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, _eba])) -> plus([n, s(mm), s(_eba)])}
(plus([_fba, _gba, _hba]) /\ plus([_fba, _gba, _iba])) -> eq_nat([_hba, _iba])
)
(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, _jba])) -> le([i, s(_jba)])}


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, _jba])) -> le([i, s(_jba)]) -> 0
(plus([n, mm, _eba])) -> plus([n, s(mm), s(_eba)]) -> 0
}


Solving took 0.112362 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_5517, q_gen_5518},
Q_f={q_gen_5517},
Delta=
{
<s>(q_gen_5518) -> q_gen_5518
<z>() -> q_gen_5518
<s, s>(q_gen_5517) -> q_gen_5517
<z, s>(q_gen_5518) -> q_gen_5517
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_5516, q_gen_5520, q_gen_5529},
Q_f={q_gen_5516},
Delta=
{
<s>(q_gen_5529) -> q_gen_5529
<z>() -> q_gen_5529
<s, s>(q_gen_5520) -> q_gen_5520
<z, s>(q_gen_5529) -> q_gen_5520
<z, z>() -> q_gen_5520
<s, s, s>(q_gen_5516) -> q_gen_5516
<s, z, s>(q_gen_5520) -> q_gen_5516
<z, s, s>(q_gen_5520) -> q_gen_5516
<z, z, s>(q_gen_5529) -> q_gen_5516
<z, z, z>() -> q_gen_5516
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.004860 s (model generation: 0.004709,  model checking: 0.000151):

Model:
|_
{
le -> 
{{{
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:
{
() -> 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, _jba])) -> le([i, s(_jba)]) -> 1
(plus([n, mm, _eba])) -> plus([n, s(mm), s(_eba)]) -> 1
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.005207 s (model generation: 0.005136,  model checking: 0.000071):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_5516},
Q_f={q_gen_5516},
Delta=
{
<z, z, z>() -> q_gen_5516
}

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

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

-------------------------------------------
Step 2, which took 0.007299 s (model generation: 0.007152,  model checking: 0.000147):

Model:
|_
{
le -> 
{{{
Q={q_gen_5517, q_gen_5518},
Q_f={q_gen_5517},
Delta=
{
<z>() -> q_gen_5518
<z, s>(q_gen_5518) -> q_gen_5517
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_5516},
Q_f={q_gen_5516},
Delta=
{
<z, z, z>() -> q_gen_5516
}

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

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

-------------------------------------------
Step 3, which took 0.008876 s (model generation: 0.008695,  model checking: 0.000181):

Model:
|_
{
le -> 
{{{
Q={q_gen_5517, q_gen_5518},
Q_f={q_gen_5517},
Delta=
{
<z>() -> q_gen_5518
<z, s>(q_gen_5518) -> q_gen_5517
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_5516, q_gen_5520},
Q_f={q_gen_5516},
Delta=
{
<z, z>() -> q_gen_5520
<z, s, s>(q_gen_5520) -> q_gen_5516
<z, z, z>() -> q_gen_5516
}

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

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

-------------------------------------------
Step 4, which took 0.011143 s (model generation: 0.010887,  model checking: 0.000256):

Model:
|_
{
le -> 
{{{
Q={q_gen_5517, q_gen_5518},
Q_f={q_gen_5517},
Delta=
{
<s>(q_gen_5518) -> q_gen_5518
<z>() -> q_gen_5518
<z, s>(q_gen_5518) -> q_gen_5517
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_5516, q_gen_5520},
Q_f={q_gen_5516},
Delta=
{
<z, z>() -> q_gen_5520
<z, s, s>(q_gen_5520) -> q_gen_5516
<z, z, z>() -> q_gen_5516
}

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

}
--
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, _jba])) -> le([i, s(_jba)]) -> 4
(plus([n, mm, _eba])) -> plus([n, s(mm), s(_eba)]) -> 4
}
Sat witness: Found: ((le([nn1, nn2])) -> le([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> s(z)
})

-------------------------------------------
Step 5, which took 0.015455 s (model generation: 0.015055,  model checking: 0.000400):

