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Proofgold Asset
asset id
554fff60b16c7dcbd6a4afb8c4d8f2369b2a6554bf32618d68e4af96219cdbf7
asset hash
136ff489f7b5b4e707e786187474f9408000b5a64cb6eba601b6fe6869ddb683
bday / block
48179
tx
68d3b..
preasset
doc published by
PrGM6..
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
5a3b5..
:=
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
)
⟶
not
(
x0
x4
x5
)
⟶
x6
)
⟶
x6
Definition
00e19..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 .
∀ x7 : ο .
(
5a3b5..
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
)
⟶
x0
x1
x6
⟶
not
(
x0
x2
x6
)
⟶
not
(
x0
x3
x6
)
⟶
not
(
x0
x4
x6
)
⟶
not
(
x0
x5
x6
)
⟶
x7
)
⟶
x7
Definition
455db..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 .
∀ x7 : ο .
(
5a3b5..
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
)
⟶
x0
x1
x6
⟶
not
(
x0
x2
x6
)
⟶
not
(
x0
x3
x6
)
⟶
x0
x4
x6
⟶
x0
x5
x6
⟶
x7
)
⟶
x7
Definition
70d65..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
455db..
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
⟶
x0
x2
x7
⟶
x0
x3
x7
⟶
not
(
x0
x4
x7
)
⟶
not
(
x0
x5
x7
)
⟶
not
(
x0
x6
x7
)
⟶
x8
)
⟶
x8
Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Param
atleastp
atleastp
:
ι
→
ι
→
ο
Definition
cdfa5..
:=
λ x0 x1 .
λ x2 :
ι →
ι → ο
.
∀ x3 .
x3
⊆
x1
⟶
atleastp
x0
x3
⟶
not
(
∀ x4 .
x4
∈
x3
⟶
∀ x5 .
x5
∈
x3
⟶
(
x4
=
x5
⟶
∀ x6 : ο .
x6
)
⟶
x2
x4
x5
)
Param
u4
:
ι
Definition
86706..
:=
cdfa5..
u4
Definition
35fb6..
:=
λ x0 .
λ x1 :
ι →
ι → ο
.
86706..
x0
(
λ x2 x3 .
not
(
x1
x2
x3
)
)
Param
SetAdjoin
SetAdjoin
:
ι
→
ι
→
ι
Param
UPair
UPair
:
ι
→
ι
→
ι
Definition
or
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Known
xm
xm
:
∀ x0 : ο .
or
x0
(
not
x0
)
Known
dneg
dneg
:
∀ x0 : ο .
not
(
not
x0
)
⟶
x0
Param
equip
equip
:
ι
→
ι
→
ο
Known
equip_atleastp
equip_atleastp
:
∀ x0 x1 .
equip
x0
x1
⟶
atleastp
x0
x1
Known
7204a..
:
∀ x0 x1 x2 x3 .
(
x0
=
x1
⟶
∀ x4 : ο .
x4
)
⟶
(
x0
=
x2
⟶
∀ x4 : ο .
x4
)
⟶
(
x1
=
x2
⟶
∀ x4 : ο .
x4
)
⟶
(
x0
=
x3
⟶
∀ x4 : ο .
x4
)
⟶
(
x1
=
x3
⟶
∀ x4 : ο .
x4
)
⟶
(
x2
=
x3
⟶
∀ x4 : ο .
x4
)
⟶
equip
u4
(
SetAdjoin
(
SetAdjoin
(
UPair
x0
x1
)
x2
)
x3
)
Known
58c12..
:
∀ x0 :
ι →
ι → ο
.
∀ x1 x2 x3 x4 .
x0
x1
x2
⟶
x0
x1
x3
⟶
x0
x1
x4
⟶
x0
x2
x3
⟶
x0
x2
x4
⟶
x0
x3
x4
⟶
(
∀ x5 .
x5
∈
SetAdjoin
(
SetAdjoin
(
UPair
x1
x2
)
x3
)
x4
⟶
∀ x6 .
x6
∈
SetAdjoin
(
SetAdjoin
(
UPair
x1
x2
)
x3
)
x4
⟶
x0
x5
x6
⟶
x0
x6
x5
)
⟶
∀ x5 .
x5
∈
SetAdjoin
(
SetAdjoin
(
UPair
x1
x2
)
x3
)
x4
⟶
∀ x6 .
x6
∈
SetAdjoin
(
SetAdjoin
(
UPair
x1
x2
)
x3
)
x4
⟶
(
x5
=
x6
⟶
∀ x7 : ο .
x7
)
⟶
x0
x5
x6
Known
c88f0..
