Solving ../../benchmarks/true/length_cons_le.smt2...

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


Learning problem is:

env: {
nat -> {s, z}  ;  natlist -> {cons, nil}
}
definition:
{
(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}
)
(length, F:
{() -> length([nil, z])
(length([ll, _vj])) -> length([cons(x, ll), s(_vj)])}
(length([_wj, _xj]) /\ length([_wj, _yj])) -> eq_nat([_xj, _yj])
)
}

properties:
{(length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak])}


over-approximation: {length}
under-approximation: {le}

Clause system for inference is:

{
() -> le([z, s(nn2)]) -> 0  ;  () -> length([nil, z]) -> 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  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 0  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 0
}


Solving took 0.048721 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_3055, q_gen_3056},
Q_f={q_gen_3055},
Delta=
{
<s>(q_gen_3056) -> q_gen_3056
<z>() -> q_gen_3056
<s, s>(q_gen_3055) -> q_gen_3055
<z, s>(q_gen_3056) -> q_gen_3055
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3054, q_gen_3058, q_gen_3064},
Q_f={q_gen_3054},
Delta=
{
<s>(q_gen_3064) -> q_gen_3064
<z>() -> q_gen_3064
<cons, s>(q_gen_3058, q_gen_3054) -> q_gen_3054
<nil, z>() -> q_gen_3054
<s, s>(q_gen_3058) -> q_gen_3058
<s, z>(q_gen_3064) -> q_gen_3058
<z, s>(q_gen_3064) -> q_gen_3058
<z, z>() -> q_gen_3058
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.004077 s (model generation: 0.003976,  model checking: 0.000101):

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

}

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

}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 0  ;  () -> length([nil, z]) -> 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  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 1  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 1
}
Sat witness: Yes: (() -> length([nil, z]), {

})

-------------------------------------------
Step 1, which took 0.003583 s (model generation: 0.003516,  model checking: 0.000067):

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

}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3054},
Q_f={q_gen_3054},
Delta=
{
<nil, z>() -> q_gen_3054
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3  ;  () -> length([nil, z]) -> 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  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 1  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 1
}
Sat witness: Yes: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.003512 s (model generation: 0.003442,  model checking: 0.000070):

Model:
|_
{
le -> 
{{{
Q={q_gen_3055, q_gen_3056},
Q_f={q_gen_3055},
Delta=
{
<z>() -> q_gen_3056
<z, s>(q_gen_3056) -> q_gen_3055
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3054},
Q_f={q_gen_3054},
Delta=
{
<nil, z>() -> q_gen_3054
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3  ;  () -> length([nil, z]) -> 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  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 1  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 4
}
Sat witness: Yes: ((length([ll, _vj])) -> length([cons(x, ll), s(_vj)]), {
_vj -> z  ;  ll -> nil  ;  x -> z
})

-------------------------------------------
Step 3, which took 0.004386 s (model generation: 0.003491,  model checking: 0.000895):

Model:
|_
{
le -> 
{{{
Q={q_gen_3055, q_gen_3056},
Q_f={q_gen_3055},
Delta=
{
<z>() -> q_gen_3056
<z, s>(q_gen_3056) -> q_gen_3055
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3054, q_gen_3058},
Q_f={q_gen_3054},
Delta=
{
<cons, s>(q_gen_3058, q_gen_3054) -> q_gen_3054
<nil, z>() -> q_gen_3054
<z, z>() -> q_gen_3058
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 3  ;  () -> length([nil, z]) -> 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  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 2  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 4
}
Sat witness: Yes: ((le([nn1, nn2])) -> le([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> s(z)
})

-------------------------------------------
Step 4, which took 0.003717 s (model generation: 0.003516,  model checking: 0.000201):

Model:
|_
{
le -> 
{{{
Q={q_gen_3055, q_gen_3056},
Q_f={q_gen_3055},
Delta=
{
<z>() -> q_gen_3056
<s, s>(q_gen_3055) -> q_gen_3055
<z, s>(q_gen_3056) -> q_gen_3055
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3054, q_gen_3058},
Q_f={q_gen_3054},
Delta=
{
<cons, s>(q_gen_3058, q_gen_3054) -> q_gen_3054
<nil, z>() -> q_gen_3054
<z, z>() -> q_gen_3058
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6  ;  () -> length([nil, z]) -> 4  ;  (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  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 3  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 4
}
Sat witness: Yes: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 5, which took 0.004554 s (model generation: 0.004305,  model checking: 0.000249):

