Solving ../../benchmarks/true/plus_succ2_le.smt2...

Inference procedure has parameters:
Ice fuel: 200
Timeout: 30s
Convolution: complete


Learning problem is:

env: {
nat -> {s, z}
}
definition:
{
(plus, F:
{() -> plus([n, z, n])
(plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)])}
(plus([_kk, _lk, _mk]) /\ plus([_kk, _lk, _nk])) -> eq_nat([_mk, _nk])
)
(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([n, s(m), _ok])) -> le([n, _ok])}


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([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 0  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 0
}


Solving took 0.328642 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5085, q_gen_5086, q_gen_5093, q_gen_5094, q_gen_5101},
Q_f={q_gen_5082, q_gen_5085},
Delta=
{
<s>(q_gen_5093) -> q_gen_5093
<z>() -> q_gen_5093
<s, s>(q_gen_5086) -> q_gen_5086
<z, z>() -> q_gen_5086
<s, s>(q_gen_5101) -> q_gen_5101
<z, s>(q_gen_5093) -> q_gen_5101
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, s, s>(q_gen_5085) -> q_gen_5085
<s, z, s>(q_gen_5101) -> q_gen_5085
<z, s, s>(q_gen_5086) -> q_gen_5085
<z, s, s>(q_gen_5101) -> q_gen_5085
<z, z, s>(q_gen_5093) -> q_gen_5085
<s, s, s>(q_gen_5082) -> q_gen_5094
<s, s, s>(q_gen_5094) -> q_gen_5094
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.006899 s (model generation: 0.006598,  model checking: 0.000301):

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

}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={},
Q_f={},
Delta=
{

}

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

}
--
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([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 1  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 1
}
Sat witness: Yes: (() -> plus([n, z, n]), {
n -> z
})

-------------------------------------------
Step 1, which took 0.010605 s (model generation: 0.010480,  model checking: 0.000125):

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

}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082},
Q_f={q_gen_5082},
Delta=
{
<z, z, z>() -> q_gen_5082
}

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

}
--
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([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 1  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 1
}
Sat witness: Yes: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.013386 s (model generation: 0.013073,  model checking: 0.000313):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084},
Q_f={q_gen_5083},
Delta=
{
<z>() -> q_gen_5084
<z, s>(q_gen_5084) -> q_gen_5083
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082},
Q_f={q_gen_5082},
Delta=
{
<z, z, z>() -> q_gen_5082
}

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

}
--
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([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 4  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 2
}
Sat witness: Yes: ((plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]), {
_jk -> z  ;  mm -> z  ;  n -> z
})

-------------------------------------------
Step 3, which took 0.012050 s (model generation: 0.011785,  model checking: 0.000265):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084},
Q_f={q_gen_5083},
Delta=
{
<z>() -> q_gen_5084
<z, s>(q_gen_5084) -> q_gen_5083
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086},
Q_f={q_gen_5082},
Delta=
{
<z, z>() -> q_gen_5086
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
}

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

}
--
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([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 4  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 2
}
Sat witness: Yes: ((le([nn1, nn2])) -> le([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> s(z)
})

-------------------------------------------
Step 4, which took 0.012369 s (model generation: 0.011730,  model checking: 0.000639):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084},
Q_f={q_gen_5083},
Delta=
{
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086},
Q_f={q_gen_5082},
Delta=
{
<z, z>() -> q_gen_5086
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
}

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

}
--
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([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 4  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 3
}
Sat witness: Yes: (() -> plus([n, z, n]), {
n -> s(z)
})

-------------------------------------------
Step 5, which took 0.013539 s (model generation: 0.011525,  model checking: 0.002014):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084},
Q_f={q_gen_5083},
Delta=
{
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086},
Q_f={q_gen_5082},
Delta=
{
<z, z>() -> q_gen_5086
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
}

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

}
--
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([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 4  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 4
}
Sat witness: Yes: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 6, which took 0.012733 s (model generation: 0.012240,  model checking: 0.000493):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086},
Q_f={q_gen_5082},
Delta=
{
<z, z>() -> q_gen_5086
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
}

