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, _ne])) -> plus([n, s(mm), s(_ne)])}
(plus([_oe, _pe, _qe]) /\ plus([_oe, _pe, _re])) -> eq_nat([_qe, _re])
)
(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:
{(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se])}


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

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, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 0
(le([z, z])) -> BOT -> 0
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 0
}


Solving took 0.254782 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1208
<z, z>() -> q_gen_1208
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1196, q_gen_1197, q_gen_1206, q_gen_1207, q_gen_1212},
Q_f={q_gen_1193, q_gen_1196},
Delta=
{
<s>(q_gen_1206) -> q_gen_1206
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, s>(q_gen_1212) -> q_gen_1212
<z, s>(q_gen_1206) -> q_gen_1212
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
<s, s, s>(q_gen_1196) -> q_gen_1196
<s, z, s>(q_gen_1212) -> q_gen_1196
<z, s, s>(q_gen_1197) -> q_gen_1196
<z, s, s>(q_gen_1212) -> q_gen_1196
<z, z, s>(q_gen_1206) -> q_gen_1196
<s, s, s>(q_gen_1193) -> q_gen_1207
<s, s, s>(q_gen_1207) -> q_gen_1207
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.010330 s (model generation: 0.009653,  model checking: 0.000677):

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, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 1
(le([z, z])) -> BOT -> 1
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 1
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.007180 s (model generation: 0.007120,  model checking: 0.000060):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193},
Q_f={q_gen_1193},
Delta=
{
<z, z, z>() -> q_gen_1193
}

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, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 1
(le([z, z])) -> BOT -> 1
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 1
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.007097 s (model generation: 0.007001,  model checking: 0.000096):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195},
Q_f={q_gen_1194},
Delta=
{
<z>() -> q_gen_1195
<z, s>(q_gen_1195) -> q_gen_1194
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193},
Q_f={q_gen_1193},
Delta=
{
<z, z, z>() -> q_gen_1193
}

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, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 1
(le([z, z])) -> BOT -> 1
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 4
}
Sat witness: Found: ((plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]), {
_ne -> z  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 3, which took 0.007795 s (model generation: 0.007602,  model checking: 0.000193):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195},
Q_f={q_gen_1194},
Delta=
{
<z>() -> q_gen_1195
<z, s>(q_gen_1195) -> q_gen_1194
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1197},
Q_f={q_gen_1193},
Delta=
{
<z, z>() -> q_gen_1197
<z, s, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
}

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, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 2
(le([z, z])) -> BOT -> 2
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 4
}
Sat witness: Found: ((le([nn1, nn2])) -> le([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> s(z)
})

-------------------------------------------
Step 4, which took 0.007734 s (model generation: 0.007501,  model checking: 0.000233):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195},
Q_f={q_gen_1194},
Delta=
{
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1197},
Q_f={q_gen_1193},
Delta=
{
<z, z>() -> q_gen_1197
<z, s, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
}

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, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 3
(le([z, z])) -> BOT -> 3
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 4
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 5, which took 0.014167 s (model generation: 0.013902,  model checking: 0.000265):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195},
Q_f={q_gen_1194},
Delta=
{
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1197},
Q_f={q_gen_1193},
Delta=
{
<s, s>(q_gen_1197) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, s, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
}

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, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 4
(le([z, z])) -> BOT -> 4
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 4
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 6, which took 0.011860 s (model generation: 0.011610,  model checking: 0.000250):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1197},
Q_f={q_gen_1193},
Delta=
{
<s, s>(q_gen_1197) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, s, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
}

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, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 4
(le([z, z])) -> BOT -> 4
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 7
}
Sat witness: Found: ((plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]), {
_ne -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 7, which took 0.009072 s (model generation: 0.008957,  model checking: 0.000115):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1197, q_gen_1206},
Q_f={q_gen_1193},
Delta=
{
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, s, s>(q_gen_1193) -> q_gen_1193
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, s, s>(q_gen_1197) -> q_gen_1193
<z, z, s>(q_gen_1206) -> q_gen_1193
<z, z, z>() -> q_gen_1193
}

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, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 7
(le([z, z])) -> BOT -> 5
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 7
}
Sat witness: Found: ((le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]), {
_se -> s(z)  ;  m -> s(z)  ;  n -> s(z)
})