Model:
|_
{
le -> 
{{{
Q={q_gen_3055, q_gen_3056},
Q_f={q_gen_3055},
Delta=
{
<s>(q_gen_3056) -> q_gen_3056
<z>() -> q_gen_3056
<s, s>(q_gen_3055) -> q_gen_3055
<z, s>(q_gen_3056) -> q_gen_3055
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3054, q_gen_3058},
Q_f={q_gen_3054},
Delta=
{
<cons, s>(q_gen_3058, q_gen_3054) -> q_gen_3054
<nil, z>() -> q_gen_3054
<z, z>() -> q_gen_3058
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 6  ;  () -> length([nil, z]) -> 4  ;  (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  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 4  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 7
}
Sat witness: Yes: ((length([ll, _vj])) -> length([cons(x, ll), s(_vj)]), {
_vj -> z  ;  ll -> nil  ;  x -> s(z)
})

-------------------------------------------
Step 6, which took 0.007706 s (model generation: 0.006900,  model checking: 0.000806):

Model:
|_
{
le -> 
{{{
Q={q_gen_3055, q_gen_3056},
Q_f={q_gen_3055},
Delta=
{
<s>(q_gen_3056) -> q_gen_3056
<z>() -> q_gen_3056
<s, s>(q_gen_3055) -> q_gen_3055
<z, s>(q_gen_3056) -> q_gen_3055
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3054, q_gen_3058, q_gen_3064},
Q_f={q_gen_3054},
Delta=
{
<z>() -> q_gen_3064
<cons, s>(q_gen_3058, q_gen_3054) -> q_gen_3054
<nil, z>() -> q_gen_3054
<s, z>(q_gen_3064) -> q_gen_3058
<z, z>() -> q_gen_3058
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 7  ;  () -> length([nil, z]) -> 5  ;  (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  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 5  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 10
}
Sat witness: Yes: ((length([ll, _vj])) -> length([cons(x, ll), s(_vj)]), {
_vj -> z  ;  ll -> nil  ;  x -> s(s(z))
})

-------------------------------------------
Step 7, which took 0.007233 s (model generation: 0.006497,  model checking: 0.000736):

Model:
|_
{
le -> 
{{{
Q={q_gen_3055, q_gen_3056},
Q_f={q_gen_3055},
Delta=
{
<s>(q_gen_3056) -> q_gen_3056
<z>() -> q_gen_3056
<s, s>(q_gen_3055) -> q_gen_3055
<z, s>(q_gen_3056) -> q_gen_3055
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3054, q_gen_3058, q_gen_3064},
Q_f={q_gen_3054},
Delta=
{
<s>(q_gen_3064) -> q_gen_3064
<z>() -> q_gen_3064
<cons, s>(q_gen_3058, q_gen_3054) -> q_gen_3054
<nil, z>() -> q_gen_3054
<s, z>(q_gen_3064) -> q_gen_3058
<z, z>() -> q_gen_3058
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 8  ;  () -> length([nil, z]) -> 6  ;  (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  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 6  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 13
}
Sat witness: Yes: ((length([ll, _vj])) -> length([cons(x, ll), s(_vj)]), {
_vj -> s(z)  ;  ll -> cons(z, nil)  ;  x -> s(z)
})

-------------------------------------------
Step 8, which took 0.004891 s (model generation: 0.004347,  model checking: 0.000544):

Model:
|_
{
le -> 
{{{
Q={q_gen_3055, q_gen_3056},
Q_f={q_gen_3055},
Delta=
{
<s>(q_gen_3056) -> q_gen_3056
<z>() -> q_gen_3056
<s, s>(q_gen_3055) -> q_gen_3055
<z, s>(q_gen_3056) -> q_gen_3055
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3054, q_gen_3058, q_gen_3064},
Q_f={q_gen_3054},
Delta=
{
<s>(q_gen_3064) -> q_gen_3064
<z>() -> q_gen_3064
<cons, s>(q_gen_3058, q_gen_3054) -> q_gen_3054
<nil, z>() -> q_gen_3054
<s, s>(q_gen_3058) -> q_gen_3058
<s, z>(q_gen_3064) -> q_gen_3058
<z, z>() -> q_gen_3058
}

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

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

Teacher's answer:
New clause system:
{
() -> le([z, s(nn2)]) -> 9  ;  () -> length([nil, z]) -> 7  ;  (le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 7  ;  (le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 7  ;  (le([s(nn1), z])) -> BOT -> 7  ;  (le([z, z])) -> BOT -> 7  ;  (length([l, _zj]) /\ length([cons(x, l), _ak])) -> le([_zj, _ak]) -> 7  ;  (length([ll, _vj])) -> length([cons(x, ll), s(_vj)]) -> 16
}
Sat witness: Yes: ((length([ll, _vj])) -> length([cons(x, ll), s(_vj)]), {
_vj -> s(z)  ;  ll -> cons(z, nil)  ;  x -> z
})

Total time: 0.048721
Reason for stopping: Proved