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, _ju])) -> plus([n, s(mm), s(_ju)])}
(plus([_ku, _lu, _mu]) /\ plus([_ku, _lu, _nu])) -> eq_nat([_mu, _nu])
)
(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([i, m, _ou])) -> le([i, s(_ou)])}


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([i, m, _ou])) -> le([i, s(_ou)]) -> 0
(plus([n, mm, _ju])) -> plus([n, s(mm), s(_ju)]) -> 0
}


Solving took 0.150105 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_5501, q_gen_5502},
Q_f={q_gen_5501},
Delta=
{
<s>(q_gen_5502) -> q_gen_5502
<z>() -> q_gen_5502
<s, s>(q_gen_5501) -> q_gen_5501
<z, s>(q_gen_5502) -> q_gen_5501
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_5500, q_gen_5504, q_gen_5513},
Q_f={q_gen_5500},
Delta=
{
<s>(q_gen_5513) -> q_gen_5513
<z>() -> q_gen_5513
<s, s>(q_gen_5504) -> q_gen_5504
<z, s>(q_gen_5513) -> q_gen_5504
<z, z>() -> q_gen_5504
<s, s, s>(q_gen_5500) -> q_gen_5500
<s, z, s>(q_gen_5504) -> q_gen_5500
<z, s, s>(q_gen_5504) -> q_gen_5500
<z, z, s>(q_gen_5513) -> q_gen_5500
<z, z, z>() -> q_gen_5500
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.026778 s (model generation: 0.026526,  model checking: 0.000252):

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

-------------------------------------------
Step 1, which took 0.014380 s (model generation: 0.014302,  model checking: 0.000078):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_5500},
Q_f={q_gen_5500},
Delta=
{
<z, z, z>() -> q_gen_5500
}

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

-------------------------------------------
Step 2, which took 0.012451 s (model generation: 0.012326,  model checking: 0.000125):

Model:
|_
{
le -> 
{{{
Q={q_gen_5501, q_gen_5502},
Q_f={q_gen_5501},
Delta=
{
<z>() -> q_gen_5502
<z, s>(q_gen_5502) -> q_gen_5501
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_5500},
Q_f={q_gen_5500},
Delta=
{
<z, z, z>() -> q_gen_5500
}

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

-------------------------------------------
Step 3, which took 0.010747 s (model generation: 0.010647,  model checking: 0.000100):

Model:
|_
{
le -> 
{{{
Q={q_gen_5501, q_gen_5502},
Q_f={q_gen_5501},
Delta=
{
<z>() -> q_gen_5502
<z, s>(q_gen_5502) -> q_gen_5501
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_5500, q_gen_5504},
Q_f={q_gen_5500},
Delta=
{
<z, z>() -> q_gen_5504
<z, s, s>(q_gen_5504) -> q_gen_5500
<z, z, z>() -> q_gen_5500
}

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([i, m, _ou])) -> le([i, s(_ou)]) -> 4
(plus([n, mm, _ju])) -> plus([n, s(mm), s(_ju)]) -> 4
}
Sat witness: Found: ((plus([i, m, _ou])) -> le([i, s(_ou)]), {
_ou -> s(z)  ;  i -> z  ;  m -> s(z)
})

-------------------------------------------
Step 4, which took 0.016060 s (model generation: 0.012872,  model checking: 0.003188):

Model:
|_
{
le -> 
{{{
Q={q_gen_5501, q_gen_5502},
Q_f={q_gen_5501},
Delta=
{
<s>(q_gen_5502) -> q_gen_5502
<z>() -> q_gen_5502
<z, s>(q_gen_5502) -> q_gen_5501
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_5500, q_gen_5504},
Q_f={q_gen_5500},
Delta=
{
<z, z>() -> q_gen_5504
<z, s, s>(q_gen_5504) -> q_gen_5500
<z, z, z>() -> q_gen_5500
}

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

-------------------------------------------
Step 5, which took 0.015994 s (model generation: 0.015755,  model checking: 0.000239):

