Search for blocks/addresses/...
Proofgold Asset
asset id
39b12bdc0ce18b6004a3f11ffeab7bc6a478f993d27502e3d1f09cf36095d104
asset hash
017f0cfe5d843c288bd722d3d00ba9fce6786d9d420679dec3b3224df2c1fdf8
bday / block
11884
tx
f88f6..
preasset
doc published by
PrGVS..
Known
8106d..
notI
:
∀ x0 : ο .
(
x0
⟶
False
)
⟶
not
x0
Known
TrueI
TrueI
:
True
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Theorem
ee98d..
:
not
(
∀ x0 :
(
(
ι →
(
ι →
ι → ι
)
→ ι
)
→
(
ι →
ι →
ι → ι
)
→
ι → ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ο
.
∀ x1 :
(
(
ι →
(
ι →
ι → ι
)
→ ι
)
→
ι →
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
)
→ ο
.
∀ x2 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
(
ι → ι
)
→
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ο
.
∀ x3 :
(
ι → ι
)
→
(
ι →
ι →
ι →
ι → ι
)
→ ο
.
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x5 .
∀ x6 :
ι →
ι →
ι →
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x3
(
λ x8 .
Inj0
(
setsum
(
Inj0
0
)
x5
)
)
(
λ x8 x9 x10 x11 .
x10
)
⟶
x3
(
λ x8 .
Inj1
(
Inj0
(
Inj1
(
Inj1
0
)
)
)
)
(
λ x8 x9 x10 x11 .
x11
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
In
(
x6
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
Inj0
(
Inj1
(
x9
0
)
)
)
)
(
x4
(
λ x8 :
(
ι → ι
)
→ ι
.
λ x9 :
ι → ι
.
λ x10 .
0
)
)
⟶
x3
(
λ x8 .
Inj1
(
x6
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
)
)
(
λ x8 x9 x10 x11 .
0
)
⟶
x3
(
λ x8 .
0
)
(
λ x8 x9 x10 x11 .
x11
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x2
(
λ x8 :
ι →
ι → ι
.
λ x9 :
ι → ι
.
λ x10 .
Inj0
x7
)
(
λ x8 :
(
ι → ι
)
→
ι →
ι → ι
.
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
λ x11 .
setsum
(
Inj0
0
)
0
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x2
(
λ x8 :
ι →
ι → ι
.
λ x9 :
ι → ι
.
λ x10 .
x9
(
setsum
0
0
)
)
(
λ x8 :
(
ι → ι
)
→
ι →
ι → ι
.
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
λ x11 .
x9
(
λ x12 .
x9
(
λ x13 .
0
)
(
x10
(
setsum
0
0
)
)
)
(
setsum
(
Inj0
(
x9
(
λ x12 .
0
)
0
)
)
(
setsum
(
setsum
0
0
)
x11
)
)
)
⟶
x3
(
λ x8 .
x7
0
)
(
λ x8 x9 x10 x11 .
x8
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x8 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x9 :
ι →
ι →
ι → ι
.
λ x10 .
0
)
(
λ x8 :
(
ι → ι
)
→ ι
.
setsum
(
setsum
x7
x5
)
(
setsum
(
Inj1
x5
)
0
)
)
⟶
x1
(
λ x8 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
(
λ x8 :
(
ι → ι
)
→ ι
.
λ x9 x10 x11 .
setsum
0
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
ι →
ι → ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
ι →
ι → ι
.
x1
(
λ x8 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
(
λ x8 :
(
ι → ι
)
→ ι
.
λ x9 x10 x11 .
setsum
(
setsum
(
Inj0
(
Inj0
0
)
)
(
setsum
(
Inj0
0
)
0
)
)
x10
)
⟶
False
)
⟶
(
∀ x4 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 x7 .
x0
(
λ x8 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x9 :
ι →
ι →
ι → ι
.
λ x10 .
x9
(
Inj0
(
Inj0
x10
)
)
(
x9
x10
0
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
)
x10
)
(
λ x8 :
(
ι → ι
)
→ ι
.
x6
)
⟶
x0
(
λ x8 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x9 :
ι →
ι →
ι → ι
.
λ x10 .
x9
0
(
setsum
(
setsum
(
Inj1
0
)
(
Inj1
0
)
)
0
)
(
setsum
(
setsum
0
0
)
(
Inj1
(
setsum
0
0
)
)
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
setsum
(
setsum
x5
(
Inj0
0
)
)
(
Inj1
x7
)
)
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
ι →
ι → ι
.
∀ x6 .
∀ x7 :
(
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
In
(
Inj1
(
x5
(
λ x8 .
0
)
(
setsum
(
setsum
0
0
)
x6
)
(
setsum
(
Inj1
0
)
(
x5
(
λ x8 .
0
)
0
0
)
)
)
)
(
Inj1
(
Inj0
0
)
)
⟶
x0
(
λ x8 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x9 :
ι →
ι →
ι → ι
.
λ x10 .
0
)
(
λ x8 :
(
ι → ι
)
→ ι
.
setsum
(
setsum
0
(
x5
(
λ x9 .
0
)
(
Inj0
0
)
0
)
)
(
x8
(
λ x9 .
x8
(
λ x10 .
0
)
)
)
)
⟶
x1
(
λ x8 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
x8
(
setsum
(
Inj1
(
setsum
0
0
)
)
0
)
(
λ x12 x13 .
setsum
(
Inj1
x12
)
(
setsum
0
(
Inj1
0
)
)
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
λ x9 x10 x11 .
setsum
0
(
setsum
(
Inj1
(
setsum
0
0
)
)
(
setsum
(
setsum
0
0
)
x11
)
)
)
)
⟶
False
)
(proof)
Theorem
536b2..
:
not
(
∀ x0 :
(
(
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→ ι
)
→
ι →
(
ι →
(
ι → ι
)
→ ι
)
→ ο
.
∀ x1 :
(
ι → ι
)
→
(
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι → ο
.
∀ x2 :
(
ι → ι
)
→
ι →
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ο
.
∀ x3 :
(
(
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι →
ι → ο
.
(
∀ x4 x5 .
∀ x6 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
x2
(
λ x8 .
0
)
(
x6
(
λ x8 .
λ x9 :
ι → ι
.
Inj1
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
)
)
(
λ x8 x9 :
ι → ι
.
λ x10 .
x7
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj1
(
Inj0
(
setsum
(
setsum
0
0
)
0
)
)
)
⟶
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 .
setsum
x7
0
)
(
setsum
(
setsum
(
Inj0
(
setsum
0
0
)
)
x5
)
(
Inj0
0
)
)
(
setsum
0
x4
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 .
Inj1
(
x7
(
λ x11 :
ι → ι
.
setsum
(
setsum
0
0
)
(
x11
0
)
)
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
(
λ x11 .
x9
(
λ x12 x13 .
Inj0
0
)
(
λ x12 .
Inj1
0
)
0
)
)
)
(
setsum
0
(
Inj0
(
Inj0
(
setsum
0
0
)
)
)
)
(
Inj1
x6
)
⟶
In
(
setsum
(
Inj1
x6
)
x4
)
(
Inj1
x5
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
.
In
(
Inj1
x5
)
(
setsum
0
(
Inj1
(
x7
(
λ x8 .
Inj1
0
)
(
λ x8 x9 .
setsum
0
0
)
0
0
)
)
)
⟶
x0
(
λ x8 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
0
)
(
Inj0
0
)
(
λ x8 .
λ x9 :
ι → ι
.
setsum
0
(
x7
(
λ x10 .
x10
)
(
λ x10 x11 .
x11
)
(
Inj0
(
x6
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
(
Inj0
0
)
)
)
⟶
x2
(
λ x8 .
x8
)
0
(
λ x8 x9 :
ι → ι
.
λ x10 .
0
)
(
λ x8 :
ι → ι
.
λ x9 .
x7
Inj1
(
λ x10 x11 .
x11
)
(
x8
(
setsum
(
x6
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
x9
)
)
0
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
In
(
Inj0
0
)
(
setsum
(
Inj1
0
)
(
Inj0
(
setsum
(
Inj1
0
)
x4
)
)
)
⟶
x2
(
λ x8 .
0
)
0
(
λ x8 x9 :
ι → ι
.
