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Proofgold Asset
asset id
f039b7a94f47907045f0cee60c7cfb80800b48049ad0ce3753240175767cb861
asset hash
aa084f55a1a706974e40ddcfe452d8ceb64567b0d635a8c9812c33791d955a62
bday / block
11849
tx
6b899..
preasset
doc published by
PrGVS..
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
notE
notE
:
∀ x0 : ο .
not
x0
⟶
x0
⟶
False
Known
e3ec9..
neq_0_1
:
not
(
0
=
1
)
Theorem
ad662..
:
∀ x0 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x1 :
(
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι → ι
.
∀ x2 :
(
(
(
(
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
.
∀ x3 :
(
(
ι →
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
(
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
(
∀ x4 x5 x6 .
∀ x7 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
x11
(
λ x14 :
ι → ι
.
x0
(
λ x15 .
x13
)
(
λ x15 x16 .
x14
0
)
(
λ x15 .
λ x16 :
ι → ι
.
λ x17 .
setsum
0
0
)
)
)
(
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
x13
)
(
x3
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x10
0
(
λ x11 :
ι → ι
.
0
)
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
setsum
0
0
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
=
Inj0
x4
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
(
ι →
ι →
ι → ι
)
→ ι
.
∀ x7 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
setsum
0
0
)
=
x4
(
λ x9 .
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
setsum
x12
x12
)
(
x2
(
λ x10 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x11 .
0
)
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
(
λ x10 x11 x12 .
0
)
(
λ x10 .
x2
(
λ x11 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
setsum
0
0
)
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
(
λ x11 x12 x13 .
0
)
(
λ x11 .
x9
)
(
Inj1
0
)
(
setsum
0
0
)
)
(
x2
(
λ x10 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x11 .
Inj1
0
)
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
(
λ x10 x11 x12 .
0
)
(
λ x10 .
setsum
0
0
)
0
(
x3
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
)
)
(
x2
(
λ x10 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x11 .
x3
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
0
)
)
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x11
0
)
(
λ x10 x11 x12 .
setsum
0
0
)
(
λ x10 .
0
)
0
(
x3
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
)
)
)
)
(
λ x9 x10 .
x7
(
λ x11 :
ι → ι
.
setsum
0
0
)
(
λ x11 .
setsum
(
x7
(
λ x12 :
ι → ι
.
x9
)
(
λ x12 .
x3
(
λ x13 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
0
)
)
)
x10
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
0
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
x11
)
(
λ x9 x10 x11 .
x9
)
(
λ x9 .
x7
)
x7
(
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 x12 .
x3
(
λ x13 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x10
(
λ x14 :
ι → ι
.
x14
0
)
)
(
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
0
)
)
(
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
Inj0
(
Inj1
0
)
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
x11
)
(
λ x9 x10 x11 .
0
)
(
λ x9 .
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
x1
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x15 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 x17 .
0
)
0
)
(
setsum
0
0
)
)
(
x6
(
λ x9 .
0
)
)
(
x0
(
λ x9 .
x3
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
λ x9 x10 .
setsum
0
0
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
Inj0
0
)
)
)
)
=
setsum
(
x6
(
λ x9 .
setsum
(
x6
(
λ x10 .
0
)
)
(
x0
(
λ x10 .
x10
)
(
λ x10 x11 .
x7
)
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
setsum
0
0
)
)
)
)
(
setsum
(
x0
(
λ x9 .
Inj0
(
setsum
0
0
)
)
(
λ x9 x10 .
x9
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
)
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
(
ι → ι
)
→
ι →
ι →
ι → ι
.
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
x6
(
setsum
(
setsum
(
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
)
(
Inj1
0
)
)
(
x0
(
λ x11 .
x7
(
λ x12 .
0
)
0
0
0
)
(
λ x11 x12 .
setsum
0
0
)
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
Inj1
0
)
)
)
(
Inj0
(
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x10
)
(
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
setsum
0
0
)
)
)
(
x0
(
λ x11 .
x3
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x9
(
λ x13 x14 :
ι → ι
.
0
)
(
λ x13 x14 .
0
)
)
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
0
)
)
(
λ x11 x12 .
x11
)
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x12
(
x1
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x15 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 x17 .
0
)
0
)
)
)
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
Inj1
(
x10
(
Inj0
0
)
)
)
(
λ x9 x10 x11 .
setsum
0
0
)
(
λ x9 .
Inj0
(
Inj0
0
)
)
0
x5
=
x6
(
Inj1
0
)
(
x7
(
λ x9 .
x9
)
(
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x7
(
λ x10 .
x2
(
λ x11 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
0
)
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
(
λ x11 x12 x13 .
0
)
(
λ x11 .
0
)
0
0
)
0
(
Inj1
0
)
(
x2
(
λ x10 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x11 .
0
)
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
(
λ x10 x11 x12 .
0
)
(
λ x10 .
0
)
0
0
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
x4
x5
)
(
setsum
0
(
Inj0
0
)
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
ι →
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 x12 .
0
)
0
=
x4
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
x9
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x7 .
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 x12 .
x0
(
setsum
(
x1
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 x16 .
x2
(
λ x17 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x18 .
0
)
(
λ x17 .
λ x18 :
ι → ι
.
λ x19 .
0
)
(
λ x17 x18 x19 .
0
)
(
λ x17 .
0
)
0
0
)
(
setsum
0
0
)
)
)
(
λ x13 x14 .
x14
)
(
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
x12
)
)
(
setsum
(
Inj0
(
x6
(
λ x9 .
λ x10 :
ι → ι
.
x2
(
λ x11 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
0
)
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
(
λ x11 x12 x13 .
0
)
(
λ x11 .
0
)
0
0
)
0
)
)
(
x5
(
λ x9 .
x0
(
λ x10 .
x2
(
λ x11 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
0
)
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
(
λ x11 x12 x13 .
0
)
(
λ x11 .
0
)
0
0
)
(
λ x10 x11 .
0
)
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x1
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 x16 .
0
)
0
)
)
(
λ x9 :
ι → ι
.
Inj0
(
x9
0
)
)
(
x5
(
λ x9 .
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
0
)
0
)
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
0
)
0
)
0
x7
)
0
)
)
=
x0
(
λ x9 .
x5
(
λ x10 .
Inj0
(
setsum
(
setsum
0
0
)
0
)
)
(
λ x10 :
ι → ι
.
x10
x9
)
(
x3
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x11 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
x3
(
λ x13 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
0
)
)
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x11
)
(
λ x11 x12 x13 .
setsum
0
0
)
(
λ x11 .
Inj0
0
)
(
Inj0
0
)
(
Inj0
0
)
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x3
(
λ x14 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj0
0
)
(
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
λ x17 .
setsum
0
0
)
)
)
(
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
x3
(
λ x14 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
λ x17 .
0
)
)
0
)
)
(
λ x9 x10 .
x9
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
setsum
(
setsum
0
(
Inj0
(
x0
(
λ x12 .
0
)
(
λ x12 x13 .
0
)
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
)
)
)
x7
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x0
(
λ x9 .
x0
(
λ x10 .
setsum
0
(
x7
x6
(
λ x11 :
ι → ι
.
λ x12 .
Inj1
0
)
)
)
(
λ x10 x11 .
