Inference procedure has parameters:
Ice fuel: 200
Timeout: 60s
Convolution: left


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, _hl])) -> length([snoc(ll, x), s(_hl)])}
(length([_il, _jl]) /\ length([_il, _kl])) -> eq_nat([_jl, _kl])
)
}

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


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, _ll]) /\ length([snoc(l, x), _ml])) -> le([_ll, _ml]) -> 0
(length([ll, _hl])) -> length([snoc(ll, x), s(_hl)]) -> 0
}


Solving took 0.067130 seconds.
Proved
Model:
|_
{
le -> 
{{{
Q={q_gen_3006, q_gen_3007},
Q_f={q_gen_3006},
Delta=
{
<s>(q_gen_3007) -> q_gen_3007
<z>() -> q_gen_3007
<s, s>(q_gen_3006) -> q_gen_3006
<z, s>(q_gen_3007) -> q_gen_3006
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  length -> 
{{{
Q={q_gen_3005, q_gen_3009},
Q_f={q_gen_3005},
Delta=
{
<s>(q_gen_3009) -> q_gen_3009
<z>() -> q_gen_3009
<nil, z>() -> q_gen_3005
<snoc, s>(q_gen_3005, q_gen_3009) -> q_gen_3005
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.012359 s (model generation: 0.012165,  model checking: 0.000194):

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

}

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

}

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

}
--
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, _ll]) /\ length([snoc(l, x), _ml])) -> le([_ll, _ml]) -> 1
(length([ll, _hl])) -> length([snoc(ll, x), s(_hl)]) -> 1
}
Sat witness: Found: (() -> length([nil, z]), {

})

-------------------------------------------
Step 1, which took 0.009776 s (model generation: 0.009702,  model checking: 0.000074):

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

}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  length -> 
{{{
Q={q_gen_3005},
Q_f={q_gen_3005},
Delta=
{
<nil, z>() -> q_gen_3005
}

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

}
--
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, _ll]) /\ length([snoc(l, x), _ml])) -> le([_ll, _ml]) -> 1
(length([ll, _hl])) -> length([snoc(ll, x), s(_hl)]) -> 1
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.008966 s (model generation: 0.008853,  model checking: 0.000113):

Model:
|_
{
le -> 
{{{
Q={q_gen_3006, q_gen_3007},
Q_f={q_gen_3006},
Delta=
{
<z>() -> q_gen_3007
<z, s>(q_gen_3007) -> q_gen_3006
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  length -> 
{{{
Q={q_gen_3005},
Q_f={q_gen_3005},
Delta=
{
<nil, z>() -> q_gen_3005
}

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

}
--
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, _ll]) /\ length([snoc(l, x), _ml])) -> le([_ll, _ml]) -> 1
(length([ll, _hl])) -> length([snoc(ll, x), s(_hl)]) -> 4
}
Sat witness: Found: ((length([ll, _hl])) -> length([snoc(ll, x), s(_hl)]), {
_hl -> z  ;  ll -> nil  ;  x -> z
})

-------------------------------------------
Step 3, which took 0.007663 s (model generation: 0.007467,  model checking: 0.000196):

Model:
|_
{
le -> 
{{{
Q={q_gen_3006, q_gen_3007},
Q_f={q_gen_3006},
Delta=
{
<z>() -> q_gen_3007
<z, s>(q_gen_3007) -> q_gen_3006
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  length -> 
{{{
Q={q_gen_3005, q_gen_3009},
Q_f={q_gen_3005},
Delta=
{
<z>() -> q_gen_3009
<nil, z>() -> q_gen_3005
<snoc, s>(q_gen_3005, q_gen_3009) -> q_gen_3005
}

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

}
--
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, _ll]) /\ length([snoc(l, x), _ml])) -> le([_ll, _ml]) -> 4
(length([ll, _hl])) -> length([snoc(ll, x), s(_hl)]) -> 4
}
Sat witness: Found: ((length([l, _ll]) /\ length([snoc(l, x), _ml])) -> le([_ll, _ml]), {
_ll -> s(z)  ;  _ml -> s(s(z))  ;  l -> snoc(nil, z)  ;  x -> z
})

-------------------------------------------
Step 4, which took 0.010040 s (model generation: 0.007928,  model checking: 0.002112):

Model:
|_
{
le -> 
{{{
Q={q_gen_3006, q_gen_3007},
Q_f={q_gen_3006},
Delta=
{
<z>() -> q_gen_3007
<s, s>(q_gen_3006) -> q_gen_3006
<z, s>(q_gen_3007) -> q_gen_3006
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  length -> 
{{{
Q={q_gen_3005, q_gen_3009},
Q_f={q_gen_3005},
Delta=
{
<z>() -> q_gen_3009
<nil, z>() -> q_gen_3005
<snoc, s>(q_gen_3005, q_gen_3009) -> q_gen_3005
}

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

}
--
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)]) -> 2
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 2
(le([s(nn1), z])) -> BOT -> 2
(le([z, z])) -> BOT -> 2
(length([l, _ll]) /\ length([snoc(l, x), _ml])) -> le([_ll, _ml]) -> 4
(length([ll, _hl])) -> length([snoc(ll, x), s(_hl)]) -> 4
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 5, which took 0.008674 s (model generation: 0.008430,  model checking: 0.000244):

Model:
|_
{
le -> 
{{{
Q={q_gen_3006, q_gen_3007},
Q_f={q_gen_3006},
Delta=
{
<s>(q_gen_3007) -> q_gen_3007
<z>() -> q_gen_3007
<s, s>(q_gen_3006) -> q_gen_3006
<z, s>(q_gen_3007) -> q_gen_3006
}

Datatype: <nat, nat>
Convolution form: left
}}}
  ;  length -> 
{{{
Q={q_gen_3005, q_gen_3009},
Q_f={q_gen_3005},
Delta=
{
<z>() -> q_gen_3009
<nil, z>() -> q_gen_3005
<snoc, s>(q_gen_3005, q_gen_3009) -> q_gen_3005
}

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

}
--
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)]) -> 3
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 3
(le([s(nn1), z])) -> BOT -> 3
(le([z, z])) -> BOT -> 3
(length([l, _ll]) /\ length([snoc(l, x), _ml])) -> le([_ll, _ml]) -> 4
(length([ll, _hl])) -> length([snoc(ll, x), s(_hl)]) -> 7
}
Sat witness: Found: ((length([ll, _hl])) -> length([snoc(ll, x), s(_hl)]), {
_hl -> z  ;  ll -> nil  ;  x -> s(z)
})

Total time: 0.067130
Reason for stopping: Proved