λ x10 .
Inj0
(
Inj0
(
Inj0
(
Inj1
0
)
)
)
)
(
λ x8 :
ι → ι
.
λ x9 .
setsum
x6
(
setsum
(
x7
(
λ x10 .
0
)
(
setsum
0
0
)
(
λ x10 .
setsum
0
0
)
(
Inj0
0
)
)
0
)
)
⟶
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 .
setsum
(
setsum
(
x9
(
λ x11 x12 .
0
)
(
λ x11 .
x8
(
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
0
)
)
(
Inj1
0
)
)
(
setsum
(
setsum
0
0
)
0
)
)
(
x9
(
λ x11 x12 .
x10
)
(
λ x11 .
Inj1
(
setsum
0
0
)
)
(
Inj1
(
Inj0
0
)
)
)
)
(
setsum
(
x7
(
λ x8 .
Inj1
(
x7
(
λ x9 .
0
)
0
(
λ x9 .
0
)
0
)
)
x5
(
λ x8 .
Inj1
0
)
(
Inj1
(
Inj1
0
)
)
)
x6
)
0
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
In
(
Inj0
(
Inj1
(
setsum
0
(
Inj0
0
)
)
)
)
(
Inj1
(
setsum
(
setsum
(
setsum
0
0
)
0
)
(
x7
(
λ x8 .
x7
(
λ x9 .
0
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj0
0
)
)
)
)
⟶
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 .
setsum
(
setsum
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
0
)
(
x7
(
λ x11 .
x10
)
(
λ x11 :
ι → ι
.
λ x12 .
x9
(
λ x13 x14 .
0
)
(
λ x13 .
setsum
0
0
)
(
setsum
0
0
)
)
)
)
(
Inj1
(
Inj1
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
)
)
(
Inj0
x4
)
⟶
x1
(
λ x8 .
Inj1
(
x5
0
(
λ x9 :
ι → ι
.
λ x10 .
x10
)
)
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
Inj0
(
Inj1
(
setsum
(
setsum
0
0
)
(
Inj1
0
)
)
)
)
(
λ x8 x9 .
Inj0
(
x7
(
λ x10 .
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
(
λ x10 :
ι → ι
.
λ x11 .
Inj1
(
x10
0
)
)
)
)
(
x7
(
λ x8 .
0
)
(
λ x8 :
ι → ι
.
λ x9 .
setsum
(
x6
(
λ x10 .
0
)
)
0
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x1
(
λ x8 .
0
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x8 x9 .
Inj1
(
setsum
x6
x8
)
)
0
⟶
In
(
Inj1
(
setsum
0
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
(
Inj0
(
setsum
(
Inj1
x6
)
(
x4
(
λ x8 .
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
In
x7
(
setsum
(
Inj0
(
setsum
0
(
setsum
0
0
)
)
)
x6
)
⟶
x2
(
λ x8 .
Inj0
(
setsum
(
Inj0
x8
)
(
setsum
(
setsum
0
0
)
0
)
)
)
(
Inj0
(
setsum
x5
0
)
)
(
λ x8 x9 :
ι → ι
.
λ x10 .
Inj0
(
Inj1
(
setsum
(
setsum
0
0
)
x7
)
)
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj1
(
setsum
0
(
setsum
0
(
setsum
0
0
)
)
)
)
⟶
x0
(
λ x8 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
Inj0
x7
)
(
Inj1
x5
)
(
λ x8 .
λ x9 :
ι → ι
.
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
x0
(
λ x8 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
setsum
(
setsum
(
setsum
0
(
Inj0
0
)
)
0
)
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
)
(
setsum
(
Inj1
(
setsum
x5
0
)
)
x7
)
(
λ x8 .
λ x9 :
ι → ι
.
setsum
(
Inj1
(
setsum
(
x6
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
x8
)
)
x7
)
⟶
x1
(
setsum
(
Inj1
(
Inj1
x7
)
)
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x8 x9 .
setsum
x9
(
setsum
(
Inj1
(
setsum
0
0
)
)
x9
)
)
(
Inj0
(
x6
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
setsum
(
setsum
0
0
)
0
)
)
)
)
⟶
False
)
(proof)
Theorem
57e52..
:
not
(
∀ x0 :
(
ι →
ι →
(
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ο
.
∀ x1 :
(
ι →
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ο
.
∀ x2 :
(
ι → ι
)
→
(
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ο
.
∀ x3 :
(
ι → ι
)
→
ι → ο
.
(
∀ x4 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x0
(
λ x8 x9 .
λ x10 :
ι →
ι → ι
.
Inj1
0
)
(
λ x8 .
setsum
(
x7
(
setsum
0
x8
)
)
(
Inj0
0
)
)
(
λ x8 .
x8
)
(
x7
(
setsum
0
(
Inj0
0
)
)
)
⟶
x3
(
λ x8 .
setsum
x6
(
Inj1
0
)
)
(
Inj0
(
setsum
(
x7
0
)
(
setsum
(
Inj1
0
)
(
Inj1
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
In
(
x4
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
0
)
(
λ x8 :
ι → ι
.
λ x9 .
x8
0
)
(
λ x8 .
0
)
0
)
(
Inj1
(
Inj0
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
⟶
x3
(
λ x8 .
Inj1
0
)
0
⟶
x0
(
λ x8 x9 .
λ x10 :
ι →
ι → ι
.
Inj1
(
setsum
0
x9
)
)
(
λ x8 .
setsum
(
Inj1
(
Inj0
(
Inj1
0
)
)
)
(
Inj0
(
setsum
0
(
Inj1
0
)
)
)
)
(
λ x8 .
x8
)
(
Inj1
(
x4
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
Inj1
(
Inj1
0
)
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj0
0
)
(
λ x8 .
0
)
x6
)
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 x6 x7 .
x3
(
λ x8 .
setsum
(
setsum
x6
x6
)
x7
)
(
Inj1
0
)
⟶
x2
(
λ x8 .
setsum
0
0
)
(
λ x8 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x8 .
x6
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x2
(
λ x8 .
setsum
x6
(
setsum
0
(
x5
0
)
)
)
(
λ x8 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x8 .
setsum
0
(
x7
(
setsum
(
Inj1
0
)
(
Inj0
0
)
)
)
)
⟶
In
(
setsum
(
Inj0
(
x5
x4
)
)
0
)
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 :
(
ι →
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
(
ι →
ι →
ι → ι
)
→
ι → ι
.
In
(
Inj0
(
Inj1
(
x5
(
λ x8 x9 .
0
)
)
)
)
(
x5
(
λ x8 x9 .
x6
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
setsum
(
x11
0
)
(
Inj0
0
)
)
)
)
⟶
x1
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x10
)
(
λ x8 :
ι → ι
.
Inj0
(
x5
(
λ x9 x10 .
x9
)
)
)
⟶
x1
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
Inj0
x10
)
(
λ x8 :
ι → ι
.
Inj1
(
Inj0
(
x5
(
λ x9 x10 .
x8
0
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
ι →
ι → ι
.
∀ x6 .
∀ x7 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
In
(
x5
(
Inj1
(
setsum
(
setsum
0
0
)
(
x4
0
)
)
)
(
x7
(
λ x8 .
setsum
(
Inj0
0
)
(
x5
0
0
)
)
(
λ x8 x9 .
Inj1
(
setsum
0
0
)
)
)
)
x6
⟶
x1
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
setsum
(
Inj0
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
(
Inj1
(
x9
x10
(
λ x13 .
0
)
(
setsum
0
0
)
)
)
)
(
λ x8 :
ι → ι
.
Inj0
0
)
⟶
x0
(
λ x8 x9 .
λ x10 :
ι →
ι → ι
.
0
)
(
λ x8 .
x8
)
(
λ x8 .
x5
(
x7
(
λ x9 .
x9
)
(
λ x9 x10 .
x8
)
)
(
setsum
0
(
x7
(
λ x9 .
Inj0
0
)
(
λ x9 x10 .
setsum
0
0
)
)
)
)
0
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 :
ι →
ι →
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
In
(
x7
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
)
(
Inj1
(
Inj1
(
x7
(
λ x8 :
(
ι → ι
)
→ ι
.
setsum
0
0
)
)
)
)
⟶
x1
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
Inj1
0
)
(
λ x8 :
ι → ι
.
