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


Learning problem is:

env: {
elt -> {a, b}  ;  eltlist -> {cons, nil}  ;  nat -> {s, z}
}
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, _es])) -> length([cons(x, ll), s(_es)])}
(length([_fs, _gs]) /\ length([_fs, _hs])) -> eq_nat([_gs, _hs])
)
(append, F:
{() -> append([nil, l2, l2])
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)])}
(append([_js, _ks, _ls]) /\ append([_js, _ks, _ms])) -> eq_eltlist([_ls, _ms])
)
}

properties:
{(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps])}


over-approximation: {append, length}
under-approximation: {}

Clause system for inference is:

{
() -> append([nil, l2, l2]) -> 0
() -> le([z, s(nn2)]) -> 0
() -> length([nil, z]) -> 0
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 0
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 0
}


Solving took 0.277456 seconds.
Proved
Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548, q_gen_3559, q_gen_3560, q_gen_3561},
Q_f={q_gen_3542},
Delta=
{
<cons>(q_gen_3560, q_gen_3559) -> q_gen_3559
<nil>() -> q_gen_3559
<a>() -> q_gen_3560
<b>() -> q_gen_3560
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<a, b>() -> q_gen_3548
<b, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, cons, cons>(q_gen_3561, q_gen_3542) -> q_gen_3542
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
<a, a, a>() -> q_gen_3561
<a, b, a>() -> q_gen_3561
<b, a, b>() -> q_gen_3561
<b, b, b>() -> q_gen_3561
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<a>() -> q_gen_3544
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.005541 s (model generation: 0.005061,  model checking: 0.000480):

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

}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={},
Q_f={},
Delta=
{

}

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

}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 0
() -> le([z, s(nn2)]) -> 0
() -> length([nil, z]) -> 3
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 1
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 1
(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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 1
}
Sat witness: Found: (() -> length([nil, z]), {

})

-------------------------------------------
Step 1, which took 0.004623 s (model generation: 0.004466,  model checking: 0.000157):

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

}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={},
Q_f={},
Delta=
{

}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539},
Q_f={q_gen_3539},
Delta=
{
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 0
() -> le([z, s(nn2)]) -> 3
() -> length([nil, z]) -> 3
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 1
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 1
(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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 1
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.005170 s (model generation: 0.004780,  model checking: 0.000390):

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

}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<z>() -> q_gen_3541
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539},
Q_f={q_gen_3539},
Delta=
{
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 3
() -> le([z, s(nn2)]) -> 3
() -> length([nil, z]) -> 3
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 1
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 1
(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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 1
}
Sat witness: Found: (() -> append([nil, l2, l2]), {
l2 -> nil
})

-------------------------------------------
Step 3, which took 0.004995 s (model generation: 0.004725,  model checking: 0.000270):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542},
Q_f={q_gen_3542},
Delta=
{
<nil, nil, nil>() -> q_gen_3542
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<z>() -> q_gen_3541
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539},
Q_f={q_gen_3539},
Delta=
{
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 3
() -> le([z, s(nn2)]) -> 3
() -> length([nil, z]) -> 3
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 1
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 1
(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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 4
}
Sat witness: Found: ((length([ll, _es])) -> length([cons(x, ll), s(_es)]), {
_es -> z  ;  ll -> nil  ;  x -> b
})

-------------------------------------------
Step 4, which took 0.005627 s (model generation: 0.005513,  model checking: 0.000114):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542},
Q_f={q_gen_3542},
Delta=
{
<nil, nil, nil>() -> q_gen_3542
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<z>() -> q_gen_3541
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 3
() -> le([z, s(nn2)]) -> 3
() -> length([nil, z]) -> 3
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 1
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 1
(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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 4
}
Sat witness: Found: ((le([nn1, nn2])) -> le([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> s(z)
})

-------------------------------------------
Step 5, which took 0.005545 s (model generation: 0.005316,  model checking: 0.000229):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542},
Q_f={q_gen_3542},
Delta=
{
<nil, nil, nil>() -> q_gen_3542
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 3
() -> le([z, s(nn2)]) -> 3
() -> length([nil, z]) -> 3
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 1
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 4
(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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 4
}
Sat witness: Found: ((append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]), {
_is -> nil  ;  h1 -> b  ;  l2 -> nil  ;  t1 -> nil
})

-------------------------------------------
Step 6, which took 0.008604 s (model generation: 0.008035,  model checking: 0.000569):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548},
Q_f={q_gen_3542},
Delta=
{
<nil, nil>() -> q_gen_3547
<b, b>() -> q_gen_3548
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 3
() -> le([z, s(nn2)]) -> 6
() -> length([nil, z]) -> 4
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 2
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 4
}
Sat witness: Found: (() -> le([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 7, which took 0.009570 s (model generation: 0.009118,  model checking: 0.000452):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548},
Q_f={q_gen_3542},
Delta=
{
<nil, nil>() -> q_gen_3547
<b, b>() -> q_gen_3548
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 6
() -> le([z, s(nn2)]) -> 6
() -> length([nil, z]) -> 4
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 3
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 4
}
Sat witness: Found: (() -> append([nil, l2, l2]), {
l2 -> cons(a, cons(a, nil))
})

