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Proofgold Asset
asset id
2edb8445bb6a7e6db9cc669ba0ba34f43f9f23678e6742de51f58b9b50ab1fbb
asset hash
101d2f90ad2b73ffdc2299fbbb216314b046fe29cce1df3f2f09dcc98f33f5fb
bday / block
34278
tx
d3abf..
preasset
doc published by
Pr4zB..
Definition
False
False
:=
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Definition
8b6ad..
:=
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
)
⟶
not
(
x0
x3
x4
)
⟶
x5
)
⟶
x5
Definition
c5756..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 .
∀ x6 : ο .
(
8b6ad..
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
)
⟶
not
(
x0
x2
x5
)
⟶
x0
x3
x5
⟶
x0
x4
x5
⟶
x6
)
⟶
x6
Definition
2de86..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 .
∀ x7 : ο .
(
c5756..
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
⟶
not
(
x0
x3
x6
)
⟶
x0
x4
x6
⟶
not
(
x0
x5
x6
)
⟶
x7
)
⟶
x7
Definition
36d58..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
2de86..
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
)
⟶
x0
x3
x7
⟶
x0
x4
x7
⟶
not
(
x0
x5
x7
)
⟶
not
(
x0
x6
x7
)
⟶
x8
)
⟶
x8
Definition
af16d..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
36d58..
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
)
⟶
x0
x4
x8
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
34ae8..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 x9 .
∀ x10 : ο .
(
af16d..
x0
x1
x2
x3
x4
x5
x6
x7
x8
⟶
(
x1
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x2
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x3
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x4
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x5
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x6
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x7
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
(
x8
=
x9
⟶
∀ x11 : ο .
x11
)
⟶
x0
x1
x9
⟶
x0
x2
x9
⟶
x0
x3
x9
⟶
not
(
x0
x4
x9
)
⟶
not
(
x0
x5
x9
)
⟶
not
(
x0
x6
x9
)
⟶
not
(
x0
x7
x9
)
⟶
x0
x8
x9
⟶
x10
)
⟶
x10
Known
neq_i_sym
neq_i_sym
:
∀ x0 x1 .
(
x0
=
x1
⟶
∀ x2 : ο .
x2
)
⟶
x1
=
x0
⟶
∀ x2 : ο .
x2
Theorem
1e1a7..
:
∀ 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
⟶
∀ x9 .
x9
∈
x0
⟶
∀ x10 .
x10
∈
x0
⟶
34ae8..
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
⟶
34ae8..
x1
x6
x7
x8
x10
x2
x3
x4
x9
x5
(proof)