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
setsum
(
Inj1
0
)
0
)
)
⟶
x0
(
λ x8 x9 .
λ x10 :
ι →
ι → ι
.
0
)
(
λ x8 .
0
)
(
λ x8 .
Inj1
(
setsum
(
setsum
(
setsum
0
0
)
(
x5
0
0
(
λ x9 .
0
)
)
)
(
Inj1
(
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
)
(
setsum
0
(
x7
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj0
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 :
(
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x8 x9 .
λ x10 :
ι →
ι → ι
.
setsum
x9
(
Inj0
(
x10
(
setsum
0
0
)
0
)
)
)
(
λ x8 .
x5
(
λ x9 :
ι → ι
.
setsum
(
Inj1
0
)
(
setsum
(
setsum
0
0
)
0
)
)
(
λ x9 x10 .
0
)
(
Inj1
(
setsum
(
setsum
0
0
)
(
x6
0
)
)
)
0
)
(
λ x8 .
0
)
(
Inj0
(
x5
(
λ x8 :
ι → ι
.
0
)
(
λ x8 x9 .
Inj0
0
)
(
setsum
0
(
x4
(
λ x8 .
0
)
)
)
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
⟶
x1
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
setsum
x12
(
Inj1
x12
)
)
(
λ x8 :
ι → ι
.
Inj0
(
setsum
(
Inj0
0
)
0
)
)
)
⟶
False
)
(proof)
Theorem
ead0e..
:
not
(
∀ x0 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ο
.
∀ x1 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→
ι → ο
.
∀ x2 :
(
ι →
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ο
.
∀ x3 :
(
(
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
ι → ο
.
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
x1
(
λ x8 :
ι → ι
.
λ x9 x10 .
Inj1
(
Inj1
(
Inj1
(
Inj0
0
)
)
)
)
0
⟶
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x9 .
Inj0
(
x6
(
λ x10 .
setsum
(
setsum
0
0
)
0
)
)
)
(
Inj0
0
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x6 x7 .
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x9 .
x9
)
(
setsum
0
(
setsum
(
setsum
(
Inj0
0
)
0
)
(
x5
(
λ x8 .
x5
(
λ x9 .
0
)
(
λ x9 x10 .
0
)
)
(
λ x8 x9 .
0
)
)
)
)
⟶
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x9 .
x9
)
(
Inj1
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x7 :
ι → ι
.
x2
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 x11 .
x8
)
(
setsum
(
Inj1
0
)
(
setsum
(
setsum
(
setsum
0
0
)
0
)
(
Inj1
(
setsum
0
0
)
)
)
)
(
λ x8 :
ι → ι
.
λ x9 .
setsum
(
x8
(
Inj0
(
Inj0
0
)
)
)
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
)
(
Inj0
(
Inj1
(
setsum
(
setsum
0
0
)
x4
)
)
)
⟶
x2
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 x11 .
0
)
(
setsum
(
x7
(
setsum
x4
(
Inj1
0
)
)
)
x4
)
(
λ x8 :
ι → ι
.
λ x9 .
x9
)
(
Inj1
(
setsum
0
(
Inj0
(
x7
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x2
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 x11 .
setsum
(
setsum
x8
(
x9
(
Inj1
0
)
(
λ x12 .
x11
)
0
)
)
0
)
(
Inj1
(
setsum
(
x4
(
λ x8 .
0
)
(
λ x8 .
x8
)
)
0
)
)
(
λ x8 :
ι → ι
.
λ x9 .
setsum
(
setsum
(
x7
(
Inj1
0
)
)
(
x7
(
Inj0
0
)
)
)
(
Inj1
(
x8
0
)
)
)
(
x5
(
λ x8 .
Inj1
0
)
)
⟶
x1
(
λ x8 :
ι → ι
.
λ x9 x10 .
0
)
(
Inj0
(
Inj1
(
setsum
0
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
x2
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 x11 .
0
)
0
(
λ x8 :
ι → ι
.
λ x9 .
0
)
(
setsum
0
0
)
⟶
x1
(
λ x8 :
ι → ι
.
λ x9 x10 .
Inj1
0
)
(
x7
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 :
ι → ι
.
λ x10 .
setsum
(
Inj0
0
)
0
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→
ι →
ι → ι
.
∀ x7 .
In
(
Inj0
x4
)
(
setsum
0
(
x6
(
λ x8 .
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
(
setsum
x4
(
Inj1
0
)
)
0
)
)
⟶
x1
(
λ x8 :
ι → ι
.
λ x9 x10 .
x10
)
0
⟶
x2
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 x11 .
setsum
0
x8
)
(
x6
(
λ x8 .
setsum
(
setsum
x8
x8
)
0
)
0
(
setsum
x4
(
Inj0
x5
)
)
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj1
0
)
(
setsum
0
(
Inj1
x7
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
In
(
x6
0
(
λ x8 :
ι → ι
.
Inj0
0
)
)
x7
⟶
x0
(
λ x8 x9 .
setsum
x9
x9
)
(
λ x8 :
ι → ι
.
λ x9 :
(
ι → ι
)
→ ι
.
x9
(
λ x10 .
setsum
(
Inj1
(
Inj1
0
)
)
0
)
)
⟶
x0
(
λ x8 x9 .
setsum
x9
(
setsum
(
Inj0
(
setsum
0
0
)
)
(
Inj1
(
x6
0
(
λ x10 :
ι → ι
.
0
)
)
)
)
)
(
λ x8 :
ι → ι
.
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 x7 .
x0
(
λ x8 x9 .
Inj0
0
)
(
λ x8 :
ι → ι
.
λ x9 :
(
ι → ι
)
→ ι
.
Inj0
0
)
⟶
In
(
Inj1
(
setsum
0
(
x4
(
λ x8 :
ι → ι
.
x6
)
(
λ x8 .
x5
0
)
)
)
)
x7
)
⟶
False
)
(proof)
Theorem
bc887..
:
not
(
∀ x0 :
(
ι →
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ο
.
∀ x1 :
(
ι →
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι → ο
.
∀ x2 :
(
ι → ι
)
→
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
ι → ο
.
∀ x3 :
(
ι →
(
ι →
ι → ι
)
→ ι
)
→
ι →
ι → ο
.
(
∀ x4 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 x6 x7 .
In
(
Inj0
x6
)
(
setsum
(
setsum
0
0
)
(
setsum
x6
(
Inj0
(
setsum
0
0
)
)
)
)
⟶
x2
(
λ x8 .
Inj0
(
Inj0
0
)
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
ι →
ι → ι
.
λ x10 x11 .
x10
)
(
setsum
(
x4
(
λ x8 :
ι →
ι → ι
.
λ x9 :
ι → ι
.
λ x10 .
Inj0
(
setsum
0
0
)
)
)
0
)
⟶
x3
(
λ x8 .
λ x9 :
ι →
ι → ι
.
Inj0
x8
)
x5
(
setsum
(
setsum
(
setsum
0
(
setsum
0
0
)
)
(
Inj0
0
)
)
0
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 x7 .
x3
(
λ x8 .
λ x9 :
ι →
ι → ι
.
0
)
x7
(
setsum
(
setsum
(
setsum
(
setsum
0
0
)
0
)
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
(
setsum
0
(
Inj1
(
setsum
0
0
)
)
)
)
⟶
x3
(
λ x8 .
λ x9 :
ι →
ι → ι
.
x7
)
(
setsum
(
x4
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
x7
)
)
x6
)
(
Inj0
(
Inj1
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x7 .
In
(
Inj0
0
)
(
setsum
0
(
x5
(
Inj0
(
setsum
0
0
)
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj0
0
)
)
)
⟶
x0
(
λ x8 x9 .
λ x10 :
(
ι → ι
)
→
ι → ι
.
x8
)
x7
(
x6
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 :
ι → ι
.
λ x10 .
Inj1
0
)
(
λ x8 x9 .
0
)
)
⟶
x2
(
λ x8 .
x8
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
ι →
ι → ι
.
λ x10 x11 .
x10
)
(
setsum
(
Inj0
0
)
(
x6
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 :
ι → ι
.
λ x10 .