-------------------------------------------
Step 8, which took 0.005068 s (model generation: 0.004921,  model checking: 0.000147):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548},
Q_f={q_gen_3542},
Delta=
{
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 6
() -> le([z, s(nn2)]) -> 6
() -> length([nil, z]) -> 4
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 4
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 7
}
Sat witness: Found: ((length([ll, _es])) -> length([cons(x, ll), s(_es)]), {
_es -> z  ;  ll -> nil  ;  x -> a
})

-------------------------------------------
Step 9, which took 0.013021 s (model generation: 0.012381,  model checking: 0.000640):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548},
Q_f={q_gen_3542},
Delta=
{
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<a>() -> q_gen_3544
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 6
() -> le([z, s(nn2)]) -> 6
() -> length([nil, z]) -> 4
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 4
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 7
(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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 7
}
Sat witness: Found: ((append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]), {
_is -> cons(b, nil)  ;  h1 -> b  ;  l2 -> cons(b, nil)  ;  t1 -> nil
})

-------------------------------------------
Step 10, which took 0.013035 s (model generation: 0.011933,  model checking: 0.001102):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548, q_gen_3559, q_gen_3560, q_gen_3561},
Q_f={q_gen_3542},
Delta=
{
<nil>() -> q_gen_3559
<b>() -> q_gen_3560
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, cons, cons>(q_gen_3561, q_gen_3542) -> q_gen_3542
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
<b, b, b>() -> q_gen_3561
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<a>() -> q_gen_3544
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 7
() -> le([z, s(nn2)]) -> 7
() -> length([nil, z]) -> 5
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 5
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 10
(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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 8
}
Sat witness: Found: ((append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]), {
_is -> cons(b, nil)  ;  h1 -> b  ;  l2 -> nil  ;  t1 -> nil
})

-------------------------------------------
Step 11, which took 0.013207 s (model generation: 0.009657,  model checking: 0.003550):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548, q_gen_3559, q_gen_3560, q_gen_3561},
Q_f={q_gen_3542},
Delta=
{
<nil>() -> q_gen_3559
<b>() -> q_gen_3560
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, cons, cons>(q_gen_3561, q_gen_3542) -> q_gen_3542
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
<b, b, b>() -> q_gen_3561
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<a>() -> q_gen_3544
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 8
() -> le([z, s(nn2)]) -> 8
() -> length([nil, z]) -> 6
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 6
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 13
(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([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 9
}
Sat witness: Found: ((append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]), {
_is -> cons(b, nil)  ;  h1 -> a  ;  l2 -> cons(b, nil)  ;  t1 -> nil
})

-------------------------------------------
Step 12, which took 0.019151 s (model generation: 0.014934,  model checking: 0.004217):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548, q_gen_3559, q_gen_3560, q_gen_3561},
Q_f={q_gen_3542},
Delta=
{
<nil>() -> q_gen_3559
<b>() -> q_gen_3560
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, cons, cons>(q_gen_3561, q_gen_3542) -> q_gen_3542
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
<a, b, a>() -> q_gen_3561
<b, b, b>() -> q_gen_3561
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<a>() -> q_gen_3544
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 9
() -> le([z, s(nn2)]) -> 9
() -> length([nil, z]) -> 7
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 7
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 16
(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
(length([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 10
}
Sat witness: Found: ((append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]), {
_is -> cons(a, nil)  ;  h1 -> b  ;  l2 -> cons(a, nil)  ;  t1 -> nil
})

-------------------------------------------
Step 13, which took 0.020869 s (model generation: 0.017674,  model checking: 0.003195):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548, q_gen_3559, q_gen_3560, q_gen_3561},
Q_f={q_gen_3542},
Delta=
{
<nil>() -> q_gen_3559
<a>() -> q_gen_3560
<b>() -> q_gen_3560
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, cons, cons>(q_gen_3561, q_gen_3542) -> q_gen_3542
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
<a, b, a>() -> q_gen_3561
<b, a, b>() -> q_gen_3561
<b, b, b>() -> q_gen_3561
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<a>() -> q_gen_3544
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 10
() -> le([z, s(nn2)]) -> 10
() -> length([nil, z]) -> 8
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 8
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 19
(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
(length([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 11
}
Sat witness: Found: ((append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]), {
_is -> cons(a, nil)  ;  h1 -> a  ;  l2 -> cons(a, nil)  ;  t1 -> nil
})

