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Proofgold Asset
asset id
49dc465320b552fd0941b74ee2e0e1aee5a4542632fdaa2f7d13bfac7fd2f49e
asset hash
5e9baee40d9da4970dc934a6bc400e50d2f942ef79ef9ee9d3f698e8a67ce48c
bday / block
11884
tx
5e03b..
preasset
doc published by
PrGVS..
Known
8106d..
notI
:
∀ x0 : ο .
(
x0
⟶
False
)
⟶
not
x0
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Theorem
7a4d9..
:
not
(
∀ x0 :
(
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
)
→ ι
)
→ ο
.
∀ x2 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ο
.
∀ x3 :
(
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
)
→
ι → ο
.
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x0
(
λ x8 .
λ x9 :
ι → ι
.
0
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
setsum
0
x5
)
⟶
x3
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
x8
(
λ x9 :
ι → ι
.
Inj1
0
)
x6
)
(
setsum
(
Inj0
0
)
(
setsum
(
Inj1
(
x4
0
)
)
(
Inj1
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 :
ι →
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x7 .
x3
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
x8
(
λ x9 :
ι → ι
.
Inj0
x7
)
0
)
(
Inj1
x7
)
⟶
False
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
(
ι →
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x8 :
ι → ι
.
λ x9 .
setsum
(
x8
(
setsum
0
(
setsum
0
0
)
)
)
x7
)
(
λ x8 .
λ x9 :
(
ι → ι
)
→
ι → ι
.
x6
(
λ x10 x11 .
setsum
(
setsum
x11
(
setsum
0
0
)
)
x11
)
)
⟶
x2
(
λ x8 :
ι → ι
.
λ x9 .
0
)
(
λ x8 .
λ x9 :
(
ι → ι
)
→
ι → ι
.
Inj1
(
Inj1
0
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 x6 x7 .
x2
(
λ x8 :
ι → ι
.
Inj1
)
(
λ x8 .
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
⟶
False
)
⟶
(
∀ x4 :
ι →
ι →
ι →
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
In
(
Inj1
0
)
(
setsum
(
Inj0
(
x4
(
setsum
0
0
)
(
Inj1
0
)
(
Inj1
0
)
(
setsum
0
0
)
)
)
0
)
⟶
x1
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 x11 .
setsum
(
Inj1
(
setsum
0
(
Inj1
0
)
)
)
(
x9
(
Inj1
0
)
)
)
(
λ x8 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x9 x10 x11 .
x11
)
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
⟶
x1
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 x11 .
0
)
(
λ x8 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x9 x10 x11 .
setsum
(
setsum
0
(
Inj0
0
)
)
(
setsum
(
Inj0
0
)
0
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
setsum
(
Inj0
(
x7
(
setsum
0
0
)
)
)
0
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x1
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 x11 .
setsum
(
Inj0
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
(
setsum
(
setsum
(
setsum
0
0
)
0
)
(
Inj1
0
)
)
)
(
λ x8 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x9 x10 x11 .
0
)
(
λ x8 :
(
ι → ι
)
→ ι
.
setsum
0
(
Inj0
x6
)
)
⟶
In
(
Inj1
(
Inj1
0
)
)
x7
)
⟶
(
∀ x4 x5 :
ι →
ι → ι
.
∀ x6 x7 .
x0
(
λ x8 .
λ x9 :
ι → ι
.
0
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
x7
)
⟶
x0
(
λ x8 .
λ x9 :
ι → ι
.
x6
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 .
∀ x6 x7 :
ι → ι
.
x0
(
λ x8 .
λ x9 :
ι → ι
.
Inj0
(
setsum
0
(
Inj1
0
)
)
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
0
)
⟶
In
(
setsum
0
0
)
(
Inj0
0
)
)
⟶
False
)
(proof)
Known
TrueI
TrueI
:
True
Theorem
9033d..
:
not
(
∀ x0 :
(
(
ι →
ι →
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι → ο
.
∀ x1 :
(
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
)
→ ο
.
(
∀ x4 :
ι →
ι →
ι →
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x3
(
λ x8 .
λ x9 :
ι → ι
.
x9
(
Inj1
(
Inj0
x6
)
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
setsum
(
Inj1
(
setsum
x6
0
)
)
(
Inj0
0
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
∀ x7 .
In
(
setsum
(
Inj0
(
setsum
(
Inj0
0
)
0
)
)
(
Inj1
0
)
)
(
Inj0
x7
)
⟶
x3
(
λ x8 .
λ x9 :
ι → ι
.
x8
)
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj0
(
setsum
(
setsum
(
setsum
0
0
)
0
)
(
setsum
(
setsum
0
0
)
0
)
)
)
⟶
x3
(
λ x8 .
λ x9 :
ι → ι
.
0
)
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι →
ι → ι
.
∀ x7 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x8 .
x5
)
(
setsum
x5
(
x6
(
Inj1
0
)
(
Inj1
(
Inj0
0
)
)
)
)
⟶
x2
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x6
(
x8
(
λ x9 :
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
0
)
(
x6
0
x4
)
(
λ x8 x9 .
Inj1
(
x7
(
λ x10 .
Inj1
x9
)
0
)
)
)
⟶
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
x2
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
Inj1
0
)
0
(
λ x8 x9 .
setsum
(
Inj0
0
)
x7
)
⟶
In
(
Inj0
(
Inj0
(
setsum
x6
(
Inj0
0
)
)
)
)
(
Inj0
(
setsum
(
Inj0
(
setsum
0
0
)
)
(
setsum
(
x4
0
0
(
λ x8 .
0
)
0
)
x7
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x7 .
In
(
Inj1
x5
)
(
Inj1
(
setsum
(
Inj1
(
x6
(
λ x8 x9 :
ι → ι
.
λ x10 .
0
)
(
λ x8 x9 .
0
)
)
)
(
Inj1
(
setsum
0
0
)
)
)
)
⟶
x1
(
λ x8 .
x6
(
λ x9 x10 :
ι → ι
.
λ x11 .
Inj0
0
)
(
λ x9 x10 .
setsum
(
Inj0
0
)
0
)
)
(
setsum
0
(
setsum
(
setsum
0
(
setsum
0
0
)
)
(
setsum
0
0
)
)
)
⟶
x1
(
λ x8 .
0
)
(
setsum
0
(
Inj0
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x1
(
λ x8 .
0
)
(
Inj1
(
setsum
(
Inj1
(
setsum
0
0
)
)
(
Inj1
0
)
)
)
⟶
False
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
In
x5
(
setsum
(
setsum
0
0
)
(
Inj1
(
setsum
(
Inj0
0
)
x4
)
)
)
⟶
x1
(
λ x8 .
x7
)
(
Inj1
0
)
⟶
x0
(
λ x8 :
ι →
ι →
(
ι → ι
)
→ ι
.
setsum
(
setsum
x7
0
)
(
x6
0
)
)
0
(
setsum
(
Inj0
(
setsum
0
(
Inj0
0
)
)
)
0
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x0
(
λ x8 :
ι →
ι →
(
ι → ι
)
→ ι
.
0
)
(
setsum
0
(
Inj0
(
setsum
x5
(
Inj1
0
)
)
)
)
(
setsum
(
Inj0
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
(
Inj0
(
Inj1
0
)
)
)
⟶
In
(
Inj1
(
Inj1
(
Inj1
0
)
)
)
(
setsum
0
(
setsum
0
(
Inj1
0
)
)
)
)
⟶
False
)
(proof)
Theorem
8e3ed..