Inj0
0
)
(
λ x8 x9 .
x9
)
)
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
In
(
Inj0
(
Inj1
0
)
)
(
setsum
(
setsum
(
Inj0
0
)
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
(
setsum
0
(
setsum
(
Inj0
0
)
(
x4
0
0
)
)
)
)
⟶
x2
(
λ x8 .
0
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
ι →
ι → ι
.
λ x10 x11 .
0
)
(
setsum
(
setsum
0
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
(
Inj1
x6
)
)
⟶
x1
(
λ x8 .
λ x9 :
(
ι → ι
)
→ ι
.
Inj0
0
)
(
x4
0
(
x5
(
setsum
(
Inj1
0
)
0
)
)
)
)
⟶
(
∀ x4 .
∀ x5 x6 :
ι → ι
.
∀ x7 .
x3
(
λ x8 .
λ x9 :
ι →
ι → ι
.
0
)
0
0
⟶
x1
(
λ x8 .
λ x9 :
(
ι → ι
)
→ ι
.
x9
(
λ x10 .
setsum
(
Inj1
(
Inj0
0
)
)
(
setsum
0
0
)
)
)
0
)
⟶
(
∀ x4 x5 x6 x7 .
In
(
Inj0
x6
)
(
setsum
(
setsum
x4
(
setsum
x5
(
setsum
0
0
)
)
)
(
Inj1
(
setsum
x5
0
)
)
)
⟶
x1
(
λ x8 .
λ x9 :
(
ι → ι
)
→ ι
.
x7
)
0
⟶
x2
(
λ x8 .
0
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
ι →
ι → ι
.
λ x10 x11 .
setsum
0
0
)
x5
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 x7 .
In
(
Inj1
0
)
x5
⟶
x1
(
λ x8 .
λ x9 :
(
ι → ι
)
→ ι
.
0
)
0
⟶
x0
(
λ x8 x9 .
λ x10 :
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
(
Inj1
(
Inj1
(
x4
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
(
Inj1
(
setsum
(
setsum
(
Inj0
0
)
0
)
(
Inj1
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x8 x9 .
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
(
x4
0
)
x7
⟶
False
)
⟶
False
)
(proof)
Theorem
1cb9d..
:
not
(
∀ x0 :
(
(
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ο
.
∀ x1 :
(
(
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι →
ι →
ι → ι
)
→
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ο
.
∀ x2 :
(
ι → ι
)
→
(
ι → ι
)
→
(
ι →
(
ι → ι
)
→ ι
)
→ ο
.
∀ x3 :
(
ι →
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι →
ι →
ι →
(
ι → ι
)
→
ι → ο
.
(
∀ x4 x5 x6 x7 .
x2
(
λ x8 .
setsum
x5
0
)
(
λ x8 .
0
)
(
λ x8 .
λ x9 :
ι → ι
.
setsum
0
x8
)
⟶
x3
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 .
x7
)
x5
(
Inj0
0
)
(
Inj1
(
Inj0
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
)
)
(
λ x8 .
0
)
x7
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
x3
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 .
setsum
x8
(
Inj0
0
)
)
0
(
setsum
0
0
)
0
(
λ x8 .
Inj1
(
Inj0
(
Inj0
(
setsum
0
0
)
)
)
)
(
setsum
(
setsum
(
x5
(
λ x8 x9 x10 .
x10
)
(
setsum
0
0
)
0
)
0
)
(
setsum
(
x5
(
λ x8 x9 x10 .
Inj0
0
)
0
0
)
0
)
)
⟶
In
(
x5
(
λ x8 x9 x10 .
Inj1
(
Inj0
(
setsum
0
0
)
)
)
(
Inj1
(
x7
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
x6
(
λ x9 .
0
)
)
(
setsum
0
0
)
(
x7
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
)
(
x7
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
0
)
(
x5
(
λ x8 x9 x10 .
0
)
0
(
Inj0
0
)
)
(
Inj1
x4
)
)
)
(
Inj1
(
Inj0
(
Inj0
x4
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
x0
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
setsum
(
x7
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
Inj1
0
)
(
λ x8 :
ι → ι
.
setsum
(
x8
0
)
0
)
(
setsum
(
setsum
0
0
)
)
(
setsum
(
x4
0
)
(
Inj1
0
)
)
)
(
Inj1
(
setsum
(
setsum
0
0
)
x5
)
)
)
⟶
x2
(
λ x8 .
Inj1
0
)
(
λ x8 .
0
)
(
λ x8 .
λ x9 :
ι → ι
.
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 x7 :
ι → ι
.
x2
(
λ x8 .
0
)
(
λ x8 .
0
)
(
λ x8 .
λ x9 :
ι → ι
.
x8
)
⟶
x2
(
λ x8 .
Inj0
(
Inj0
(
x6
(
setsum
0
0
)
)
)
)
(
λ x8 .
0
)
(
λ x8 .
λ x9 :
ι → ι
.
0
)
)
⟶
(
∀ x4 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x5 .
∀ x6 x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
In
(
Inj0
(
x4
(
Inj1
(
Inj1
0
)
)
(
λ x8 x9 .
x6
0
(
λ x10 :
ι → ι
.
λ x11 .
Inj1
0
)
)
(
setsum
0
(
Inj1
0
)
)
0
)
)
(
x7
(
setsum
(
x7
(
setsum
0
0
)
(
λ x8 :
ι → ι
.
λ x9 .
setsum
0
0
)
)
(
setsum
(
x4
0
(
λ x8 x9 .
0
)
0
0
)
(
Inj0
0
)
)
)
(
λ x8 :
ι → ι
.
λ x9 .
setsum
(
Inj0
(
Inj1
0
)
)
0
)
)
⟶
x0
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
x7
0
(
λ x9 :
ι → ι
.
λ x10 .
Inj0
(
Inj1
0
)
)
)
(
setsum
(
x7
(
setsum
(
setsum
0
0
)
(
x4
0
(
λ x8 x9 .
0
)
0
0
)
)
(
λ x8 :
ι → ι
.
λ x9 .
x6
0
(
λ x10 :
ι → ι
.
λ x11 .
setsum
0
0
)
)
)
x5
)
⟶
x1
(
λ x8 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
setsum
(
Inj1
(
Inj1
0
)
)
(
Inj1
(
Inj1
0
)
)
)
(
λ x8 x9 x10 x11 .
0
)
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
setsum
(
setsum
(
Inj0
0
)
0
)
(
Inj1
(
Inj1
x10
)
)
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 x6 :
ι →
ι → ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
x1
(
λ x8 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x6
(
Inj1
(
setsum
(
Inj0
0
)
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
0
0
)
)
)
(
setsum
(
Inj0
(
setsum
0
0
)
)
(
setsum
(
Inj0
0
)
0
)
)
)
(
λ x8 x9 x10 x11 .
x8
)
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
0
)
⟶
x3
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
Inj0
)
(
Inj0
(
x7
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
setsum
(
Inj0
0
)
(
Inj1
0
)
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj1
0
)
0
(
Inj0
0
)
)
)
(
setsum
(
setsum
(
x6
(
Inj0
0
)
(
x4
(
λ x8 x9 .
0
)
(
λ x8 :
ι → ι
.
λ x9 .
0
)
)
)
0
)
(
x6
0
(
Inj1
(
Inj1
0
)
)
)
)
(
Inj0
(
Inj1
(
setsum
(
x6
0
0
)
(
Inj0
0
)
)
)
)
(
λ x8 .
x8
)
(
setsum
(
setsum
0
(
Inj0
(
setsum
0
0
)
)
)
(
x7
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
setsum
0
0
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj0
(
Inj0
0
)
)
(
Inj1
(
x4
(
λ x8 x9 .
0
)
(
λ x8 :
ι → ι
.
λ x9 .
0
)
)
)
(
setsum
(
setsum
0
0
)
(
x4
(
λ x8 x9 .
0
)
(
λ x8 :
ι → ι
.
λ x9 .
0
)
)
)
)
)
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 .
In
(
setsum
(
setsum
0
(
Inj0
(
Inj0
0
)
)
)
(
x4
(
λ x8 :
ι → ι
.
λ x9 x10 .
Inj0
(
Inj0
0
)
)
)
)
(
x4
(
λ x8 :
ι → ι
.