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

}
--
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([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 7  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 5
}
Sat witness: Yes: ((plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]), {
_jk -> s(z)  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 7, which took 0.013108 s (model generation: 0.012663,  model checking: 0.000445):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086, q_gen_5093},
Q_f={q_gen_5082},
Delta=
{
<z>() -> q_gen_5093
<z, z>() -> q_gen_5086
<s, s, s>(q_gen_5082) -> q_gen_5082
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, s>(q_gen_5093) -> q_gen_5082
<z, z, z>() -> q_gen_5082
}

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

}
--
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, z])) -> BOT -> 5  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 7  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 8
}
Sat witness: Yes: ((plus([n, s(m), _ok])) -> le([n, _ok]), {
_ok -> s(z)  ;  m -> z  ;  n -> s(z)
})

-------------------------------------------
Step 8, which took 0.014394 s (model generation: 0.014362,  model checking: 0.000032):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<z, z>() -> q_gen_5083
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086, q_gen_5093},
Q_f={q_gen_5082},
Delta=
{
<z>() -> q_gen_5093
<z, z>() -> q_gen_5086
<s, s, s>(q_gen_5082) -> q_gen_5082
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, s>(q_gen_5093) -> q_gen_5082
<z, z, z>() -> q_gen_5082
}

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

}
--
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, z])) -> BOT -> 8  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 7  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 8
}
Sat witness: Yes: ((le([z, z])) -> BOT, {

})

-------------------------------------------
Step 9, which took 0.013951 s (model generation: 0.013788,  model checking: 0.000163):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5096},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<s, s>(q_gen_5096) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<z, z>() -> q_gen_5096
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086, q_gen_5093},
Q_f={q_gen_5082},
Delta=
{
<z>() -> q_gen_5093
<z, z>() -> q_gen_5086
<s, s, s>(q_gen_5082) -> q_gen_5082
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, s>(q_gen_5093) -> q_gen_5082
<z, z, z>() -> q_gen_5082
}

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

}
--
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, z])) -> BOT -> 8  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 7  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 8
}
Sat witness: Yes: ((le([s(nn1), s(nn2)])) -> le([nn1, nn2]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 10, which took 0.014501 s (model generation: 0.013977,  model checking: 0.000524):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086, q_gen_5092, q_gen_5093},
Q_f={q_gen_5082},
Delta=
{
<z>() -> q_gen_5093
<z, z>() -> q_gen_5086
<s, s, s>(q_gen_5092) -> q_gen_5082
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, s, s>(q_gen_5082) -> q_gen_5092
<z, z, s>(q_gen_5093) -> q_gen_5092
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6  ;  () -> plus([n, z, n]) -> 9  ;  (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, z])) -> BOT -> 8  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 7  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 8
}
Sat witness: Yes: (() -> plus([n, z, n]), {
n -> s(s(z))
})

-------------------------------------------
Step 11, which took 0.015546 s (model generation: 0.013909,  model checking: 0.001637):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086, q_gen_5092, q_gen_5093},
Q_f={q_gen_5082},
Delta=
{
<z>() -> q_gen_5093
<s, s>(q_gen_5086) -> q_gen_5086
<z, z>() -> q_gen_5086
<s, s, s>(q_gen_5092) -> q_gen_5082
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, s, s>(q_gen_5082) -> q_gen_5092
<z, z, s>(q_gen_5093) -> q_gen_5092
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 7  ;  () -> plus([n, z, n]) -> 9  ;  (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, z])) -> BOT -> 8  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 10  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 8
}
Sat witness: Yes: ((plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]), {
_jk -> s(s(z))  ;  mm -> s(z)  ;  n -> s(z)
})