0
)
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x10
)
)
(
λ x9 x10 .
x2
(
λ x11 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
x9
)
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x2
(
λ x14 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x15 .
x2
(
λ x16 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x17 .
x1
(
λ x18 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x19 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x20 x21 .
0
)
0
)
(
λ x16 .
λ x17 :
ι → ι
.
λ x18 .
0
)
(
λ x16 x17 x18 .
x18
)
(
λ x16 .
setsum
0
0
)
0
(
x14
(
λ x16 x17 :
ι → ι
.
0
)
(
λ x16 x17 .
0
)
)
)
(
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
setsum
(
x3
(
λ x17 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x17 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x18 .
λ x19 :
ι → ι
.
λ x20 .
0
)
)
(
setsum
0
0
)
)
(
λ x14 x15 x16 .
x1
(
λ x17 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x18 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x19 x20 .
Inj1
0
)
(
setsum
0
0
)
)
(
λ x14 .
x13
)
(
Inj1
(
Inj1
0
)
)
(
setsum
0
x10
)
)
(
λ x11 x12 x13 .
0
)
(
λ x11 .
setsum
(
Inj1
0
)
(
x7
(
x0
(
λ x12 .
0
)
(
λ x12 x13 .
0
)
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
)
(
λ x12 :
ι → ι
.
λ x13 .
x3
(
λ x14 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
λ x17 .
0
)
)
)
)
(
Inj0
(
Inj1
(
Inj1
0
)
)
)
x9
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
=
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
x6
)
(
λ x9 .
λ x10 :
ι → ι
.
setsum
0
)
(
λ x9 x10 x11 .
Inj1
(
Inj1
x11
)
)
(
λ x9 .
x3
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x9
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x13
)
)
(
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
0
)
0
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
)
(
x4
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
(
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
x6
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
(
λ x9 x10 x11 .
Inj1
(
x0
(
λ x12 .
0
)
(
λ x12 x13 .
0
)
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
)
)
(
λ x9 .
setsum
x9
(
x0
(
λ x10 .
0
)
(
λ x10 x11 .
0
)
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
)
)
0
(
setsum
(
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
0
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
(
λ x9 x10 x11 .
0
)
(
λ x9 .
0
)
0
0
)
(
x0
(
λ x9 .
0
)
(
λ x9 x10 .
0
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
ι → ι
)
→ ι
.
∀ x6 x7 .
x0
(
λ x9 .
0
)
(
λ x9 x10 .
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
x11
(
λ x15 :
ι → ι
.
λ x16 .
x0
(
λ x17 .
x2
(
λ x18 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x19 .
0
)
(
λ x18 .
λ x19 :
ι → ι
.
λ x20 .
0
)
(
λ x18 x19 x20 .
0
)
(
λ x18 .
0
)
0
0
)
(
λ x17 x18 .
x0
(
λ x19 .
0
)
(
λ x19 x20 .
0
)
(
λ x19 .
λ x20 :
ι → ι
.
λ x21 .
0
)
)
(
λ x17 .
λ x18 :
ι → ι
.
λ x19 .
x16
)
)
(
x3
(
λ x15 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
0
0
)
(
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x16 .
λ x17 :
ι → ι
.
λ x18 .
setsum
0
0
)
)
(
Inj1
0
)
)
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
setsum
x9
(
Inj0
(
x3
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
0
0
)
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
x2
(
λ x16 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x17 .
0
)
(
λ x16 .
λ x17 :
ι → ι
.
λ x18 .
0
)
(
λ x16 x17 x18 .
0
)
(
λ x16 .
0
)
0
0
)
)
)
)
=
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
(
x3
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x7
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
x5
(
λ x10 x11 .
x3
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 x16 .
0
)
0
)
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
setsum
0
0
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x10
)
)
⟶
False
(proof)
Theorem
c3fef..
:
∀ x0 :
(
ι →
ι → ι
)
→
ι →
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ 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
.
∀ x2 :
(
ι → ι
)
→
(
ι →
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x3 :
(
(
ι →
ι →
ι →
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x3
(
λ x9 :
ι →
ι →
ι →
ι → ι
.
λ x10 .
Inj0
0
)
(
λ x9 .
x2
(
λ x10 .
x7
(
x7
0
)
)
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 x13 .
x3
(
λ x14 :
ι →
ι →
ι →
ι → ι
.
λ x15 .
setsum
0
(
x3
(
λ x16 :
ι →
ι →
ι →
ι → ι
.
λ x17 .
0
)
(
λ x16 .
0
)
0
)
)
(
λ x14 .
0
)
(
x3
(
λ x14 :
ι →
ι →
ι →
ι → ι
.
λ x15 .
setsum
0
0
)
(
λ x14 .
x12
)
(
x0
(
λ x14 x15 .
0
)
0
(
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
λ x14 x15 .
0
)
)
)
)
)
(
x5
(
x2
(
λ x9 .
x3
(
λ x10 :
ι →
ι →
ι →
ι → ι
.
λ x11 .
x7
0
)
(
λ x10 .
x1
(
λ x11 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x11 :
ι → ι
.
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 x13 .
0
)
)
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
0
)
)
)
=
x5
(
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
setsum
(
x9
(
λ x13 .
x0
(
λ x14 x15 .
0
)
0
(
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
λ x14 x15 .
0
)
)
(
λ x13 x14 .
0
)
)
(
setsum
0
(
x3
(
λ x13 :
ι →
ι →
ι →
ι → ι
.
λ x14 .
0
)
(
λ x13 .
0
)
0
)
)
)
(
λ x9 :
ι → ι
.
setsum
x6
(
x9
(
x3
(
λ x10 :
ι →
ι →
ι →
ι → ι
.
λ x11 .
0
)
(
λ x10 .
0
)
0
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 .
Inj0
x10
)
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
ι →
ι →
(
ι → ι
)
→ ι
.
x3
(
λ x9 :
ι →
ι →
ι →
ι → ι
.
λ x10 .
x10
)
(
λ x9 .
x7
0
0
(
λ x10 .
0
)
)
(
x7
(
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x2
(
λ x13 .
x2
(
λ x14 .
0
)
(
λ x14 .
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 x17 .
0
)
)
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 x16 .
x2
(
λ x17 .
0
)
(
λ x17 .
λ x18 :
(
ι → ι
)
→ ι
.
λ x19 x20 .
0
)
)
)
(
λ x9 :
ι → ι
.
Inj0
(
x7
0
0
(
λ x10 .
0
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 .
x11
)
)
0
(
λ x9 .
x5
)
)
=
setsum
0
0
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
ι →
ι → ι
.
∀ x6 x7 :
ι → ι
.
x2
(
λ x9 .
setsum
x9
(
x7
x9
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
0
)
=
x6
0
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x2
(
λ x9 .
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
x0
(
λ x13 x14 .
Inj0
(
Inj0
(
x2
(
λ x15 .
0
)
(
λ x15 .
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 x18 .
0
)
)
)
)
(
x0
(
λ x13 x14 .
0
)
x12
(
λ x13 :
(
ι → ι
)
→ ι
.
x12
)
(
λ x13 x14 .
Inj0
x12
)
)
(
λ x13 :
(
ι → ι
)
→ ι
.
setsum
(
Inj0
x11
)
0
)
(
λ x13 x14 .