:
not
(
∀ x0 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ο
.
∀ x1 :
(
ι → ι
)
→
(
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
)
→ ο
.
∀ x2 :
(
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 :
(
ι →
ι → ι
)
→ ι
.
∀ x6 x7 .
x3
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
0
)
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
(
λ x8 :
ι → ι
.
λ x9 x10 .
Inj1
(
setsum
(
Inj0
x9
)
0
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x3
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
setsum
(
Inj1
0
)
(
setsum
(
Inj0
(
setsum
0
0
)
)
x7
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
x6
)
(
λ x8 :
ι → ι
.
λ x9 x10 .
x10
)
⟶
x3
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
Inj1
(
x10
(
λ x11 .
setsum
(
setsum
0
0
)
0
)
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
(
λ x8 :
ι → ι
.
λ x9 x10 .
setsum
x10
x7
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x5 x6 x7 .
In
(
x4
(
λ x8 .
λ x9 :
ι → ι
.
x6
)
(
λ x8 :
ι → ι
.
λ x9 .
Inj1
(
Inj0
(
setsum
0
0
)
)
)
(
setsum
0
(
setsum
(
setsum
0
0
)
0
)
)
x5
)
(
setsum
0
(
setsum
(
setsum
(
setsum
0
0
)
0
)
x7
)
)
⟶
x0
(
λ x8 :
ι →
ι → ι
.
x7
)
(
Inj1
(
setsum
x5
(
Inj1
x6
)
)
)
⟶
x2
(
λ x8 .
λ x9 :
ι →
ι →
ι → ι
.
Inj1
0
)
(
λ x8 .
x5
)
(
λ x8 .
0
)
x6
0
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→ ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 .
In
(
x6
(
Inj0
(
setsum
(
Inj1
0
)
(
Inj1
0
)
)
)
)
(
setsum
(
x5
(
x5
0
(
λ x8 :
ι → ι
.
setsum
0
0
)
(
λ x8 .
Inj1
0
)
(
setsum
0
0
)
)
(
λ x8 :
ι → ι
.
setsum
(
x6
0
)
0
)
(
λ x8 .
x5
0
(
λ x9 :
ι → ι
.
0
)
(
λ x9 .
x7
)
(
setsum
0
0
)
)
0
)
(
x4
(
λ x8 x9 .
Inj1
(
setsum
0
0
)
)
)
)
⟶
x2
(
λ x8 .
λ x9 :
ι →
ι →
ι → ι
.
x6
(
Inj1
x7
)
)
(
λ x8 .
x8
)
(
λ x8 .
setsum
0
0
)
(
x5
0
(
λ x8 :
ι → ι
.
x5
0
(
λ x9 :
ι → ι
.
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
(
λ x9 .
Inj1
(
x6
0
)
)
0
)
(
λ x8 .
setsum
(
x5
(
setsum
0
0
)
(
λ x9 :
ι → ι
.
Inj0
0
)
(
λ x9 .
Inj0
0
)
(
Inj1
0
)
)
(
Inj1
0
)
)
(
x4
(
λ x8 x9 .
0
)
)
)
(
x6
0
)
⟶
x0
(
λ x8 :
ι →
ι → ι
.
x5
(
Inj1
0
)
(
λ x9 :
ι → ι
.
Inj0
0
)
Inj1
(
Inj1
0
)
)
(
Inj1
(
x6
(
Inj0
x7
)
)
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
In
x5
x5
⟶
x1
(
λ x8 .
Inj1
x5
)
(
λ x8 :
(
ι →
ι → ι
)
→
ι → ι
.
0
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
(
ι → ι
)
→ ι
.
In
(
Inj0
(
setsum
(
Inj0
(
Inj1
0
)
)
0
)
)
(
Inj0
(
setsum
(
x4
(
Inj1
0
)
)
x6
)
)
⟶
x1
(
λ x8 .
0
)
(
λ x8 :
(
ι →
ι → ι
)
→
ι → ι
.
x5
)
⟶
x3
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
setsum
(
Inj1
(
setsum
0
0
)
)
(
Inj1
x8
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj1
(
setsum
(
Inj1
0
)
0
)
)
(
λ x8 :
ι → ι
.
λ x9 x10 .
x10
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x5 .
∀ x6 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
ι → ι
.
In
(
Inj0
0
)
(
setsum
(
x6
(
setsum
(
setsum
0
0
)
(
Inj1
0
)
)
(
λ x8 :
ι → ι
.
x5
)
)
(
x6
0
(
λ x8 :
ι → ι
.
x5
)
)
)
⟶
x0
(
λ x8 :
ι →
ι → ι
.
0
)
(
setsum
0
(
Inj1
(
setsum
(
Inj0
0
)
(
x4
(
λ x8 :
ι →
ι → ι
.
λ x9 :
ι → ι
.
0
)
(
λ x8 :
ι → ι
.
0
)
0
0
)
)
)
)
)
⟶
(
∀ x4 :
ι →
(
ι → ι
)
→ ι
.
∀ x5 :
ι →
(
ι →
ι → ι
)
→ ι
.
∀ x6 :
ι →
ι → ι
.
∀ x7 .
x0
(
λ x8 :
ι →
ι → ι
.
Inj1
(
setsum
(
setsum
(
setsum
0
0
)
0
)
(
setsum
(
Inj0
0
)
0
)
)
)
0
⟶
x3
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
setsum
0
(
x10
(
λ x11 .
setsum
0
0
)
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj0
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
)
(
λ x8 :
ι → ι
.
λ x9 x10 .
setsum
(
x8
0
)
x7
)
)
⟶
False
)
(proof)
Theorem
29cbb..
:
not
(
∀ x0 :
(
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
)
→ ι
)
→
ι →
(
(
ι → ι
)
→ ι
)
→ ο
.
∀ x3 :
(
ι → ι
)
→
ι →
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ο
.
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 .
In
(
setsum
0
(
setsum
(
x6
(
λ x8 .
Inj0
0
)
)
0
)
)
(
setsum
(
x6
(
λ x8 .
Inj0
(
setsum
0
0
)
)
)
(
Inj0
(
setsum
(
setsum
0
0
)
(
x6
(
λ x8 .
0
)
)
)
)
)
⟶
x0
(
λ x8 .
setsum
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
(
x6
(
λ x9 .
Inj0
x8
)
)
)
(
x4
(
Inj1
(
Inj1
(
Inj0
0
)
)
)
(
setsum
0
(
Inj1
(
x6
(
λ x8 .
0
)
)
)
)
)
⟶
x3
(
λ x8 .
x7
)
(
x5
(
λ x8 :
ι →
ι → ι
.
λ x9 .
setsum
0
(
Inj1
(
x8
0
0
)
)
)
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 .
setsum
(
Inj1
x7
)
(
x6
(
λ x10 .
x10
)
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x5 :
ι →
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
∀ x7 .
In
(
x4
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
)
(
Inj1
(
setsum
(
Inj1
0
)
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
)
)
⟶
x3
(
λ x8 .
Inj0
0
)
(
Inj0
(
Inj0
(
x5
0
(
λ x8 .
0
)
)
)
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 .
0
)
⟶
x0
(
λ x8 .