λ x9 x10 .
0
)
)
⟶
x3
(
λ x8 .
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 .
Inj0
(
setsum
x8
x7
)
)
(
Inj0
x7
)
(
Inj0
(
setsum
x7
0
)
)
(
Inj0
(
setsum
(
Inj1
0
)
(
setsum
x5
x7
)
)
)
(
λ x8 .
x8
)
0
⟶
x0
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
x5
)
(
setsum
(
Inj0
0
)
(
Inj0
0
)
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x6 x7 .
In
(
Inj0
(
Inj1
(
x5
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 x10 .
Inj1
0
)
)
)
)
(
Inj1
(
x4
(
Inj1
0
)
(
setsum
(
setsum
0
0
)
0
)
)
)
⟶
x0
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
Inj1
x7
)
⟶
x1
(
λ x8 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
setsum
x7
0
)
(
λ x8 x9 x10 x11 .
setsum
(
Inj0
x9
)
x9
)
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
Inj0
(
setsum
(
x9
0
)
0
)
)
)
⟶
False
)
(proof)
Theorem
44176..
:
not
(
∀ x0 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι →
ι → ι
)
→
ι → ο
.
∀ x1 :
(
ι →
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι →
ι → ι
)
→ ο
.
∀ x2 :
(
ι →
ι → ι
)
→
ι → ο
.
∀ x3 :
(
(
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι → ο
.
(
∀ x4 .
∀ x5 :
ι →
(
ι → ι
)
→ ι
.
∀ x6 x7 .
In
x6
(
setsum
x6
(
Inj1
(
Inj1
(
Inj1
0
)
)
)
)
⟶
x3
(
λ x8 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
setsum
(
Inj0
0
)
0
)
x4
(
Inj0
0
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
In
(
Inj1
(
setsum
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
x7
)
)
(
Inj0
(
setsum
(
x6
x7
)
x7
)
)
⟶
x3
(
λ x8 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
Inj1
(
setsum
0
0
)
)
0
(
Inj0
(
Inj0
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
)
⟶
x2
(
λ x8 x9 .
0
)
(
setsum
(
Inj0
0
)
(
setsum
(
x6
0
)
(
x6
(
x4
(
λ x8 x9 .
0
)
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι →
ι → ι
.
∀ x5 :
ι →
(
ι → ι
)
→ ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
In
(
x5
0
Inj1
)
(
Inj1
(
x4
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
Inj0
(
Inj1
0
)
)
(
λ x8 .
0
)
(
setsum
0
0
)
0
)
)
⟶
x2
(
λ x8 x9 .
Inj1
(
setsum
(
x6
(
λ x10 .
λ x11 :
ι → ι
.
Inj1
0
)
(
λ x10 x11 .
0
)
)
(
x6
(
λ x10 .
λ x11 :
ι → ι
.
0
)
(
λ x10 x11 .
x9
)
)
)
)
(
setsum
(
setsum
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
(
setsum
(
Inj1
0
)
(
Inj1
0
)
)
)
0
)
)
⟶
(
∀ x4 :
ι →
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x8 x9 .
setsum
x9
(
Inj1
(
setsum
(
setsum
0
0
)
0
)
)
)
(
setsum
(
x4
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
(
λ x8 .
Inj0
x5
)
)
(
setsum
(
x4
(
Inj1
0
)
(
λ x8 .
Inj0
0
)
)
(
Inj1
0
)
)
)
⟶
x3
(
λ x8 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
Inj0
(
x8
(
λ x10 .
0
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
0
(
Inj1
(
Inj1
0
)
)
)
)
(
setsum
(
setsum
(
Inj0
(
setsum
0
0
)
)
x7
)
x5
)
0
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
(
ι → ι
)
→
ι → ι
.
In
(
Inj0
(
x7
(
λ x8 .
0
)
0
)
)
(
Inj1
(
Inj0
x4
)
)
⟶
x1
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x8 x9 x10 .
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 x7 :
ι → ι
.
In
(
setsum
0
(
setsum
(
x7
0
)
x5
)
)
(
Inj1
0
)
⟶
x1
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x8 x9 x10 .
Inj1
0
)
⟶
x1
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x8 x9 x10 .
Inj1
x8
)
)
⟶
(
∀ x4 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
(
ι → ι
)
→ ι
.
In
(
x4
(
Inj0
(
Inj1
x5
)
)
(
λ x8 :
ι → ι
.
setsum
(
x7
(
λ x9 .
x8
0
)
)
(
x8
(
setsum
0
0
)
)
)
(
λ x8 .
0
)
)
(
setsum
(
Inj1
x6
)
x6
)
⟶
x2
(
λ x8 .
Inj1
)
(
x7
(
λ x8 .
x7
(
λ x9 .
0
)
)
)
⟶
x0
(
λ x8 .
0
)
(
λ x8 :
ι → ι
.
λ x9 x10 .
setsum
(
setsum
(
Inj0
x9
)
(
Inj1
(
Inj1
0
)
)
)
(
Inj0
(
setsum
(
Inj0
0
)
(
x8
0
)
)
)
)
(
setsum
x5
(
setsum
(
setsum
0
(
Inj1
0
)
)
(
x4
0
(
λ x8 :
ι → ι
.
Inj0
0
)
(
λ x8 .
x6
)
)
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 x7 .
x0
(
λ x8 .
0
)
(
λ x8 :
ι → ι
.
λ x9 .
setsum
0
)
x6
⟶
x0
(
λ x8 .
x7
)
(
λ x8 :
ι → ι
.
λ x9 x10 .
x7
)
(
setsum
(
Inj0
(
Inj1
(
x4
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
(
λ x8 .
0
)
)
)
)
(
x4
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
Inj1
)
)
)
⟶
False
)
(proof)
Theorem
c6930..
:
not
(
∀ x0 :
(
ι →
ι → ι
)
→
(
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ο
.
∀ x1 :
(
ι → ι
)
→
(
(
ι →
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ο
.
∀ x2 :
(
ι → ι
)
→
(
ι →
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ο
.
∀ x3 :
(
(
ι → ι
)
→
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ο
.
(
∀ x4 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 x7 .
In
(
setsum
x5
(
Inj0
x6
)
)
(
setsum
0
(
x4
(
λ x8 :
ι →
ι → ι
.
Inj0
(
Inj0
0
)
)
)
)
⟶
x0
(
λ x8 x9 .
0
)
(
λ x8 :
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
setsum
0
(
Inj0
(
setsum
x6
(
Inj0
0
)
)
)
)
⟶
x3
(
λ x8 :
ι → ι
.
λ x9 .
λ x10 :
(
ι → ι
)
→
ι → ι
.
x7
)
(
λ x8 :
(
ι →
ι → ι
)
→ ι
.
x6
)
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj1
0
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
x3
(
λ x8 :
ι → ι
.
λ x9 .
λ x10 :
(
ι → ι
)
→
ι → ι
.
x9
)
(
λ x8 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj1
x6
)
⟶
In
(
Inj0
0
)
(
setsum
(
setsum
x6
(
setsum
x5
0
)
)
(
Inj0
(
setsum
x6
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
∀ x7 :
(
ι →
ι →
ι → ι
)
→
ι → ι
.
In
(
x4
(
λ x8 :
ι →
ι → ι
.
x7
(
λ x9 x10 x11 .
x10
)
0
)
)
(
Inj1
(
x7
(
λ x8 x9 x10 .
0
)
(
setsum
0
(
Inj0
0
)
)
)
)
⟶
x1
(
λ x8 .
Inj1
(
Inj1
0
)
)
(
λ x8 :
ι →
ι →
ι → ι
.
setsum
0
(
Inj1
(
setsum
(
setsum
0
0
)
(
x8
0
0
0
)
)
)
)
(
λ x8 .
setsum
(
x7
(
λ x9 x10 x11 .
0
)
(
Inj1
x8
)
)
(
Inj1
x8
)
)
(
λ x8 :
ι → ι
.
0
)
(
λ x8 .
x5
)
⟶
x2
(
λ x8 .
setsum
(
setsum
0
(
Inj0
x5
)
)
(
Inj1
0
)
)
(
λ x8 .
λ x9 :
(
ι → ι
)
→ ι
.
Inj1
(
x9
(
λ x10 .
