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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
SNo
x0
.
Apply minus_SNo_prop1 with
x0
,
SNoCutP
{
minus_SNo
x1
|x1 ∈
SNoR
x0
}
{
minus_SNo
x1
|x1 ∈
SNoL
x0
}
leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1:
and
(
and
(
SNo
(
minus_SNo
x0
)
)
(
∀ x1 .
x1
∈
SNoL
x0
⟶
SNoLt
(
minus_SNo
x0
)
(
minus_SNo
x1
)
)
)
(
∀ x1 .
x1
∈
SNoR
x0
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
)
.
Apply H1 with
SNoCutP
(
prim5
(
SNoR
x0
)
minus_SNo
)
(
prim5
(
SNoL
x0
)
minus_SNo
)
⟶
SNoCutP
{
minus_SNo
x1
|x1 ∈
SNoR
x0
}
{
minus_SNo
x1
|x1 ∈
SNoL
x0
}
.
Assume H2:
and
(
SNo
(
minus_SNo
x0
)
)
(
∀ x1 .
x1
∈
SNoL
x0
⟶
SNoLt
(
minus_SNo
x0
)
(
minus_SNo
x1
)
)
.
Apply H2 with
(
∀ x1 .
x1
∈
SNoR
x0
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
)
⟶
SNoCutP
(
prim5
(
SNoR
x0
)
minus_SNo
)
(
prim5
(
SNoL
x0
)
minus_SNo
)
⟶
SNoCutP
{
minus_SNo
x1
|x1 ∈
SNoR
x0
}
{
minus_SNo
x1
|x1 ∈
SNoL
x0
}
.
Assume H3:
SNo
(
minus_SNo
x0
)
.
Assume H4:
∀ x1 .
x1
∈
SNoL
x0
⟶
SNoLt
(
minus_SNo
x0
)
(
minus_SNo
x1
)
.
Assume H5:
∀ x1 .
x1
∈
SNoR
x0
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
.
Assume H6:
SNoCutP
(
prim5
(
SNoR
x0
)
minus_SNo
)
(
prim5
(
SNoL
x0
)
minus_SNo
)
.
The subproof is completed by applying H6.
■