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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Assume H0:
368eb..
x0
.
Apply H0 with
368eb..
x1
⟶
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
368eb..
(
add_SNo
x0
x1
)
.
Let x2 of type
ι
be given.
Assume H1:
(
λ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
or
(
x0
=
mul_SNo
(
eps_
x3
)
x4
)
(
x0
=
minus_SNo
(
mul_SNo
(
eps_
x3
)
x4
)
)
)
)
)
x2
.
Apply H1 with
368eb..
x1
⟶
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
368eb..
(
add_SNo
x0
x1
)
.
Assume H2:
x2
∈
omega
.
Claim L3:
...
...
Assume H4:
∃ x3 .
and
(
x3
∈
omega
)
(
or
(
x0
=
mul_SNo
(
eps_
x2
)
x3
)
(
x0
=
minus_SNo
(
mul_SNo
(
eps_
x2
)
x3
)
)
)
.
Apply H4 with
368eb..
x1
⟶
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
368eb..
(
add_SNo
x0
x1
)
.
Let x3 of type
ι
be given.
Assume H5:
(
λ x4 .
and
(
x4
∈
omega
)
(
or
(
x0
=
mul_SNo
(
eps_
x2
)
x4
)
(
x0
=
minus_SNo
(
mul_SNo
(
eps_
x2
)
x4
)
)
)
)
x3
.
Apply H5 with
368eb..
x1
⟶
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
368eb..
(
add_SNo
x0
x1
)
.
Assume H6:
x3
∈
omega
.
Assume H7:
or
(
x0
=
mul_SNo
(
eps_
x2
)
x3
)
(
x0
=
minus_SNo
(
mul_SNo
(
eps_
x2
)
x3
)
)
.
Apply H7 with
368eb..
x1
⟶
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
368eb..
(
add_SNo
x0
x1
)
leaving 2 subgoals.
Assume H8:
x0
=
mul_SNo
(
eps_
x2
)
x3
.
Assume H9:
368eb..
x1
.
Apply H9 with
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
368eb..
(
add_SNo
x0
x1
)
.
Let x4 of type
ι
be given.
Assume H10:
(
λ x5 .
and
(
x5
∈
omega
)
(
∃ x6 .
and
(
x6
∈
omega
)
(
or
(
x1
=
mul_SNo
(
eps_
x5
)
x6
)
(
x1
=
minus_SNo
(
mul_SNo
(
eps_
x5
)
x6
)
)
)
)
)
x4
.
Apply H10 with
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
368eb..
(
add_SNo
x0
x1
)
.
Assume H11:
x4
∈
omega
.
Claim L12:
...
...
Assume H13:
∃ x5 .
and
(
x5
∈
omega
)
(
or
(
x1
=
mul_SNo
(
eps_
x4
)
x5
)
(
x1
=
minus_SNo
(
mul_SNo
(
eps_
x4
)
x5
)
)
)
.
Apply H13 with
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
368eb..
(
add_SNo
x0
x1
)
.
Let x5 of type
ι
be given.
Assume H14:
(
λ x6 .
and
(
x6
∈
omega
)
(
or
(
x1
=
mul_SNo
(
eps_
x4
)
x6
)
(
x1
=
minus_SNo
(
mul_SNo
(
eps_
x4
)
x6
)
)
)
)
x5
.
Apply H14 with
...
⟶
...
⟶
368eb..
(
add_SNo
...
...
)
.
...
...
■