Inj0
(
Inj0
(
Inj1
x8
)
)
)
(
x6
(
λ x8 :
(
ι → ι
)
→ ι
.
x5
(
setsum
0
0
)
(
setsum
(
Inj0
0
)
)
)
(
setsum
(
x4
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj1
0
)
)
0
)
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→ ι
.
∀ x5 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
∀ x7 .
x3
(
λ x8 .
Inj1
0
)
0
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 .
0
)
⟶
x2
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x8 :
ι →
ι → ι
.
λ x9 :
(
ι → ι
)
→ ι
.
setsum
(
x9
(
λ x10 .
x8
(
setsum
0
0
)
0
)
)
(
setsum
0
0
)
)
(
setsum
(
Inj1
(
Inj0
(
setsum
0
0
)
)
)
(
Inj0
0
)
)
(
λ x8 :
ι → ι
.
x7
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 .
x2
(
λ x8 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
x5
)
(
λ x8 :
ι →
ι → ι
.
λ x9 :
(
ι → ι
)
→ ι
.
0
)
(
Inj1
(
Inj0
0
)
)
(
λ x8 :
ι → ι
.
setsum
(
setsum
(
Inj1
(
x6
0
0
0
)
)
(
setsum
x7
(
x8
0
)
)
)
x5
)
⟶
x3
(
λ x8 .
Inj0
(
Inj1
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
)
)
(
x6
0
0
0
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 .
Inj0
(
Inj1
0
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x3
(
λ x8 .
setsum
(
setsum
(
setsum
(
Inj1
0
)
0
)
(
Inj1
(
setsum
0
0
)
)
)
0
)
(
setsum
(
setsum
(
setsum
0
(
setsum
0
0
)
)
(
setsum
x5
x4
)
)
(
Inj1
0
)
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 .
x7
)
⟶
x1
(
λ x8 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x9 .
Inj1
x7
)
(
λ x8 x9 .
setsum
x6
0
)
(
λ x8 .
0
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x1
(
λ x8 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x9 .
Inj0
(
Inj0
(
setsum
x7
(
setsum
0
0
)
)
)
)
(
λ x8 x9 .
x7
)
(
λ x8 .
0
)
⟶
x3
(
λ x8 .
0
)
0
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
setsum
(
setsum
0
(
x8
(
λ x9 .
x7
)
0
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x8 .
x5
)
(
Inj0
0
)
⟶
x0
(
λ x8 .
setsum
x5
(
Inj1
(
Inj1
(
x6
0
)
)
)
)
(
setsum
(
Inj1
(
setsum
0
(
Inj1
0
)
)
)
(
setsum
(
x6
x7
)
(
Inj1
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 :
ι →
ι →
ι →
ι → ι
.
∀ x7 :
ι → ι
.
x0
(
λ x8 .
setsum
(
setsum
(
setsum
x8
(
x6
0
0
0
0
)
)
x8
)
(
setsum
0
0
)
)
(
setsum
(
x5
(
λ x8 :
ι → ι
.
λ x9 x10 .
setsum
x9
(
setsum
0
0
)
)
(
λ x8 .
x8
)
)
(
setsum
(
x6
(
setsum
0
0
)
(
setsum
0
0
)
(
Inj1
0
)
(
x7
0
)
)
(
setsum
0
(
x6
0
0
0
0
)
)
)
)
⟶
False
)
⟶
False
)
(proof)
Theorem
91710..
:
not
(
∀ x0 :
(
ι → ι
)
→
(
ι → ι
)
→
ι → ο
.
∀ x1 :
(
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
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→ ο
.
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→
ι → ι
.
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
x2
(
λ x8 .
x7
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x8
(
λ x9 :
ι → ι
.
λ x10 .
setsum
(
x8
(
λ x11 :
ι → ι
.
λ x12 .
setsum
0
0
)
(
λ x11 .
x8
(
λ x12 :
ι → ι
.
λ x13 .
0
)
(
λ x12 .
0
)
0
)
0
)
(
setsum
(
setsum
0
0
)
x10
)
)
(
λ x9 .
Inj1
0
)
0
)
⟶
x3
(
λ x8 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
setsum
(
Inj0
(
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
(
Inj1
0
)
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
λ x9 x10 .
x8
x7
(
λ x11 .
setsum
(
x8
(
Inj0
0
)
(
λ x12 .
x11
)
)
(
Inj1
x11
)
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x5 x6 x7 .
In
(
Inj1
x5
)
(
x4
(
λ x8 :
(
ι → ι
)
→ ι
.
λ x9 :
ι → ι
.
Inj1
(
setsum
(
setsum
0
0
)
0
)
)
(
λ x8 x9 .
x9
)
0
(
setsum
0
x6
)
)
⟶
x3
(
λ x8 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
setsum
x6
x6
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
λ x9 x10 .
setsum
(
setsum
(
Inj1
(
Inj1
0
)
)
0
)
(
Inj0
(
Inj0
x7
)
)
)
⟶
x1
(
λ x8 .
setsum
0
0
)
(
λ x8 .
setsum
(
setsum
(
setsum
(
Inj0
0
)
(
Inj1
0
)
)
x5
)
x7
)
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
0
)
)
⟶
(
∀ x4 x5 x6 x7 .
In
(
setsum
(
Inj1
(
Inj0
(
Inj1
0
)
)
)
x5
)
(
Inj1
0
)
⟶
x2
(
λ x8 .
setsum
(
Inj1
0
)
(
Inj1
(
setsum
0
(
setsum
0
0
)
)
)
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
)
⟶
(
∀ x4 :
ι →
(
ι → ι
)
→ ι
.
∀ x5 x6 x7 .
x2
(
λ x8 .
Inj0
0
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
⟶
x0
(
λ x8 .
x8
)
(
λ x8 .
0
)
(
setsum
(
Inj1
(
Inj1
x7
)
)
(
setsum
(
Inj0
(
Inj0
0
)
)
(
Inj1
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
(
ι →
ι →
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x1
(
λ x8 .
Inj0
(
Inj0
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
(
λ x8 .
x8
)
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
Inj0
(
x9
0
)
)
)
⟶
(
∀ x4 .
∀ x5 x6 :
ι →
ι → ι
.
∀ x7 .
In
x7
(
setsum
x7
(
x5
(
x5
(
x6
0
0
)
(
setsum
0
0
)
)
(
x5
(
x6
0
0
)
0
)
)
)
⟶
x1
(
λ x8 .
x7
)
(
λ x8 .
0
)
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
x8
)
⟶
x3
(
λ x8 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
Inj1
(
Inj1
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
)
(
λ x8 :
ι →
(
ι → ι
)
→ ι
.
λ x9 x10 .
setsum
x9
x7
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
∀ x6 :
ι →
ι → ι
.
∀ x7 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
x0
(
λ x8 .
Inj0
(
Inj1
0
)
)
(
λ x8 .
Inj0
(
x7
(
λ x9 .
setsum
0
(
Inj0
0
)
)
(
λ x9 :
ι → ι
.
Inj1
0
)
x8
0
)
)
(
Inj0
(
setsum
0
(
x4
(
λ x8 :
(
ι → ι
)
→ ι
.
Inj1
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
In
(
x7
(
λ x8 :
ι →
ι → ι
.
λ x9 :
ι → ι
.
x8
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
(
Inj1
(
x6
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
0
0
)
(
Inj1
(
x6
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
)
)
)
⟶
x0
(
λ x8 .
0
)
(
λ x8 .
