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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Assume H0:
SNo
x0
.
Assume H1:
SNo
x1
.
Assume H2:
SNoLt
x0
x1
.
Apply minus_SNo_prop1 with
x0
,
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H3:
and
(
and
(
SNo
(
minus_SNo
x0
)
)
(
∀ x2 .
x2
∈
SNoL
x0
⟶
SNoLt
(
minus_SNo
x0
)
(
minus_SNo
x2
)
)
)
(
∀ x2 .
x2
∈
SNoR
x0
⟶
SNoLt
(
minus_SNo
x2
)
(
minus_SNo
x0
)
)
.
Apply H3 with
SNoCutP
(
prim5
(
SNoR
x0
)
minus_SNo
)
(
prim5
(
SNoL
x0
)
minus_SNo
)
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
.
Assume H4:
and
(
SNo
(
minus_SNo
x0
)
)
(
∀ x2 .
x2
∈
SNoL
x0
⟶
SNoLt
(
minus_SNo
x0
)
(
minus_SNo
x2
)
)
.
Apply H4 with
(
∀ x2 .
x2
∈
SNoR
x0
⟶
SNoLt
(
minus_SNo
x2
)
(
minus_SNo
x0
)
)
⟶
SNoCutP
(
prim5
(
SNoR
x0
)
minus_SNo
)
(
prim5
(
SNoL
x0
)
minus_SNo
)
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
.
Assume H5:
SNo
(
minus_SNo
x0
)
.
Assume H6:
∀ x2 .
x2
∈
SNoL
x0
⟶
SNoLt
(
minus_SNo
x0
)
(
minus_SNo
x2
)
.
Assume H7:
∀ x2 .
x2
∈
SNoR
x0
⟶
SNoLt
(
minus_SNo
x2
)
(
minus_SNo
x0
)
.
Assume H8:
SNoCutP
(
prim5
(
SNoR
x0
)
minus_SNo
)
(
prim5
(
SNoL
x0
)
minus_SNo
)
.
Apply minus_SNo_prop1 with
x1
,
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H9:
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
)
)
.
Apply H9 with
SNoCutP
(
prim5
(
SNoR
x1
)
minus_SNo
)
(
prim5
(
SNoL
x1
)
minus_SNo
)
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
.
Assume H10:
and
(
SNo
(
minus_SNo
x1
)
)
(
∀ x2 .
x2
∈
SNoL
x1
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x2
)
)
.
Apply H10 with
(
∀ x2 .
x2
∈
SNoR
x1
⟶
SNoLt
(
minus_SNo
x2
)
(
minus_SNo
x1
)
)
⟶
SNoCutP
(
prim5
(
SNoR
x1
)
minus_SNo
)
(
prim5
(
SNoL
x1
)
minus_SNo
)
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
.
Assume H11:
SNo
(
minus_SNo
x1
)
.
Assume H12:
∀ x2 .
x2
∈
SNoL
x1
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x2
)
.
Assume H13:
∀ x2 .
x2
∈
SNoR
x1
⟶
SNoLt
(
minus_SNo
x2
)
(
minus_SNo
x1
)
.
Assume H14:
SNoCutP
(
prim5
(
SNoR
x1
)
minus_SNo
)
(
prim5
(
SNoL
x1
)
minus_SNo
)
.
Apply SNoLtE with
x0
,
x1
,
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
leaving 6 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x2 of type
ι
be given.
Assume H15:
SNo
x2
.
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