Solving ../../benchmarks/true/insert_length_leq.smt2...

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


Learning problem is:

env: {
elt -> {a, b}  ;  eltlist -> {cons, nil}  ;  nat -> {s, z}
}
definition:
{
(leq, P:
{() -> leq([a, y])
() -> leq([b, b])
(leq([b, a])) -> BOT}
)
(insert, F:
{() -> insert([x, nil, cons(x, nil)])
(insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)])
(leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))])}
(insert([_z, _aa, _ba]) /\ insert([_z, _aa, _ca])) -> eq_eltlist([_ba, _ca])
)
(length, F:
{() -> length([nil, z])
(length([ll, _da])) -> length([cons(x, ll), s(_da)])}
(length([_ea, _fa]) /\ length([_ea, _ga])) -> eq_nat([_fa, _ga])
)
(leqnat, P:
{() -> leqnat([z, s(nn2)])
() -> leqnat([z, z])
(leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)])
(leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2])
(leqnat([s(nn1), z])) -> BOT}
)
}

properties:
{(insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja])}


over-approximation: {insert, length}
under-approximation: {leq, leqnat}

Clause system for inference is:

{
() -> insert([x, nil, cons(x, nil)]) -> 0  ;  () -> length([nil, z]) -> 0  ;  () -> leq([a, y]) -> 0  ;  () -> leq([b, b]) -> 0  ;  () -> leqnat([z, s(nn2)]) -> 0  ;  () -> leqnat([z, z]) -> 0  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 0  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 0  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 0  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 0  ;  (leq([b, a])) -> BOT -> 0  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 0  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 0  ;  (leqnat([s(nn1), z])) -> BOT -> 0
}


Solving took 0.304789 seconds.
Proved
Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<cons>(q_gen_1083, q_gen_1082) -> q_gen_1082
<nil>() -> q_gen_1082
<a>() -> q_gen_1083
<b>() -> q_gen_1083
<cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1086
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<cons, a>(q_gen_1083, q_gen_1082) -> q_gen_1087
<cons, b>(q_gen_1083, q_gen_1082) -> q_gen_1087
<nil, a>() -> q_gen_1087
<nil, b>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<a, b>() -> q_gen_1089
<b, a>() -> q_gen_1089
<b, b>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<a, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091, q_gen_1116},
Q_f={q_gen_1080},
Delta=
{
<s>(q_gen_1116) -> q_gen_1116
<z>() -> q_gen_1116
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<a, s>(q_gen_1116) -> q_gen_1091
<a, z>() -> q_gen_1091
<b, s>(q_gen_1116) -> q_gen_1091
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

-------------------
STEPS:
-------------------------------------------
Step 0, which took 0.004450 s (model generation: 0.004118,  model checking: 0.000332):

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

}

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

}

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

}

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

}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0  ;  () -> length([nil, z]) -> 0  ;  () -> leq([a, y]) -> 0  ;  () -> leq([b, b]) -> 0  ;  () -> leqnat([z, s(nn2)]) -> 0  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1  ;  (leq([b, a])) -> BOT -> 1  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1  ;  (leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Yes: (() -> leqnat([z, z]), {

})

-------------------------------------------
Step 1, which took 0.003681 s (model generation: 0.003636,  model checking: 0.000045):

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

}

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

}

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

}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075},
Q_f={q_gen_1075},
Delta=
{
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0  ;  () -> length([nil, z]) -> 0  ;  () -> leq([a, y]) -> 0  ;  () -> leq([b, b]) -> 0  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1  ;  (leq([b, a])) -> BOT -> 1  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1  ;  (leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Yes: (() -> leqnat([z, s(nn2)]), {
nn2 -> z
})

-------------------------------------------
Step 2, which took 0.003625 s (model generation: 0.003604,  model checking: 0.000021):

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

}

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

}

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

}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0  ;  () -> length([nil, z]) -> 0  ;  () -> leq([a, y]) -> 0  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1  ;  (leq([b, a])) -> BOT -> 1  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1  ;  (leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Yes: (() -> leq([b, b]), {

