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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
368eb..
x0
.
Apply H0 with
368eb..
(
minus_SNo
x0
)
.
Let x1 of type
ι
be given.
Assume H1:
(
λ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
or
(
x0
=
mul_SNo
(
eps_
x2
)
x3
)
(
x0
=
minus_SNo
(
mul_SNo
(
eps_
x2
)
x3
)
)
)
)
)
x1
.
Apply H1 with
368eb..
(
minus_SNo
x0
)
.
Assume H2:
x1
∈
omega
.
Assume H3:
∃ x2 .
and
(
x2
∈
omega
)
(
or
(
x0
=
mul_SNo
(
eps_
x1
)
x2
)
(
x0
=
minus_SNo
(
mul_SNo
(
eps_
x1
)
x2
)
)
)
.
Apply H3 with
368eb..
(
minus_SNo
x0
)
.
Let x2 of type
ι
be given.
Assume H4:
(
λ x3 .
and
(
x3
∈
omega
)
(
or
(
x0
=
mul_SNo
(
eps_
x1
)
x3
)
(
x0
=
minus_SNo
(
mul_SNo
(
eps_
x1
)
x3
)
)
)
)
x2
.
Apply H4 with
368eb..
(
minus_SNo
x0
)
.
Assume H5:
x2
∈
omega
.
Assume H6:
or
(
x0
=
mul_SNo
(
eps_
x1
)
x2
)
(
x0
=
minus_SNo
(
mul_SNo
(
eps_
x1
)
x2
)
)
.
Apply H6 with
368eb..
(
minus_SNo
x0
)
leaving 2 subgoals.
Assume H7:
x0
=
mul_SNo
(
eps_
x1
)
x2
.
Let x3 of type
ο
be given.
Assume H8:
∀ x4 .
and
(
x4
∈
omega
)
(
∃ x5 .
and
(
x5
∈
omega
)
(
or
(
minus_SNo
x0
=
mul_SNo
(
eps_
x4
)
x5
)
(
minus_SNo
x0
=
minus_SNo
(
mul_SNo
(
eps_
x4
)
x5
)
)
)
)
⟶
x3
.
Apply H8 with
x1
.
Apply andI with
x1
∈
omega
,
∃ x4 .
and
(
x4
∈
omega
)
(
or
(
minus_SNo
x0
=
mul_SNo
(
eps_
x1
)
x4
)
(
minus_SNo
x0
=
minus_SNo
(
mul_SNo
(
eps_
x1
)
x4
)
)
)
leaving 2 subgoals.
The subproof is completed by applying H2.
Let x4 of type
ο
be given.
Assume H9:
∀ x5 .
and
(
x5
∈
omega
)
(
or
(
minus_SNo
x0
=
mul_SNo
(
eps_
x1
)
x5
)
(
minus_SNo
x0
=
minus_SNo
(
mul_SNo
(
eps_
x1
)
x5
)
)
)
⟶
x4
.
Apply H9 with
x2
.
Apply andI with
x2
∈
omega
,
or
(
minus_SNo
x0
=
mul_SNo
(
eps_
x1
)
x2
)
(
minus_SNo
x0
=
minus_SNo
(
mul_SNo
(
eps_
x1
)
x2
)
)
leaving 2 subgoals.
The subproof is completed by applying H5.
Apply orIR with
minus_SNo
x0
=
mul_SNo
(
eps_
x1
)
x2
,
minus_SNo
x0
=
minus_SNo
(
mul_SNo
(
eps_
x1
)
x2
)
.
set y5 to be
minus_SNo
(
mul_SNo
(
eps_
x1
)
x2
)
Claim L10:
∀ x6 :
ι → ο
.
x6
y5
⟶
x6
(
minus_SNo
x0
)
Let x6 of type
ι
→
ο
be given.
The subproof is completed by applying H7 with
λ x7 x8 .
(
λ x9 .
x6
)
(
minus_SNo
x7
)
(
minus_SNo
x8
)
.
Let x6 of type
ι
→
ι
→
ο
be given.
Apply L10 with
λ x7 .
x6
x7
y5
⟶
x6
y5
x7
.
Assume H11:
x6
y5
y5
.
The subproof is completed by applying H11.
Assume H7:
x0
=
minus_SNo
(
mul_SNo
(
eps_
x1
)
x2
)
.
...
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