-------------------------------------------
Step 8, which took 0.011112 s (model generation: 0.010978,  model checking: 0.000134):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<z, z>() -> q_gen_1194
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1197, q_gen_1206},
Q_f={q_gen_1193},
Delta=
{
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, s, s>(q_gen_1193) -> q_gen_1193
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, s, s>(q_gen_1197) -> q_gen_1193
<z, z, s>(q_gen_1206) -> q_gen_1193
<z, z, z>() -> q_gen_1193
}

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)]) -> 5
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 5
(le([s(nn1), z])) -> BOT -> 5
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 7
(le([z, z])) -> BOT -> 8
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 7
}
Sat witness: Found: ((le([z, z])) -> BOT, {

})

-------------------------------------------
Step 9, which took 0.010388 s (model generation: 0.010315,  model checking: 0.000073):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1209},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<s, s>(q_gen_1209) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<z, z>() -> q_gen_1209
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1197, q_gen_1206},
Q_f={q_gen_1193},
Delta=
{
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, s, s>(q_gen_1193) -> q_gen_1193
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, s, s>(q_gen_1197) -> q_gen_1193
<z, z, s>(q_gen_1206) -> q_gen_1193
<z, z, z>() -> q_gen_1193
}

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)]) -> 5
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 8
(le([s(nn1), z])) -> BOT -> 6
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 7
(le([z, z])) -> BOT -> 8
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 7
}
Sat witness: Found: ((le([s(nn1), s(nn2)])) -> le([nn1, nn2]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 10, which took 0.012132 s (model generation: 0.011119,  model checking: 0.001013):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1208
<z, z>() -> q_gen_1208
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1197, q_gen_1205, q_gen_1206},
Q_f={q_gen_1193},
Delta=
{
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, s, s>(q_gen_1205) -> q_gen_1193
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, s, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
<s, s, s>(q_gen_1193) -> q_gen_1205
<z, z, s>(q_gen_1206) -> q_gen_1205
}

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)]) -> 6
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 8
(le([s(nn1), z])) -> BOT -> 7
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 7
(le([z, z])) -> BOT -> 8
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 10
}
Sat witness: Found: ((plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]), {
_ne -> s(s(z))  ;  mm -> s(z)  ;  n -> s(z)
})

-------------------------------------------
Step 11, which took 0.011133 s (model generation: 0.010905,  model checking: 0.000228):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1208
<z, z>() -> q_gen_1208
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1196, q_gen_1197, q_gen_1206, q_gen_1207},
Q_f={q_gen_1193, q_gen_1196},
Delta=
{
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, s>(q_gen_1206) -> q_gen_1197
<z, z>() -> q_gen_1197
<z, z, z>() -> q_gen_1193
<s, s, s>(q_gen_1196) -> q_gen_1196
<s, z, s>(q_gen_1197) -> q_gen_1196
<z, s, s>(q_gen_1197) -> q_gen_1196
<z, z, s>(q_gen_1206) -> q_gen_1196
<s, s, s>(q_gen_1193) -> q_gen_1207
}

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)]) -> 7
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 8
(le([s(nn1), z])) -> BOT -> 7
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 10
(le([z, z])) -> BOT -> 8
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 10
}
Sat witness: Found: ((le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]), {
_se -> s(s(z))  ;  m -> s(z)  ;  n -> s(s(z))
})

-------------------------------------------
Step 12, which took 0.013726 s (model generation: 0.012424,  model checking: 0.001302):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1208
<z, z>() -> q_gen_1208
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1196, q_gen_1197, q_gen_1206, q_gen_1207},
Q_f={q_gen_1193, q_gen_1196},
Delta=
{
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, s>(q_gen_1206) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, s, s>(q_gen_1196) -> q_gen_1193
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
<z, s, s>(q_gen_1197) -> q_gen_1196
<z, z, s>(q_gen_1206) -> q_gen_1196
<s, s, s>(q_gen_1193) -> q_gen_1207
}

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)]) -> 8
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 9
(le([s(nn1), z])) -> BOT -> 8
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 10
(le([z, z])) -> BOT -> 9
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 13
}
Sat witness: Found: ((plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]), {
_ne -> s(s(z))  ;  mm -> z  ;  n -> s(s(z))
})