})

-------------------------------------------
Step 3, which took 0.003421 s (model generation: 0.003395,  model checking: 0.000026):

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

}

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

}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078},
Q_f={q_gen_1078},
Delta=
{
<b, b>() -> q_gen_1078
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0  ;  () -> length([nil, z]) -> 0  ;  () -> leq([a, y]) -> 3  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1  ;  (leq([b, a])) -> BOT -> 1  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1  ;  (leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Yes: (() -> leq([a, y]), {
y -> b
})

-------------------------------------------
Step 4, which took 0.003796 s (model generation: 0.003732,  model checking: 0.000064):

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

}

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

}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078},
Q_f={q_gen_1078},
Delta=
{
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 0  ;  () -> length([nil, z]) -> 3  ;  () -> leq([a, y]) -> 3  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1  ;  (leq([b, a])) -> BOT -> 1  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1  ;  (leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Yes: (() -> length([nil, z]), {

})

-------------------------------------------
Step 5, which took 0.006125 s (model generation: 0.005926,  model checking: 0.000199):

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

}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080},
Q_f={q_gen_1080},
Delta=
{
<nil, z>() -> q_gen_1080
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078},
Q_f={q_gen_1078},
Delta=
{
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3  ;  () -> length([nil, z]) -> 3  ;  () -> leq([a, y]) -> 3  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1  ;  (leq([b, a])) -> BOT -> 1  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 1  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 1  ;  (leqnat([s(nn1), z])) -> BOT -> 1
}
Sat witness: Yes: (() -> insert([x, nil, cons(x, nil)]), {
x -> b
})

-------------------------------------------
Step 6, which took 0.008517 s (model generation: 0.008266,  model checking: 0.000251):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<b>() -> q_gen_1083
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080},
Q_f={q_gen_1080},
Delta=
{
<nil, z>() -> q_gen_1080
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078},
Q_f={q_gen_1078},
Delta=
{
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3  ;  () -> length([nil, z]) -> 3  ;  () -> leq([a, y]) -> 3  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 1  ;  (leq([b, a])) -> BOT -> 1  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 2  ;  (leqnat([s(nn1), z])) -> BOT -> 2
}
Sat witness: Yes: ((leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]), {
nn1 -> z  ;  nn2 -> z
})

-------------------------------------------
Step 7, which took 0.009005 s (model generation: 0.008363,  model checking: 0.000642):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<b>() -> q_gen_1083
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080},
Q_f={q_gen_1080},
Delta=
{
<nil, z>() -> q_gen_1080
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078},
Q_f={q_gen_1078},
Delta=
{
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3  ;  () -> length([nil, z]) -> 3  ;  () -> leq([a, y]) -> 3  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 1  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4  ;  (leq([b, a])) -> BOT -> 2  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 2  ;  (leqnat([s(nn1), z])) -> BOT -> 2
}
Sat witness: Yes: ((leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]), {
x -> a  ;  y -> b  ;  z -> nil
})

-------------------------------------------
Step 8, which took 0.010172 s (model generation: 0.009893,  model checking: 0.000279):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<b>() -> q_gen_1083
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<nil, a>() -> q_gen_1087
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, a>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080},
Q_f={q_gen_1080},
Delta=
{
<nil, z>() -> q_gen_1080
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078},
Q_f={q_gen_1078},
Delta=
{
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3  ;  () -> length([nil, z]) -> 3  ;  () -> leq([a, y]) -> 3  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 1  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4  ;  (leq([b, a])) -> BOT -> 2  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 2  ;  (leqnat([s(nn1), z])) -> BOT -> 2
}
Sat witness: Yes: ((length([ll, _da])) -> length([cons(x, ll), s(_da)]), {
_da -> z  ;  ll -> nil  ;  x -> b
})

