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Proofgold Asset
asset id
545c21ba88d68530c4d19e674d5b4d652709311459c6675973881a2e8d5c63bb
asset hash
24972ba760ba61cb9853c9383e2db665aefd1b16678bd922564d0249b8bef1d1
bday / block
36778
tx
befe5..
preasset
doc published by
Pr4zB..
Param
4402e..
:
ι
→
(
ι
→
ι
→
ο
) →
ο
Param
cf2df..
:
ι
→
(
ι
→
ι
→
ο
) →
ο
Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Param
setminus
setminus
:
ι
→
ι
→
ι
Param
Sing
Sing
:
ι
→
ι
Definition
False
False
:=
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Definition
2f869..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 .
∀ x5 : ο .
(
(
x1
=
x2
⟶
∀ x6 : ο .
x6
)
⟶
(
x1
=
x3
⟶
∀ x6 : ο .
x6
)
⟶
(
x2
=
x3
⟶
∀ x6 : ο .
x6
)
⟶
(
x1
=
x4
⟶
∀ x6 : ο .
x6
)
⟶
(
x2
=
x4
⟶
∀ x6 : ο .
x6
)
⟶
(
x3
=
x4
⟶
∀ x6 : ο .
x6
)
⟶
not
(
x0
x1
x2
)
⟶
not
(
x0
x1
x3
)
⟶
not
(
x0
x2
x3
)
⟶
not
(
x0
x1
x4
)
⟶
not
(
x0
x2
x4
)
⟶
x0
x3
x4
⟶
x5
)
⟶
x5
Definition
87c36..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 .
∀ x6 : ο .
(
2f869..
x0
x1
x2
x3
x4
⟶
(
x1
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x2
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x3
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x4
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
not
(
x0
x1
x5
)
⟶
x0
x2
x5
⟶
not
(
x0
x3
x5
)
⟶
x0
x4
x5
⟶
x6
)
⟶
x6
Definition
6648a..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 .
∀ x7 : ο .
(
87c36..
x0
x1
x2
x3
x4
x5
⟶
(
x1
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x2
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x3
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x4
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x5
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
not
(
x0
x1
x6
)
⟶
x0
x2
x6
⟶
x0
x3
x6
⟶
not
(
x0
x4
x6
)
⟶
not
(
x0
x5
x6
)
⟶
x7
)
⟶
x7
Definition
c9184..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
6648a..
x0
x1
x2
x3
x4
x5
x6
⟶
(
x1
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x2
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x3
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x4
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x5
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x6
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
x0
x1
x7
⟶
not
(
x0
x2
x7
)
⟶
not
(
x0
x3
x7
)
⟶
not
(
x0
x4
x7
)
⟶
not
(
x0
x5
x7
)
⟶
not
(
x0
x6
x7
)
⟶
x8
)
⟶
x8
Definition
389f9..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
c9184..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
not
(
x0
x1
x8
)
⟶
not
(
x0
x2
x8
)
⟶
not
(
x0
x3
x8
)
⟶
not
(
x0
x4
x8
)
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
df271..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
6648a..
x0
x1
x2
x3
x4
x5
x6
⟶
(
x1
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x2
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x3
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x4
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x5
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x6
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
x0
x1
x7
⟶
not
(
x0
x2
x7
)
⟶
not
(
x0
x3
x7
)
⟶
not
(
x0
x4
x7
)
⟶
not
(
x0
x5
x7
)
⟶
x0
x6
x7
⟶
x8
)
⟶
x8
Definition
279d8..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
df271..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
x0
x1
x8
⟶
not
(
x0
x2
x8
)
⟶
not
(
x0
x3
x8
)
⟶
not
(
x0
x4
x8
)
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
836ee..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
6648a..
x0
x1
x2
x3
x4
x5
x6
⟶
(
x1
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x2
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x3
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x4
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x5
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x6
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
x0
x1
x7
⟶
not
(
x0
x2
x7
)
⟶
not
(
x0
x3
x7
)
⟶
not
(
x0
x4
x7
)
⟶
x0
x5
x7
⟶
x0
x6
x7
⟶
x8
)
⟶
x8
Definition
725c8..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
836ee..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
x0
x1
x8
⟶
not
(
x0
x2
x8
)
⟶
not
(
x0
x3
x8
)
⟶
not
(
x0
x4
x8
)
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
3f579..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
c9184..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
not
(
x0
x1
x8
)
⟶
not
(
x0
x2
x8
)
⟶
not
(
x0
x3
x8
)
⟶
not
(
x0
x4
x8
)
⟶
not
(
x0
x5
x8
)
⟶
x0
x6
x8
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
24aad..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
c9184..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
not
(
x0
x1
x8
)
⟶
not
(
x0
x2
x8
)
⟶
not
(
x0
x3
x8
)
⟶
not
(
x0
x4
x8
)
⟶
x0
x5
x8
⟶
x0
x6
x8
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
810e7..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
c9184..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
not
(
x0
x1
x8
)
⟶
not
(
x0
x2
x8
)
⟶
not
(
x0
x3
x8
)
⟶
not
(
x0
x4
x8
)
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
x0
x7
x8
⟶
x9
)
⟶
x9
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Definition
nIn
nIn
:=
λ x0 x1 .
not
(
x0
∈
x1
)
Known
setminusE
setminusE
:
∀ x0 x1 x2 .
x2
∈
setminus
x0
x1
⟶
and
(
x2
∈
x0
)
(
nIn
x2
x1
)
Known
37eb7..