-------------------------------------------
Step 13, which took 0.014785 s (model generation: 0.014127,  model checking: 0.000658):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1208
<z, z>() -> q_gen_1208
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1197, q_gen_1205, q_gen_1206, q_gen_1212},
Q_f={q_gen_1193},
Delta=
{
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, z>() -> q_gen_1197
<z, s>(q_gen_1206) -> q_gen_1212
<s, s, s>(q_gen_1205) -> q_gen_1193
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, s, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
<s, s, s>(q_gen_1193) -> q_gen_1205
<s, z, s>(q_gen_1212) -> q_gen_1205
<z, s, s>(q_gen_1212) -> q_gen_1205
<z, z, s>(q_gen_1206) -> q_gen_1205
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 9
() -> plus([n, z, n]) -> 9
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 9
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 10
(le([s(nn1), z])) -> BOT -> 9
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 13
(le([z, z])) -> BOT -> 10
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 13
}
Sat witness: Found: ((le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]), {
_se -> s(s(z))  ;  m -> s(s(z))  ;  n -> s(s(z))
})

-------------------------------------------
Step 14, which took 0.015393 s (model generation: 0.014779,  model checking: 0.000614):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208, q_gen_1209},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1209) -> q_gen_1208
<z, z>() -> q_gen_1209
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1196, q_gen_1197, q_gen_1206, q_gen_1207},
Q_f={q_gen_1193, q_gen_1196},
Delta=
{
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, s>(q_gen_1206) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, s, s>(q_gen_1207) -> q_gen_1193
<z, z, z>() -> q_gen_1193
<s, s, s>(q_gen_1196) -> q_gen_1196
<s, z, s>(q_gen_1197) -> q_gen_1196
<z, s, s>(q_gen_1197) -> q_gen_1196
<z, z, s>(q_gen_1206) -> q_gen_1196
<s, s, s>(q_gen_1193) -> q_gen_1207
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 10
() -> plus([n, z, n]) -> 10
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 10
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13
(le([s(nn1), z])) -> BOT -> 10
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 13
(le([z, z])) -> BOT -> 11
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 13
}
Sat witness: Found: ((le([s(nn1), s(nn2)])) -> le([nn1, nn2]), {
nn1 -> s(z)  ;  nn2 -> s(z)
})

-------------------------------------------
Step 15, which took 0.014272 s (model generation: 0.014012,  model checking: 0.000260):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1208
<z, z>() -> q_gen_1208
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1196, q_gen_1197, q_gen_1200, q_gen_1206, q_gen_1207},
Q_f={q_gen_1193, q_gen_1196},
Delta=
{
<z>() -> q_gen_1206
<z, z>() -> q_gen_1197
<s, s>(q_gen_1197) -> q_gen_1200
<z, s>(q_gen_1206) -> q_gen_1200
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
<s, s, s>(q_gen_1196) -> q_gen_1196
<s, z, s>(q_gen_1200) -> q_gen_1196
<z, s, s>(q_gen_1197) -> q_gen_1196
<z, s, s>(q_gen_1200) -> q_gen_1196
<z, z, s>(q_gen_1206) -> q_gen_1196
<s, s, s>(q_gen_1193) -> q_gen_1207
<s, s, s>(q_gen_1207) -> q_gen_1207
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 10
() -> plus([n, z, n]) -> 13
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 11
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13
(le([s(nn1), z])) -> BOT -> 11
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 13
(le([z, z])) -> BOT -> 11
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 13
}
Sat witness: Found: (() -> plus([n, z, n]), {
n -> s(s(s(z)))
})