:
∀ x0 x1 .
x1
∈
x0
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
SetAdjoin
(
SetAdjoin
(
UPair
x1
x2
)
x3
)
x4
⊆
x0
Theorem
35fcd..
:
∀ x0 :
ι →
ι → ο
.
∀ x1 x2 .
x2
∈
x1
⟶
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
∀ x5 .
x5
∈
x1
⟶
∀ x6 .
x6
∈
x1
⟶
∀ x7 .
x7
∈
x1
⟶
∀ x8 .
x8
∈
x1
⟶
∀ x9 .
x9
∈
x1
⟶
∀ x10 .
x10
∈
x1
⟶
∀ x11 .
x11
∈
x1
⟶
∀ x12 .
x12
∈
x1
⟶
∀ x13 .
x13
∈
x1
⟶
∀ x14 .
x14
∈
x1
⟶
(
∀ x15 .
x15
∈
x1
⟶
∀ x16 .
x16
∈
x1
⟶
x0
x15
x16
⟶
x0
x16
x15
)
⟶
(
x2
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
00e19..
x0
x2
x3
x4
x5
x6
x7
⟶
70d65..
(
λ x15 x16 .
not
(
x0
x15
x16
)
)
x8
x9
x10
x11
x12
x13
x14
⟶
86706..
x1
x0
⟶
35fb6..
x1
x0
⟶
(
not
(
x0
x2
x11
)
⟶
not
(
x0
x2
x12
)
⟶
not
(
x0
x2
x10
)
⟶
not
(
x0
x2
x9
)
⟶
x0
x3
x10
⟶
not
(
x0
x3
x9
)
⟶
False
)
⟶
(
x0
x6
x8
⟶
not
(
x0
x3
x8
)
⟶
False
)
⟶
(
x0
x6
x8
⟶
not
(
x0
x2
x8
)
⟶
False
)
⟶
(
x0
x5
x8
⟶
not
(
x0
x4
x8
)
⟶
False
)
⟶
(
x0
x5
x8
⟶
not
(
x0
x2
x8
)
⟶
False
)
⟶
(
x0
x4
x8
⟶
not
(
x0
x3
x8
)
⟶
False
)
⟶
(
x0
x2
x12
⟶
not
(
x0
x2
x9
)
⟶
False
)
⟶
(
x0
x2
x11
⟶
not
(
x0
x2
x9
)
⟶
False
)
⟶
(
x0
x2
x10
⟶
not
(
x0
x2
x9
)
⟶
False
)
⟶
False
...
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
122f9..
:
∀ 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
⟶
70d65..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
70d65..
x1
x2
x6
x5
x4
x3
x8
x7
Theorem
0e2b5..
:
∀ x0 :
ι →
ι → ο
.
∀ x1 x2 .
x2
∈
x1
⟶
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
∀ x5 .
x5
∈
x1
⟶
∀ x6 .
x6
∈
x1
⟶
∀ x7 .
x7
∈
x1
⟶
∀ x8 .
x8
∈
x1
⟶
∀ x9 .
x9
∈
x1
⟶
∀ x10 .
x10
∈
x1
⟶
∀ x11 .
x11
∈
x1
⟶
∀ x12 .
x12
∈
x1
⟶
∀ x13 .
x13
∈
x1
⟶
∀ x14 .
x14
∈
x1
⟶
(
∀ x15 .
x15
∈
x1
⟶
∀ x16 .
x16
∈
x1
⟶
x0
x15
x16
⟶
x0
x16
x15
)
⟶
(
x2
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
00e19..
x0
x2
x3
x4
x5
x6
x7
⟶
70d65..
(
λ x15 x16 .
not
(
x0
x15
x16
)
)
x8
x9
x10
x11
x12
x13
x14
⟶
86706..
x1
x0
⟶
35fb6..
x1
x0
⟶
(
not
(
x0
x2
x10
)
⟶
not
(
x0
x2
x9
)
⟶
not
(
x0
x2
x11
)
⟶
not
(
x0
x2
x12
)
⟶
x0
x3
x11
⟶
not
(
x0
x3
x12
)
⟶
False
)
⟶
(
x0
x2
x10
⟶
not
(
x0
x2
x12
)
⟶
False
)
⟶
(
x0
x2
x11
⟶
not
(
x0
x2
x12
)
⟶
False
)
⟶
(
x0
x6
x8
⟶
not
(
x0
x3
x8
)
⟶
False
)
⟶
(
x0
x6
x8
⟶
not
(
x0
x2
x8
)
⟶
False
)
⟶
(
x0
x5
x8
⟶
not
(
x0
x4
x8
)
⟶
False
)
⟶
(
x0
x5
x8
⟶
not
(
x0
x2
x8
)
⟶
False
)
⟶
(
x0
x4
x8
⟶
not
(
x0
x3
x8
)
⟶
False
)
⟶
False
...