Inj0
0
)
)
=
x0
(
λ x9 .
setsum
(
Inj1
(
setsum
(
x6
0
)
(
x2
(
λ x10 .
0
)
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 x13 .
0
)
)
)
)
)
(
x2
(
λ x9 .
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
setsum
(
setsum
x9
(
x0
(
λ x13 x14 .
0
)
0
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 .
0
)
)
)
(
setsum
(
x10
(
λ x13 .
0
)
)
x12
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
setsum
(
setsum
(
Inj1
(
x6
0
)
)
0
)
(
setsum
(
Inj0
(
x7
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 x12 .
0
)
)
)
0
)
)
(
λ x9 x10 .
Inj1
(
Inj0
(
x1
(
λ x11 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
x3
(
λ x15 :
ι →
ι →
ι →
ι → ι
.
λ x16 .
0
)
(
λ x15 .
0
)
0
)
(
λ x11 :
ι → ι
.
setsum
0
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 x13 .
x12
)
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
x2
(
λ x10 .
x9
(
x2
(
λ x11 .
x3
(
λ x12 :
ι →
ι →
ι →
ι → ι
.
λ x13 .
0
)
(
λ x12 .
0
)
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 x14 .
setsum
0
0
)
)
)
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 x13 .
x0
(
λ x14 x15 .
0
)
0
(
λ x14 :
(
ι → ι
)
→ ι
.
Inj0
(
x2
(
λ x15 .
0
)
(
λ x15 .
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 x18 .
0
)
)
)
(
λ x14 x15 .
x13
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 .
0
)
=
x2
(
λ x9 .
Inj0
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
x10
(
λ x13 .
0
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x5 x6 x7 .
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
setsum
0
x10
)
(
λ x9 :
ι → ι
.
x7
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 .
0
)
=
x7
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x0
(
λ x9 x10 .
x3
(
λ x11 :
ι →
ι →
ι →
ι → ι
.
λ x12 .
x11
(
Inj0
(
x1
(
λ x13 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 :
ι → ι
.
0
)
(
λ x13 :
ι → ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 x15 .
0
)
)
)
x10
x9
(
x2
(
λ x13 .
x0
(
λ x14 x15 .
0
)
0
(
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
λ x14 x15 .
0
)
)
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 x16 .
0
)
)
)
(
λ x11 .
setsum
(
Inj0
0
)
0
)
(
x1
(
λ x11 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x11 :
ι → ι
.
x0
(
λ x12 x13 .
x11
0
)
0
(
λ x12 :
(
ι → ι
)
→ ι
.
Inj0
0
)
(
λ x12 x13 .
x10
)
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 x13 .
x13
)
)
)
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
Inj1
x7
)
(
λ x9 x10 .
x10
)
=
x3
(
λ x9 :
ι →
ι →
ι →
ι → ι
.
λ x10 .
setsum
(
setsum
0
0
)
0
)
(
λ x9 .
x0
(
λ x10 x11 .
x2
(
λ x12 .
x11
)
(
λ x12 .
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 x15 .
0
)
)
0
(
λ x10 :
(
ι → ι
)
→ ι
.
setsum
(
Inj1
0
)
0
)
(
λ x10 x11 .
x0
(
λ x12 x13 .
x12
)
(
x1
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 x14 .
x13
)
)
(
λ x12 :
(
ι → ι
)
→ ι
.
setsum
0
x11
)
(
λ x12 x13 .
x13
)
)
)
(
Inj0
x7
)
)
⟶
(
∀ x4 :
ι →
(
ι → ι
)
→ ι
.
∀ x5 x6 x7 .
x0
(
λ x9 x10 .
x1
(
λ x11 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x11 :
ι → ι
.
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 x13 .
setsum
(
Inj1
(
Inj0
0
)
)
(
x11
(
λ x14 .
x11
(
λ x15 .
0
)
)
)
)
)
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
(
λ x9 x10 .
x6
)
=
setsum
x7
0
)
⟶
False
(proof)
Theorem
d6666..
:
∀ x0 :
(
ι → ι
)
→
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x1 :
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x2 :
(
ι → ι
)
→
ι → ι
.
∀ x3 :
(
(
ι → ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
.
(
∀ x4 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
∀ x5 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
ι →
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x6
(
x10
(
λ x11 :
ι → ι
.
λ x12 .
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
(
λ x11 .
x0
(
λ x12 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 x14 .
setsum
0
0
)
)
0
)
(
λ x11 .
setsum
(
x1
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x13 .
0
)
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 x15 .
0
)
)
(
λ x12 .
0
)
)
0
)
(
x2
(
λ x11 .
setsum
0
(
x0
(
λ x12 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 x14 .
0
)
)
)
0
)
)
(
x4
(
λ x9 :
ι →
ι → ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x10
(
λ x11 :
ι → ι
.
x1
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 .
0
)
)
)
(
λ x10 .
x0
(
λ x11 .
setsum
0
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
setsum
0
0
)
)
)
(
λ x9 x10 .
x7
)
(
setsum
0
(
x6
x7
(
λ x9 .
Inj1
0
)
0
)
)
)
=
setsum
x7
(
x4
(
λ x9 :
ι →
ι → ι
.
setsum
(
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj1
0
)
(
λ x10 .
setsum
0
0
)
)
(
x3
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
(
setsum
0
0
)
)
)
(
λ x9 x10 .
x1
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
(
Inj0
0
)
)
(
λ x11 .
x7
)
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
setsum
(
x6
0
(
λ x11 .
0
)
0
)
(
setsum
0
0
)
)
(
x4
(
λ x9 :
ι →
ι → ι
.
setsum
0
0
)
(
λ x9 x10 .
x9
)
(
Inj1
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
Inj0
0
)
(
x10
(
λ x11 :
ι → ι
.
λ x12 .
0
)
(
λ x11 .
x0
(
λ x12 .
x1
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 .
0
)
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 x14 .
Inj1
0
)
)
0
)
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
x5
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
x7
)
(
λ x9 .
setsum
(
x2
(
λ x10 .
0
)
0
)
(
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 .
0
)
)
)
(
λ x9 .
Inj0
x6
)
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
=
Inj1
(
setsum
0
0
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
ι → ι
.
x2
(
λ x9 .
0
)
(
setsum
(
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x1
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 .
0
)
)
(
Inj0
0
)
)
)
(
setsum
0
(
Inj0
(
x2
(
λ x9 .
0
)
0
)
)
)
)
=
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
x0
(
λ x10 .
x7
0
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 x12 .
setsum
0
(
x9
(
λ x13 .
setsum
0
0
)
0
)
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 .
x7
)
(
Inj0
0
)
=
Inj1
0
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
ι →
(
ι → ι
)
→ ι
.
∀ x6 x7 .
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x2
(
λ x11 .
0
)
(
x1
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 .
setsum
0
0
)
)
)
)
=
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
setsum
0
(
setsum
(
setsum
0
0
)
(
setsum
x7
0
)
)
)
(
setsum
(
x4
(
x5
(
setsum
0
0
)
(
λ x9 .
x2
(
λ x10 .
0
)
0
)
)
)
x7
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x9
(
λ x10 :
ι → ι
.
x10
(
setsum
(
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
0
)
(
Inj0
0
)
)
)
)
Inj0
=
setsum
(
x2
(
λ x9 .
