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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
SNo
x0
.
Assume H1:
∀ x1 .
x1
∈
omega
⟶
∃ x2 .
and
(
x2
∈
SNoS_
omega
)
(
and
(
SNoLt
x2
x0
)
(
SNoLt
x0
(
add_SNo
x2
(
eps_
x1
)
)
)
)
.
Let x1 of type
ι
be given.
Assume H2:
x1
∈
omega
.
Apply H1 with
x1
,
∃ x2 .
and
(
x2
∈
SNoS_
omega
)
(
and
(
SNoLt
x2
(
minus_SNo
x0
)
)
(
SNoLt
(
minus_SNo
x0
)
(
add_SNo
x2
(
eps_
x1
)
)
)
)
leaving 2 subgoals.
The subproof is completed by applying H2.
Let x2 of type
ι
be given.
Assume H3:
(
λ x3 .
and
(
x3
∈
SNoS_
omega
)
(
and
(
SNoLt
x3
x0
)
(
SNoLt
x0
(
add_SNo
x3
(
eps_
x1
)
)
)
)
)
x2
.
Apply H3 with
∃ x3 .
and
(
x3
∈
SNoS_
omega
)
(
and
(
SNoLt
x3
(
minus_SNo
x0
)
)
(
SNoLt
(
minus_SNo
x0
)
(
add_SNo
x3
(
eps_
x1
)
)
)
)
.
Assume H4:
x2
∈
SNoS_
omega
.
Assume H5:
and
(
SNoLt
x2
x0
)
(
SNoLt
x0
(
add_SNo
x2
(
eps_
x1
)
)
)
.
Apply H5 with
∃ x3 .
and
(
x3
∈
SNoS_
omega
)
(
and
(
SNoLt
x3
(
minus_SNo
x0
)
)
(
SNoLt
(
minus_SNo
x0
)
(
add_SNo
x3
(
eps_
x1
)
)
)
)
.
Assume H6:
SNoLt
x2
x0
.
Assume H7:
SNoLt
x0
(
add_SNo
x2
(
eps_
x1
)
)
.
Apply SNoS_E2 with
omega
,
x2
,
∃ x3 .
and
(
x3
∈
SNoS_
omega
)
(
and
(
SNoLt
x3
(
minus_SNo
x0
)
)
(
SNoLt
(
minus_SNo
x0
)
(
add_SNo
x3
(
eps_
x1
)
)
)
)
leaving 3 subgoals.
The subproof is completed by applying omega_ordinal.
The subproof is completed by applying H4.
Assume H8:
SNoLev
x2
∈
omega
.
Assume H9:
ordinal
(
SNoLev
x2
)
.
Assume H10:
SNo
x2
.
Assume H11:
SNo_
(
SNoLev
x2
)
x2
.
Let x3 of type
ο
be given.
Assume H12:
∀ x4 .
and
(
x4
∈
SNoS_
omega
)
(
and
(
SNoLt
x4
(
minus_SNo
x0
)
)
(
SNoLt
(
minus_SNo
x0
)
(
add_SNo
x4
(
eps_
x1
)
)
)
)
⟶
x3
.
Apply H12 with
minus_SNo
(
add_SNo
x2
(
eps_
x1
)
)
.
Apply andI with
minus_SNo
(
add_SNo
x2
(
eps_
x1
)
)
∈
SNoS_
omega
,
and
(
SNoLt
(
minus_SNo
(
add_SNo
x2
(
eps_
x1
)
)
)
(
minus_SNo
x0
)
)
(
SNoLt
(
minus_SNo
x0
)
(
add_SNo
(
minus_SNo
(
add_SNo
x2
(
eps_
x1
)
)
)
(
eps_
x1
)
)
)
leaving 2 subgoals.
Apply minus_SNo_SNoS_omega with
add_SNo
x2
(
eps_
x1
)
.
Apply add_SNo_SNoS_omega with
x2
,
eps_
x1
leaving 2 subgoals.
The subproof is completed by applying H4.
Apply SNo_eps_SNoS_omega with
x1
.
The subproof is completed by applying H2.
Apply andI with
SNoLt
(
minus_SNo
(
add_SNo
x2
(
eps_
x1
)
)
)
(
minus_SNo
x0
)
,
SNoLt
(
minus_SNo
x0
)
(
add_SNo
(
minus_SNo
(
add_SNo
x2
(
eps_
x1
)
)
)
(
eps_
x1
)
)
leaving 2 subgoals.
Apply minus_SNo_Lt_contra with
x0
,
add_SNo
x2
(
eps_
x1
)
leaving 3 subgoals.
The subproof is completed by applying H0.
Apply SNo_add_SNo with
x2
,
eps_
x1
leaving 2 subgoals.
The subproof is completed by applying H10.
Apply SNo_eps_ with
x1
.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
Apply minus_add_SNo_distr with
x2
,
eps_
x1
,
λ x4 x5 .
SNoLt
(
minus_SNo
x0
)
...
leaving 3 subgoals.
...
...
...
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