0
)
(
Inj0
0
)
⟶
x2
(
λ x8 .
setsum
0
(
setsum
(
Inj1
0
)
(
setsum
(
Inj0
0
)
(
Inj0
0
)
)
)
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
Inj1
(
setsum
(
setsum
0
0
)
0
)
)
(
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
setsum
0
(
x10
0
)
)
(
setsum
(
Inj0
0
)
(
Inj0
0
)
)
)
)
)
⟶
False
)
(proof)
Theorem
05d15..
:
not
(
∀ x0 :
(
(
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
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ο
.
∀ x2 :
(
ι → ι
)
→
(
ι → ι
)
→ ο
.
∀ x3 :
(
ι → ι
)
→
ι → ο
.
(
∀ x4 x5 x6 x7 .
In
(
setsum
(
Inj0
0
)
0
)
(
Inj0
(
Inj1
x5
)
)
⟶
x3
(
λ x8 .
setsum
x8
(
setsum
(
setsum
(
Inj0
0
)
0
)
(
setsum
0
x5
)
)
)
x4
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
x3
(
λ x8 .
setsum
(
Inj0
(
Inj1
(
x6
(
λ x9 :
ι →
ι → ι
.
λ x10 :
ι → ι
.
λ x11 .
0
)
0
(
λ x9 .
0
)
)
)
)
(
Inj1
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
)
(
x5
0
)
⟶
x1
(
λ x8 :
ι → ι
.
λ x9 x10 .
λ x11 :
ι → ι
.
λ x12 .
Inj0
x9
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
x9
(
λ x11 .
setsum
0
(
Inj1
0
)
)
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 :
ι → ι
.
Inj1
(
setsum
(
setsum
(
x6
(
λ x10 :
ι →
ι → ι
.
λ x11 :
ι → ι
.
λ x12 .
0
)
0
(
λ x10 .
0
)
)
(
setsum
0
0
)
)
0
)
)
)
⟶
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
In
(
Inj1
(
setsum
0
(
Inj1
(
x4
0
0
(
λ x8 .
0
)
0
)
)
)
)
(
setsum
(
setsum
(
Inj0
(
Inj1
0
)
)
0
)
x5
)
⟶
x1
(
λ x8 :
ι → ι
.
λ x9 x10 .
λ x11 :
ι → ι
.
λ x12 .
Inj1
0
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
Inj1
(
x10
(
setsum
(
x10
0
)
(
setsum
0
0
)
)
)
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 :
ι → ι
.
0
)
⟶
x2
(
λ x8 .
Inj1
x7
)
(
setsum
(
Inj1
(
Inj0
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
(
ι →
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
In
(
x7
(
λ x8 x9 .
0
)
(
λ x8 x9 .
Inj1
(
setsum
(
x7
(
λ x10 x11 .
0
)
(
λ x10 x11 .
0
)
)
(
setsum
0
0
)
)
)
)
(
Inj0
0
)
⟶
x2
(
λ x8 .
0
)
(
λ x8 .
0
)
⟶
x0
(
λ x8 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
Inj1
(
Inj1
0
)
)
(
λ x8 .
λ x9 x10 :
ι → ι
.
λ x11 .
setsum
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
(
x9
0
)
)
(
λ x8 .
0
)
(
Inj1
(
x4
(
λ x8 .
x8
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x8 .
0
)
(
λ x8 .
0
)
⟶
x1
(
λ x8 :
ι → ι
.
λ x9 x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
0
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 :
ι → ι
.
setsum
(
setsum
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
(
Inj0
0
)
)
(
Inj1
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x1
(
λ x8 :
ι → ι
.
λ x9 x10 .
λ x11 :
ι → ι
.
λ x12 .
x11
(
Inj1
(
Inj0
x10
)
)
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
0
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
λ x9 :
ι → ι
.
0
)
⟶
x3
(
λ x8 .
0
)
(
Inj0
(
Inj1
(
x5
(
λ x8 x9 .
0
)
(
λ x8 :
ι → ι
.
setsum
0
0
)
(
λ x8 .
0
)
)
)
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
ι →
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
In
(
Inj0
(
Inj1
0
)
)
(
x5
(
Inj0
(
setsum
0
0
)
)
(
λ x8 .
0
)
)
⟶
x0
(
λ x8 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x8 .
λ x9 x10 :
ι → ι
.
λ x11 .
setsum
(
Inj0
(
x10
(
setsum
0
0
)
)
)
(
Inj0
x11
)
)
(
λ x8 .
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
setsum
(
setsum
(
x7
0
)
(
Inj0
0
)
)
0
)
0
(
λ x9 .
setsum
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
(
setsum
x9
(
x6
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
0
(
λ x10 .
0
)
)
)
)
)
(
setsum
(
setsum
0
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
(
Inj0
0
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x8 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
Inj1
x6
)
(
λ x8 .
λ x9 x10 :
ι → ι
.
λ x11 .
0
)
(
λ x8 .
Inj0
(
setsum
(
Inj1
(
setsum
0
0
)
)
0
)
)
(
Inj0
x5
)
⟶
x2
(
λ x8 .
x6
)
(
λ x8 .
x7
(
λ x9 :
ι →
ι → ι
.
setsum
(
setsum
x6
(
setsum
0
0
)
)
0
)
)
)
⟶
False
)
(proof)
Theorem
6993e..
:
not
(
∀ x0 :
(
(
(
(
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
)
→ 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
)
→ ο
.
(
∀ x4 x5 x6 .
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
In
(
Inj0
0
)
x6
⟶
x0
(
λ x8 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
setsum
(
setsum
(
setsum
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
)
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
)
)
(
setsum
0
0
)
)
x5
)
(
setsum
(
Inj1
0
)
(
x7
(
λ x8 .
λ x9 :
ι → ι
.
λ x10 .
Inj1
(
Inj1
0
)
)
)
)
(
Inj1
0
)
(
λ x8 :
ι → ι
.
x7
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
)
(
Inj0
x5
)
⟶
x3
(
λ x8 :
ι →
ι → ι
.
λ x9 .
x8
(
Inj0
0
)
(
Inj0
0
)
)
x5
(
λ x8 :
(
ι → ι
)
→ ι
.
0
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 x7 .
x3
(
λ x8 :
ι →
ι → ι
.
λ x9 .
Inj0
x6
)
(
setsum
(
setsum
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
0
)
(
setsum
0
(
x5
(
λ x8 .
setsum
0
0
)
)
)
)
(
λ x8 :
(
ι → ι
)
→ ι
.
x8
(
λ x9 .
Inj1
(
setsum
0
(
setsum
0
0
)
)
)
)
⟶
x1
(
λ x8 x9 .
λ x10 :
ι → ι
.
x7
)
(
λ x8 .
setsum
(
Inj0
(
setsum
(
Inj0
0
)
0
)
)
x7
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
ι →
(
ι → ι
)
→ ι
.
x2
(
λ x8 x9 .
Inj1
0
)
(
λ x8 :
ι → ι
.
setsum
0
(
x7
(
Inj1
(
setsum
0
0
)
)
(
λ x9 .
Inj1
(
setsum
0
0
)
)
)
)
(
λ x8 :
ι →
ι → ι
.
λ x9 x10 .
x7
(
Inj0
x9
)
(
λ x11 .
x8
(
setsum
(
Inj0
0
)
(
Inj1
0
)
)
x10
)
)
0
⟶
x2
(
λ x8 x9 .