-------------------------------------------
Step 9, which took 0.012603 s (model generation: 0.011703,  model checking: 0.000900):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<b>() -> q_gen_1083
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<nil, a>() -> q_gen_1087
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, a>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078},
Q_f={q_gen_1078},
Delta=
{
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3  ;  () -> length([nil, z]) -> 3  ;  () -> leq([a, y]) -> 3  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 1  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4  ;  (leq([b, a])) -> BOT -> 2  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 2  ;  (leqnat([s(nn1), z])) -> BOT -> 2
}
Sat witness: Yes: ((insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]), {
_y -> cons(b, nil)  ;  x -> b  ;  y -> a  ;  z -> nil
})

-------------------------------------------
Step 10, which took 0.013145 s (model generation: 0.012723,  model checking: 0.000422):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<b>() -> q_gen_1083
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<nil, a>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078},
Q_f={q_gen_1078},
Delta=
{
<a, b>() -> q_gen_1078
<b, a>() -> q_gen_1078
<b, b>() -> q_gen_1078
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3  ;  () -> length([nil, z]) -> 3  ;  () -> leq([a, y]) -> 3  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 3  ;  () -> leqnat([z, z]) -> 3  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 2  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4  ;  (leq([b, a])) -> BOT -> 5  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 3  ;  (leqnat([s(nn1), z])) -> BOT -> 3
}
Sat witness: Yes: ((leq([b, a])) -> BOT, {

})

-------------------------------------------
Step 11, which took 0.011037 s (model generation: 0.010703,  model checking: 0.000334):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<b>() -> q_gen_1083
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<nil, a>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3  ;  () -> length([nil, z]) -> 3  ;  () -> leq([a, y]) -> 3  ;  () -> leq([b, b]) -> 3  ;  () -> leqnat([z, s(nn2)]) -> 6  ;  () -> leqnat([z, z]) -> 4  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 3  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4  ;  (leq([b, a])) -> BOT -> 5  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 4  ;  (leqnat([s(nn1), z])) -> BOT -> 4
}
Sat witness: Yes: (() -> leqnat([z, s(nn2)]), {
nn2 -> s(z)
})

-------------------------------------------
Step 12, which took 0.012432 s (model generation: 0.012035,  model checking: 0.000397):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<b>() -> q_gen_1083
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<nil, a>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 3  ;  () -> length([nil, z]) -> 3  ;  () -> leq([a, y]) -> 6  ;  () -> leq([b, b]) -> 4  ;  () -> leqnat([z, s(nn2)]) -> 6  ;  () -> leqnat([z, z]) -> 4  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4  ;  (leq([b, a])) -> BOT -> 5  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 4  ;  (leqnat([s(nn1), z])) -> BOT -> 4
}
Sat witness: Yes: (() -> leq([a, y]), {
y -> a
})

-------------------------------------------
Step 13, which took 0.012106 s (model generation: 0.011571,  model checking: 0.000535):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<b>() -> q_gen_1083
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<nil, a>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 6  ;  () -> length([nil, z]) -> 4  ;  () -> leq([a, y]) -> 6  ;  () -> leq([b, b]) -> 4  ;  () -> leqnat([z, s(nn2)]) -> 6  ;  () -> leqnat([z, z]) -> 4  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 4  ;  (leq([b, a])) -> BOT -> 5  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 4  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 4  ;  (leqnat([s(nn1), z])) -> BOT -> 4
}
Sat witness: Yes: (() -> insert([x, nil, cons(x, nil)]), {
x -> a
})

-------------------------------------------
Step 14, which took 0.020755 s (model generation: 0.008740,  model checking: 0.012015):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<a>() -> q_gen_1083
<b>() -> q_gen_1083
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<nil, a>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<a, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 6  ;  () -> length([nil, z]) -> 4  ;  () -> leq([a, y]) -> 6  ;  () -> leq([b, b]) -> 4  ;  () -> leqnat([z, s(nn2)]) -> 6  ;  () -> leqnat([z, z]) -> 4  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 4  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 7  ;  (leq([b, a])) -> BOT -> 5  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 5  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 5  ;  (leqnat([s(nn1), z])) -> BOT -> 5
}
Sat witness: Yes: ((leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]), {
x -> b  ;  y -> b  ;  z -> nil
})

