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, _qga])) -> plus([n, s(mm), s(_qga)])}
(plus([_rga, _sga, _tga]) /\ plus([_rga, _sga, _uga])) -> eq_nat([_tga, _uga])
)
(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:
{(plus([m, i, _vga])) -> le([i, s(_vga)])}


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

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, z])) -> BOT -> 0
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 0
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 0
}


Solving took 0.185419 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_8619, q_gen_8620},
Q_f={q_gen_8619},
Delta=
{
<s>(q_gen_8620) -> q_gen_8620
<z>() -> q_gen_8620
<s, s>(q_gen_8619) -> q_gen_8619
<z, s>(q_gen_8620) -> q_gen_8619
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8618, q_gen_8622, q_gen_8631},
Q_f={q_gen_8618},
Delta=
{
<s>(q_gen_8631) -> q_gen_8631
<z>() -> q_gen_8631
<s, s>(q_gen_8622) -> q_gen_8622
<z, s>(q_gen_8631) -> q_gen_8622
<z, z>() -> q_gen_8622
<s, s, s>(q_gen_8618) -> q_gen_8618
<s, z, s>(q_gen_8622) -> q_gen_8618
<z, s, s>(q_gen_8622) -> q_gen_8618
<z, z, s>(q_gen_8631) -> q_gen_8618
<z, z, z>() -> q_gen_8618
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.016195 s (model generation: 0.015754,  model checking: 0.000441):

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, z])) -> BOT -> 1
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 1
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 1
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.015544 s (model generation: 0.015423,  model checking: 0.000121):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8618},
Q_f={q_gen_8618},
Delta=
{
<z, z, z>() -> q_gen_8618
}

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, z])) -> BOT -> 1
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 1
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 1
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.016741 s (model generation: 0.016556,  model checking: 0.000185):

Model:
|_
{
le -> 
{{{
Q={q_gen_8619, q_gen_8620},
Q_f={q_gen_8619},
Delta=
{
<z>() -> q_gen_8620
<z, s>(q_gen_8620) -> q_gen_8619
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8618},
Q_f={q_gen_8618},
Delta=
{
<z, z, z>() -> q_gen_8618
}

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, z])) -> BOT -> 1
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 1
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 4
}
Sat witness: Found: ((plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]), {
_qga -> z  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 3, which took 0.017470 s (model generation: 0.015140,  model checking: 0.002330):

Model:
|_
{
le -> 
{{{
Q={q_gen_8619, q_gen_8620},
Q_f={q_gen_8619},
Delta=
{
<z>() -> q_gen_8620
<z, s>(q_gen_8620) -> q_gen_8619
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8618, q_gen_8622},
Q_f={q_gen_8618},
Delta=
{
<z, z>() -> q_gen_8622
<z, s, s>(q_gen_8622) -> q_gen_8618
<z, z, z>() -> q_gen_8618
}

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, z])) -> BOT -> 1
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 4
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 4
}
Sat witness: Found: ((plus([m, i, _vga])) -> le([i, s(_vga)]), {
_vga -> s(z)  ;  i -> s(z)  ;  m -> z
})

-------------------------------------------
Step 4, which took 0.016962 s (model generation: 0.016435,  model checking: 0.000527):

Model:
|_
{
le -> 
{{{
Q={q_gen_8619, q_gen_8620},
Q_f={q_gen_8619},
Delta=
{
<z>() -> q_gen_8620
<s, s>(q_gen_8619) -> q_gen_8619
<z, s>(q_gen_8620) -> q_gen_8619
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8618, q_gen_8622},
Q_f={q_gen_8618},
Delta=
{
<z, z>() -> q_gen_8622
<z, s, s>(q_gen_8622) -> q_gen_8618
<z, z, z>() -> q_gen_8618
}

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)]) -> 2
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 2
(le([s(nn1), z])) -> BOT -> 2
(le([z, z])) -> BOT -> 2
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 4
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 4
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 5, which took 0.017335 s (model generation: 0.016903,  model checking: 0.000432):