-------------------------------------------
Step 16, which took 0.015576 s (model generation: 0.015075,  model checking: 0.000501):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1208
<z, z>() -> q_gen_1208
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1196, q_gen_1197, q_gen_1200, q_gen_1206, q_gen_1207},
Q_f={q_gen_1193, q_gen_1196},
Delta=
{
<z>() -> q_gen_1206
<z, z>() -> q_gen_1197
<s, s>(q_gen_1197) -> q_gen_1200
<s, s>(q_gen_1200) -> q_gen_1200
<z, s>(q_gen_1206) -> q_gen_1200
<s, s, s>(q_gen_1196) -> q_gen_1193
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
<s, z, s>(q_gen_1200) -> q_gen_1196
<z, s, s>(q_gen_1197) -> q_gen_1196
<z, s, s>(q_gen_1200) -> q_gen_1196
<z, z, s>(q_gen_1206) -> q_gen_1196
<s, s, s>(q_gen_1193) -> q_gen_1207
<s, s, s>(q_gen_1207) -> q_gen_1207
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 11
() -> plus([n, z, n]) -> 13
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 12
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13
(le([s(nn1), z])) -> BOT -> 12
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 13
(le([z, z])) -> BOT -> 12
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 16
}
Sat witness: Found: ((plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]), {
_ne -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 17, which took 0.018330 s (model generation: 0.018092,  model checking: 0.000238):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1208
<z, z>() -> q_gen_1208
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1196, q_gen_1197, q_gen_1200, q_gen_1206, q_gen_1207},
Q_f={q_gen_1193, q_gen_1196},
Delta=
{
<s>(q_gen_1206) -> q_gen_1206
<z>() -> q_gen_1206
<s, s>(q_gen_1200) -> q_gen_1197
<z, z>() -> q_gen_1197
<s, s>(q_gen_1197) -> q_gen_1200
<z, s>(q_gen_1206) -> q_gen_1200
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
<s, s, s>(q_gen_1196) -> q_gen_1196
<s, z, s>(q_gen_1200) -> q_gen_1196
<z, s, s>(q_gen_1197) -> q_gen_1196
<z, s, s>(q_gen_1200) -> q_gen_1196
<z, z, s>(q_gen_1206) -> q_gen_1196
<s, s, s>(q_gen_1193) -> q_gen_1207
<s, s, s>(q_gen_1207) -> q_gen_1207
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 12
() -> plus([n, z, n]) -> 13
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 13
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13
(le([s(nn1), z])) -> BOT -> 13
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 16
(le([z, z])) -> BOT -> 13
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 16
}
Sat witness: Found: ((le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]), {
_se -> s(s(s(z)))  ;  m -> s(z)  ;  n -> s(s(s(z)))
})

-------------------------------------------
Step 18, which took 0.020843 s (model generation: 0.019960,  model checking: 0.000883):

Model:
|_
{
le -> 
{{{
Q={q_gen_1194, q_gen_1195, q_gen_1208},
Q_f={q_gen_1194},
Delta=
{
<s>(q_gen_1195) -> q_gen_1195
<z>() -> q_gen_1195
<s, s>(q_gen_1194) -> q_gen_1194
<z, s>(q_gen_1195) -> q_gen_1194
<s, s>(q_gen_1208) -> q_gen_1208
<z, z>() -> q_gen_1208
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  plus -> 
{{{
Q={q_gen_1193, q_gen_1196, q_gen_1197, q_gen_1206, q_gen_1207, q_gen_1212},
Q_f={q_gen_1193, q_gen_1196},
Delta=
{
<s>(q_gen_1206) -> q_gen_1206
<z>() -> q_gen_1206
<s, s>(q_gen_1197) -> q_gen_1197
<z, z>() -> q_gen_1197
<z, s>(q_gen_1206) -> q_gen_1212
<s, z, s>(q_gen_1197) -> q_gen_1193
<z, z, z>() -> q_gen_1193
<s, s, s>(q_gen_1196) -> q_gen_1196
<s, z, s>(q_gen_1212) -> q_gen_1196
<z, s, s>(q_gen_1197) -> q_gen_1196
<z, s, s>(q_gen_1212) -> q_gen_1196
<z, z, s>(q_gen_1206) -> q_gen_1196
<s, s, s>(q_gen_1193) -> q_gen_1207
<s, s, s>(q_gen_1207) -> q_gen_1207
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 13
() -> plus([n, z, n]) -> 14
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 14
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 14
(le([s(nn1), z])) -> BOT -> 14
(le([z, m]) /\ le([z, n]) /\ plus([n, m, _se])) -> le([n, _se]) -> 16
(le([z, z])) -> BOT -> 14
(plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]) -> 19
}
Sat witness: Found: ((plus([n, mm, _ne])) -> plus([n, s(mm), s(_ne)]), {
_ne -> s(s(z))  ;  mm -> s(z)  ;  n -> z
})

Total time: 0.254782
Reason for stopping: Proved