0
)
)
)
(
Inj0
(
x4
(
λ x8 :
ι →
ι → ι
.
0
)
)
)
(
setsum
0
0
)
(
setsum
(
x7
(
λ x8 x9 x10 .
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
(
x6
(
λ x8 :
ι →
ι → ι
.
setsum
0
0
)
(
λ x8 x9 .
x7
(
λ x10 x11 x12 .
0
)
0
)
(
setsum
0
0
)
)
)
(
x4
(
λ x8 :
ι →
ι → ι
.
x7
(
λ x9 x10 x11 .
setsum
0
0
)
(
x6
(
λ x9 :
ι →
ι → ι
.
0
)
(
λ x9 x10 .
0
)
0
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
ι →
ι → ι
.
In
(
Inj0
(
Inj1
(
setsum
(
x4
0
)
(
setsum
0
0
)
)
)
)
(
Inj0
0
)
⟶
x2
(
λ x8 .
Inj0
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
setsum
(
Inj1
0
)
0
)
0
(
x6
(
Inj1
0
)
)
0
)
)
(
λ x8 .
λ x9 :
(
ι → ι
)
→ ι
.
0
)
(
Inj1
(
x4
(
Inj0
0
)
)
)
(
x6
(
setsum
(
x7
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
setsum
0
0
)
(
Inj1
0
)
(
x7
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
0
)
0
0
0
)
0
)
(
x7
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
setsum
0
0
)
(
setsum
0
0
)
(
x6
0
)
0
)
)
)
(
setsum
(
x7
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
Inj0
0
)
(
x4
(
x6
0
)
)
(
setsum
(
Inj1
0
)
(
Inj1
0
)
)
0
)
(
Inj0
(
x4
0
)
)
)
⟶
x2
(
λ x8 .
setsum
0
(
Inj0
(
x6
(
setsum
0
0
)
)
)
)
(
λ x8 .
λ x9 :
(
ι → ι
)
→ ι
.
x8
)
(
Inj0
(
x6
(
setsum
0
(
setsum
0
0
)
)
)
)
(
Inj0
x5
)
(
Inj0
(
setsum
x5
(
x4
(
x4
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 .
x1
(
λ x8 .
Inj0
(
setsum
(
Inj1
(
Inj1
0
)
)
x8
)
)
(
λ x8 :
ι →
ι →
ι → ι
.
Inj1
(
x5
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 x11 .
0
)
0
0
0
)
)
(
λ x8 .
setsum
0
(
Inj1
(
Inj1
x7
)
)
)
(
λ x8 :
ι → ι
.
0
)
(
λ x8 .
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι → ι
)
→ ι
.
∀ x6 :
ι →
ι → ι
.
∀ x7 :
ι → ι
.
x1
(
λ x8 .
x5
(
Inj1
(
setsum
0
(
Inj1
0
)
)
)
(
λ x9 .
setsum
(
setsum
(
Inj1
0
)
x9
)
x8
)
)
(
λ x8 :
ι →
ι →
ι → ι
.
Inj0
(
Inj0
(
Inj0
(
x7
0
)
)
)
)
(
λ x8 .
Inj1
(
Inj0
(
setsum
(
Inj0
0
)
(
x6
0
0
)
)
)
)
(
λ x8 :
ι → ι
.
x6
(
setsum
(
x6
0
(
setsum
0
0
)
)
(
x6
(
Inj0
0
)
(
x8
0
)
)
)
(
Inj0
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
)
(
λ x8 .
0
)
⟶
x1
(
λ x8 .
x6
(
Inj1
(
x5
0
(
λ x9 .
x8
)
)
)
0
)
(
λ x8 :
ι →
ι →
ι → ι
.
setsum
(
Inj0
(
Inj0
(
Inj1
0
)
)
)
0
)
(
λ x8 .
Inj0
(
setsum
(
setsum
(
Inj0
0
)
(
x7
0
)
)
0
)
)
(
λ x8 :
ι → ι
.
0
)
(
λ x8 .
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 x7 .
In
x7
x4
⟶
x0
(
λ x8 .
Inj0
)
(
λ x8 :
ι →
(
ι → ι
)
→
ι → ι
.
Inj0
(
setsum
x7
0
)
)
(
Inj0
(
x5
(
Inj0
x7
)
(
λ x8 x9 .
0
)
(
λ x8 .
0
)
)
)
⟶
x0
(
λ x8 x9 .
Inj0
x8
)
(
λ x8 :
ι →
(
ι → ι
)
→
ι → ι
.
Inj1
x6
)
(
setsum
(
setsum
0
(
Inj1
x6
)
)
(
setsum
(
Inj1
0
)
(
setsum
0
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 x7 .
x0
(
λ x8 x9 .
setsum
x7
x6
)
(
λ x8 :
ι →
(
ι → ι
)
→
ι → ι
.
Inj1
(
x8
x7
(
λ x9 .
setsum
x6
(
x8
0
(
λ x10 .
0
)
0
)
)
0
)
)
(
Inj0
(
x4
(
x4
(
Inj1
0
)
)
)
)
⟶
In
(
Inj1
(
setsum
(
x4
(
setsum
0
0
)
)
(
x4
(
setsum
0
0
)
)
)
)
(
Inj1
(
Inj0
0
)
)
)
⟶
False
)
(proof)
Theorem
002b6..
:
not
(
∀ x0 :
(
(
ι → ι
)
→
ι →
(
ι →
ι → ι
)
→ ι
)
→
ι → ο
.
∀ x1 :
(
ι →
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ο
.
∀ x2 :
(
(
(
ι →
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
ι → ο
.
∀ x3 :
(
(
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→ ι
)
→ ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ο
.
(
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x2
(
λ x8 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
Inj1
0
)
(
Inj1
(
setsum
(
Inj0
0
)
(
x5
0
(
Inj0
0
)
)
)
)
⟶
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x6
)
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
)
⟶
(
∀ x4 .
∀ x5 x6 :
ι → ι
.
∀ x7 .
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
setsum
x7
(
setsum
(
x6
(
setsum
0
0
)
)
0
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
setsum
(
x8
(
λ x9 .
0
)
)
(
Inj1
(
Inj0
(
Inj0
0
)
)
)
)
⟶
x0
(
λ x8 :
ι → ι
.
λ x9 .
λ x10 :
ι →
ι → ι
.
setsum
0
x7
)
(
setsum
(
Inj1
0
)
(
setsum
(
Inj0
0
)
(
Inj1
(
Inj1
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
In
(
Inj1
(
setsum
0
(
setsum
0
0
)
)
)
(
Inj1
0
)
⟶
x2
(
λ x8 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
0
)
(
x7
(
λ x8 :
ι →
ι → ι
.
λ x9 :
ι → ι
.
λ x10 .
x8
(
Inj0
(
setsum
0
0
)
)
(
x9
(
setsum
0
0
)
)
)
)
⟶
x2
(
λ x8 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
setsum
(
setsum
(
setsum
(
x7
(
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
setsum
0
0
)
)
(
Inj1
(
setsum
0
0
)
)
)
(
x9
(
λ x11 .
setsum
(
Inj1
0
)
(
x8
(
λ x12 .
λ x13 :
ι → ι
.
0
)
)
)
(
λ x11 .
0
)
)
)
(
x4
(
λ x8 x9 :
ι → ι
.
λ x10 .
x8
0
)
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι →
ι → ι
.
x2
(
λ x8 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
0
)
0
⟶
x3
(
λ x8 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
)
⟶
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→ ι
.
∀ x5 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x6 x7 .
In
(
Inj0
(
x4
(
λ x8 x9 x10 .
0
)
)
)
(
Inj0
0
)
⟶
x2
(
λ x8 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
Inj1
x7
)
0
⟶
x1
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
Inj1
(
x9
x8
)
)
(
λ x8 :
ι → ι
.
Inj1
(
setsum
(
x8
(
setsum
0
0
)
)
(
Inj0
(
Inj0
0
)
)
)
)
(
λ x8 .
Inj1
(
Inj0
0
)
)
(
λ x8 x9 .
0
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x1
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
Inj0
(
x10
(
λ x12 .
