> q_gen_2049
}

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([i, z])) -> eq_nat([i, z]) -> 4
(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, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 7
}
Sat witness: Found: ((plus([n, mm, _mh])) -> plus([n, s(mm), s(_mh)]), {
_mh -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 7, which took 0.009253 s (model generation: 0.008889,  model checking: 0.000364):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2050, q_gen_2057},
Q_f={q_gen_2050},
Delta=
{
<z>() -> q_gen_2057
<s, s>(q_gen_2050) -> q_gen_2050
<z, s>(q_gen_2057) -> q_gen_2050
<z, z>() -> q_gen_2050
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049, q_gen_2052, q_gen_2061},
Q_f={q_gen_2049},
Delta=
{
<z>() -> q_gen_2061
<s, s>(q_gen_2052) -> q_gen_2052
<z, z>() -> q_gen_2052
<s, s, s>(q_gen_2049) -> q_gen_2049
<s, z, s>(q_gen_2052) -> q_gen_2049
<z, s, s>(q_gen_2052) -> q_gen_2049
<z, z, s>(q_gen_2061) -> q_gen_2049
<z, z, z>() -> q_gen_2049
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 9
() -> plus([n, z, n]) -> 7
(leq([i, z])) -> eq_nat([i, z]) -> 5
(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, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 7
}
Sat witness: Found: (() -> leq([z, n2]), {
n2 -> s(s(z))
})

-------------------------------------------
Step 8, which took 0.009919 s (model generation: 0.008875,  model checking: 0.001044):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2050, q_gen_2057},
Q_f={q_gen_2050},
Delta=
{
<s>(q_gen_2057) -> q_gen_2057
<z>() -> q_gen_2057
<s, s>(q_gen_2050) -> q_gen_2050
<z, s>(q_gen_2057) -> q_gen_2050
<z, z>() -> q_gen_2050
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049, q_gen_2052, q_gen_2061},
Q_f={q_gen_2049},
Delta=
{
<z>() -> q_gen_2061
<s, s>(q_gen_2052) -> q_gen_2052
<z, z>() -> q_gen_2052
<s, s, s>(q_gen_2049) -> q_gen_2049
<s, z, s>(q_gen_2052) -> q_gen_2049
<z, s, s>(q_gen_2052) -> q_gen_2049
<z, z, s>(q_gen_2061) -> q_gen_2049
<z, z, z>() -> q_gen_2049
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 9
() -> plus([n, z, n]) -> 7
(leq([i, z])) -> eq_nat([i, z]) -> 6
(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, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _mh])) -> plus([n, s(mm), s(_mh)]), {
_mh -> s(z)  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 9, which took 0.012705 s (model generation: 0.011943,  model checking: 0.000762):

Model:
|_
{
leq -> 
{{{
Q={q_gen_2050, q_gen_2057},
Q_f={q_gen_2050},
Delta=
{
<s>(q_gen_2057) -> q_gen_2057
<z>() -> q_gen_2057
<s, s>(q_gen_2050) -> q_gen_2050
<z, s>(q_gen_2057) -> q_gen_2050
<z, z>() -> q_gen_2050
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_2049, q_gen_2052, q_gen_2061},
Q_f={q_gen_2049},
Delta=
{
<z>() -> q_gen_2061
<s, s>(q_gen_2052) -> q_gen_2052
<z, s>(q_gen_2061) -> q_gen_2052
<z, z>() -> q_gen_2052
<s, s, s>(q_gen_2049) -> q_gen_2049
<s, z, s>(q_gen_2052) -> q_gen_2049
<z, s, s>(q_gen_2052) -> q_gen_2049
<z, z, s>(q_gen_2061) -> q_gen_2049
<z, z, z>() -> q_gen_2049
}

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

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

Teacher's answer:
New clause system:
{
() -> leq([z, n2]) -> 10
() -> plus([n, z, n]) -> 8
(leq([i, z])) -> eq_nat([i, z]) -> 7
(leq([nn1, nn2])) -> leq([s(nn1), s(nn2)]) -> 7
(leq([s(nn1), s(nn2)])) -> leq([nn1, nn2]) -> 7
(leq([s(nn1), z])) -> BOT -> 7
(plus([n, mm, _mh])) -> plus([n, s(mm), s(_mh)]) -> 13
}
Sat witness: Found: ((plus([n, mm, _mh])) -> plus([n, s(mm), s(_mh)]), {
_mh -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

Total time: 0.105596
Reason for stopping: Proved