Known
3ad8d..
:
∀ 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
⟶
70d65..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
70d65..
x1
x2
x5
x6
x3
x4
x8
x7
Theorem
aec09..
:
∀ x0 :
ι →
ι → ο
.
∀ x1 x2 .
x2
∈
x1
⟶
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
∀ x5 .
x5
∈
x1
⟶
∀ x6 .
x6
∈
x1
⟶
∀ x7 .
x7
∈
x1
⟶
∀ x8 .
x8
∈
x1
⟶
∀ x9 .
x9
∈
x1
⟶
∀ x10 .
x10
∈
x1
⟶
∀ x11 .
x11
∈
x1
⟶
∀ x12 .
x12
∈
x1
⟶
∀ x13 .
x13
∈
x1
⟶
∀ x14 .
x14
∈
x1
⟶
(
∀ x15 .
x15
∈
x1
⟶
∀ x16 .
x16
∈
x1
⟶
x0
x15
x16
⟶
x0
x16
x15
)
⟶
(
x2
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
00e19..
x0
x2
x3
x4
x5
x6
x7
⟶
70d65..
(
λ x15 x16 .
not
(
x0
x15
x16
)
)
x8
x9
x10
x11
x12
x13
x14
⟶
86706..
x1
x0
⟶
35fb6..
x1
x0
⟶
(
not
(
x0
x2
x12
)
⟶
not
(
x0
x2
x11
)
⟶
not
(
x0
x2
x9
)
⟶
not
(
x0
x2
x10
)
⟶
x0
x3
x9
⟶
not
(
x0
x3
x10
)
⟶
False
)
⟶
(
x0
x2
x9
⟶
not
(
x0
x2
x10
)
⟶
False
)
⟶
(
x0
x6
x8
⟶
not
(
x0
x3
x8
)
⟶
False
)
⟶
(
x0
x6
x8
⟶
not
(
x0
x2
x8
)
⟶
False
)
⟶
(
x0
x5
x8
⟶
not
(
x0
x4
x8
)
⟶
False
)
⟶
(
x0
x5
x8
⟶
not
(
x0
x2
x8
)
⟶
False
)
⟶
(
x0
x4
x8
⟶
not
(
x0
x3
x8
)
⟶
False
)
⟶
False
...
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Known
e99aa..
:
∀ 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
⟶
70d65..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
70d65..
x1
x2
x4
x3
x6
x5
x7
x8
Known
andI
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Theorem
382f7..
:
∀ x0 :
ι →
ι → ο
.
∀ x1 x2 .
x2
∈
x1
⟶
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
∀ x5 .
x5
∈
x1
⟶
∀ x6 .
x6
∈
x1
⟶
∀ x7 .
x7
∈
x1
⟶
∀ x8 .
x8
∈
x1
⟶
∀ x9 .
x9
∈
x1
⟶
∀ x10 .
x10
∈
x1
⟶
∀ x11 .
x11
∈
x1
⟶
∀ x12 .
x12
∈
x1
⟶
∀ x13 .
x13
∈
x1
⟶
∀ x14 .
x14
∈
x1
⟶
(
∀ x15 .
x15
∈
x1
⟶
∀ x16 .
x16
∈
x1
⟶
x0
x15
x16
⟶
x0
x16
x15
)
⟶
(
x2
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
00e19..
x0
x2
x3
x4
x5
x6
x7
⟶
70d65..
(
λ x15 x16 .
not
(
x0
x15
x16
)
)
x8
x9
x10
x11
x12
x13
x14
⟶
86706..
x1
x0
⟶
35fb6..
x1
x0
⟶
(
x0
x6
x8
⟶
not
(
x0
x3
x8
)
⟶
False
)
⟶
(
x0
x6
x8
⟶
not
(
x0
x2
x8
)
⟶
False
)
⟶
(
x0
x5
x8
⟶
not
(
x0
x4
x8
)
⟶
False
)
⟶
(
x0
x5
x8
⟶
not
(
x0
x2
x8
)
⟶
False
)
⟶
(
x0
x4
x8
⟶
not
(
x0
x3
x8
)
⟶
False
)
⟶
False
...
Known
8f85a..
:
∀ x0 :
ι →
ι → ο
.
∀ x1 :
ι →
ι →
ι →
ι →
ι →
ι →
ι → ο
.