-------------------------------------------
Step 12, which took 0.015504 s (model generation: 0.015091,  model checking: 0.000413):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5085, q_gen_5086, q_gen_5093, q_gen_5094},
Q_f={q_gen_5082, q_gen_5085},
Delta=
{
<z>() -> q_gen_5093
<s, s>(q_gen_5086) -> q_gen_5086
<z, s>(q_gen_5093) -> q_gen_5086
<z, z>() -> q_gen_5086
<z, z, z>() -> q_gen_5082
<s, s, s>(q_gen_5085) -> q_gen_5085
<s, z, s>(q_gen_5086) -> q_gen_5085
<z, s, s>(q_gen_5086) -> q_gen_5085
<z, z, s>(q_gen_5093) -> q_gen_5085
<s, s, s>(q_gen_5082) -> q_gen_5094
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 8  ;  () -> plus([n, z, n]) -> 9  ;  (le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 8  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 8  ;  (le([s(nn1), z])) -> BOT -> 8  ;  (le([z, z])) -> BOT -> 8  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 10  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 11
}
Sat witness: Yes: ((plus([n, s(m), _ok])) -> le([n, _ok]), {
_ok -> s(s(z))  ;  m -> z  ;  n -> s(s(z))
})

-------------------------------------------
Step 13, which took 0.017648 s (model generation: 0.016452,  model checking: 0.001196):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5085, q_gen_5086, q_gen_5093, q_gen_5094},
Q_f={q_gen_5082, q_gen_5085},
Delta=
{
<z>() -> q_gen_5093
<s, s>(q_gen_5086) -> q_gen_5086
<z, s>(q_gen_5093) -> q_gen_5086
<z, z>() -> q_gen_5086
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, s, s>(q_gen_5085) -> q_gen_5085
<z, s, s>(q_gen_5086) -> q_gen_5085
<z, z, s>(q_gen_5093) -> q_gen_5085
<s, s, s>(q_gen_5082) -> q_gen_5094
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 9  ;  () -> plus([n, z, n]) -> 10  ;  (le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 9  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 9  ;  (le([s(nn1), z])) -> BOT -> 9  ;  (le([z, z])) -> BOT -> 9  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 13  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 11
}
Sat witness: Yes: ((plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]), {
_jk -> s(s(z))  ;  mm -> z  ;  n -> s(s(z))
})

-------------------------------------------
Step 14, which took 0.017425 s (model generation: 0.016825,  model checking: 0.000600):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5086, q_gen_5092, q_gen_5093, q_gen_5101},
Q_f={q_gen_5082},
Delta=
{
<z>() -> q_gen_5093
<s, s>(q_gen_5086) -> q_gen_5086
<z, z>() -> q_gen_5086
<z, s>(q_gen_5093) -> q_gen_5101
<s, s, s>(q_gen_5092) -> q_gen_5082
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, s, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, s, s>(q_gen_5082) -> q_gen_5092
<s, z, s>(q_gen_5101) -> q_gen_5092
<z, s, s>(q_gen_5101) -> q_gen_5092
<z, z, s>(q_gen_5093) -> q_gen_5092
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 10  ;  () -> plus([n, z, n]) -> 11  ;  (le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 10  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 10  ;  (le([s(nn1), z])) -> BOT -> 10  ;  (le([z, z])) -> BOT -> 10  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 13  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 14
}
Sat witness: Yes: ((plus([n, s(m), _ok])) -> le([n, _ok]), {
_ok -> s(s(z))  ;  m -> s(z)  ;  n -> s(s(z))
})

-------------------------------------------
Step 15, which took 0.011104 s (model generation: 0.011031,  model checking: 0.000073):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095, q_gen_5096},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5096) -> q_gen_5095
<z, z>() -> q_gen_5096
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5085, q_gen_5086, q_gen_5093, q_gen_5094},
Q_f={q_gen_5082, q_gen_5085},
Delta=
{
<z>() -> q_gen_5093
<s, s>(q_gen_5086) -> q_gen_5086
<z, s>(q_gen_5093) -> q_gen_5086
<z, z>() -> q_gen_5086
<s, s, s>(q_gen_5094) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, s, s>(q_gen_5085) -> q_gen_5085
<s, z, s>(q_gen_5086) -> q_gen_5085
<z, s, s>(q_gen_5086) -> q_gen_5085
<z, z, s>(q_gen_5093) -> q_gen_5085
<s, s, s>(q_gen_5082) -> q_gen_5094
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 10  ;  () -> plus([n, z, n]) -> 11  ;  (le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 10  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13  ;  (le([s(nn1), z])) -> BOT -> 11  ;  (le([z, z])) -> BOT -> 11  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 13  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 14
}
Sat witness: Yes: ((le([s(nn1), s(nn2)])) -> le([nn1, nn2]), {
nn1 -> s(z)  ;  nn2 -> s(z)
})