-------------------------------------------
Step 15, which took 0.015367 s (model generation: 0.015049,  model checking: 0.000318):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<a>() -> q_gen_1083
<b>() -> q_gen_1083
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<nil, a>() -> q_gen_1087
<nil, b>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<b, b>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<a, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 6  ;  () -> length([nil, z]) -> 4  ;  () -> leq([a, y]) -> 6  ;  () -> leq([b, b]) -> 4  ;  () -> leqnat([z, s(nn2)]) -> 6  ;  () -> leqnat([z, z]) -> 4  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 4  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 7  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 7  ;  (leq([b, a])) -> BOT -> 5  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 5  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 5  ;  (leqnat([s(nn1), z])) -> BOT -> 5
}
Sat witness: Yes: ((length([ll, _da])) -> length([cons(x, ll), s(_da)]), {
_da -> z  ;  ll -> nil  ;  x -> a
})

-------------------------------------------
Step 16, which took 0.030975 s (model generation: 0.015529,  model checking: 0.015446):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<nil>() -> q_gen_1082
<a>() -> q_gen_1083
<b>() -> q_gen_1083
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<nil, a>() -> q_gen_1087
<nil, b>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<b, b>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<a, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<a, z>() -> q_gen_1091
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 6  ;  () -> length([nil, z]) -> 4  ;  () -> leq([a, y]) -> 6  ;  () -> leq([b, b]) -> 4  ;  () -> leqnat([z, s(nn2)]) -> 6  ;  () -> leqnat([z, z]) -> 4  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 4  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 7  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 7  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 7  ;  (leq([b, a])) -> BOT -> 5  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 5  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 5  ;  (leqnat([s(nn1), z])) -> BOT -> 5
}
Sat witness: Yes: ((insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]), {
_y -> cons(b, cons(b, nil))  ;  x -> b  ;  y -> a  ;  z -> cons(b, nil)
})

-------------------------------------------
Step 17, which took 0.019645 s (model generation: 0.011945,  model checking: 0.007700):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<cons>(q_gen_1083, q_gen_1082) -> q_gen_1082
<nil>() -> q_gen_1082
<a>() -> q_gen_1083
<b>() -> q_gen_1083
<cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1086
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<cons, a>(q_gen_1083, q_gen_1082) -> q_gen_1087
<nil, a>() -> q_gen_1087
<nil, b>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<b, b>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<a, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<a, z>() -> q_gen_1091
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 7  ;  () -> length([nil, z]) -> 5  ;  () -> leq([a, y]) -> 7  ;  () -> leq([b, b]) -> 5  ;  () -> leqnat([z, s(nn2)]) -> 7  ;  () -> leqnat([z, z]) -> 5  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 5  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 7  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 7  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 10  ;  (leq([b, a])) -> BOT -> 6  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 6  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 6  ;  (leqnat([s(nn1), z])) -> BOT -> 6
}
Sat witness: Yes: ((leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]), {
x -> b  ;  y -> b  ;  z -> cons(b, nil)
})

-------------------------------------------
Step 18, which took 0.016532 s (model generation: 0.015328,  model checking: 0.001204):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<cons>(q_gen_1083, q_gen_1082) -> q_gen_1082
<nil>() -> q_gen_1082
<a>() -> q_gen_1083
<b>() -> q_gen_1083
<cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1086
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<cons, a>(q_gen_1083, q_gen_1082) -> q_gen_1087
<cons, b>(q_gen_1083, q_gen_1082) -> q_gen_1087
<nil, a>() -> q_gen_1087
<nil, b>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<b, b>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<a, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091},
Q_f={q_gen_1080},
Delta=
{
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<a, z>() -> q_gen_1091
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 7  ;  () -> length([nil, z]) -> 6  ;  () -> leq([a, y]) -> 7  ;  () -> leq([b, b]) -> 6  ;  () -> leqnat([z, s(nn2)]) -> 7  ;  () -> leqnat([z, z]) -> 6  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 6  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 7  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 10  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 10  ;  (leq([b, a])) -> BOT -> 7  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 7  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 7  ;  (leqnat([s(nn1), z])) -> BOT -> 7
}
Sat witness: Yes: ((length([ll, _da])) -> length([cons(x, ll), s(_da)]), {
_da -> s(z)  ;  ll -> cons(a, nil)  ;  x -> b
})

