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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 mul_SNoCutP_lem with
x0
,
x1
,
x2
,
x3
,
x4
,
x5
,
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
}
)
leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Assume H4:
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
}
{
...
|x6 ∈
...
}
)
...
)
.
...
■