-------------------------------------------
Step 16, which took 0.013198 s (model generation: 0.012654,  model checking: 0.000544):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5085, q_gen_5086, q_gen_5093, q_gen_5094, q_gen_5098},
Q_f={q_gen_5082, q_gen_5085},
Delta=
{
<z>() -> q_gen_5093
<z, z>() -> q_gen_5086
<s, s>(q_gen_5086) -> q_gen_5098
<z, s>(q_gen_5093) -> q_gen_5098
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, s, s>(q_gen_5085) -> q_gen_5085
<s, z, s>(q_gen_5098) -> q_gen_5085
<z, s, s>(q_gen_5086) -> q_gen_5085
<z, s, s>(q_gen_5098) -> q_gen_5085
<z, z, s>(q_gen_5093) -> q_gen_5085
<s, s, s>(q_gen_5082) -> q_gen_5094
<s, s, s>(q_gen_5094) -> q_gen_5094
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 11  ;  () -> plus([n, z, n]) -> 14  ;  (le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 11  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13  ;  (le([s(nn1), z])) -> BOT -> 12  ;  (le([z, z])) -> BOT -> 12  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 13  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 14
}
Sat witness: Yes: (() -> plus([n, z, n]), {
n -> s(s(s(z)))
})

-------------------------------------------
Step 17, which took 0.017321 s (model generation: 0.016351,  model checking: 0.000970):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5085, q_gen_5086, q_gen_5093, q_gen_5094, q_gen_5098},
Q_f={q_gen_5082, q_gen_5085},
Delta=
{
<z>() -> q_gen_5093
<s, s>(q_gen_5098) -> q_gen_5086
<z, z>() -> q_gen_5086
<s, s>(q_gen_5086) -> q_gen_5098
<z, s>(q_gen_5093) -> q_gen_5098
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, s, s>(q_gen_5085) -> q_gen_5085
<s, z, s>(q_gen_5098) -> q_gen_5085
<z, s, s>(q_gen_5086) -> q_gen_5085
<z, s, s>(q_gen_5098) -> q_gen_5085
<z, z, s>(q_gen_5093) -> q_gen_5085
<s, s, s>(q_gen_5082) -> q_gen_5094
<s, s, s>(q_gen_5094) -> q_gen_5094
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 12  ;  () -> plus([n, z, n]) -> 14  ;  (le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 12  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 13  ;  (le([s(nn1), z])) -> BOT -> 13  ;  (le([z, z])) -> BOT -> 13  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 16  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 14
}
Sat witness: Yes: ((plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]), {
_jk -> s(s(z))  ;  mm -> z  ;  n -> s(z)
})

-------------------------------------------
Step 18, which took 0.018308 s (model generation: 0.018016,  model checking: 0.000292):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5085, q_gen_5086, q_gen_5093, q_gen_5094, q_gen_5098},
Q_f={q_gen_5082, q_gen_5085},
Delta=
{
<s>(q_gen_5093) -> q_gen_5093
<z>() -> q_gen_5093
<s, s>(q_gen_5098) -> q_gen_5086
<z, z>() -> q_gen_5086
<s, s>(q_gen_5086) -> q_gen_5098
<z, s>(q_gen_5093) -> q_gen_5098
<s, s, s>(q_gen_5085) -> q_gen_5082
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, z, s>(q_gen_5098) -> q_gen_5085
<z, s, s>(q_gen_5086) -> q_gen_5085
<z, s, s>(q_gen_5098) -> q_gen_5085
<z, z, s>(q_gen_5093) -> q_gen_5085
<s, s, s>(q_gen_5082) -> q_gen_5094
<s, s, s>(q_gen_5094) -> q_gen_5094
}