0
)
x8
)
)
(
λ x8 :
ι → ι
.
setsum
x7
(
setsum
(
setsum
0
(
setsum
0
0
)
)
x7
)
)
(
λ x8 .
0
)
(
λ x8 x9 .
0
)
⟶
In
(
Inj1
(
setsum
(
x5
(
λ x8 x9 :
ι → ι
.
λ x10 .
Inj0
0
)
x4
(
λ x8 .
setsum
0
0
)
x7
)
(
Inj0
0
)
)
)
(
setsum
(
setsum
x7
(
setsum
0
(
setsum
0
0
)
)
)
(
setsum
(
setsum
(
setsum
0
0
)
x4
)
x6
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
In
x7
(
setsum
0
0
)
⟶
x2
(
λ x8 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
Inj1
(
setsum
0
(
Inj0
0
)
)
)
(
x6
(
setsum
(
setsum
(
x6
0
)
0
)
(
x6
(
Inj1
0
)
)
)
)
⟶
x0
(
λ x8 :
ι → ι
.
λ x9 .
λ x10 :
ι →
ι → ι
.
Inj1
0
)
(
x6
(
Inj0
(
setsum
(
setsum
0
0
)
0
)
)
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
ι → ι
)
→
ι → ι
.
∀ x7 :
(
ι →
ι →
ι → ι
)
→ ι
.
In
(
Inj0
(
Inj1
0
)
)
x5
⟶
x0
(
λ x8 :
ι → ι
.
λ x9 .
λ x10 :
ι →
ι → ι
.
x8
x9
)
(
x4
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
0
)
)
⟶
x0
(
λ x8 :
ι → ι
.
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
0
)
⟶
False
)
(proof)
Theorem
751ed..
:
not
(
∀ x0 :
(
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
)
→
(
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ο
.
∀ x1 :
(
(
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
(
ι →
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ο
.
∀ x2 :
(
ι → ι
)
→
ι → ο
.
∀ x3 :
(
ι → ι
)
→
ι →
ι → ο
.
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x6 x7 :
(
ι →
ι → ι
)
→ ι
.
In
(
setsum
(
setsum
(
setsum
(
x4
0
)
(
setsum
0
0
)
)
(
Inj0
0
)
)
(
setsum
(
x6
(
λ x8 x9 .
Inj1
0
)
)
0
)
)
(
Inj1
(
setsum
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
0
)
)
⟶
x3
(
λ x8 .
0
)
0
(
setsum
(
setsum
(
x7
(
λ x8 x9 .
setsum
0
0
)
)
0
)
(
Inj1
(
Inj1
(
Inj0
0
)
)
)
)
⟶
x3
(
λ x8 .
setsum
(
Inj1
(
setsum
(
x7
(
λ x9 x10 .
0
)
)
(
setsum
0
0
)
)
)
(
Inj1
(
setsum
0
(
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
)
(
x4
(
x7
(
λ x8 x9 .
setsum
0
(
Inj1
0
)
)
)
)
0
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
In
(
Inj0
0
)
x7
⟶
x3
(
λ x8 .
Inj0
0
)
0
x6
⟶
x3
(
λ x8 .
0
)
(
setsum
0
0
)
(
Inj0
(
Inj1
0
)
)
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x1
(
λ x8 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x9 .
0
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
setsum
x7
)
⟶
x2
(
λ x8 .
0
)
(
x4
(
λ x8 x9 :
ι → ι
.
Inj1
(
Inj0
0
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 .
In
(
setsum
x4
(
setsum
(
Inj0
(
setsum
0
0
)
)
(
setsum
0
x7
)
)
)
(
Inj1
0
)
⟶
x2
(
λ x8 .
Inj1
(
Inj1
0
)
)
(
setsum
0
x7
)
⟶
x2
(
λ x8 .
x7
)
(
Inj0
(
Inj1
0
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
In
(
Inj1
(
setsum
(
Inj1
0
)
(
Inj1
(
x6
(
λ x8 :
ι → ι
.
0
)
)
)
)
)
(
Inj0
(
setsum
0
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
⟶
x2
(
λ x8 .
Inj0
(
x6
(
λ x9 :
ι → ι
.
setsum
(
Inj0
0
)
0
)
)
)
x5
⟶
x1
(
λ x8 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x9 .
Inj0
(
Inj0
(
x6
(
λ x10 :
ι → ι
.
x7
)
)
)
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
λ x9 .
setsum
0
(
setsum
(
setsum
(
Inj1
0
)
x9
)
(
Inj0
x7
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
x1
(
λ x8 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x9 .
setsum
(
setsum
0
(
setsum
x9
0
)
)
0
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
setsum
0
)
⟶
False
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x0
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
setsum
(
Inj0
0
)
x6
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x9 .
0
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x7 .
x0
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
x7
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x9 .
x6
(
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
setsum
0
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
)
(
λ x10 x11 .
setsum
x9
(
x8
(
λ x12 :
ι → ι
.
setsum
0
0
)
(
λ x12 .
Inj0
0
)
x9
)
)
x9
(
setsum
(
Inj0
0
)
(
Inj0
(
setsum
0
0
)
)
)
)
⟶
x0
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
setsum
(
x6
(
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
setsum
x12
0
)
(
λ x10 x11 .
x10
)
x7
(
Inj0
(
setsum
0
0
)
)
)
0
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x9 .
setsum
(
setsum
(
setsum
(
setsum
0
0
)
0
)
(
setsum
(
setsum
0
0
)
0
)
)
(
setsum
(
Inj1
(
setsum
0
0
)
)
(
Inj1
(
setsum
0
0
)
)
)
)
)
⟶
False
)
(proof)
Theorem
a3378..
:
not
(
∀ x0 :
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι →
ι → ο
.
∀ x1 :
(
(
(
ι →
ι →
ι → ι
)
→
ι → ι
)
→
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι →
ι →
ι → ι
)
→ ο
.
∀ x2 :
(
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
ι → ο
.
∀ x3 :
(
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
(
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ο
.
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 :
ι →
ι → ι
.
In
(
Inj1
(
setsum
x4
(
Inj0
(
Inj0
0
)
)
)
)
x4
⟶
x0
(
λ x8 .
0
)
(
λ x8 .
Inj1
(
Inj1
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
(
λ x8 .
setsum
(
setsum
0
x8
)
(
setsum
(
Inj1
(
Inj0
0
)
)
(
Inj0
0
)
)
)
(
setsum
(
Inj1
(
Inj0
(
Inj1
0
)
)
)
x4
)
0
⟶
x3
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
x5
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
x6
(
Inj1
(
x6
0
)
)
)
x5
(
λ x8 .
x8
)
0
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 .
x3
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
Inj0
0
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
x7
)
(
Inj1
(
setsum
(
Inj1
(
Inj1
0
)
)
(
setsum
(
Inj0
0
)
x5
)
)
)
(
λ x8 .
0
)
(
setsum
(
Inj1
(
Inj0
(
Inj1
0
)
)
)
(
Inj0
x7
)
)
⟶
In
(
Inj0
x4
)
x5
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
In
(
setsum
(
Inj1
(
Inj0
(
x7
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
0
)
(
Inj1
(
Inj1
0
)
)
⟶
x2
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
Inj1
0
)
⟶
x2
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x6
(
λ x9 x10 .
0
)
(
λ x9 :
ι → ι
.
λ x10 .
Inj1
(
Inj0
(
Inj0
0
)
)
)
)
(
Inj1
0
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι →
ι → ι
.
x2
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
0
⟶
x2
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x5
)
(
Inj0
x5
)
)
⟶
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
x3
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
setsum
(
Inj0
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
)
(
setsum
(
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
setsum
0
0
)
)
(
setsum
0
0
)
)
)
(
x5
(
λ x8 .
Inj1
(
setsum
(
Inj0
0
)
(
Inj0
0
)
)
)
(
x7
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj0
(
Inj1
0
)
)
)
(
λ x8 .
0
)
(
Inj0
(
Inj0
(
Inj0
0
)
)
)
)
Inj1
(
setsum
(
Inj1
(
x5
(
λ x8 .
x8
)
(
setsum
0
0
)
(
λ x8 .
0
)
0
)
)
(
Inj1
(
Inj1
(
x7
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
)
⟶
x1
(
λ x8 :
(
ι →
ι →
ι → ι
)
→
ι → ι
.