0
)
(
x2
(
λ x9 .
Inj1
(
x3
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
0
)
)
(
setsum
0
(
setsum
0
0
)
)
)
)
0
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 .
x0
(
λ x9 .
0
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 x11 .
x10
)
=
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
setsum
(
x2
(
λ x10 .
x7
)
(
Inj1
0
)
)
(
x0
(
λ x10 .
x9
(
λ x11 .
Inj1
0
)
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 x12 .
x3
(
λ x13 :
ι → ι
.
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
(
x3
(
λ x13 :
ι → ι
.
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x9 .
x2
(
λ x10 .
setsum
0
x9
)
x9
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 x11 .
x11
)
=
x2
(
λ x9 .
Inj1
(
x7
(
λ x10 :
ι →
ι → ι
.
setsum
(
Inj0
0
)
(
x0
(
λ x11 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
0
)
)
)
)
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x0
(
λ x11 .
x2
(
λ x12 .
x2
(
λ x13 .
0
)
0
)
(
setsum
0
0
)
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
x3
(
λ x14 :
ι → ι
.
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
(
setsum
0
0
)
)
)
0
)
)
⟶
False
(proof)
Theorem
098cf..
:
∀ x0 :
(
(
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
.
∀ x1 :
(
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
ι → ι
.
∀ x2 :
(
ι → ι
)
→
(
ι →
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x3 :
(
(
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι →
ι →
ι → ι
)
→
ι →
ι →
ι → ι
)
→ ι
.
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
Inj1
(
Inj1
(
setsum
x6
(
setsum
0
0
)
)
)
)
(
λ x9 :
ι →
ι →
ι → ι
.
λ x10 x11 x12 .
x11
)
=
Inj0
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
x9
(
λ x10 :
ι →
ι → ι
.
λ x11 .
x0
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
0
)
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
x3
(
λ x15 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x15 :
ι →
ι →
ι → ι
.
λ x16 x17 x18 .
0
)
)
(
λ x12 :
(
ι → ι
)
→ ι
.
x0
(
λ x13 :
ι → ι
.
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
λ x17 .
0
)
(
λ x13 :
ι → ι
.
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
(
λ x9 :
ι →
ι →
ι → ι
.
λ x10 x11 x12 .
x3
(
λ x13 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
x1
(
λ x14 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
0
0
)
(
setsum
0
0
)
)
(
λ x13 :
ι →
ι →
ι → ι
.
λ x14 x15 x16 .
x14
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x5 .
∀ x6 :
(
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x7 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x9 :
ι →
ι →
ι → ι
.
λ x10 x11 x12 .
0
)
=
setsum
x5
x5
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
ι →
ι → ι
.
∀ x6 .
∀ x7 :
ι →
ι →
ι →
ι → ι
.
x2
(
λ x9 .
setsum
(
x7
(
x1
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
(
setsum
0
0
)
)
x9
0
(
x5
(
Inj1
0
)
(
setsum
0
0
)
)
)
(
x0
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
x13
)
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x0
(
λ x13 :
ι → ι
.
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
λ x17 .
x3
(
λ x18 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x18 :
ι →
ι →
ι → ι
.
λ x19 x20 x21 .
0
)
)
(
λ x13 :
ι → ι
.
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
x0
(
λ x16 :
ι → ι
.
λ x17 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x18 .
λ x19 :
ι → ι
.
λ x20 .
0
)
(
λ x16 :
ι → ι
.
λ x17 :
(
ι → ι
)
→
ι → ι
.
λ x18 :
ι → ι
.
0
)
(
λ x16 :
(
ι → ι
)
→ ι
.
0
)
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
)
(
λ x10 :
(
ι → ι
)
→ ι
.
Inj1
(
setsum
0
0
)
)
)
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
λ x12 .
Inj0
0
)
=
Inj1
(
x7
(
setsum
(
setsum
(
setsum
0
0
)
(
Inj1
0
)
)
(
x4
(
x2
(
λ x9 .
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
)
)
)
0
(
x0
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x3
(
λ x14 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x14 :
ι →
ι →
ι → ι
.
λ x15 x16 x17 .
x3
(
λ x18 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x18 :
ι →
ι →
ι → ι
.
λ x19 x20 x21 .
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x0
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x7
0
0
0
0
)
)
)
(
Inj1
(
x1
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj0
0
)
(
x1
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 .
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
λ x12 .
Inj0
0
)
=
setsum
(
x0
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x2
(
λ x14 .
x14
)
(
λ x14 x15 .
λ x16 :
ι → ι
.
λ x17 .
x15
)
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
x9
(
x1
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
x1
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
(
x4
(
λ x9 .
setsum
(
x0
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
setsum
0
0
)
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x3
(
λ x13 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
λ x14 x15 x16 .
0
)
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
)
x5
)
(
x2
(
λ x9 .
x9
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
λ x12 .
x3
(
λ x13 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
λ x14 x15 x16 .
x0
(
λ x17 :
ι → ι
.
λ x18 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x19 .
λ x20 :
ι → ι
.
λ x21 .
0
)
(
λ x17 :
ι → ι
.
λ x18 :
(
ι → ι
)
→
ι → ι
.
λ x19 :
ι → ι
.
0
)
(
λ x17 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x1
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x7
)
(
x4
(
setsum
0
(
x5
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
(
x0
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
)
=
setsum
(
setsum
(
x2
(
setsum
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
λ x12 .
setsum
(
setsum
0
0
)
x10
)
)
(
x0
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x3
(
λ x14 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
x0
(
λ x15 :
ι → ι
.
λ x16 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x17 .
λ x18 :
ι → ι
.
λ x19 .
0
)
(
λ x15 :
ι → ι
.
λ x16 :
(
ι → ι
)
→
ι → ι
.
λ x17 :
ι → ι
.
0
)
(
λ x15 :
(
ι → ι
)
→ ι
.
0
)
)
(
λ x14 :
ι →
ι →
ι → ι
.
λ x15 x16 x17 .
0
)
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
x7
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
)
x7
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x9
(
λ x10 :
ι →
ι → ι
.
x10
0
(
x7
0
)
)
)
x5
=
x5
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
ι →
ι →
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x7 .
x0
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
Inj0
(
Inj0
(
x3
(
λ x14 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
x13
)
(
λ x14 :
ι →
ι →
ι → ι
.
λ x15 x16 x17 .
x2
(
λ x18 .
0
)
(
λ x18 x19 .
λ x20 :
ι → ι
.
λ x21 .
0
)
)
)
)
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
=
x6
(
λ x9 :
(
ι → ι
)
→ ι
.
x6
(
λ x10 :
(
ι → ι
)
→ ι
.
x1
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj0
0
)
(
Inj0
(
x0
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
0
)
(
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
(
λ x10 x11 .
Inj1
(
x3
(
λ x12 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x12 :
ι →
ι →
ι → ι
.
λ x13 x14 x15 .
0
)
)
)
)
(
λ x9 x10 .
x10
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
x0
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
setsum
(
x1
(
λ x14 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
(
x3
(
λ x15 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x15 :
ι →
ι →
ι → ι
.
λ x16 x17 x18 .
