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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.
Assume H0:
SNoCutP
x0
x1
.
Assume H1:
SNoCutP
x2
x3
.
Assume H2:
x4
=
SNoCut
x0
x1
.
Assume H3:
x5
=
SNoCut
x2
x3
.
Apply H0 with
and
(
and
(
SNoCutP
(
binunion
{
add_SNo
(
mul_SNo
(
ap
x6
0
)
x5
)
(
add_SNo
(
mul_SNo
x4
(
ap
x6
1
)
)
(
minus_SNo
(
mul_SNo
(
ap
x6
0
)
(
ap
x6
1
)
)
)
)
|x6 ∈
setprod
x0
x2
}
{
add_SNo
(
mul_SNo
(
ap
x6
0
)
x5
)
(
add_SNo
(
mul_SNo
x4
(
ap
x6
1
)
)
(
minus_SNo
(
mul_SNo
(
ap
x6
0
)
(
ap
x6
1
)
)
)
)
|x6 ∈
setprod
x1
x3
}
)
(
binunion
{
add_SNo
(
mul_SNo
(
ap
x6
0
)
x5
)
(
add_SNo
(
mul_SNo
x4
(
ap
x6
1
)
)
(
minus_SNo
(
mul_SNo
(
ap
x6
0
)
(
ap
x6
1
)
)
)
)
|x6 ∈
setprod
x0
x3
}
{
add_SNo
(
mul_SNo
(
ap
x6
0
)
x5
)
(
add_SNo
(
mul_SNo
x4
(
ap
x6
1
)
)
(
minus_SNo
(
mul_SNo
(
ap
x6
0
)
(
ap
x6
1
)
)
)
)
|x6 ∈
setprod
x1
x2
}
)
)
(
mul_SNo
x4
x5
=
SNoCut
(
binunion
{
add_SNo
(
mul_SNo
(
ap
x6
0
)
x5
)
(
add_SNo
(
mul_SNo
x4
(
ap
x6
1
)
)
(
minus_SNo
(
mul_SNo
(
ap
x6
0
)
(
ap
x6
1
)
)
)
)
|x6 ∈
setprod
x0
x2
}
{
add_SNo
(
mul_SNo
(
ap
x6
0
)
x5
)
(
add_SNo
(
mul_SNo
x4
(
ap
x6
1
)
)
(
minus_SNo
(
mul_SNo
(
ap
x6
0
)
(
ap
x6
1
)
)
)
)
|x6 ∈
setprod
x1
x3
}
)
(
binunion
{
add_SNo
(
mul_SNo
(
ap
x6
0
)
x5
)
(
add_SNo
(
mul_SNo
x4
(
ap
x6
1
)
)
(
minus_SNo
(
mul_SNo
(
ap
x6
0
)
(
ap
x6
1
)
)
)
)
|x6 ∈
setprod
x0
x3
}
{
add_SNo
(
mul_SNo
(
ap
x6
0
)
x5
)
(
add_SNo
(
mul_SNo
x4
(
ap
x6
1
)
)
(
minus_SNo
(
mul_SNo
(
ap
x6
0
)
(
ap
x6
1
)
)
)
)
|x6 ∈
setprod
x1
x2
}
)
)
)
(
∀ x6 : ο .
(
∀ x7 x8 x9 x10 .
...
⟶
...
⟶
...
⟶
...
⟶
...
⟶
...
⟶
...
⟶
(
∀ x11 .
...
⟶
∀ x12 .
...
⟶
add_SNo
(
mul_SNo
x11
x5
)
(
add_SNo
(
mul_SNo
x4
x12
)
...
)
∈
...
)
⟶
SNoCutP
(
binunion
x7
x8
)
(
binunion
x9
x10
)
⟶
mul_SNo
x4
x5
=
SNoCut
(
binunion
x7
x8
)
(
binunion
x9
x10
)
⟶
x6
)
⟶
x6
)
.
...
■