(
∀ x2 x3 .
x3
∈
x2
⟶
∀ x4 .
x4
∈
x2
⟶
∀ x5 .
x5
∈
x2
⟶
∀ x6 .
x6
∈
x2
⟶
∀ x7 .
x7
∈
x2
⟶
∀ x8 .
x8
∈
x2
⟶
∀ x9 .
x9
∈
x2
⟶
∀ x10 .
x10
∈
x2
⟶
∀ x11 .
x11
∈
x2
⟶
∀ x12 .
x12
∈
x2
⟶
∀ x13 .
x13
∈
x2
⟶
∀ x14 .
x14
∈
x2
⟶
∀ x15 .
x15
∈
x2
⟶
(
∀ x16 .
x16
∈
x2
⟶
∀ x17 .
x17
∈
x2
⟶
x0
x16
x17
⟶
x0
x17
x16
)
⟶
(
x3
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
00e19..
x0
x3
x4
x5
x6
x7
x8
⟶
x1
x9
x10
x11
x12
x13
x14
x15
⟶
86706..
x2
x0
⟶
35fb6..
x2
x0
⟶
(
x0
x7
x9
⟶
not
(
x0
x4
x9
)
⟶
False
)
⟶
(
x0
x7
x9
⟶
not
(
x0
x3
x9
)
⟶
False
)
⟶
(
x0
x6
x9
⟶
not
(
x0
x5
x9
)
⟶
False
)
⟶
(
x0
x6
x9
⟶
not
(
x0
x4
x9
)
⟶
False
)
⟶
(
x0
x6
x9
⟶
not
(
x0
x3
x9
)
⟶
False
)
⟶
(
x0
x5
x9
⟶
not
(
x0
x4
x9
)
⟶
False
)
⟶
False
)
⟶
∀ x2 x3 .
x3
∈
x2
⟶
∀ x4 .
x4
∈
x2
⟶
∀ x5 .
x5
∈
x2
⟶
∀ x6 .
x6
∈
x2
⟶
∀ x7 .
x7
∈
x2
⟶
∀ x8 .
x8
∈
x2
⟶
∀ x9 .
x9
∈
x2
⟶
∀ x10 .
x10
∈
x2
⟶
∀ x11 .
x11
∈
x2
⟶
∀ x12 .
x12
∈
x2
⟶
∀ x13 .
x13
∈
x2
⟶
∀ x14 .
x14
∈
x2
⟶
∀ x15 .
x15
∈
x2
⟶
(
∀ x16 .
x16
∈
x2
⟶
∀ x17 .
x17
∈
x2
⟶
x0
x16
x17
⟶
x0
x17
x16
)
⟶
(
x3
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x9
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x10
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x11
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x12
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x13
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x14
⟶
∀ x16 : ο .
x16
)
⟶
(
x3
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x4
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x5
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x6
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x7
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
(
x8
=
x15
⟶
∀ x16 : ο .
x16
)
⟶
00e19..
x0
x3
x4
x5
x6
x7
x8
⟶
x1
x9
x10
x11
x12
x13
x14
x15
⟶
86706..
x2
x0
⟶
35fb6..
x2
x0
⟶
False
Theorem
06402..
:
∀ x0 :
ι →
ι → ο
.
∀ x1 x2 .
x2
∈
x1
⟶
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
∀ x5 .
x5
∈
x1
⟶
∀ x6 .
x6
∈
x1
⟶
∀ x7 .
x7
∈
x1
⟶
∀ x8 .
x8
∈
x1
⟶
∀ x9 .
x9
∈
x1
⟶
∀ x10 .
x10
∈
x1
⟶
∀ x11 .
x11
∈
x1
⟶
∀ x12 .
x12
∈
x1
⟶
∀ x13 .
x13
∈
x1
⟶
∀ x14 .
x14
∈
x1
⟶
(
∀ x15 .
x15
∈
x1
⟶
∀ x16 .
x16
∈
x1
⟶
x0
x15
x16
⟶
x0
x16
x15
)
⟶
(
x2
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x8
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x9
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x10
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x11
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x12
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x13
⟶
∀ x15 : ο .
x15
)
⟶
(
x2
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x3
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x4
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x5
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x6
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
(
x7
=
x14
⟶
∀ x15 : ο .
x15
)
⟶
00e19..
x0
x2
x3
x4
x5
x6
x7
⟶
70d65..
(
λ x15 x16 .
not
(
x0
x15
x16
)
)
x8
x9
x10
x11
x12
x13
x14
⟶
86706..
x1
x0
⟶
35fb6..
x1
x0
⟶
False
...