0
)
)
0
)
(
Inj0
x11
)
)
(
x1
(
λ x14 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x12
0
)
0
)
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
x11
(
x11
(
x9
(
x11
0
)
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x5
(
x6
(
x6
0
)
)
(
λ x10 .
0
)
(
λ x10 .
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
Inj0
(
Inj1
0
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
λ x12 x13 x14 .
0
)
)
(
x2
(
λ x10 .
x1
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x9
(
λ x12 .
0
)
)
0
)
(
λ x10 x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
)
)
=
x5
(
setsum
0
(
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 .
x2
(
λ x12 .
x9
(
λ x13 .
0
)
)
(
λ x12 x13 .
λ x14 :
ι → ι
.
λ x15 .
Inj1
0
)
)
(
Inj0
(
x5
0
(
λ x9 .
0
)
(
λ x9 .
0
)
0
)
)
(
λ x9 .
x9
)
)
)
(
λ x9 .
Inj0
(
x0
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
setsum
x14
0
)
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
)
)
(
λ x9 .
Inj0
(
x2
(
λ x10 .
0
)
(
λ x10 x11 .
λ x12 :
ι → ι
.
λ x13 .
x1
(
λ x14 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
)
)
x4
)
⟶
False
(proof)
Theorem
c1833..
:
∀ x0 :
(
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
(
ι →
ι →
ι → ι
)
→
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x1 :
(
(
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι →
ι →
ι → ι
)
→ ι
)
→
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x2 :
(
ι →
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
ι →
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
(
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
.
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 .
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x6
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x12
(
x10
(
λ x14 :
ι → ι
.
x1
(
λ x15 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x16 .
λ x17 :
ι → ι
.
λ x18 .
0
)
(
λ x15 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x15 .
0
)
0
)
(
λ x14 .
x1
(
λ x15 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x16 .
λ x17 :
ι → ι
.
λ x18 .
0
)
(
λ x15 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x15 .
0
)
0
)
)
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
x2
(
λ x15 .
λ x16 :
(
ι →
ι → ι
)
→ ι
.
λ x17 :
ι → ι
.
λ x18 .
x0
(
λ x19 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x19 x20 x21 .
x1
(
λ x22 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x23 .
λ x24 :
ι → ι
.
λ x25 .
0
)
(
λ x22 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x22 .
0
)
0
)
(
Inj0
0
)
(
λ x19 :
ι → ι
.
Inj0
0
)
)
(
λ x15 .
λ x16 :
ι →
ι → ι
.
x13
(
x2
(
λ x17 .
λ x18 :
(
ι →
ι → ι
)
→ ι
.
λ x19 :
ι → ι
.
λ x20 .
0
)
(
λ x17 .
λ x18 :
ι →
ι → ι
.
0
)
)
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
Inj0
(
Inj1
0
)
)
(
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
x11
(
λ x15 x16 x17 .
0
)
(
λ x15 .
setsum
0
0
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
x10
(
λ x12 .
setsum
0
0
)
)
(
setsum
0
0
)
0
(
λ x11 .
0
)
0
)
0
(
λ x11 .
x0
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj1
(
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x13 .
0
)
0
)
)
(
λ x12 x13 x14 .
0
)
(
x10
(
λ x12 .
x3
(
λ x13 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x14 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 :
ι → ι
.
0
)
0
)
)
(
λ x12 :
ι → ι
.
0
)
)
(
x2
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x13
(
Inj1
0
)
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
x0
(
λ x17 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x17 x18 x19 .
0
)
0
(
λ x17 :
ι → ι
.
0
)
)
(
λ x13 :
ι →
ι →
ι → ι
.
x12
0
0
)
0
0
(
λ x13 .
setsum
0
0
)
(
x0
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 x14 x15 .
0
)
0
(
λ x13 :
ι → ι
.
0
)
)
)
)
)
(
λ x9 x10 :
ι → ι
.
x2
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x12
(
λ x15 .
Inj1
)
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
Inj0
(
x15
0
)
)
(
λ x13 :
ι →
ι →
ι → ι
.
x0
(
λ x14 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x14 x15 x16 .
x16
)
(
Inj1
0
)
(
λ x14 :
ι → ι
.
0
)
)
(
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
0
)
(
setsum
0
0
)
0
(
λ x13 .
Inj1
0
)
(
Inj1
0
)
)
x11
(
λ x13 .
x11
)
0
)
)
0
=
Inj1
0
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 x7 .
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
x0
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 x13 x14 .
x3
(
λ x15 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x16 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
Inj1
0
)
(
λ x15 .
λ x16 :
(
ι → ι
)
→ ι
.
0
)
(
λ x15 x16 :
ι → ι
.
0
)
0
)
x7
(
λ x12 :
ι → ι
.
x0
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x12
0
)
(
λ x13 x14 x15 .
x3
(
λ x16 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x17 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x16 .
λ x17 :
(
ι → ι
)
→ ι
.
0
)
(
λ x16 x17 :
ι → ι
.
0
)
0
)
(
x0
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 x14 x15 .
0
)
0
(
λ x13 :
ι → ι
.
0
)
)
(
λ x13 :
ι → ι
.
0
)
)
)
(
λ x11 x12 x13 .
x12
)
(
Inj1
(
x2
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x13
0
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
x10
(
λ x13 .
0
)
)
)
)
(
λ x11 :
ι → ι
.
x9
)
)
(
λ x9 x10 :
ι → ι
.
Inj1
0
)
(
Inj1
(
x4
x6
(
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
x7
)
(
λ x9 x10 :
ι → ι
.
x6
)
(
setsum
0
0
)
)
)
)
=
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
(
x2
(
λ x10 .
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
setsum
0
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
x13
0
0
0
)
0
0
(
λ x13 .
x12
0
)
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
x7
)
)
(
x3
(
λ x10 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x0
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x13 .
0
)
0
)
(
λ x12 x13 x14 .
0
)
(
x0
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 x13 x14 .
0
)
0
(
λ x12 :
ι → ι
.
0
)
)
(
λ x12 :
ι → ι
.
x11
(
λ x13 :
ι → ι
.
0
)
(
λ x13 .
0
)
)
)
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
x7
)
(
λ x10 x11 :
ι → ι
.
0
)
(
setsum
(
Inj1
0
)
0
)
)
)
(
λ x9 x10 x11 .
Inj1
(
setsum
(
x0
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x10
)
(
λ x12 x13 x14 .
0
)
x10
(
λ x12 :
ι → ι
.
x12
0
)
)
(
x3
(
λ x12 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x2
(
λ x14 .
λ x15 :
(
ι →
ι → ι
)
→ ι
.
λ x16 :
ι → ι
.
λ x17 .
0
)
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
)
(
λ x12 .
λ x13 :
(
ι → ι
)
→ ι
.
x10
)
(
λ x12 x13 :
ι → ι
.
0
)
(
setsum
0
0
)
)
)
)
(
Inj1
(
x5
(
λ x9 :
ι → ι
.
λ x10 .
0
)
(
λ x9 :
ι → ι
.
λ x10 .
setsum
(
x2
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
)
0
)
)
)
(
λ x9 :
ι → ι
.
x7
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
ι →
(
ι → ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 .
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
setsum
x9
0
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
=
x6
0
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 .
∀ x6 :
ι →
ι →
ι →
ι → ι
.