Model:
|_
{
le -> 
{{{
Q={q_gen_5517, q_gen_5518},
Q_f={q_gen_5517},
Delta=
{
<s>(q_gen_5518) -> q_gen_5518
<z>() -> q_gen_5518
<s, s>(q_gen_5517) -> q_gen_5517
<z, s>(q_gen_5518) -> q_gen_5517
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_5516, q_gen_5520},
Q_f={q_gen_5516},
Delta=
{
<z, z>() -> q_gen_5520
<z, s, s>(q_gen_5520) -> q_gen_5516
<z, z, z>() -> q_gen_5516
}

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

}
--
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, _jba])) -> le([i, s(_jba)]) -> 4
(plus([n, mm, _eba])) -> plus([n, s(mm), s(_eba)]) -> 4
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 6, which took 0.012477 s (model generation: 0.011940,  model checking: 0.000537):

Model:
|_
{
le -> 
{{{
Q={q_gen_5517, q_gen_5518},
Q_f={q_gen_5517},
Delta=
{
<s>(q_gen_5518) -> q_gen_5518
<z>() -> q_gen_5518
<s, s>(q_gen_5517) -> q_gen_5517
<z, s>(q_gen_5518) -> q_gen_5517
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_5516, q_gen_5520},
Q_f={q_gen_5516},
Delta=
{
<s, s>(q_gen_5520) -> q_gen_5520
<z, z>() -> q_gen_5520
<s, z, s>(q_gen_5520) -> q_gen_5516
<z, s, s>(q_gen_5520) -> q_gen_5516
<z, z, z>() -> q_gen_5516
}

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

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

-------------------------------------------
Step 7, which took 0.015304 s (model generation: 0.012402,  model checking: 0.002902):

Model:
|_
{
le -> 
{{{
Q={q_gen_5517, q_gen_5518},
Q_f={q_gen_5517},
Delta=
{
<s>(q_gen_5518) -> q_gen_5518
<z>() -> q_gen_5518
<s, s>(q_gen_5517) -> q_gen_5517
<z, s>(q_gen_5518) -> q_gen_5517
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_5516, q_gen_5520, q_gen_5529},
Q_f={q_gen_5516},
Delta=
{
<z>() -> q_gen_5529
<s, s>(q_gen_5520) -> q_gen_5520
<z, z>() -> q_gen_5520
<s, s, s>(q_gen_5516) -> q_gen_5516
<s, z, s>(q_gen_5520) -> q_gen_5516
<z, s, s>(q_gen_5520) -> q_gen_5516
<z, z, s>(q_gen_5529) -> q_gen_5516
<z, z, z>() -> q_gen_5516
}

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

}
--
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, _jba])) -> le([i, s(_jba)]) -> 5
(plus([n, mm, _eba])) -> plus([n, s(mm), s(_eba)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _eba])) -> plus([n, s(mm), s(_eba)]), {
_eba -> s(z)  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 8, which took 0.014924 s (model generation: 0.013174,  model checking: 0.001750):

Model:
|_
{
le -> 
{{{
Q={q_gen_5517, q_gen_5518},
Q_f={q_gen_5517},
Delta=
{
<s>(q_gen_5518) -> q_gen_5518
<z>() -> q_gen_5518
<s, s>(q_gen_5517) -> q_gen_5517
<z, s>(q_gen_5518) -> q_gen_5517
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_5516, q_gen_5520, q_gen_5529},
Q_f={q_gen_5516},
Delta=
{
<z>() -> q_gen_5529
<s, s>(q_gen_5520) -> q_gen_5520
<z, s>(q_gen_5529) -> q_gen_5520
<z, z>() -> q_gen_5520
<s, s, s>(q_gen_5516) -> q_gen_5516
<s, z, s>(q_gen_5520) -> q_gen_5516
<z, s, s>(q_gen_5520) -> q_gen_5516
<z, z, s>(q_gen_5529) -> q_gen_5516
<z, z, z>() -> q_gen_5516
}

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

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

Total time: 0.112362
Reason for stopping: Proved