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, _a])) -> plus([n, s(mm), s(_a)])}
(plus([_b, _c, _d]) /\ plus([_b, _c, _e])) -> eq_nat([_d, _e])
)
(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, _f])) -> le([i, s(_f)])}


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, _f])) -> le([i, s(_f)]) -> 0
(plus([n, mm, _a])) -> plus([n, s(mm), s(_a)]) -> 0
}


Solving took 0.137023 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_1, q_gen_2},
Q_f={q_gen_1},
Delta=
{
<s>(q_gen_2) -> q_gen_2
<z>() -> q_gen_2
<s, s>(q_gen_1) -> q_gen_1
<z, s>(q_gen_2) -> q_gen_1
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_0, q_gen_13, q_gen_15},
Q_f={q_gen_0},
Delta=
{
<s>(q_gen_13) -> q_gen_13
<z>() -> q_gen_13
<s, s>(q_gen_15) -> q_gen_15
<z, s>(q_gen_13) -> q_gen_15
<z, z>() -> q_gen_15
<s, s, s>(q_gen_0) -> q_gen_0
<s, z, s>(q_gen_15) -> q_gen_0
<z, s, s>(q_gen_15) -> q_gen_0
<z, z, s>(q_gen_13) -> q_gen_0
<z, z, z>() -> q_gen_0
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.010620 s (model generation: 0.010221,  model checking: 0.000399):

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

-------------------------------------------
Step 1, which took 0.017971 s (model generation: 0.010929,  model checking: 0.007042):

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

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_0},
Q_f={q_gen_0},
Delta=
{
<z, z, z>() -> q_gen_0
}

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

-------------------------------------------
Step 2, which took 0.011242 s (model generation: 0.011023,  model checking: 0.000219):

Model:
|_
{
le -> 
{{{
Q={q_gen_1, q_gen_2},
Q_f={q_gen_1},
Delta=
{
<z>() -> q_gen_2
<z, s>(q_gen_2) -> q_gen_1
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_0},
Q_f={q_gen_0},
Delta=
{
<z, z, z>() -> q_gen_0
}

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

-------------------------------------------
Step 3, which took 0.011748 s (model generation: 0.011533,  model checking: 0.000215):

Model:
|_
{
le -> 
{{{
Q={q_gen_1, q_gen_2},
Q_f={q_gen_1},
Delta=
{
<z>() -> q_gen_2
<z, s>(q_gen_2) -> q_gen_1
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_0, q_gen_4},
Q_f={q_gen_0},
Delta=
{
<z, z>() -> q_gen_4
<z, s, s>(q_gen_4) -> q_gen_0
<z, z, z>() -> q_gen_0
}

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

-------------------------------------------
Step 4, which took 0.012028 s (model generation: 0.011313,  model checking: 0.000715):

Model:
|_
{
le -> 
{{{
Q={q_gen_1, q_gen_2},
Q_f={q_gen_1},
Delta=
{
<z>() -> q_gen_2
<s, s>(q_gen_1) -> q_gen_1
<z, s>(q_gen_2) -> q_gen_1
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_0, q_gen_4},
Q_f={q_gen_0},
Delta=
{
<z, z>() -> q_gen_4
<z, s, s>(q_gen_4) -> q_gen_0
<z, z, z>() -> q_gen_0
}

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

-------------------------------------------
Step 5, which took 0.013972 s (model generation: 0.011599,  model checking: 0.002373):

Model:
|_
{
le -> 
{{{
Q={q_gen_1, q_gen_2},
Q_f={q_gen_1},
Delta=
{
<z>() -> q_gen_2
<s, s>(q_gen_1) -> q_gen_1
<z, s>(q_gen_2) -> q_gen_1
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_0, q_gen_4},
Q_f={q_gen_0},
Delta=
{
<s, s>(q_gen_4) -> q_gen_4
<z, z>() -> q_gen_4
<s, z, s>(q_gen_4) -> q_gen_0
<z, s, s>(q_gen_4) -> q_gen_0
<z, z, z>() -> q_gen_0
}

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]) -> 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, _f])) -> le([i, s(_f)]) -> 4
(plus([n, mm, _a])) -> plus([n, s(mm), s(_a)]) -> 4
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 6, which took 0.012301 s (model generation: 0.011750,  model checking: 0.000551):

Model:
|_
{
le -> 
{{{
Q={q_gen_1, q_gen_2},
Q_f={q_gen_1},
Delta=
{
<s>(q_gen_2) -> q_gen_2
<z>() -> q_gen_2
<s, s>(q_gen_1) -> q_gen_1
<z, s>(q_gen_2) -> q_gen_1
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_0, q_gen_4},
Q_f={q_gen_0},
Delta=
{
<s, s>(q_gen_4) -> q_gen_4
<z, z>() -> q_gen_4
<s, z, s>(q_gen_4) -> q_gen_0
<z, s, s>(q_gen_4) -> q_gen_0
<z, z, z>() -> q_gen_0
}

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

-------------------------------------------
Step 7, which took 0.014732 s (model generation: 0.012704,  model checking: 0.002028):

Model:
|_
{
le -> 
{{{
Q={q_gen_1, q_gen_2},
Q_f={q_gen_1},
Delta=
{
<s>(q_gen_2) -> q_gen_2
<z>() -> q_gen_2
<s, s>(q_gen_1) -> q_gen_1
<z, s>(q_gen_2) -> q_gen_1
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_0, q_gen_13, q_gen_4},
Q_f={q_gen_0},
Delta=
{
<z>() -> q_gen_13
<s, s>(q_gen_4) -> q_gen_4
<z, z>() -> q_gen_4
<s, s, s>(q_gen_0) -> q_gen_0
<s, z, s>(q_gen_4) -> q_gen_0
<z, s, s>(q_gen_4) -> q_gen_0
<z, z, s>(q_gen_13) -> q_gen_0
<z, z, z>() -> q_gen_0
}

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

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

-------------------------------------------
Step 8, which took 0.015497 s (model generation: 0.013864,  model checking: 0.001633):

Model:
|_
{
le -> 
{{{
Q={q_gen_1, q_gen_2},
Q_f={q_gen_1},
Delta=
{
<s>(q_gen_2) -> q_gen_2
<z>() -> q_gen_2
<s, s>(q_gen_1) -> q_gen_1
<z, s>(q_gen_2) -> q_gen_1
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  plus -> 
{{{
Q={q_gen_0, q_gen_13, q_gen_15},
Q_f={q_gen_0},
Delta=
{
<z>() -> q_gen_13
<s, s>(q_gen_15) -> q_gen_15
<z, s>(q_gen_13) -> q_gen_15
<z, z>() -> q_gen_15
<s, s, s>(q_gen_0) -> q_gen_0
<s, z, s>(q_gen_15) -> q_gen_0
<z, s, s>(q_gen_15) -> q_gen_0
<z, z, s>(q_gen_13) -> q_gen_0
<z, z, z>() -> q_gen_0
}

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

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

Total time: 0.137023
Reason for stopping: Proved