∀ x7 .
x2
(
λ x9 .
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
Inj0
(
setsum
x12
0
)
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x9
)
=
x6
x5
(
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
setsum
(
Inj0
(
x2
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
)
)
(
x10
(
λ x11 :
ι → ι
.
Inj0
0
)
(
λ x11 .
Inj1
0
)
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
x7
)
(
λ x9 x10 :
ι → ι
.
setsum
0
(
x10
(
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x11 .
0
)
0
)
)
)
0
)
0
0
)
⟶
(
∀ x4 :
ι →
(
ι → ι
)
→ ι
.
∀ x5 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 x7 .
x1
(
λ x9 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
(
λ x9 :
ι →
ι →
ι → ι
.
x5
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x12
)
)
(
x5
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
x10
(
Inj1
0
)
)
)
x6
Inj1
0
=
x6
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 x7 .
x1
(
λ x9 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x10
)
(
λ x9 :
ι →
ι →
ι → ι
.
x3
(
λ x10 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
x9
(
x2
(
λ x12 .
λ x13 :
(
ι →
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
x13
(
λ x16 x17 .
0
)
)
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
)
x7
(
Inj1
(
Inj0
0
)
)
)
(
λ x10 x11 :
ι → ι
.
x11
(
Inj0
(
x0
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 x13 x14 .
0
)
0
(
λ x12 :
ι → ι
.
0
)
)
)
)
(
setsum
x7
x6
)
)
0
(
setsum
x5
(
x1
(
λ x9 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x10
)
(
λ x9 :
ι →
ι →
ι → ι
.
x0
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
0
0
)
(
λ x10 x11 x12 .
x11
)
(
Inj1
0
)
(
λ x10 :
ι → ι
.
0
)
)
x5
(
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj1
0
)
(
λ x9 x10 x11 .
x10
)
(
x2
(
λ x9 .
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
0
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
)
(
λ x9 :
ι → ι
.
x2
(
λ x10 .
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
0
)
)
)
(
λ x9 .
0
)
(
setsum
(
setsum
0
0
)
x5
)
)
)
(
λ x9 .
Inj0
(
Inj0
0
)
)
(
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
setsum
(
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
x11
(
λ x15 x16 x17 .
0
)
(
λ x15 .
0
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
(
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 :
ι → ι
.
0
)
0
)
(
x2
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
)
(
λ x11 .
setsum
0
0
)
x6
)
(
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
Inj0
0
)
(
Inj0
0
)
(
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x11 .
0
)
0
)
(
λ x11 .
x2
(
λ x12 .
λ x13 :
(
ι →
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
0
)
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
)
(
x0
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 x12 x13 .
0
)
0
(
λ x11 :
ι → ι
.
0
)
)
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
x9
(
setsum
x9
(
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x11 .
0
)
0
)
)
(
λ x11 .
x10
(
λ x12 .
x9
)
)
(
setsum
x6
x6
)
)
(
λ x9 x10 :
ι → ι
.
setsum
(
Inj0
(
Inj0
0
)
)
(
setsum
(
x10
0
)
(
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 :
ι → ι
.
0
)
0
)
)
)
(
Inj1
0
)
)
=
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
Inj1
x7
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
x3
(
λ x14 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x15 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
setsum
(
x0
(
λ x16 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x16 x17 x18 .
0
)
0
(
λ x16 :
ι → ι
.
0
)
)
(
x14
0
(
λ x16 :
ι → ι
.
λ x17 .
0
)
0
)
)
(
λ x14 .
λ x15 :
(
ι → ι
)
→ ι
.
x1
(
λ x16 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x17 .
λ x18 :
ι → ι
.
λ x19 .
x18
0
)
(
λ x16 :
ι →
ι →
ι → ι
.
x15
(
λ x17 .
0
)
)
(
x0
(
λ x16 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x16 x17 x18 .
0
)
0
(
λ x16 :
ι → ι
.
0
)
)
(
x0
(
λ x16 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x16 x17 x18 .
0
)
0
(
λ x16 :
ι → ι
.
0
)
)
(
λ x16 .
x14
)
x14
)
(
λ x14 x15 :
ι → ι
.
0
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
setsum
(
x2
(
λ x12 .
λ x13 :
(
ι →
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
0
)
(
λ x12 .
λ x13 :
ι →
ι → ι
.
x13
0
0
)
)
(
x0
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x10
(
λ x13 .
0
)
)
(
λ x12 x13 x14 .
x0
(
λ x15 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x15 x16 x17 .
0
)
0
(
λ x15 :
ι → ι
.
0
)
)
(
Inj1
0
)
(
λ x12 :
ι → ι
.
0
)
)
)
(
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x10
(
λ x13 .
x1
(
λ x14 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
ι → ι
.
λ x17 .
0
)
(
λ x14 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x14 .
0
)
0
)
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
x3
(
λ x13 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x14 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x12
(
λ x15 .
0
)
)
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 :
ι → ι
.
x3
(
λ x15 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x16 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x15 .
λ x16 :
(
ι → ι
)
→ ι
.
0
)
(
λ x15 x16 :
ι → ι
.
0
)
0
)
x11
)
(
λ x11 x12 :
ι → ι
.
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
x13
(
λ x17 x18 x19 .
0
)
(
λ x17 .
0
)
)
(
λ x13 :
ι →
ι →
ι → ι
.
0
)
(
setsum
0
0
)
(
x12
0
)
(
λ x13 .
0
)
0
)
(
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 :
ι → ι
.
x2
(
λ x13 .
λ x14 :
(
ι →
ι → ι
)
→ ι
.
λ x15 :
ι → ι
.
λ x16 .
0
)
(
λ x13 .
λ x14 :
ι →
ι → ι
.
0
)
)
x6
)
)
(
x2
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
x3
(
λ x13 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x14 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
Inj1
0
)
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 :
ι → ι
.
x3
(
λ x15 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x16 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x15 .
λ x16 :
(
ι → ι
)
→ ι
.
0
)
(
λ x15 x16 :
ι → ι
.
0
)
0
)
0
)
)
(
λ x11 .
setsum
x9
(
Inj0
(
x10
(
λ x12 .
0
)
)
)
)
(
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
x11
(
λ x15 x16 x17 .
x16
)
(
λ x15 .
setsum
0
0
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
(
setsum
(
Inj0
0
)
(
x0
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 x12 x13 .
0
)
0
(
λ x11 :
ι → ι
.
0
)
)
)
x6
(
λ x11 .
setsum
0
(
setsum
0
0
)
)
(
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
x3
(
λ x13 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x14 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 :
ι → ι
.
0
)
0
)
(
λ x11 x12 :
ι → ι
.
x11
0
)
x9
)
)
)
(
λ x9 x10 :
ι → ι
.
setsum
(
setsum
x6
(
setsum
(
x1
(
λ x11 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x11 .
0
)
0
)
(
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 :
ι → ι
.
0
)
0
)
)
)
(
x9
(
x9
(
setsum
0
0
)
)
)
)
(
x3
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x9
(
x0
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 x12 x13 .
x13
)
(
x0
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 x12 x13 .
0
)
0
(
λ x11 :
ι → ι
.
0
)
)
(
λ x11 :
ι → ι
.
x2
(
λ x12 .