-------------------------------------------
Step 14, which took 0.022780 s (model generation: 0.018618,  model checking: 0.004162):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548, q_gen_3559, q_gen_3560, q_gen_3561},
Q_f={q_gen_3542},
Delta=
{
<nil>() -> q_gen_3559
<a>() -> q_gen_3560
<b>() -> q_gen_3560
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, cons, cons>(q_gen_3561, q_gen_3542) -> q_gen_3542
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
<a, a, a>() -> q_gen_3561
<a, b, a>() -> q_gen_3561
<b, a, b>() -> q_gen_3561
<b, b, b>() -> q_gen_3561
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<a>() -> q_gen_3544
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 11
() -> le([z, s(nn2)]) -> 11
() -> length([nil, z]) -> 9
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 9
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 22
(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
(length([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 12
}
Sat witness: Found: ((append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]), {
_is -> cons(b, cons(b, nil))  ;  h1 -> a  ;  l2 -> cons(b, nil)  ;  t1 -> nil
})

-------------------------------------------
Step 15, which took 0.032313 s (model generation: 0.022460,  model checking: 0.009853):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548, q_gen_3559, q_gen_3560, q_gen_3561},
Q_f={q_gen_3542},
Delta=
{
<cons>(q_gen_3560, q_gen_3559) -> q_gen_3559
<nil>() -> q_gen_3559
<a>() -> q_gen_3560
<b>() -> q_gen_3560
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, cons, cons>(q_gen_3561, q_gen_3542) -> q_gen_3542
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
<a, a, a>() -> q_gen_3561
<a, b, a>() -> q_gen_3561
<b, a, b>() -> q_gen_3561
<b, b, b>() -> q_gen_3561
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<a>() -> q_gen_3544
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 12
() -> le([z, s(nn2)]) -> 12
() -> length([nil, z]) -> 10
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 10
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 25
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 11
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 11
(le([s(nn1), z])) -> BOT -> 11
(le([z, z])) -> BOT -> 11
(length([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 13
}
Sat witness: Found: ((append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]), {
_is -> cons(a, cons(b, nil))  ;  h1 -> b  ;  l2 -> cons(a, cons(b, nil))  ;  t1 -> nil
})

-------------------------------------------
Step 16, which took 0.038456 s (model generation: 0.027507,  model checking: 0.010949):

Model:
|_
{
append -> 
{{{
Q={q_gen_3542, q_gen_3547, q_gen_3548, q_gen_3559, q_gen_3560, q_gen_3561},
Q_f={q_gen_3542},
Delta=
{
<cons>(q_gen_3560, q_gen_3559) -> q_gen_3559
<nil>() -> q_gen_3559
<a>() -> q_gen_3560
<b>() -> q_gen_3560
<cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3547
<nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3547
<nil, nil>() -> q_gen_3547
<a, a>() -> q_gen_3548
<b, a>() -> q_gen_3548
<b, b>() -> q_gen_3548
<cons, cons, cons>(q_gen_3561, q_gen_3542) -> q_gen_3542
<cons, nil, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, cons, cons>(q_gen_3548, q_gen_3547) -> q_gen_3542
<nil, nil, cons>(q_gen_3560, q_gen_3559) -> q_gen_3542
<nil, nil, nil>() -> q_gen_3542
<a, a, a>() -> q_gen_3561
<a, b, a>() -> q_gen_3561
<b, a, b>() -> q_gen_3561
<b, b, b>() -> q_gen_3561
}

Datatype: <eltlist, eltlist, eltlist>
Convolution form: right
}}}
  ;  le -> 
{{{
Q={q_gen_3540, q_gen_3541},
Q_f={q_gen_3540},
Delta=
{
<s>(q_gen_3541) -> q_gen_3541
<z>() -> q_gen_3541
<s, s>(q_gen_3540) -> q_gen_3540
<z, s>(q_gen_3541) -> q_gen_3540
}

Datatype: <nat, nat>
Convolution form: right
}}}
  ;  length -> 
{{{
Q={q_gen_3539, q_gen_3544},
Q_f={q_gen_3539},
Delta=
{
<a>() -> q_gen_3544
<b>() -> q_gen_3544
<cons, s>(q_gen_3544, q_gen_3539) -> q_gen_3539
<nil, z>() -> q_gen_3539
}

Datatype: <eltlist, nat>
Convolution form: right
}}}

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

Teacher's answer:
New clause system:
{
() -> append([nil, l2, l2]) -> 13
() -> le([z, s(nn2)]) -> 13
() -> length([nil, z]) -> 11
(append([l1, l2, _os]) /\ le([z, _ns]) /\ length([_os, _ps]) /\ length([l1, _ns])) -> le([z, _ps]) -> 11
(append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]) -> 28
(le([nn1, nn2])) -> le([s(nn1), s(nn2)]) -> 12
(le([s(nn1), s(nn2)])) -> le([nn1, nn2]) -> 12
(le([s(nn1), z])) -> BOT -> 12
(le([z, z])) -> BOT -> 12
(length([ll, _es])) -> length([cons(x, ll), s(_es)]) -> 14
}
Sat witness: Found: ((append([t1, l2, _is])) -> append([cons(h1, t1), l2, cons(h1, _is)]), {
_is -> cons(b, cons(a, nil))  ;  h1 -> a  ;  l2 -> cons(b, cons(a, nil))  ;  t1 -> nil
})

Total time: 0.277456
Reason for stopping: Proved