Solving ../../benchmarks/true/length_cons_le_tsil.smt2...

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


Learning problem is:

env: {
nat -> {s, z}  ;  nattsil -> {nil, snoc}
}
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, _by])) -> length([snoc(ll, x), s(_by)])}
(length([_cy, _dy]) /\ length([_cy, _ey])) -> eq_nat([_dy, _ey])
)
}

properties:
{(length([l, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy])}


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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 0  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 0
}


Solving took 0.092659 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_3073, q_gen_3074},
Q_f={q_gen_3073},
Delta=
{
<s>(q_gen_3074) -> q_gen_3074
<z>() -> q_gen_3074
<s, s>(q_gen_3073) -> q_gen_3073
<z, s>(q_gen_3074) -> q_gen_3073
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3072, q_gen_3076, q_gen_3082},
Q_f={q_gen_3072},
Delta=
{
<s>(q_gen_3082) -> q_gen_3082
<z>() -> q_gen_3082
<nil, z>() -> q_gen_3072
<snoc, s>(q_gen_3072, q_gen_3076) -> q_gen_3072
<s, s>(q_gen_3076) -> q_gen_3076
<s, z>(q_gen_3082) -> q_gen_3076
<z, s>(q_gen_3082) -> q_gen_3076
<z, z>() -> q_gen_3076
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.008404 s (model generation: 0.008012,  model checking: 0.000392):

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

}

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

}

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

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

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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 1  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 1
}
Sat witness: Yes: (() -> length([nil, z]), {

})

-------------------------------------------
Step 1, which took 0.006102 s (model generation: 0.005969,  model checking: 0.000133):

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

}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3072},
Q_f={q_gen_3072},
Delta=
{
<nil, z>() -> q_gen_3072
}

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

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

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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 1  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 1
}
Sat witness: Yes: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.007830 s (model generation: 0.007620,  model checking: 0.000210):

Model:
|_
{
le -> 
{{{
Q={q_gen_3073, q_gen_3074},
Q_f={q_gen_3073},
Delta=
{
<z>() -> q_gen_3074
<z, s>(q_gen_3074) -> q_gen_3073
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3072},
Q_f={q_gen_3072},
Delta=
{
<nil, z>() -> q_gen_3072
}

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

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

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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 1  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 4
}
Sat witness: Yes: ((length([ll, _by])) -> length([snoc(ll, x), s(_by)]), {
_by -> z  ;  ll -> nil  ;  x -> z
})

-------------------------------------------
Step 3, which took 0.011377 s (model generation: 0.010466,  model checking: 0.000911):

Model:
|_
{
le -> 
{{{
Q={q_gen_3073, q_gen_3074},
Q_f={q_gen_3073},
Delta=
{
<z>() -> q_gen_3074
<z, s>(q_gen_3074) -> q_gen_3073
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3072, q_gen_3076},
Q_f={q_gen_3072},
Delta=
{
<nil, z>() -> q_gen_3072
<snoc, s>(q_gen_3072, q_gen_3076) -> q_gen_3072
<z, z>() -> q_gen_3076
}

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

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

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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 2  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 4
}
Sat witness: Yes: ((le([nn1, nn2])) -> le([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> s(z)
})

-------------------------------------------
Step 4, which took 0.006963 s (model generation: 0.006744,  model checking: 0.000219):

Model:
|_
{
le -> 
{{{
Q={q_gen_3073, q_gen_3074},
Q_f={q_gen_3073},
Delta=
{
<z>() -> q_gen_3074
<s, s>(q_gen_3073) -> q_gen_3073
<z, s>(q_gen_3074) -> q_gen_3073
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3072, q_gen_3076},
Q_f={q_gen_3072},
Delta=
{
<nil, z>() -> q_gen_3072
<snoc, s>(q_gen_3072, q_gen_3076) -> q_gen_3072
<z, z>() -> q_gen_3076
}

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

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

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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 3  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 4
}
Sat witness: Yes: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 5, which took 0.005288 s (model generation: 0.004689,  model checking: 0.000599):