Model:
|_
{
le -> 
{{{
Q={q_gen_5501, q_gen_5502},
Q_f={q_gen_5501},
Delta=
{
<s>(q_gen_5502) -> q_gen_5502
<z>() -> q_gen_5502
<s, s>(q_gen_5501) -> q_gen_5501
<z, s>(q_gen_5502) -> q_gen_5501
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_5500, q_gen_5504},
Q_f={q_gen_5500},
Delta=
{
<z, z>() -> q_gen_5504
<z, s, s>(q_gen_5504) -> q_gen_5500
<z, z, z>() -> q_gen_5500
}

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

-------------------------------------------
Step 6, which took 0.013218 s (model generation: 0.012584,  model checking: 0.000634):

Model:
|_
{
le -> 
{{{
Q={q_gen_5501, q_gen_5502},
Q_f={q_gen_5501},
Delta=
{
<s>(q_gen_5502) -> q_gen_5502
<z>() -> q_gen_5502
<s, s>(q_gen_5501) -> q_gen_5501
<z, s>(q_gen_5502) -> q_gen_5501
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_5500, q_gen_5504},
Q_f={q_gen_5500},
Delta=
{
<s, s>(q_gen_5504) -> q_gen_5504
<z, z>() -> q_gen_5504
<s, z, s>(q_gen_5504) -> q_gen_5500
<z, s, s>(q_gen_5504) -> q_gen_5500
<z, z, z>() -> q_gen_5500
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 4
() -> 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([i, m, _ou])) -> le([i, s(_ou)]) -> 4
(plus([n, mm, _ju])) -> plus([n, s(mm), s(_ju)]) -> 7
}
Sat witness: Found: ((plus([n, mm, _ju])) -> plus([n, s(mm), s(_ju)]), {
_ju -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 7, which took 0.013446 s (model generation: 0.012938,  model checking: 0.000508):

Model:
|_
{
le -> 
{{{
Q={q_gen_5501, q_gen_5502},
Q_f={q_gen_5501},
Delta=
{
<s>(q_gen_5502) -> q_gen_5502
<z>() -> q_gen_5502
<s, s>(q_gen_5501) -> q_gen_5501
<z, s>(q_gen_5502) -> q_gen_5501
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_5500, q_gen_5504, q_gen_5513},
Q_f={q_gen_5500},
Delta=
{
<z>() -> q_gen_5513
<s, s>(q_gen_5504) -> q_gen_5504
<z, z>() -> q_gen_5504
<s, s, s>(q_gen_5500) -> q_gen_5500
<s, z, s>(q_gen_5504) -> q_gen_5500
<z, s, s>(q_gen_5504) -> q_gen_5500
<z, z, s>(q_gen_5513) -> q_gen_5500
<z, z, z>() -> q_gen_5500
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 5
() -> 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([i, m, _ou])) -> le([i, s(_ou)]) -> 5
(plus([n, mm, _ju])) -> plus([n, s(mm), s(_ju)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _ju])) -> plus([n, s(mm), s(_ju)]), {
_ju -> s(z)  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 8, which took 0.012575 s (model generation: 0.012081,  model checking: 0.000494):

Model:
|_
{
le -> 
{{{
Q={q_gen_5501, q_gen_5502},
Q_f={q_gen_5501},
Delta=
{
<s>(q_gen_5502) -> q_gen_5502
<z>() -> q_gen_5502
<s, s>(q_gen_5501) -> q_gen_5501
<z, s>(q_gen_5502) -> q_gen_5501
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_5500, q_gen_5504, q_gen_5513},
Q_f={q_gen_5500},
Delta=
{
<z>() -> q_gen_5513
<s, s>(q_gen_5504) -> q_gen_5504
<z, s>(q_gen_5513) -> q_gen_5504
<z, z>() -> q_gen_5504
<s, s, s>(q_gen_5500) -> q_gen_5500
<s, z, s>(q_gen_5504) -> q_gen_5500
<z, s, s>(q_gen_5504) -> q_gen_5500
<z, z, s>(q_gen_5513) -> q_gen_5500
<z, z, z>() -> q_gen_5500
}

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]) -> 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([i, m, _ou])) -> le([i, s(_ou)]) -> 6
(plus([n, mm, _ju])) -> plus([n, s(mm), s(_ju)]) -> 13
}
Sat witness: Found: ((plus([n, mm, _ju])) -> plus([n, s(mm), s(_ju)]), {
_ju -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

Total time: 0.150105
Reason for stopping: Proved