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Proofgold Proof
pf
Apply SNoLev_ind with
λ x0 .
and
(
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
)
)
)
(
SNoCutP
{
minus_SNo
x1
|x1 ∈
SNoR
x0
}
{
minus_SNo
x1
|x1 ∈
SNoL
x0
}
)
.
Let x0 of type
ι
be given.
Assume H0:
SNo
x0
.
Assume H1:
∀ x1 .
x1
∈
SNoS_
(
SNoLev
x0
)
⟶
and
(
and
(
and
(
SNo
(
minus_SNo
x1
)
)
(
∀ x2 .
x2
∈
SNoL
x1
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x2
)
)
)
(
∀ x2 .
x2
∈
SNoR
x1
⟶
SNoLt
(
minus_SNo
x2
)
(
minus_SNo
x1
)
)
)
(
SNoCutP
{
minus_SNo
x2
|x2 ∈
SNoR
x1
}
{
minus_SNo
x2
|x2 ∈
SNoL
x1
}
)
.
Claim L2:
...
...
Claim L3:
...
...
Claim L4:
...
...
Claim L5:
...
...
Apply andI with
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
)
)
,
SNoCutP
{
minus_SNo
x1
|x1 ∈
SNoR
x0
}
{
minus_SNo
x1
|x1 ∈
SNoL
x0
}
leaving 2 subgoals.
Apply minus_SNo_eq with
x0
,
λ x1 x2 .
and
(
and
(
SNo
x2
)
(
∀ x3 .
x3
∈
SNoL
x0
⟶
SNoLt
x2
(
minus_SNo
x3
)
)
)
(
∀ x3 .
x3
∈
SNoR
x0
⟶
SNoLt
(
minus_SNo
x3
)
x2
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Apply and3I with
SNo
(
SNoCut
(
prim5
(
SNoR
x0
)
minus_SNo
)
(
prim5
(
SNoL
x0
)
minus_SNo
)
)
,
∀ x1 .
x1
∈
SNoL
x0
⟶
SNoLt
(
SNoCut
(
prim5
(
SNoR
x0
)
minus_SNo
)
(
prim5
(
SNoL
x0
)
minus_SNo
)
)
(
minus_SNo
x1
)
,
∀ x1 .
x1
∈
SNoR
x0
⟶
SNoLt
(
minus_SNo
x1
)
(
SNoCut
(
prim5
(
SNoR
x0
)
minus_SNo
)
(
prim5
(
SNoL
x0
)
minus_SNo
)
)
leaving 3 subgoals.
The subproof is completed by applying L5.
Let x1 of type
ι
be given.
Assume H6:
x1
∈
SNoL
x0
.
Apply SNoL_E with
x0
,
x1
,
SNoLt
(
SNoCut
{
minus_SNo
x2
|x2 ∈
SNoR
x0
}
{
minus_SNo
x2
|x2 ∈
SNoL
x0
}
)
(
minus_SNo
x1
)
leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H6.
Assume H7:
SNo
x1
.
Assume H8:
SNoLev
x1
∈
SNoLev
x0
.
Assume H9:
SNoLt
x1
x0
.
Claim L10:
minus_SNo
x1
∈
{
minus_SNo
x2
|x2 ∈
SNoL
x0
}
Apply ReplI with
SNoL
x0
,
minus_SNo
,
x1
.
The subproof is completed by applying H6.
Apply SNoCutP_SNoCut_R with
{
minus_SNo
x2
|x2 ∈
SNoR
x0
}
,
{
minus_SNo
x2
|x2 ∈
SNoL
x0
}
,
minus_SNo
x1
leaving 2 subgoals.
The subproof is completed by applying L4.
The subproof is completed by applying L10.
Let x1 of type
ι
be given.
Assume H6:
x1
∈
SNoR
x0
.
Apply SNoR_E with
x0
,
x1
,
SNoLt
(
minus_SNo
x1
)
(
SNoCut
{
minus_SNo
x2
|x2 ∈
SNoR
x0
}
{
...
|x2 ∈
SNoL
...
}
)
leaving 3 subgoals.
...
...
...
...
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