Model:
|_
{
le -> 
{{{
Q={q_gen_3073, q_gen_3074},
Q_f={q_gen_3073},
Delta=
{
<s>(q_gen_3074) -> q_gen_3074
<z>() -> q_gen_3074
<s, s>(q_gen_3073) -> q_gen_3073
<z, s>(q_gen_3074) -> q_gen_3073
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3072, q_gen_3076},
Q_f={q_gen_3072},
Delta=
{
<nil, z>() -> q_gen_3072
<snoc, s>(q_gen_3072, q_gen_3076) -> q_gen_3072
<z, z>() -> q_gen_3076
}

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

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

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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 4  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 7
}
Sat witness: Yes: ((length([ll, _by])) -> length([snoc(ll, x), s(_by)]), {
_by -> z  ;  ll -> nil  ;  x -> s(z)
})

-------------------------------------------
Step 6, which took 0.012330 s (model generation: 0.012025,  model checking: 0.000305):

Model:
|_
{
le -> 
{{{
Q={q_gen_3073, q_gen_3074},
Q_f={q_gen_3073},
Delta=
{
<s>(q_gen_3074) -> q_gen_3074
<z>() -> q_gen_3074
<s, s>(q_gen_3073) -> q_gen_3073
<z, s>(q_gen_3074) -> q_gen_3073
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3072, q_gen_3076, q_gen_3082},
Q_f={q_gen_3072},
Delta=
{
<z>() -> q_gen_3082
<nil, z>() -> q_gen_3072
<snoc, s>(q_gen_3072, q_gen_3076) -> q_gen_3072
<s, z>(q_gen_3082) -> q_gen_3076
<z, z>() -> q_gen_3076
}

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

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

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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 5  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 10
}
Sat witness: Yes: ((length([ll, _by])) -> length([snoc(ll, x), s(_by)]), {
_by -> z  ;  ll -> nil  ;  x -> s(s(z))
})

-------------------------------------------
Step 7, which took 0.008582 s (model generation: 0.006980,  model checking: 0.001602):

Model:
|_
{
le -> 
{{{
Q={q_gen_3073, q_gen_3074},
Q_f={q_gen_3073},
Delta=
{
<s>(q_gen_3074) -> q_gen_3074
<z>() -> q_gen_3074
<s, s>(q_gen_3073) -> q_gen_3073
<z, s>(q_gen_3074) -> q_gen_3073
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3072, q_gen_3076, q_gen_3082},
Q_f={q_gen_3072},
Delta=
{
<s>(q_gen_3082) -> q_gen_3082
<z>() -> q_gen_3082
<nil, z>() -> q_gen_3072
<snoc, s>(q_gen_3072, q_gen_3076) -> q_gen_3072
<s, z>(q_gen_3082) -> q_gen_3076
<z, z>() -> q_gen_3076
}

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

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

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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 6  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 13
}
Sat witness: Yes: ((length([ll, _by])) -> length([snoc(ll, x), s(_by)]), {
_by -> s(z)  ;  ll -> snoc(nil, z)  ;  x -> s(z)
})

-------------------------------------------
Step 8, which took 0.014245 s (model generation: 0.013024,  model checking: 0.001221):

Model:
|_
{
le -> 
{{{
Q={q_gen_3073, q_gen_3074},
Q_f={q_gen_3073},
Delta=
{
<s>(q_gen_3074) -> q_gen_3074
<z>() -> q_gen_3074
<s, s>(q_gen_3073) -> q_gen_3073
<z, s>(q_gen_3074) -> q_gen_3073
}

Datatype: <nat, nat>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_3072, q_gen_3076, q_gen_3082},
Q_f={q_gen_3072},
Delta=
{
<s>(q_gen_3082) -> q_gen_3082
<z>() -> q_gen_3082
<nil, z>() -> q_gen_3072
<snoc, s>(q_gen_3072, q_gen_3076) -> q_gen_3072
<s, s>(q_gen_3076) -> q_gen_3076
<s, z>(q_gen_3082) -> q_gen_3076
<z, z>() -> q_gen_3076
}

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

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

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, _fy]) /\ length([snoc(l, x), _gy])) -> le([_fy, _gy]) -> 7  ;  (length([ll, _by])) -> length([snoc(ll, x), s(_by)]) -> 16
}
Sat witness: Yes: ((length([ll, _by])) -> length([snoc(ll, x), s(_by)]), {
_by -> s(z)  ;  ll -> snoc(nil, z)  ;  x -> z
})

Total time: 0.092659
Reason for stopping: Proved