-------------------------------------------
Step 19, which took 0.039925 s (model generation: 0.018032,  model checking: 0.021893):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<cons>(q_gen_1083, q_gen_1082) -> q_gen_1082
<nil>() -> q_gen_1082
<a>() -> q_gen_1083
<b>() -> q_gen_1083
<cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1086
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<cons, a>(q_gen_1083, q_gen_1082) -> q_gen_1087
<cons, b>(q_gen_1083, q_gen_1082) -> q_gen_1087
<nil, a>() -> q_gen_1087
<nil, b>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<b, a>() -> q_gen_1089
<b, b>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<a, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091, q_gen_1116},
Q_f={q_gen_1080},
Delta=
{
<z>() -> q_gen_1116
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<a, z>() -> q_gen_1091
<b, s>(q_gen_1116) -> q_gen_1091
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 8  ;  () -> length([nil, z]) -> 7  ;  () -> leq([a, y]) -> 8  ;  () -> leq([b, b]) -> 7  ;  () -> leqnat([z, s(nn2)]) -> 8  ;  () -> leqnat([z, z]) -> 7  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 7  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 8  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 10  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 13  ;  (leq([b, a])) -> BOT -> 8  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 8  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 8  ;  (leqnat([s(nn1), z])) -> BOT -> 8
}
Sat witness: Yes: ((leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]), {
x -> b  ;  y -> b  ;  z -> cons(a, nil)
})

-------------------------------------------
Step 20, which took 0.015696 s (model generation: 0.006846,  model checking: 0.008850):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<cons>(q_gen_1083, q_gen_1082) -> q_gen_1082
<nil>() -> q_gen_1082
<a>() -> q_gen_1083
<b>() -> q_gen_1083
<cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1086
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<cons, a>(q_gen_1083, q_gen_1082) -> q_gen_1087
<cons, b>(q_gen_1083, q_gen_1082) -> q_gen_1087
<nil, a>() -> q_gen_1087
<nil, b>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<a, b>() -> q_gen_1089
<b, a>() -> q_gen_1089
<b, b>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<a, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091, q_gen_1116},
Q_f={q_gen_1080},
Delta=
{
<z>() -> q_gen_1116
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<a, z>() -> q_gen_1091
<b, s>(q_gen_1116) -> q_gen_1091
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 9  ;  () -> length([nil, z]) -> 8  ;  () -> leq([a, y]) -> 9  ;  () -> leq([b, b]) -> 8  ;  () -> leqnat([z, s(nn2)]) -> 9  ;  () -> leqnat([z, z]) -> 8  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 8  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 9  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 13  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 13  ;  (leq([b, a])) -> BOT -> 9  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 9  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 9  ;  (leqnat([s(nn1), z])) -> BOT -> 9
}
Sat witness: Yes: ((length([ll, _da])) -> length([cons(x, ll), s(_da)]), {
_da -> s(z)  ;  ll -> cons(b, nil)  ;  x -> a
})

-------------------------------------------
Step 21, which took 0.015706 s (model generation: 0.006930,  model checking: 0.008776):