0
)
(
λ x8 :
ι → ι
.
Inj0
0
)
(
λ x8 :
ι →
ι → ι
.
λ x9 x10 .
Inj1
(
Inj0
(
Inj0
(
Inj1
0
)
)
)
)
(
x7
(
x4
(
λ x8 x9 .
setsum
0
(
Inj0
0
)
)
(
λ x8 :
ι → ι
.
λ x9 .
x6
0
(
λ x10 :
ι → ι
.
x7
0
(
λ x11 .
0
)
)
)
(
λ x8 .
Inj1
0
)
)
(
λ x8 .
Inj1
(
Inj0
(
Inj1
0
)
)
)
)
)
⟶
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 :
ι →
ι →
ι →
ι → ι
.
∀ x7 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
x2
(
λ x8 x9 .
setsum
x9
x9
)
(
λ x8 :
ι → ι
.
0
)
(
λ x8 :
ι →
ι → ι
.
λ x9 x10 .
Inj0
(
Inj0
(
setsum
0
(
x7
0
(
λ x11 x12 .
0
)
0
0
)
)
)
)
(
Inj0
(
Inj0
(
Inj1
(
setsum
0
0
)
)
)
)
⟶
x1
(
λ x8 x9 .
λ x10 :
ι → ι
.
Inj1
x8
)
(
λ x8 .
setsum
(
Inj0
(
x5
(
Inj0
0
)
)
)
(
x6
0
x8
x8
(
Inj0
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 x7 .
x2
(
λ x8 x9 .
setsum
0
(
Inj1
0
)
)
(
λ x8 :
ι → ι
.
x6
)
(
λ x8 :
ι →
ι → ι
.
λ x9 x10 .
x9
)
0
⟶
x1
(
λ x8 x9 .
λ x10 :
ι → ι
.
setsum
(
Inj0
(
setsum
(
x10
0
)
(
setsum
0
0
)
)
)
(
setsum
(
setsum
x7
(
Inj1
0
)
)
0
)
)
(
λ x8 .
0
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 :
ι →
(
ι → ι
)
→ ι
.
In
(
Inj1
0
)
(
setsum
0
(
x4
(
λ x8 .
x8
)
)
)
⟶
x1
(
λ x8 x9 .
λ x10 :
ι → ι
.
x7
(
Inj1
0
)
(
λ x11 .
0
)
)
(
λ x8 .
0
)
⟶
x1
(
λ x8 x9 .
λ x10 :
ι → ι
.
x7
(
setsum
(
setsum
x9
0
)
x9
)
(
λ x11 .
setsum
0
x9
)
)
(
setsum
(
setsum
0
(
Inj1
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
(
ι → ι
)
→ ι
)
→ ι
.
In
(
Inj0
0
)
(
setsum
0
(
setsum
(
setsum
(
Inj0
0
)
(
x7
(
λ x8 :
ι → ι
.
0
)
)
)
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
⟶
x0
(
λ x8 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x8
(
λ x9 :
ι →
ι → ι
.
λ x10 :
ι → ι
.
λ x11 .
0
)
(
λ x9 :
ι → ι
.
λ x10 .
Inj0
(
Inj1
(
setsum
0
0
)
)
)
)
(
setsum
0
(
Inj1
(
setsum
0
(
Inj0
0
)
)
)
)
(
x7
(
λ x8 :
ι → ι
.
Inj1
(
Inj1
x6
)
)
)
(
λ x8 :
ι → ι
.
Inj0
(
x8
x5
)
)
0
⟶
x0
(
λ x8 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
Inj0
(
setsum
x5
x6
)
)
0
(
λ x8 :
ι → ι
.
setsum
(
Inj1
(
setsum
0
0
)
)
(
Inj0
x6
)
)
0
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
ι →
ι → ι
.
In
(
setsum
0
0
)
(
Inj1
(
Inj1
(
Inj0
(
Inj0
0
)
)
)
)
⟶
x0
(
λ x8 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
Inj0
0
)
(
Inj1
(
setsum
0
(
Inj1
(
Inj0
0
)
)
)
)
x5
(
λ x8 :
ι → ι
.
setsum
x6
x5
)
(
setsum
(
setsum
(
setsum
0
0
)
0
)
0
)
⟶
x3
(
λ x8 :
ι →
ι → ι
.
λ x9 .
Inj0
0
)
0
(
λ x8 :
(
ι → ι
)
→ ι
.
setsum
0
(
setsum
(
x7
(
x7
0
0
)
(
setsum
0
0
)
)
(
x7
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
)
⟶
False
)
(proof)
Theorem
30bb5..
:
not
(
∀ x0 :
(
(
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
)
→ 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 → ο
.
(
∀ x4 x5 x6 .
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
x3
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
setsum
(
setsum
(
setsum
(
Inj1
0
)
(
Inj0
0
)
)
(
x9
(
λ x11 x12 .
x10
)
(
λ x11 .
Inj1
0
)
)
)
(
x7
(
λ x11 :
(
ι → ι
)
→ ι
.
x9
(
λ x12 x13 .
Inj0
0
)
(
λ x12 .
0
)
)
)
)
(
setsum
(
setsum
(
Inj1
(
setsum
0
0
)
)
0
)
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
)
(
setsum
(
setsum
(
Inj0
(
setsum
0
0
)
)
(
Inj1
0
)
)
(
Inj0
(
Inj0
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x3
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
setsum
0
(
setsum
(
setsum
x8
(
Inj0
0
)
)
(
Inj1
(
setsum
0
0
)
)
)
)
(
Inj1
x7
)
(
x5
x4
(
λ x8 x9 .
x7
)
(
λ x8 .
setsum
x6
0
)
(
x5
(
x5
(
setsum
0
0
)
(
λ x8 x9 .
Inj1
0
)
(
λ x8 .
Inj1
0
)
(
Inj0
0
)
)
(
λ x8 x9 .
Inj1
(
setsum
0
0
)
)
(
λ x8 .
setsum
0
0
)
0
)
)
⟶
x1
(
λ x8 .
Inj1
0
)
(
λ x8 :
ι →
ι →
ι → ι
.
x5
(
Inj0
(
Inj0
0
)
)
(
λ x9 x10 .
setsum
0
0
)
(
λ x9 .
x6
)
(
setsum
(
x8
0
x7
(
Inj0
0
)
)
(
x5
0
(
λ x9 x10 .
Inj1
0
)
(
λ x9 .
Inj1
0
)
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x8 :
ι →
ι → ι
.
λ x9 x10 x11 .
0
)
(
λ x8 .
x7
)
(
x6
(
setsum
(
Inj0
0
)
(
setsum
(
setsum
0
0
)
0
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
In
(
Inj0
(
Inj1
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
(
setsum
0
0
)
⟶
x2
(
λ x8 :
ι →
ι → ι
.
λ x9 x10 x11 .
Inj0
(
setsum
(
Inj0
(
Inj0
0
)
)
0
)
)
(
λ x8 .
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
Inj1
(
setsum
0
0
)
)
(
λ x9 x10 .
Inj1
x9
)
)
(
setsum
x5
0
)
⟶
x1
(
λ x8 .
setsum
0
(
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
x6
(
setsum
0
0
)
)
(
λ x9 x10 .
Inj1
(
setsum
0
0
)
)
)
)
(
λ x8 :
ι →
ι →
ι → ι
.
Inj0
(
Inj0
(
Inj0
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
In
(
setsum
(
setsum
(
x4
(
λ x8 x9 :
ι → ι
.