:
∀ x0 x1 .
∀ x2 :
ι →
ι → ο
.
(
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
x2
x3
x4
⟶
x2
x4
x3
)
⟶
4402e..
x1
x2
⟶
cf2df..
x1
x2
⟶
∀ x3 .
x3
∈
x1
⟶
x0
⊆
setminus
x1
(
Sing
x3
)
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
∀ x9 .
x9
∈
x0
⟶
∀ x10 .
x10
∈
x0
⟶
c9184..
x2
x4
x5
x6
x7
x8
x9
x10
⟶
∀ x11 : ο .
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
x2
x10
x3
⟶
x11
)
⟶
(
not
(
x2
x4
x3
)
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
x2
x9
x3
⟶
x2
x10
x3
⟶
x11
)
⟶
x11
Known
13473..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
c9184..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
c9184..
x1
x2
x7
x6
x5
x4
x3
x8
Known
268a2..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
c9184..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
c9184..
x1
x8
x3
x4
x5
x6
x7
x2
Known
e5d54..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
c9184..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
c9184..
x1
x2
x3
x5
x4
x7
x6
x8
Known
d4057..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
6648a..
x1
x2
x3
x4
x5
x6
x7
⟶
6648a..
x1
x2
x7
x6
x5
x4
x3
Known
de43b..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
6648a..
x1
x2
x3
x4
x5
x6
x7
⟶
6648a..
x1
x2
x3
x5
x4
x7
x6
Known
neq_i_sym
neq_i_sym
:
∀ x0 x1 .
(
x0
=
x1
⟶
∀ x2 : ο .
x2
)
⟶
x1
=
x0
⟶
∀ x2 : ο .
x2
Known
a91b6..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
c9184..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
c9184..
x1
x2
x5
x7
x3
x6
x4
x8
Known
4c77b..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
6648a..
x1
x2
x3
x4
x5
x6
x7
⟶
6648a..
x1
x2
x5
x7
x3
x6
x4
Known
6698d..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
c9184..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
c9184..
x1
x2
x4
x3
x6
x5
x7
x8
Known
a1242..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
6648a..
x1
x2
x3
x4
x5
x6
x7
⟶
6648a..
x1
x2
x4
x3
x6
x5
x7
Known
5be00..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
c9184..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
c9184..
x1
x2
x6
x4
x7
x3
x5
x8
Known
a246a..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
6648a..
x1
x2
x3
x4
x5
x6
x7
⟶
6648a..
x1
x2
x6
x4
x7
x3
x5
Known
Subq_tra
Subq_tra
:
∀ x0 x1 x2 .
x0
⊆
x1
⟶
x1
⊆
x2
⟶
x0
⊆
x2
Known
setminus_Subq
setminus_Subq
:
∀ x0 x1 .
setminus
x0
x1
⊆
x0
Known
SingI
SingI
:
∀ x0 .
x0
∈
Sing
x0
Theorem
9c256..
:
∀ x0 x1 .
∀ x2 :
ι →
ι → ο
.
(
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
x2
x3
x4
⟶
x2
x4
x3
)
⟶
4402e..
x1
x2
⟶
cf2df..
x1
x2
⟶
∀ x3 .
x3
∈
x1
⟶
x0
⊆
setminus
x1
(
Sing
x3
)
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
∀ x9 .
x9
∈
x0
⟶
∀ x10 .
x10
∈
x0
⟶
c9184..
x2
x4
x5
x6
x7
x8
x9
x10
⟶
∀ x11 : ο .
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
389f9..
x2
x12
x13
x14
x15
x16
x17
x18
x3
⟶
x11
)
⟶
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
279d8..
x2
x12
x13
x14
x15
x16
x17
x3
x18
⟶
x11
)
⟶
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
725c8..
x2
x12
x13
x14
x15
x16
x17
x3
x18
⟶
x11
)
⟶
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
3f579..
x2
x12
x13
x14
x15
x16
x17
x18
x3
⟶
x11
)
⟶
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
24aad..
x2
x12
x13
x14
x15
x16
x17
x18
x3
⟶
x11
)
⟶
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
810e7..
x2
x12
x13
x14
x15
x16
x17
x18
x3
⟶
x11
)
⟶
x11
(proof)