Model:
|_
{
le -> 
{{{
Q={q_gen_8619, q_gen_8620},
Q_f={q_gen_8619},
Delta=
{
<z>() -> q_gen_8620
<s, s>(q_gen_8619) -> q_gen_8619
<z, s>(q_gen_8620) -> q_gen_8619
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8618, q_gen_8622},
Q_f={q_gen_8618},
Delta=
{
<s, s>(q_gen_8622) -> q_gen_8622
<z, z>() -> q_gen_8622
<s, z, s>(q_gen_8622) -> q_gen_8618
<z, s, s>(q_gen_8622) -> q_gen_8618
<z, z, z>() -> q_gen_8618
}

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)]) -> 3
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 3
(le([s(nn1), z])) -> BOT -> 3
(le([z, z])) -> BOT -> 3
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 4
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 4
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 6, which took 0.018851 s (model generation: 0.016763,  model checking: 0.002088):

Model:
|_
{
le -> 
{{{
Q={q_gen_8619, q_gen_8620},
Q_f={q_gen_8619},
Delta=
{
<s>(q_gen_8620) -> q_gen_8620
<z>() -> q_gen_8620
<s, s>(q_gen_8619) -> q_gen_8619
<z, s>(q_gen_8620) -> q_gen_8619
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8618, q_gen_8622},
Q_f={q_gen_8618},
Delta=
{
<s, s>(q_gen_8622) -> q_gen_8622
<z, z>() -> q_gen_8622
<s, z, s>(q_gen_8622) -> q_gen_8618
<z, s, s>(q_gen_8622) -> q_gen_8618
<z, z, z>() -> q_gen_8618
}

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, z])) -> BOT -> 4
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 4
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 7
}
Sat witness: Found: ((plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]), {
_qga -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 7, which took 0.019965 s (model generation: 0.018971,  model checking: 0.000994):

Model:
|_
{
le -> 
{{{
Q={q_gen_8619, q_gen_8620},
Q_f={q_gen_8619},
Delta=
{
<s>(q_gen_8620) -> q_gen_8620
<z>() -> q_gen_8620
<s, s>(q_gen_8619) -> q_gen_8619
<z, s>(q_gen_8620) -> q_gen_8619
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8618, q_gen_8622, q_gen_8631},
Q_f={q_gen_8618},
Delta=
{
<z>() -> q_gen_8631
<s, s>(q_gen_8622) -> q_gen_8622
<z, z>() -> q_gen_8622
<s, s, s>(q_gen_8618) -> q_gen_8618
<s, z, s>(q_gen_8622) -> q_gen_8618
<z, s, s>(q_gen_8622) -> q_gen_8618
<z, z, s>(q_gen_8631) -> q_gen_8618
<z, z, z>() -> q_gen_8618
}

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)]) -> 5
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 5
(le([s(nn1), z])) -> BOT -> 5
(le([z, z])) -> BOT -> 5
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 5
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]), {
_qga -> s(z)  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 8, which took 0.020866 s (model generation: 0.019789,  model checking: 0.001077):

Model:
|_
{
le -> 
{{{
Q={q_gen_8619, q_gen_8620},
Q_f={q_gen_8619},
Delta=
{
<s>(q_gen_8620) -> q_gen_8620
<z>() -> q_gen_8620
<s, s>(q_gen_8619) -> q_gen_8619
<z, s>(q_gen_8620) -> q_gen_8619
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_8618, q_gen_8622, q_gen_8631},
Q_f={q_gen_8618},
Delta=
{
<z>() -> q_gen_8631
<s, s>(q_gen_8622) -> q_gen_8622
<z, s>(q_gen_8631) -> q_gen_8622
<z, z>() -> q_gen_8622
<s, s, s>(q_gen_8618) -> q_gen_8618
<s, z, s>(q_gen_8622) -> q_gen_8618
<z, s, s>(q_gen_8622) -> q_gen_8618
<z, z, s>(q_gen_8631) -> q_gen_8618
<z, z, z>() -> q_gen_8618
}

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)]) -> 6
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 6
(le([s(nn1), z])) -> BOT -> 6
(le([z, z])) -> BOT -> 6
(plus([m, i, _vga])) -> le([i, s(_vga)]) -> 6
(plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]) -> 13
}
Sat witness: Found: ((plus([n, mm, _qga])) -> plus([n, s(mm), s(_qga)]), {
_qga -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

Total time: 0.185419
Reason for stopping: Proved