λ x10 .
0
)
x5
)
(
setsum
(
Inj0
0
)
x6
)
)
0
)
(
Inj0
(
setsum
(
Inj0
(
Inj1
0
)
)
x5
)
)
⟶
x3
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
0
)
0
x7
⟶
x1
(
λ x8 .
0
)
(
λ x8 :
ι →
ι →
ι → ι
.
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
(
ι → ι
)
→
ι →
ι → ι
.
x1
(
λ x8 .
0
)
(
λ x8 :
ι →
ι →
ι → ι
.
x6
)
⟶
x0
(
λ x8 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x7
(
λ x9 .
Inj0
(
Inj1
0
)
)
(
x8
(
Inj1
(
setsum
0
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
(
x8
(
setsum
0
0
)
(
λ x9 :
ι → ι
.
λ x10 .
x9
0
)
0
)
)
(
x7
(
λ x9 .
setsum
(
setsum
0
0
)
(
Inj1
0
)
)
(
Inj0
(
setsum
0
0
)
)
x6
)
)
(
λ x8 x9 .
Inj0
)
(
λ x8 :
ι →
ι → ι
.
Inj0
(
setsum
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
(
setsum
(
Inj1
0
)
0
)
)
)
(
λ x8 :
ι → ι
.
x6
)
(
λ x8 .
x6
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 .
In
(
setsum
0
(
setsum
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
0
)
)
(
Inj1
(
Inj1
x7
)
)
⟶
x0
(
λ x8 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
(
λ x8 x9 x10 .
Inj0
(
Inj1
x8
)
)
(
λ x8 :
ι →
ι → ι
.
0
)
(
λ x8 :
ι → ι
.
x6
(
setsum
0
(
Inj1
(
x8
0
)
)
)
)
(
λ x8 .
Inj1
(
Inj1
(
Inj0
(
x5
0
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
x0
(
λ x8 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x8
(
x8
(
setsum
(
Inj1
0
)
(
x8
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
Inj0
(
x8
0
(
λ x11 :
ι → ι
.
λ x12 .
0
)
0
)
)
(
setsum
0
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
(
setsum
(
Inj1
x6
)
x6
)
)
(
λ x8 x9 x10 .
Inj1
(
setsum
(
Inj0
(
Inj1
0
)
)
x9
)
)
(
λ x8 :
ι →
ι → ι
.
0
)
(
λ x8 :
ι → ι
.
x8
x6
)
(
λ x8 .
setsum
x8
(
Inj1
(
setsum
x5
(
setsum
0
0
)
)
)
)
⟶
x0
(
λ x8 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x8
0
(
λ x9 :
ι → ι
.
λ x10 .
Inj0
(
setsum
0
0
)
)
(
setsum
x6
(
Inj1
(
Inj1
0
)
)
)
)
(
λ x8 x9 x10 .
0
)
(
λ x8 :
ι →
ι → ι
.
Inj1
(
setsum
x5
0
)
)
(
λ x8 :
ι → ι
.
0
)
(
λ x8 .
setsum
0
0
)
)
⟶
False
)
(proof)
Theorem
6c860..
:
not
(
∀ x0 :
(
ι →
ι → ι
)
→
ι → ο
.
∀ x1 :
(
ι → ι
)
→
ι → ο
.
∀ x2 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
)
→
ι → ο
.
∀ x3 :
(
ι →
ι →
ι → ι
)
→
ι → ο
.
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x8 x9 .
setsum
0
(
setsum
(
setsum
x8
(
x6
(
λ x10 .
0
)
0
(
λ x10 .
0
)
)
)
x7
)
)
(
λ x8 :
ι → ι
.
λ x9 :
ι →
ι → ι
.
x6
(
λ x10 .
Inj0
(
setsum
0
0
)
)
(
setsum
(
Inj1
0
)
(
setsum
(
Inj1
0
)
(
x8
0
)
)
)
(
λ x10 .
0
)
)
x7
⟶
x3
(
λ x8 x9 x10 .
x9
)
0
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 .
In
(
Inj0
0
)
(
Inj1
(
setsum
0
(
setsum
0
(
setsum
0
0
)
)
)
)
⟶
x3
(
λ x8 x9 x10 .
Inj1
(
Inj0
0
)
)
(
Inj1
0
)
⟶
x1
(
λ x8 .
0
)
(
Inj1
(
Inj0
(
Inj0
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
In
(
Inj0
0
)
(
Inj0
(
Inj0
(
Inj1
(
Inj0
0
)
)
)
)
⟶
x2
(
λ x8 x9 .
0
)
(
λ x8 :
ι → ι
.
λ x9 :
ι →
ι → ι
.
setsum
(
Inj1
(
setsum
(
Inj0
0
)
0
)
)
x6
)
0
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
ι →
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
In
(
Inj0
(
setsum
0
(
setsum
0
(
Inj0
0
)
)
)
)
x6
⟶
x2
(
λ x8 x9 .
setsum
0
x6
)
(
λ x8 :
ι → ι
.
λ x9 :
ι →
ι → ι
.
Inj0
(
x9
0
0
)
)
(
setsum
(
x5
(
λ x8 x9 .
Inj1
0
)
(
x5
(
λ x8 x9 .
x9
)
(
setsum
0
0
)
)
)
0
)
⟶
x0
(
λ x8 x9 .
setsum
(
setsum
x9
(
setsum
(
setsum
0
0
)
0
)
)
x6
)
(
setsum
(
setsum
(
x7
(
Inj1
0
)
(
λ x8 x9 .
setsum
0
0
)
(
λ x8 .
setsum
0
0
)
0
)
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
)
(
setsum
(
Inj1
(
Inj1
0
)
)
(
setsum
(
x4
0
)
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x5 :
ι →
(
ι → ι
)
→ ι
.
∀ x6 x7 .
x1
(
λ x8 .
0
)
(
Inj1
0
)
)
⟶
(
∀ x4 x5 x6 x7 .
x1
(
λ x8 .
0
)
x5
⟶
x1
(
λ x8 .
x6
)
0
)
⟶
(
∀ x4 x5 x6 x7 .
In
(
Inj1
(
Inj0
(
Inj1
0
)
)
)
(
Inj1
(
Inj1
0
)
)
⟶
x0
(
λ x8 x9 .
x7
)
(
setsum
0
0
)
)
⟶
(
∀ x4 .
∀ x5 x6 :
ι → ι
.
∀ x7 .
x0
(
λ x8 x9 .
0
)
(
x5
(
Inj0
(
Inj0
0
)
)
)
⟶
x3
(
λ x8 x9 x10 .
0
)
0
)
⟶
False
)
(proof)
Theorem
1b96e..
:
not
(
∀ x0 :
(
(
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
)
→
(
ι →
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ο
.
∀ x1 :
(
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
)
→ ο
.
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x8 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
λ x9 :
ι →
ι →
ι → ι
.
setsum
0
(
x9
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
(
Inj1
0
)
0
)
)
(
λ x8 .
x7
)
⟶
x3
(
λ x8 .
Inj0
(
x6
(
setsum
(
Inj0
0
)
(
x6
0
)
)
)
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x9 .
x7
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x6 .
∀ x7 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x8 .
setsum
(
Inj0
0
)
(
Inj0
0
)
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x9 .
0
)
⟶
x1
(
λ x8 .
λ x9 :
ι → ι
.
