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Proofgold Proof
pf
Apply SNoLev_ind with
λ x0 .
SNoLt
0
x0
⟶
and
(
SNo
(
recip_SNo_pos
x0
)
)
(
mul_SNo
x0
(
recip_SNo_pos
x0
)
=
1
)
.
Let x0 of type
ι
be given.
Assume H0:
SNo
x0
.
Assume H1:
∀ x1 .
x1
∈
SNoS_
(
SNoLev
x0
)
⟶
SNoLt
0
x1
⟶
and
(
SNo
(
recip_SNo_pos
x1
)
)
(
mul_SNo
x1
(
recip_SNo_pos
x1
)
=
1
)
.
Assume H2:
SNoLt
0
x0
.
Apply recip_SNo_pos_eq with
x0
,
λ x1 x2 .
and
(
SNo
x2
)
(
mul_SNo
x0
x2
=
1
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Apply SNo_recipaux_lem2 with
x0
,
recip_SNo_pos
,
and
(
SNo
(
(
λ x1 .
λ x2 :
ι → ι
.
SNoCut
(
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x1
x2
x3
)
0
)
)
(
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x1
x2
x3
)
1
)
)
)
x0
recip_SNo_pos
)
)
(
mul_SNo
x0
(
(
λ x1 .
λ x2 :
ι → ι
.
SNoCut
(
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x1
x2
x3
)
0
)
)
(
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x1
x2
x3
)
1
)
)
)
x0
recip_SNo_pos
)
=
1
)
leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Assume H3:
and
(
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
ap
(
SNo_recipaux
x0
recip_SNo_pos
x2
)
0
)
⟶
SNo
x1
)
(
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
ap
(
SNo_recipaux
x0
recip_SNo_pos
x2
)
1
)
⟶
SNo
x1
)
.
Apply H3 with
(
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
ap
(
SNo_recipaux
x0
recip_SNo_pos
x2
)
0
)
⟶
∀ x2 .
x2
∈
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x0
recip_SNo_pos
x3
)
1
)
⟶
SNoLt
x1
x2
)
⟶
and
(
SNo
(
(
λ x1 .
λ x2 :
ι → ι
.
SNoCut
(
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x1
x2
x3
)
0
)
)
(
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x1
x2
x3
)
1
)
)
)
x0
recip_SNo_pos
)
)
(
mul_SNo
x0
(
(
λ x1 .
λ x2 :
ι → ι
.
SNoCut
(
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x1
x2
x3
)
0
)
)
(
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x1
x2
x3
)
1
)
)
)
x0
recip_SNo_pos
)
=
1
)
.
Assume H4:
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
ap
(
SNo_recipaux
x0
recip_SNo_pos
x2
)
0
)
⟶
SNo
x1
.
Assume H5:
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
ap
(
SNo_recipaux
x0
recip_SNo_pos
x2
)
1
)
⟶
SNo
x1
.
Assume H6:
∀ x1 .
...
⟶
∀ x2 .
x2
∈
famunion
omega
(
λ x3 .
ap
(
SNo_recipaux
x0
recip_SNo_pos
x3
)
1
)
⟶
SNoLt
x1
x2
.
...
■