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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Let x2 of type
ι
be given.
Let x3 of type
ι
be given.
Let x4 of type
ι
be given.
Let x5 of type
ι
be given.
Let x6 of type
ι
be given.
Apply Repl_Empty with
λ x7 .
SetAdjoin
x7
(
Sing
8
)
,
λ x7 x8 .
binunion
(
binunion
(
binunion
(
binunion
(
f4b0e..
x0
x1
x2
x3
)
{
(
λ x10 .
SetAdjoin
x10
(
Sing
5
)
)
x9
|x9 ∈
x4
}
)
{
(
λ x10 .
SetAdjoin
x10
(
Sing
6
)
)
x9
|x9 ∈
x5
}
)
{
(
λ x10 .
SetAdjoin
x10
(
Sing
7
)
)
x9
|x9 ∈
x6
}
)
x8
=
binunion
(
binunion
(
binunion
(
f4b0e..
x0
x1
x2
x3
)
{
(
λ x10 .
SetAdjoin
x10
(
Sing
5
)
)
x9
|x9 ∈
x4
}
)
{
(
λ x10 .
SetAdjoin
x10
(
Sing
6
)
)
x9
|x9 ∈
x5
}
)
{
(
λ x10 .
SetAdjoin
x10
(
Sing
7
)
)
x9
|x9 ∈
x6
}
.
Apply binunion_idr with
binunion
(
binunion
(
binunion
(
f4b0e..
x0
x1
x2
x3
)
{
(
λ x8 .
SetAdjoin
x8
(
Sing
5
)
)
x7
|x7 ∈
x4
}
)
{
(
λ x8 .
SetAdjoin
x8
(
Sing
6
)
)
x7
|x7 ∈
x5
}
)
{
(
λ x8 .
SetAdjoin
x8
(
Sing
7
)
)
x7
|x7 ∈
x6
}
,
λ x7 x8 .
x8
=
binunion
(
binunion
(
binunion
(
f4b0e..
x0
x1
x2
x3
)
{
(
λ x10 .
SetAdjoin
x10
(
Sing
5
)
)
x9
|x9 ∈
x4
}
)
{
(
λ x10 .
SetAdjoin
x10
(
Sing
6
)
)
x9
|x9 ∈
x5
}
)
{
(
λ x10 .
SetAdjoin
x10
(
Sing
7
)
)
x9
|x9 ∈
x6
}
.
Let x7 of type
ι
→
ι
→
ο
be given.
Assume H0:
x7
(
binunion
(
binunion
(
binunion
(
f4b0e..
x0
x1
x2
x3
)
{
(
λ x9 .
SetAdjoin
x9
(
Sing
5
)
)
x8
|x8 ∈
x4
}
)
{
(
λ x9 .
SetAdjoin
x9
(
Sing
6
)
)
x8
|x8 ∈
x5
}
)
{
(
λ x9 .
SetAdjoin
x9
(
Sing
7
)
)
x8
|x8 ∈
x6
}
)
(
binunion
(
binunion
(
binunion
(
f4b0e..
x0
x1
x2
x3
)
{
(
λ x9 .
SetAdjoin
x9
(
Sing
5
)
)
x8
|x8 ∈
x4
}
)
{
(
λ x9 .
SetAdjoin
x9
(
Sing
6
)
)
x8
|x8 ∈
x5
}
)
{
(
λ x9 .
SetAdjoin
x9
(
Sing
7
)
)
x8
|x8 ∈
x6
}
)
.
The subproof is completed by applying H0.
■