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

}
--
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)]) -> 13  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 14  ;  (le([s(nn1), z])) -> BOT -> 14  ;  (le([z, z])) -> BOT -> 14  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 16  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 17
}
Sat witness: Yes: ((plus([n, s(m), _ok])) -> le([n, _ok]), {
_ok -> s(s(s(z)))  ;  m -> z  ;  n -> s(s(s(z)))
})

-------------------------------------------
Step 19, which took 0.017852 s (model generation: 0.016915,  model checking: 0.000937):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5085, q_gen_5086, q_gen_5093, q_gen_5094, q_gen_5101},
Q_f={q_gen_5082, q_gen_5085},
Delta=
{
<s>(q_gen_5093) -> q_gen_5093
<z>() -> q_gen_5093
<s, s>(q_gen_5086) -> q_gen_5086
<z, z>() -> q_gen_5086
<z, s>(q_gen_5093) -> q_gen_5101
<s, s, s>(q_gen_5085) -> q_gen_5082
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, z, s>(q_gen_5101) -> q_gen_5085
<z, s, s>(q_gen_5086) -> q_gen_5085
<z, s, s>(q_gen_5101) -> q_gen_5085
<z, z, s>(q_gen_5093) -> q_gen_5085
<s, s, s>(q_gen_5082) -> q_gen_5094
<s, s, s>(q_gen_5094) -> q_gen_5094
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 14  ;  () -> plus([n, z, n]) -> 15  ;  (le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 14  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 15  ;  (le([s(nn1), z])) -> BOT -> 15  ;  (le([z, z])) -> BOT -> 15  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 19  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 17
}
Sat witness: Yes: ((plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]), {
_jk -> s(s(z))  ;  mm -> s(z)  ;  n -> z
})

-------------------------------------------
Step 20, which took 0.024021 s (model generation: 0.023093,  model checking: 0.000928):

Model:
|_
{
le -> 
{{{
Q={q_gen_5083, q_gen_5084, q_gen_5095},
Q_f={q_gen_5083},
Delta=
{
<s>(q_gen_5084) -> q_gen_5084
<z>() -> q_gen_5084
<s, s>(q_gen_5083) -> q_gen_5083
<z, s>(q_gen_5084) -> q_gen_5083
<s, s>(q_gen_5095) -> q_gen_5095
<z, z>() -> q_gen_5095
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  plus -> 
{{{
Q={q_gen_5082, q_gen_5085, q_gen_5086, q_gen_5093, q_gen_5094, q_gen_5101},
Q_f={q_gen_5082, q_gen_5085},
Delta=
{
<s>(q_gen_5093) -> q_gen_5093
<z>() -> q_gen_5093
<s, s>(q_gen_5086) -> q_gen_5086
<z, z>() -> q_gen_5086
<s, s>(q_gen_5101) -> q_gen_5101
<z, s>(q_gen_5093) -> q_gen_5101
<s, s, s>(q_gen_5085) -> q_gen_5082
<s, z, s>(q_gen_5086) -> q_gen_5082
<z, z, z>() -> q_gen_5082
<s, z, s>(q_gen_5101) -> q_gen_5085
<z, s, s>(q_gen_5086) -> q_gen_5085
<z, s, s>(q_gen_5101) -> q_gen_5085
<z, z, s>(q_gen_5093) -> q_gen_5085
<s, s, s>(q_gen_5082) -> q_gen_5094
<s, s, s>(q_gen_5094) -> q_gen_5094
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 15  ;  () -> plus([n, z, n]) -> 16  ;  (le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 15  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 16  ;  (le([s(nn1), z])) -> BOT -> 16  ;  (le([z, z])) -> BOT -> 16  ;  (plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]) -> 22  ;  (plus([n, s(m), _ok])) -> le([n, _ok]) -> 18
}
Sat witness: Yes: ((plus([n, mm, _jk])) -> plus([n, s(mm), s(_jk)]), {
_jk -> s(s(s(z)))  ;  mm -> s(z)  ;  n -> s(s(z))
})

Total time: 0.328642
Reason for stopping: Proved