0
)
(
Inj1
0
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 x7 .
In
(
Inj1
0
)
(
setsum
x7
(
Inj0
(
setsum
0
(
x4
(
λ x8 :
ι →
ι → ι
.
λ x9 :
ι → ι
.
0
)
)
)
)
)
⟶
x2
(
λ x8 .
setsum
(
Inj1
0
)
(
Inj0
0
)
)
(
Inj1
(
setsum
0
0
)
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
setsum
(
setsum
0
(
Inj1
(
Inj0
0
)
)
)
(
setsum
x6
0
)
)
⟶
x2
(
λ x8 .
0
)
x7
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
In
x7
(
Inj1
(
setsum
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
(
x5
(
setsum
0
0
)
(
setsum
0
0
)
(
λ x8 .
0
)
0
)
)
)
⟶
x2
(
λ x8 .
Inj1
x6
)
(
x5
(
Inj0
(
setsum
x6
0
)
)
x7
(
λ x8 .
x7
)
x4
)
(
λ x8 :
(
ι → ι
)
→
ι → ι
.
Inj1
x7
)
⟶
x0
(
λ x8 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
λ x9 :
ι →
ι →
ι → ι
.
0
)
(
λ x8 .
x8
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
In
(
setsum
(
setsum
(
x6
(
λ x8 :
(
ι → ι
)
→ ι
.
λ x9 .
setsum
0
0
)
(
setsum
0
0
)
(
λ x8 .
setsum
0
0
)
)
(
Inj0
(
setsum
0
0
)
)
)
(
setsum
0
(
Inj0
(
Inj1
0
)
)
)
)
(
x6
(
λ x8 :
(
ι → ι
)
→ ι
.
x7
)
0
(
λ x8 .
0
)
)
⟶
x1
(
λ x8 .
λ x9 :
ι → ι
.
x9
0
)
(
Inj1
(
setsum
(
x7
(
Inj0
0
)
)
(
x7
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x6 x7 .
In
(
setsum
x4
x7
)
(
Inj1
(
setsum
(
setsum
(
setsum
0
0
)
0
)
x4
)
)
⟶
x1
(
λ x8 .
λ x9 :
ι → ι
.
0
)
0
⟶
x3
(
λ x8 .
x7
)
(
λ x8 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x9 .
setsum
0
x6
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι →
ι →
ι →
ι → ι
.
∀ x7 .
x0
(
λ x8 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
λ x9 :
ι →
ι →
ι → ι
.
0
)
(
λ x8 .
0
)
⟶
x0
(
λ x8 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
λ x9 :
ι →
ι →
ι → ι
.
Inj0
(
setsum
x7
x7
)
)
(
λ x8 .
Inj1
(
Inj1
0
)
)
)
⟶
(
∀ x4 x5 x6 :
ι → ι
.
∀ x7 .
x0
(
λ x8 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
λ x9 :
ι →
ι →
ι → ι
.
x7
)
(
λ x8 .
Inj1
0
)
⟶
In
x7
(
x4
(
setsum
0
(
x4
(
Inj0
0
)
)
)
)
)
⟶
False
)
(proof)
Theorem
c13a6..
:
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 → 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 → ο
.
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
In
(
x7
0
)
x4
⟶
x0
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→ ι
.
Inj1
(
x6
(
λ x10 .
Inj0
0
)
(
λ x10 .
x10
)
)
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
0
)
⟶
x3
(
λ x8 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x9 .
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
Inj1
0
)
0
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι → ι
.
In
(
setsum
0
(
Inj1
x6
)
)
(
Inj1
(
Inj1
(
x5
(
Inj1
0
)
(
λ x8 :
ι → ι
.
λ x9 .
0
)
)
)
)
⟶
x3
(
λ x8 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x9 .
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
Inj1
0
)
x4
⟶
x2
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
(
setsum
(
setsum
(
setsum
0
0
)
0
)
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
0
)
0
(
λ x8 .
0
)
(
λ x8 .
setsum
(
Inj0
(
Inj1
(
x5
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
)
)
0
)
)
⟶
(
∀ x4 .
∀ x5 x6 :
ι → ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
x0
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
x7
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
setsum
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
)
)
⟶
x2
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x6
0
)
0
(
λ x8 .
0
)
(
λ x8 .
0
)
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x2
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x8
(
λ x9 :
ι → ι
.
x7
)
)
0
(
λ x8 .
Inj0
(
x6
(
λ x9 .
setsum
x8
(
Inj0
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
setsum
x7
(
setsum
0
0
)
)
(
λ x9 .
0
)
0
)
)
(
λ x8 .
setsum
x8
(
Inj0
0
)
)
⟶
False
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
ι →
ι → ι
.
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→
ι →
ι → ι
.
∀ x7 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x2
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj0
(
setsum
(
x8
(
λ x9 :
ι → ι
.
Inj1
0
)
)
0
)
)
0
(
λ x8 .
0
)
(
λ x8 .
Inj1
(
Inj0
(
setsum
0
(
setsum
0
0
)
)
)
)
⟶
x1
(
λ x8 x9 .
setsum
0
0
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 .
Inj1
(
setsum
(
setsum
(
x6
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
(
λ x10 .
0
)
0
0
)
(
setsum
0
0
)
)
(
setsum
0
(
x8
(
λ x10 :
ι → ι
.
0
)
)
)
)
)
(
λ x8 .
Inj0
(
setsum
(
x6
(
λ x9 :
(
ι → ι
)
→ ι
.
x9
(
λ x10 .
0
)
)
(
λ x9 .
0
)
0
0
)
(
x6
(
λ x9 :
(
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x9 .
x7
0
(
λ x10 x11 .
0
)
(
λ x10 .
0
)
0
)
(
x5
0
0
)
0
)
)
)
(
λ x8 .
setsum
(
Inj0
(
setsum
0
0
)
)
(
Inj1
0
)
)
(
Inj1
(
x5
(
x7
(
Inj0
0
)
(
λ x8 x9 .
setsum
0
0
)
(
λ x8 .
setsum
0
0
)
(
x5
0
0
)
)
(
Inj1
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x1
(
λ x8 x9 .
0
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x9 .
Inj0
(
x6
0
)
)
(
λ x8 .
0
)
(
λ x8 .
x7
)
(
x6
0
)
⟶
x2
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
0
(
λ x8 .
Inj1
(
x5
x8
(
λ x9 :
ι → ι
.
0
)
)
)
(
λ x8 .
setsum
(
Inj0
x8
)
(
x6
0
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
In
(
Inj0
x7
)
(
Inj0
x5
)
⟶
x2
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x5
)
x7
(
λ x8 .
Inj0
(
setsum
(
Inj1
(
setsum
0
0
)
)
0
)
)
(
λ x8 .
0
)
⟶
x0
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x9
(
λ x10 x11 .
x9
(
λ x12 x13 .
x10
)
)
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
Inj0
0
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
(
ι → ι
)
→ ι
)
→ ι
.
In
(
Inj1
0
)
(
Inj1
(
setsum
0
0
)
)
⟶
x0
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→ ι
.
Inj0
(
setsum
0
(
setsum
(
Inj0
0
)
(
x9
(
λ x10 x11 .
0
)
)
)
)
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
Inj1
0
)
⟶
x0
(
λ x8 .
λ x9 :
(
ι →
ι → ι
)
→ ι
.