λ x13 :
(
ι →
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
0
)
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
)
)
(
λ x11 :
ι → ι
.
λ x12 .
x2
(
λ x13 .
λ x14 :
(
ι →
ι → ι
)
→ ι
.
λ x15 :
ι → ι
.
λ x16 .
x0
(
λ x17 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x17 x18 x19 .
0
)
0
(
λ x17 :
ι → ι
.
0
)
)
(
λ x13 .
λ x14 :
ι →
ι → ι
.
setsum
0
0
)
)
(
Inj1
(
x3
(
λ x11 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 :
ι → ι
.
0
)
0
)
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
0
)
(
λ x9 x10 :
ι → ι
.
0
)
x7
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x5
(
λ x10 .
x6
0
)
)
(
λ x9 x10 x11 .
x10
)
(
x2
(
λ x9 .
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
Inj0
(
x3
(
λ x17 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x18 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x17 .
λ x18 :
(
ι → ι
)
→ ι
.
0
)
(
λ x17 x18 :
ι → ι
.
0
)
0
)
)
(
λ x13 :
ι →
ι →
ι → ι
.
setsum
(
setsum
0
0
)
0
)
(
x0
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x14 .
λ x15 :
(
ι →
ι → ι
)
→ ι
.
λ x16 :
ι → ι
.
λ x17 .
0
)
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
)
(
λ x13 x14 x15 .
x0
(
λ x16 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x16 x17 x18 .
0
)
0
(
λ x16 :
ι → ι
.
0
)
)
x9
(
λ x13 :
ι → ι
.
setsum
0
0
)
)
(
setsum
(
setsum
0
0
)
(
x0
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 x14 x15 .
0
)
0
(
λ x13 :
ι → ι
.
0
)
)
)
(
λ x13 .
Inj1
0
)
(
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
x15
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
x11
0
)
(
x10
(
λ x13 x14 .
0
)
)
(
setsum
0
0
)
(
λ x13 .
x11
0
)
(
x1
(
λ x13 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
λ x16 .
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x13 .
0
)
0
)
)
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
setsum
(
x0
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x12 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
0
)
(
λ x12 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x12 .
0
)
0
)
(
λ x11 x12 x13 .
x10
0
0
)
x9
(
λ x11 :
ι → ι
.
x9
)
)
0
)
)
(
λ x9 :
ι → ι
.
x2
(
λ x10 .
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
λ x13 .
x0
(
λ x14 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
0
(
Inj1
0
)
)
(
λ x14 x15 x16 .
0
)
(
Inj0
(
x0
(
λ x14 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x14 x15 x16 .
0
)
0
(
λ x14 :
ι → ι
.
0
)
)
)
(
λ x14 :
ι → ι
.
0
)
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
x2
(
λ x12 .
λ x13 :
(
ι →
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
x0
(
λ x16 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x17 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x18 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x17 .
λ x18 :
(
ι → ι
)
→ ι
.
0
)
(
λ x17 x18 :
ι → ι
.
0
)
0
)
(
λ x16 x17 x18 .
setsum
0
0
)
(
x1
(
λ x16 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x17 .
λ x18 :
ι → ι
.
λ x19 .
0
)
(
λ x16 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x16 .
0
)
0
)
(
λ x16 :
ι → ι
.
x1
(
λ x17 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x18 .
λ x19 :
ι → ι
.
λ x20 .
0
)
(
λ x17 :
ι →
ι →
ι → ι
.
0
)
0
0
(
λ x17 .
0
)
0
)
)
(
λ x12 .
λ x13 :
ι →
ι → ι
.
x12
)
)
)
=
Inj1
(
x7
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
Inj1
0
)
(
λ x9 .
setsum
(
x5
(
λ x10 .
x0
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 x12 x13 .
0
)
0
(
λ x11 :
ι → ι
.
0
)
)
)
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 x7 .
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x7
)
(
λ x9 x10 x11 .
Inj0
x10
)
(
Inj0
x6
)
(
λ x9 :
ι → ι
.
x7
)
=
x7
)
⟶
False
(proof)
Theorem
464b5..
:
∀ x0 :
(
ι → ι
)
→
ι →
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x1 :
(
(
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
(
(
ι →
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x2 :
(
ι → ι
)
→
ι →
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
ι →
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
x9
)
0
=
x5
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x6 :
(
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 .
x3
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
x0
(
λ x11 .
0
)
(
setsum
0
x7
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x3
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
Inj0
(
x2
(
λ x13 .
0
)
0
(
λ x13 .
λ x14 :
ι → ι
.
0
)
)
)
)
(
λ x11 :
ι → ι
.
x7
)
(
λ x11 .
x0
(
λ x12 .
x2
(
λ x13 .
x12
)
(
x2
(
λ x13 .
0
)
0
(
λ x13 .
λ x14 :
ι → ι
.
0
)
)
(
λ x13 .
λ x14 :
ι → ι
.
x2
(
λ x15 .
0
)
0
(
λ x15 .
λ x16 :
ι → ι
.
0
)
)
)
(
setsum
x11
x9
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
x3
(
λ x14 .
λ x15 :
(
ι → ι
)
→ ι
.
0
)
(
x0
(
λ x14 .
0
)
0
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
0
)
(
λ x14 :
ι → ι
.
0
)
(
λ x14 .
0
)
)
)
(
λ x12 :
ι → ι
.
setsum
0
(
setsum
0
0
)
)
(
λ x12 .
x2
(
λ x13 .
setsum
0
0
)
(
x0
(
λ x13 .
0
)
0
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 :
ι → ι
.
0
)
(
λ x13 .
0
)
)
(
λ x13 .
λ x14 :
ι → ι
.
setsum
0
0
)
)
)
)
(
x6
(
λ x9 x10 .
0
)
0
(
λ x9 .
x3
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
Inj0
(
x11
(
λ x12 .
0
)
)
)
(
x1
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
Inj0
0
)
(
λ x10 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x11 .
x11
)
(
λ x10 .
Inj1
0
)
(
λ x10 x11 .
0
)
)
)
)
=
x6
(
λ x9 x10 .
Inj0
0
)
(
x5
(
λ x9 :
ι →
ι → ι
.
λ x10 x11 .
x1
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x13 .
Inj0
0
)
(
λ x12 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x13 .
x10
)
(
λ x12 .
x11
)
(
λ x12 x13 .
x2
(
λ x14 .
setsum
0
0
)
(
Inj0
0
)
(
λ x14 .
λ x15 :
ι → ι
.
0
)
)
)
)
(
λ x9 .
x3
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
x2
(
λ x12 .
x11
(
λ x13 .
x1
(
λ x14 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x15 .
0
)
(
λ x14 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x15 .
0
)
(
λ x14 .
0
)
(
λ x14 x15 .
0
)
)
)
(
x11
(
λ x12 .
0
)
)
(
λ x12 .
λ x13 :
ι → ι
.
x10
)
)
(
x2
(
λ x10 .
setsum
(
x1
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
(
λ x11 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
(
λ x11 .
0
)
(
λ x11 x12 .
0
)
)
x9
)
0
(
λ x10 .
λ x11 :
ι → ι
.
x9
)
)
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 .
x9
)
(
Inj1
(
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
)
(
λ x9 .
