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Proofgold Proof
pf
Assume H0:
∀ x0 :
ι → ο
.
∀ x1 :
ι → ι
.
and
(
(
x0
(
x1
(
binintersect
(
x1
(
Power
0
)
)
(
x1
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
0
)
)
)
)
⟶
TransSet
(
x1
0
)
⟶
∀ x2 .
In
x2
0
⟶
∃ x3 .
and
(
Subq
x3
x2
)
(
exactly2
(
ordsucc
(
x1
x3
)
)
)
)
⟶
∃ x2 .
and
(
∃ x3 .
and
(
Subq
x3
x2
)
(
x2
=
x1
x2
⟶
∃ x4 .
and
(
Subq
x4
x2
)
(
not
(
x0
(
setprod
x3
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
)
)
)
)
)
(
not
(
SNo
x2
)
)
)
(
not
(
x0
(
x1
(
x1
(
x1
(
binrep
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
0
)
)
0
)
)
)
)
)
)
.
Apply and_notTrue with
(
True
⟶
TransSet
0
⟶
∀ x0 .
In
x0
0
⟶
∃ x1 .
and
(
Subq
x1
x0
)
(
exactly2
1
)
)
⟶
∃ x0 .
and
(
∃ x1 .
and
(
Subq
x1
x0
)
(
x0
=
0
⟶
∃ x2 .
and
(
Subq
x2
x0
)
(
not
True
)
)
)
(
not
(
SNo
x0
)
)
.
The subproof is completed by applying H0 with
λ x0 .
True
,
λ x0 .
0
.
■