Inj1
0
)
(
λ x8 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
setsum
0
(
x5
(
λ x9 .
0
)
)
)
)
⟶
False
)
(proof)
Theorem
29cb8..
:
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 → ο
.
∀ x2 :
(
ι → ι
)
→
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
ι → ο
.
∀ x3 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
ι → ο
.
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 :
(
ι → ι
)
→ ι
.
In
(
Inj1
0
)
(
Inj1
(
x7
(
λ x8 .
setsum
(
Inj0
0
)
(
Inj1
0
)
)
)
)
⟶
x1
(
λ x8 .
Inj1
0
)
(
λ x8 .
λ x9 :
ι →
ι → ι
.
setsum
(
setsum
(
setsum
(
Inj1
0
)
0
)
0
)
(
setsum
(
setsum
(
setsum
0
0
)
0
)
(
setsum
0
(
x9
0
0
)
)
)
)
(
λ x8 :
ι →
ι → ι
.
0
)
(
Inj0
(
Inj0
(
setsum
0
(
x5
0
)
)
)
)
⟶
x3
(
λ x8 .
λ x9 :
ι → ι
.
x6
(
λ x10 :
ι → ι
.
λ x11 .
x10
(
Inj0
(
x9
0
)
)
)
)
(
x6
(
λ x8 :
ι → ι
.
λ x9 .
x6
(
λ x10 :
ι → ι
.
λ x11 .
setsum
(
setsum
0
0
)
(
Inj1
0
)
)
)
)
(
setsum
(
Inj0
0
)
0
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x7 .
x3
(
λ x8 .
λ x9 :
ι → ι
.
setsum
(
setsum
x8
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
)
0
)
(
x5
(
λ x8 :
ι →
ι → ι
.
x6
(
λ x9 :
ι →
ι → ι
.
setsum
0
(
setsum
0
0
)
)
x7
)
)
(
x5
(
λ x8 :
ι →
ι → ι
.
Inj0
0
)
)
⟶
In
(
Inj0
(
setsum
(
setsum
0
0
)
(
Inj0
(
setsum
0
0
)
)
)
)
(
Inj0
0
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 x7 .
In
(
setsum
(
setsum
0
(
setsum
(
x4
(
λ x8 .
0
)
(
λ x8 :
ι → ι
.
0
)
)
x6
)
)
(
setsum
(
x4
(
λ x8 .
Inj1
0
)
(
λ x8 :
ι → ι
.
setsum
0
0
)
)
0
)
)
(
Inj1
(
Inj1
(
setsum
(
Inj1
0
)
0
)
)
)
⟶
x3
(
λ x8 .
λ x9 :
ι → ι
.
0
)
(
Inj0
(
setsum
x5
x5
)
)
0
⟶
x2
(
λ x8 .
0
)
(
λ x8 :
(
ι →
ι → ι
)
→ ι
.
Inj0
(
x8
(
λ x9 x10 .
0
)
)
)
x6
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
ι →
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
(
ι → ι
)
→
ι → ι
.
In
(
setsum
(
setsum
(
setsum
(
x7
(
λ x8 .
0
)
0
)
0
)
(
x7
(
λ x8 .
Inj1
0
)
(
setsum
0
0
)
)
)
x6
)
(
Inj0
x6
)
⟶
x2
(
λ x8 .
0
)
(
λ x8 :
(
ι →
ι → ι
)
→ ι
.
x5
(
λ x9 x10 x11 .
setsum
(
Inj1
x10
)
(
Inj0
(
Inj0
0
)
)
)
(
x5
(
λ x9 x10 x11 .
Inj0
0
)
0
)
)
(
Inj1
(
setsum
(
setsum
x6
(
Inj1
0
)
)
(
Inj0
(
Inj0
0
)
)
)
)
⟶
x0
(
λ x8 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
x7
(
λ x9 .
0
)
(
setsum
0
0
)
)
(
setsum
(
Inj0
0
)
x6
)
(
λ x8 .
setsum
(
Inj0
(
Inj0
(
Inj0
0
)
)
)
(
setsum
(
x5
(
λ x9 x10 x11 .
Inj1
0
)
(
setsum
0
0
)
)
(
setsum
0
0
)
)
)
(
Inj0
x6
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x8 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
setsum
(
setsum
(
x5
(
Inj1
0
)
(
λ x9 :
ι → ι
.
setsum
0
0
)
(
λ x9 .
setsum
0
0
)
)
(
x7
(
λ x9 :
ι → ι
.
setsum
0
0
)
)
)
(
setsum
(
setsum
0
(
setsum
0
0
)
)
(
setsum
(
x5
0
(
λ x9 :
ι → ι
.
0
)
(
λ x9 .
0
)
)
(
x8
0
0
(
λ x9 .
0
)
0
)
)
)
)
x6
(
λ x8 .
Inj1
(
Inj1
(
x5
(
x5
0
(
λ x9 :
ι → ι
.
0
)
(
λ x9 .
0
)
)
(
λ x9 :
ι → ι
.
x8
)
(
λ x9 .
Inj1
0
)
)
)
)
0
⟶
x1
(
λ x8 .
setsum
(
setsum
0
0
)
0
)
(
λ x8 .
λ x9 :
ι →
ι → ι
.
x9
x8
(
x9
(
Inj1
(
setsum
0
0
)
)
(
Inj1
(
x9
0
0
)
)
)
)
(
λ x8 :
ι →
ι → ι
.
x6
)
(
Inj1
0
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x8 .
x7
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
Inj1
(
x7
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x10
(
λ x11 .
0
)
0
)
)
)
)
(
λ x8 .
λ x9 :
ι →
ι → ι
.
Inj0
0
)
(
λ x8 :
ι →
ι → ι
.
Inj1
(
setsum
0
x6
)
)
(
setsum
0
(
setsum
(
Inj1
0
)
0
)
)
⟶
In
(
setsum
0
(
setsum
0
(
Inj1
(
setsum
0
0
)
)
)
)
(
Inj0
(
setsum
(
Inj1
(
Inj1
0
)
)
(
Inj0
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 x7 .
x1
(
λ x8 .
0
)
(
λ x8 .
λ x9 :
ι →
ι → ι
.
x7
)
(
λ x8 :
ι →
ι → ι
.
x5
(
setsum
x7
(
setsum
(
Inj1
0
)
0
)
)
)
x7
⟶
x0
(
λ x8 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
Inj0
(
setsum
(
setsum
x6
x6
)
x7
)
)
0
(
λ x8 .
x8
)
(
Inj0
x7
)
)
⟶
(
∀ x4 x5 :
ι → ι
.
∀ x6 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 .
x0
(
λ x8 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
setsum
(
setsum
(
Inj1
(
x5
0
)
)
0
)
(
x6
(
λ x9 :
ι → ι
.
λ x10 .
Inj0
0
)
)
)
(
Inj1
x7
)
(
λ x8 .
x6
(
λ x9 :
ι → ι
.
λ x10 .
x9
0
)
)
x7
⟶
In
(
x6
(
λ x8 :
ι → ι
.
λ x9 .
setsum
(
Inj0
(
Inj0
0
)
)
(
Inj0
(
x8
0
)
)
)
)
(
Inj1
(
setsum
(
x4
(
setsum
0
0
)
)
(
setsum
(
x6
(
λ x8 :
ι → ι
.
λ x9 .
0
)
)
(
Inj0
0
)
)
)
)
)
⟶
False
)
(proof)