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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
x0
∈
real
.
Apply real_Re_eq with
x0
,
λ x1 x2 .
SNo_pair
(
add_SNo
(
mul_SNo
(
CSNo_Re
(
SNo_pair
0
1
)
)
x2
)
(
minus_SNo
(
mul_SNo
(
CSNo_Im
(
SNo_pair
0
1
)
)
(
CSNo_Im
x0
)
)
)
)
(
add_SNo
(
mul_SNo
(
CSNo_Re
(
SNo_pair
0
1
)
)
(
CSNo_Im
x0
)
)
(
mul_SNo
(
CSNo_Im
(
SNo_pair
0
1
)
)
x2
)
)
=
SNo_pair
0
x0
leaving 2 subgoals.
The subproof is completed by applying H0.
Apply real_Im_eq with
x0
,
λ x1 x2 .
SNo_pair
(
add_SNo
(
mul_SNo
(
CSNo_Re
(
SNo_pair
0
1
)
)
x0
)
(
minus_SNo
(
mul_SNo
(
CSNo_Im
(
SNo_pair
0
1
)
)
x2
)
)
)
(
add_SNo
(
mul_SNo
(
CSNo_Re
(
SNo_pair
0
1
)
)
x2
)
(
mul_SNo
(
CSNo_Im
(
SNo_pair
0
1
)
)
x0
)
)
=
SNo_pair
0
x0
leaving 2 subgoals.
The subproof is completed by applying H0.
Apply complex_Re_eq with
0
,
1
,
λ x1 x2 .
SNo_pair
(
add_SNo
(
mul_SNo
x2
x0
)
(
minus_SNo
(
mul_SNo
(
CSNo_Im
(
SNo_pair
0
1
)
)
0
)
)
)
(
add_SNo
(
mul_SNo
x2
0
)
(
mul_SNo
(
CSNo_Im
(
SNo_pair
0
1
)
)
x0
)
)
=
SNo_pair
0
x0
leaving 3 subgoals.
The subproof is completed by applying real_0.
The subproof is completed by applying real_1.
Apply complex_Im_eq with
0
,
1
,
λ x1 x2 .
SNo_pair
(
add_SNo
(
mul_SNo
0
x0
)
(
minus_SNo
(
mul_SNo
x2
0
)
)
)
(
add_SNo
(
mul_SNo
0
0
)
(
mul_SNo
x2
x0
)
)
=
SNo_pair
0
x0
leaving 3 subgoals.
The subproof is completed by applying real_0.
The subproof is completed by applying real_1.
Apply mul_SNo_zeroL with
x0
,
λ x1 x2 .
SNo_pair
(
add_SNo
x2
(
minus_SNo
(
mul_SNo
1
0
)
)
)
(
add_SNo
(
mul_SNo
0
0
)
(
mul_SNo
1
x0
)
)
=
SNo_pair
0
x0
leaving 2 subgoals.
Apply real_SNo with
x0
.
The subproof is completed by applying H0.
Apply mul_SNo_zeroR with
1
,
λ x1 x2 .
SNo_pair
(
add_SNo
0
(
minus_SNo
x2
)
)
(
add_SNo
(
mul_SNo
0
0
)
(
mul_SNo
1
x0
)
)
=
SNo_pair
0
x0
leaving 2 subgoals.
The subproof is completed by applying SNo_1.
Apply minus_SNo_0 with
λ x1 x2 .
SNo_pair
(
add_SNo
0
x2
)
(
add_SNo
(
mul_SNo
0
0
)
(
mul_SNo
1
x0
)
)
=
SNo_pair
0
x0
.
Apply mul_SNo_zeroL with
0
,
λ x1 x2 .
SNo_pair
(
add_SNo
0
0
)
(
add_SNo
x2
(
mul_SNo
1
x0
)
)
=
SNo_pair
0
x0
leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply mul_SNo_oneL with
x0
,
λ x1 x2 .
SNo_pair
(
add_SNo
0
0
)
(
add_SNo
0
x2
)
=
SNo_pair
0
x0
leaving 2 subgoals.
Apply real_SNo with
x0
.
The subproof is completed by applying H0.
Apply add_SNo_0L with
0
,
λ x1 x2 .
SNo_pair
x2
(
add_SNo
0
x0
)
=
SNo_pair
0
x0
leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply add_SNo_0L with
x0
,
λ x1 x2 .
SNo_pair
0
x2
=
SNo_pair
0
x0
leaving 2 subgoals.
Apply real_SNo with
x0
.
The subproof is completed by applying H0.
set y1 to be
SNo_pair
0
x0
Let x2 of type
ι
→
ι
→
ο
be given.
Assume H1:
x2
y1
y1
.
The subproof is completed by applying H1.
■