Model:
|_
{
insert -> 
{{{
Q={q_gen_1081, q_gen_1082, q_gen_1083, q_gen_1086, q_gen_1087, q_gen_1088, q_gen_1089},
Q_f={q_gen_1081},
Delta=
{
<cons>(q_gen_1083, q_gen_1082) -> q_gen_1082
<nil>() -> q_gen_1082
<a>() -> q_gen_1083
<b>() -> q_gen_1083
<cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1086
<nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1086
<cons, a>(q_gen_1083, q_gen_1082) -> q_gen_1087
<cons, b>(q_gen_1083, q_gen_1082) -> q_gen_1087
<nil, a>() -> q_gen_1087
<nil, b>() -> q_gen_1087
<a, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<b, cons>(q_gen_1083, q_gen_1082) -> q_gen_1088
<a, a>() -> q_gen_1089
<a, b>() -> q_gen_1089
<b, a>() -> q_gen_1089
<b, b>() -> q_gen_1089
<a, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<a, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
<b, cons, cons>(q_gen_1089, q_gen_1088, q_gen_1087, q_gen_1086) -> q_gen_1081
<b, nil, cons>(q_gen_1083, q_gen_1082) -> q_gen_1081
}

Datatype: <elt, eltlist, eltlist>
Convolution form: complete
}}}
  ;  length -> 
{{{
Q={q_gen_1080, q_gen_1091, q_gen_1116},
Q_f={q_gen_1080},
Delta=
{
<z>() -> q_gen_1116
<cons, s>(q_gen_1091, q_gen_1080) -> q_gen_1080
<nil, z>() -> q_gen_1080
<a, s>(q_gen_1116) -> q_gen_1091
<a, z>() -> q_gen_1091
<b, s>(q_gen_1116) -> q_gen_1091
<b, z>() -> q_gen_1091
}

Datatype: <eltlist, nat>
Convolution form: complete
}}}
  ;  leq -> 
{{{
Q={q_gen_1078, q_gen_1092},
Q_f={q_gen_1078},
Delta=
{
<a, a>() -> q_gen_1078
<a, b>() -> q_gen_1078
<b, b>() -> q_gen_1078
<b, a>() -> q_gen_1092
}

Datatype: <elt, elt>
Convolution form: complete
}}}
  ;  leqnat -> 
{{{
Q={q_gen_1075, q_gen_1077},
Q_f={q_gen_1075},
Delta=
{
<s>(q_gen_1077) -> q_gen_1077
<z>() -> q_gen_1077
<s, s>(q_gen_1075) -> q_gen_1075
<z, s>(q_gen_1077) -> q_gen_1075
<z, z>() -> q_gen_1075
}

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

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

Teacher's answer:
New clause system:
{
() -> insert([x, nil, cons(x, nil)]) -> 10  ;  () -> length([nil, z]) -> 9  ;  () -> leq([a, y]) -> 10  ;  () -> leq([b, b]) -> 9  ;  () -> leqnat([z, s(nn2)]) -> 10  ;  () -> leqnat([z, z]) -> 9  ;  (insert([x, l, _ia]) /\ length([_ia, _ja]) /\ length([l, _ha])) -> leqnat([_ha, _ja]) -> 9  ;  (insert([x, z, _y]) /\ not leq([x, y])) -> insert([x, cons(y, z), cons(y, _y)]) -> 10  ;  (length([ll, _da])) -> length([cons(x, ll), s(_da)]) -> 16  ;  (leq([x, y])) -> insert([x, cons(y, z), cons(x, cons(y, z))]) -> 14  ;  (leq([b, a])) -> BOT -> 10  ;  (leqnat([nn1, nn2])) -> leqnat([s(nn1), s(nn2)]) -> 10  ;  (leqnat([s(nn1), s(nn2)])) -> leqnat([nn1, nn2]) -> 10  ;  (leqnat([s(nn1), z])) -> BOT -> 10
}
Sat witness: Yes: ((length([ll, _da])) -> length([cons(x, ll), s(_da)]), {
_da -> s(s(z))  ;  ll -> cons(b, cons(b, nil))  ;  x -> b
})

Total time: 0.304789
Reason for stopping: Proved