λ x9 :
(
ι →
ι → ι
)
→
ι → ι
.
0
)
(
λ x8 x9 x10 .
x9
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x7 .
x1
(
λ x8 :
(
ι →
ι →
ι → ι
)
→
ι → ι
.
λ x9 :
(
ι →
ι → ι
)
→
ι → ι
.
Inj1
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
(
λ x8 x9 x10 .
setsum
x9
(
Inj0
(
setsum
x8
x8
)
)
)
⟶
In
(
setsum
(
x6
(
λ x8 .
0
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj0
(
setsum
0
0
)
)
x4
)
(
setsum
0
(
Inj1
0
)
)
)
(
Inj1
x7
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
ι →
ι →
(
ι → ι
)
→ ι
.
∀ x6 x7 .
In
(
setsum
(
setsum
0
0
)
0
)
(
x5
0
0
(
λ x8 .
setsum
(
Inj0
(
Inj1
0
)
)
0
)
)
⟶
x0
(
λ x8 .
x6
)
(
λ x8 .
0
)
(
λ x8 .
0
)
x7
x7
⟶
x0
(
λ x8 .
Inj0
0
)
(
λ x8 .
Inj1
0
)
(
λ x8 .
setsum
0
(
x5
0
(
setsum
(
setsum
0
0
)
(
Inj1
0
)
)
(
λ x9 .
x8
)
)
)
0
0
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 :
ι →
ι →
ι →
ι → ι
.
∀ x7 .
In
(
setsum
(
x5
(
Inj1
0
)
)
0
)
(
setsum
(
Inj0
x7
)
(
Inj1
(
setsum
x7
(
setsum
0
0
)
)
)
)
⟶
x0
(
λ x8 .
Inj1
0
)
(
λ x8 .
0
)
(
λ x8 .
Inj1
(
x5
0
)
)
(
x5
(
Inj1
0
)
)
0
⟶
x2
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
setsum
0
(
setsum
(
setsum
0
0
)
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
)
)
(
x5
(
setsum
0
x4
)
)
)
⟶
False
)
(proof)
Theorem
619f0..
:
not
(
∀ x0 :
(
ι → ι
)
→
(
(
ι →
ι →
ι → ι
)
→ ι
)
→ ο
.
∀ x1 :
(
ι → ι
)
→
(
ι → ι
)
→ ο
.
∀ x2 :
(
(
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
ι →
ι → ι
)
→
ι → ο
.
∀ x3 :
(
ι → ι
)
→
ι → ο
.
(
∀ x4 x5 x6 x7 .
x3
(
λ x8 .
0
)
0
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι → ι
.
In
(
x7
x4
)
(
Inj0
0
)
⟶
x3
(
λ x8 .
x7
x6
)
(
setsum
(
x7
(
setsum
(
setsum
0
0
)
x6
)
)
0
)
⟶
x2
(
λ x8 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x9 x10 x11 .
setsum
(
x8
(
λ x12 x13 :
ι → ι
.
Inj0
0
)
x10
(
λ x12 .
0
)
(
Inj0
(
setsum
0
0
)
)
)
(
setsum
(
setsum
(
Inj0
0
)
(
Inj0
0
)
)
(
Inj1
x11
)
)
)
(
setsum
(
setsum
(
Inj0
(
setsum
0
0
)
)
(
setsum
(
x5
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
(
λ x8 :
ι → ι
.
λ x9 .
0
)
)
(
setsum
0
0
)
)
)
(
setsum
(
x5
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj1
0
)
(
λ x8 :
ι → ι
.
λ x9 .
x9
)
)
(
Inj1
(
Inj1
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x8 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x9 x10 x11 .
Inj1
(
Inj0
x9
)
)
(
setsum
(
Inj1
(
Inj0
(
setsum
0
0
)
)
)
(
Inj1
(
setsum
(
Inj0
0
)
0
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
In
(
Inj0
(
setsum
(
x7
0
(
λ x8 :
ι → ι
.
λ x9 .
0
)
(
setsum
0
0
)
(
x6
0
)
)
(
x6
(
setsum
0
0
)
)
)
)
(
Inj0
(
Inj1
0
)
)
⟶
x2
(
λ x8 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x9 x10 x11 .
0
)
(
setsum
(
x6
(
Inj1
(
x6
0
)
)
)
(
Inj0
(
x7
(
setsum
0
0
)
(
λ x8 :
ι → ι
.
λ x9 .
0
)
x4
(
x7
0
(
λ x8 :
ι → ι
.
λ x9 .
0
)
0
0
)
)
)
)
⟶
x3
(
λ x8 .
setsum
(
setsum
(
x6
(
Inj1
0
)
)
(
x6
x5
)
)
(
x7
(
setsum
(
Inj0
0
)
(
x7
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
0
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
Inj0
0
)
(
Inj0
(
x7
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
0
0
)
)
(
x7
x8
(
λ x9 :
ι → ι
.
λ x10 .
setsum
0
0
)
0
(
Inj1
0
)
)
)
)
(
Inj0
x4
)
)
⟶
(
∀ x4 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x5 :
(
ι →
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
In
(
Inj1
(
Inj0
0
)
)
(
setsum
(
x4
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
(
λ x8 x9 .
x8
)
(
setsum
(
Inj1
0
)
(
x4
0
(
λ x8 x9 .
0
)
0
0
)
)
x6
)
(
setsum
(
x4
(
Inj1
0
)
(
λ x8 x9 .
Inj1
0
)
(
setsum
0
0
)
0
)
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
⟶
x0
(
λ x8 .
Inj1
x6
)
(
λ x8 :
ι →
ι →
ι → ι
.
0
)
⟶
x1
(
λ x8 .
Inj1
(
setsum
0
(
setsum
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
)
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
)
)
)
)
(
λ x8 .
setsum
(
x5
(
λ x9 x10 .
x10
)
)
(
setsum
0
(
Inj1
0
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x7 :
ι →
(
ι →
ι → ι
)
→ ι
.
In
(
x7
(
setsum
0
0
)
(
λ x8 x9 .
Inj1
(
Inj1
0
)
)
)
(
setsum
(
Inj1
x4
)
(
x6
(
λ x8 :
ι → ι
.
λ x9 x10 .
setsum
x9
0
)
(
setsum
(
setsum
0
0
)
(
x6
(
λ x8 :
ι → ι
.
λ x9 x10 .
0
)
0
0
)
)
(
x6
(
λ x8 :
ι → ι
.
λ x9 x10 .
Inj0
0
)
x4
0
)
)
)
⟶
x1
(
λ x8 .
setsum
0
(
setsum
0
(
Inj1
0
)
)
)
(
λ x8 .
setsum
(
x5
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
(
setsum
(
setsum
0
(
setsum
0
0
)
)
(
Inj0
(
Inj1
0
)
)
)
)
⟶
x1
(
λ x8 .
Inj1
(
x6
(
λ x9 :
ι → ι
.
λ x10 x11 .
setsum
(
Inj1
0
)
(
x9
0
)
)
(
Inj0
(
Inj0
0
)
)
(
Inj0
(
x5
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
)
)
)
(
λ x8 .
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x7 :
ι →
ι →
ι → ι
.
In
x4
(
x7
0
(
Inj1
(
setsum
(
Inj0
0
)
(
Inj0
0
)
)
)
x5
)
⟶
x0
(
λ x8 .
setsum
x8
(
setsum
x8
0
)
)
(
λ x8 :
ι →
ι →
ι → ι
.
0
)
⟶
x0
(
λ x8 .
x7
(
setsum
0
0
)
(
setsum
x8
0
)
(
Inj1
0
)
)
(
λ x8 :
ι →
ι →
ι → ι
.
0
)
)
⟶
(
∀ x4 x5 x6 x7 .
x0
(
λ x8 .
setsum
(
setsum
0
0
)
0
)
(
λ x8 :
ι →
ι →
ι → ι
.
setsum
(
setsum
0
0
)
(
setsum
(
Inj1
0
)
0
)
)
⟶
x3
(
λ x8 .
0
)
(
Inj0
(
setsum
(
Inj0
0
)
(
Inj1
(
setsum
0
0
)
)
)
)
)
⟶
False
)
(proof)