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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
RealsStruct
x0
.
Let x1 of type
ι
be given.
Assume H1:
x1
∈
field0
x0
.
Apply RealsStruct_minus_eq with
x0
,
x1
,
λ x2 x3 .
field1b
x0
x1
x3
=
field4
x0
leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply Field_of_RealsStruct_3 with
x0
,
λ x2 x3 .
field1b
x0
x1
(
explicit_Field_minus
(
field0
x0
)
(
ap
(
Field_of_RealsStruct
x0
)
3
)
(
ap
(
Field_of_RealsStruct
x0
)
4
)
(
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
1
)
)
(
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
2
)
)
x1
)
=
x2
.
set y2 to be
field1b
x0
x1
(
explicit_Field_minus
(
field0
x0
)
(
ap
(
Field_of_RealsStruct
x0
)
3
)
(
ap
(
Field_of_RealsStruct
x0
)
4
)
(
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
1
)
)
(
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
2
)
)
x1
)
set y3 to be
ap
(
Field_of_RealsStruct
x1
)
3
Claim L2:
∀ x4 :
ι → ο
.
x4
y3
⟶
x4
y2
Let x4 of type
ι
→
ο
be given.
Assume H2:
x4
(
ap
(
Field_of_RealsStruct
y2
)
3
)
.
Apply pack_b_b_e_e_1_eq2 with
field0
y2
,
field1b
y2
,
field2b
y2
,
field4
y2
,
RealsStruct_one
y2
,
y3
,
explicit_Field_minus
(
field0
y2
)
(
ap
(
Field_of_RealsStruct
y2
)
3
)
(
ap
(
Field_of_RealsStruct
y2
)
4
)
(
decode_b
(
ap
(
Field_of_RealsStruct
y2
)
1
)
)
(
decode_b
(
ap
(
Field_of_RealsStruct
y2
)
2
)
)
y3
,
λ x5 .
x4
leaving 3 subgoals.
The subproof is completed by applying H1.
Apply explicit_Field_minus_clos with
field0
y2
,
ap
(
Field_of_RealsStruct
y2
)
3
,
ap
(
Field_of_RealsStruct
y2
)
4
,
decode_b
(
ap
(
Field_of_RealsStruct
y2
)
1
)
,
decode_b
(
ap
(
Field_of_RealsStruct
y2
)
2
)
,
y3
leaving 2 subgoals.
Apply explicit_Field_of_RealsStruct_2 with
y2
.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply explicit_Field_minus_R with
field0
y2
,
ap
(
Field_of_RealsStruct
y2
)
3
,
ap
(
Field_of_RealsStruct
y2
)
4
,
decode_b
(
ap
(
Field_of_RealsStruct
y2
)
1
)
,
decode_b
(
ap
(
Field_of_RealsStruct
y2
)
2
)
,
y3
,
λ x5 .
x4
leaving 3 subgoals.
Apply explicit_Field_of_RealsStruct_2 with
y2
.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x4 of type
ι
→
ι
→
ο
be given.
Apply L2 with
λ x5 .
x4
x5
y3
⟶
x4
y3
x5
.
Assume H3:
x4
y3
y3
.
The subproof is completed by applying H3.
■