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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Assume H0:
diadic_rational_p
x0
.
Apply H0 with
diadic_rational_p
x1
⟶
diadic_rational_p
(
add_SNo
x0
x1
)
.
Let x2 of type
ι
be given.
Assume H1:
(
λ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
int
)
(
x0
=
mul_SNo
(
eps_
x3
)
x4
)
)
)
x2
.
Apply H1 with
diadic_rational_p
x1
⟶
diadic_rational_p
(
add_SNo
x0
x1
)
.
Assume H2:
x2
∈
omega
.
Claim L3:
...
...
Assume H4:
∃ x3 .
and
(
x3
∈
int
)
(
x0
=
mul_SNo
(
eps_
x2
)
x3
)
.
Apply H4 with
diadic_rational_p
x1
⟶
diadic_rational_p
(
add_SNo
x0
x1
)
.
Let x3 of type
ι
be given.
Assume H5:
(
λ x4 .
and
(
x4
∈
int
)
(
x0
=
mul_SNo
(
eps_
x2
)
x4
)
)
x3
.
Apply H5 with
diadic_rational_p
x1
⟶
diadic_rational_p
(
add_SNo
x0
x1
)
.
Assume H6:
x3
∈
int
.
Assume H7:
x0
=
mul_SNo
(
eps_
x2
)
x3
.
Assume H8:
diadic_rational_p
x1
.
Apply H8 with
diadic_rational_p
(
add_SNo
x0
x1
)
.
Let x4 of type
ι
be given.
Assume H9:
(
λ x5 .
and
(
x5
∈
omega
)
(
∃ x6 .
and
(
x6
∈
int
)
(
x1
=
mul_SNo
(
eps_
x5
)
x6
)
)
)
x4
.
Apply H9 with
diadic_rational_p
(
add_SNo
x0
x1
)
.
Assume H10:
x4
∈
omega
.
Claim L11:
...
...
Assume H12:
∃ x5 .
and
(
x5
∈
int
)
(
x1
=
mul_SNo
(
eps_
x4
)
x5
)
.
Apply H12 with
diadic_rational_p
(
add_SNo
x0
x1
)
.
Let x5 of type
ι
be given.
Assume H13:
(
λ x6 .
and
(
x6
∈
int
)
(
x1
=
mul_SNo
(
eps_
x4
)
x6
)
)
x5
.
Apply H13 with
diadic_rational_p
(
add_SNo
x0
x1
)
.
Assume H14:
x5
∈
int
.
Assume H15:
x1
=
mul_SNo
(
eps_
x4
)
x5
.
Let x6 of type
ο
be given.
Assume H16:
∀ x7 .
and
(
x7
∈
omega
)
(
∃ x8 .
and
(
x8
∈
int
)
(
add_SNo
x0
x1
=
mul_SNo
(
eps_
x7
)
x8
)
)
⟶
x6
.
Apply H16 with
add_SNo
x2
x4
.
Apply andI with
add_SNo
x2
x4
∈
omega
,
∃ x7 .
and
(
x7
∈
int
)
(
add_SNo
x0
x1
=
mul_SNo
(
eps_
(
add_SNo
x2
x4
)
)
x7
)
leaving 2 subgoals.
Apply add_SNo_In_omega with
x2
,
x4
leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H10.
Let x7 of type
ο
be given.
Assume H17:
∀ x8 .
and
(
x8
∈
int
)
(
add_SNo
x0
x1
=
mul_SNo
(
eps_
(
add_SNo
x2
x4
)
)
x8
)
⟶
x7
.
Apply H17 with
add_SNo
(
mul_SNo
(
exp_SNo_nat
2
x4
)
x3
)
(
mul_SNo
(
exp_SNo_nat
2
x2
)
x5
)
.
Claim L18:
...
...
Claim L19:
...
...
Claim L20:
...
...
Apply andI with
add_SNo
(
mul_SNo
(
exp_SNo_nat
2
x4
)
x3
)
(
mul_SNo
(
exp_SNo_nat
2
x2
)
x5
)
∈
int
,
add_SNo
x0
x1
=
mul_SNo
(
eps_
(
add_SNo
x2
x4
)
)
(
add_SNo
(
mul_SNo
(
exp_SNo_nat
2
x4
)
x3
)
(
mul_SNo
(
exp_SNo_nat
2
x2
)
x5
)
)
leaving 2 subgoals.
Apply int_add_SNo with
mul_SNo
...
...
,
...
leaving 2 subgoals.
...
...
...
■