λ x10 :
ι → ι
.
Inj0
(
Inj1
(
setsum
(
x0
(
λ x11 .
0
)
0
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
(
λ x11 :
ι → ι
.
0
)
(
λ x11 .
0
)
)
x7
)
)
)
=
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
Inj1
(
Inj1
(
x1
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
x1
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
(
λ x12 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
(
λ x12 .
0
)
(
λ x12 x13 .
0
)
)
(
λ x10 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x11 .
x3
(
λ x12 .
λ x13 :
(
ι → ι
)
→ ι
.
0
)
0
)
(
λ x10 .
x10
)
(
λ x10 x11 .
0
)
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 .
x9
)
0
(
λ x9 .
λ x10 :
ι → ι
.
0
)
=
setsum
(
x2
(
λ x9 .
0
)
0
(
λ x9 .
λ x10 :
ι → ι
.
x7
)
)
(
Inj0
x5
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
x1
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 .
x2
(
λ x11 .
0
)
(
x7
(
Inj0
x10
)
)
(
λ x11 .
λ x12 :
ι → ι
.
x3
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
x12
(
setsum
0
0
)
)
0
)
)
(
λ x9 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x10 .
x6
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x2
(
λ x14 .
Inj1
(
x0
(
λ x15 .
0
)
0
(
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 :
ι → ι
.
0
)
(
λ x15 :
ι → ι
.
0
)
(
λ x15 .
0
)
)
)
(
setsum
(
x2
(
λ x14 .
0
)
0
(
λ x14 .
λ x15 :
ι → ι
.
0
)
)
(
setsum
0
0
)
)
(
λ x14 .
λ x15 :
ι → ι
.
0
)
)
)
(
λ x9 .
Inj0
0
)
(
λ x9 x10 .
x7
(
x7
x9
)
)
=
Inj0
(
x0
(
λ x9 .
setsum
(
setsum
x9
(
setsum
0
0
)
)
(
x0
(
λ x10 .
0
)
(
x2
(
λ x10 .
0
)
0
(
λ x10 .
λ x11 :
ι → ι
.
0
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
x0
(
λ x12 .
0
)
0
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
(
λ x12 .
0
)
)
(
λ x10 :
ι → ι
.
x6
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
λ x10 .
x1
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
(
λ x11 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
(
λ x11 .
0
)
(
λ x11 x12 .
0
)
)
)
)
0
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
0
)
(
λ x9 .
x1
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
x7
(
setsum
0
0
)
)
(
λ x10 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x11 .
x1
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x13 .
x2
(
λ x14 .
0
)
0
(
λ x14 .
λ x15 :
ι → ι
.
0
)
)
(
λ x12 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
(
λ x12 .
x0
(
λ x13 .
0
)
0
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 :
ι → ι
.
0
)
(
λ x13 .
0
)
)
(
λ x12 x13 .
0
)
)
x7
(
λ x10 x11 .
x11
)
)
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 .
x6
)
(
λ x9 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x10 .
setsum
(
Inj0
(
x0
(
λ x11 .
x0
(
λ x12 .
0
)
0
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
(
λ x12 .
0
)
)
(
Inj1
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x11
(
λ x13 .
0
)
0
)
(
λ x11 :
ι → ι
.
x11
0
)
(
λ x11 .
x1
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
(
λ x12 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
(
λ x12 .
0
)
(
λ x12 x13 .
0
)
)
)
)
(
x9
(
λ x11 x12 .
x10
)
0
)
)
(
λ x9 .
x5
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
)
(
λ x9 x10 .
Inj1
0
)
=
x6
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
(
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x0
(
λ x9 .
0
)
(
x0
(
λ x9 .
x1
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
(
λ x10 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
(
λ x10 .
0
)
(
λ x10 x11 .
x9
)
)
(
x1
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 .
x2
(
λ x11 .
0
)
x6
(
λ x11 .
λ x12 :
ι → ι
.
x10
)
)
(
λ x9 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x10 .
x3
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
setsum
0
0
)
(
x3
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
0
)
0
)
)
(
λ x9 .
Inj0
(
x0
(
λ x10 .
0
)
0
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
0
)
(
λ x10 :
ι → ι
.
0
)
(
λ x10 .
0
)
)
)
(
λ x9 x10 .
Inj0
(
Inj1
0
)
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x10
(
Inj0
(
x1
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
(
λ x11 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
(
λ x11 .
0
)
(
λ x11 x12 .
0
)
)
)
)
(
λ x9 :
ι → ι
.
setsum
(
Inj0
0
)
(
x9
(
x9
0
)
)
)
(
λ x9 .
Inj0
(
Inj0
(
Inj1
0
)
)
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x0
(
λ x11 .
x3
(
λ x12 .
λ x13 :
(
ι → ι
)
→ ι
.
0
)
0
)
(
x10
x6
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
Inj0
0
)
(
λ x11 :
ι → ι
.
x0
(
λ x12 .
0
)
(
x0
(
λ x12 .
setsum
0
0
)
(
x2
(
λ x12 .
0
)
0
(
λ x12 .
λ x13 :
ι → ι
.
0
)
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
x3
(
λ x14 .
λ x15 :
(
ι → ι
)
→ ι
.
0
)
0
)
(
λ x12 :
ι → ι
.
x2
(
λ x13 .
0
)
0
(
λ x13 .
λ x14 :
ι → ι
.
0
)
)
(
λ x12 .
0
)
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
x1
(
λ x14 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x15 .
x12
(
λ x16 .
0
)
0
)
(
λ x14 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x15 .
setsum
0
0
)
(
λ x14 .
setsum
0
0
)
(
λ x14 x15 .
0
)
)
(
λ x12 :
ι → ι
.
setsum
(
x3
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
0
)
0
)
0
)
(
λ x12 .
x12
)
)
(
λ x11 .
0
)
)
(
λ x9 :
ι → ι
.
setsum
(
x3
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
x11
(
λ x12 .
0
)
)
(
Inj0
0
)
)
(
setsum
x6
(
x5
(
λ x10 :
ι → ι
.
x9
0
)
0
(
λ x10 .
setsum
0
0
)
(
x5
(
λ x10 :
ι → ι
.
0
)
0
(
λ x10 .
0
)
0
)
)
)
)
(
λ x9 .
x6
)
=
x0
(
λ x9 .
x9
)
(
x3
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
Inj1
(
Inj0
0
)
)
x6
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
setsum
(
x2
(
λ x11 .
setsum
(
x9
(
λ x12 .
0
)
0
)
(
x0
(
λ x12 .
0
)
0
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
(
λ x12 .
0
)
)
)
(
x2
(
λ x11 .
0
)
x6
(
λ x11 .
λ x12 :
ι → ι
.
0
)
)
(
λ x11 .
λ x12 :
ι → ι
.
x0
(
λ x13 .
x1
(
λ x14 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x15 .
0
)
(
λ x14 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x15 .
0
)
(
λ x14 .
0
)
(
λ x14 x15 .
0
)
)
(
Inj1
0
)
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
Inj0
0
)
(
λ x13 :
ι → ι
.
x2
(
λ x14 .
0
)
0
(
λ x14 .
λ x15 :
ι → ι
.
0
)
)
(
λ x13 .
0
)
)
)
(
Inj1
0
)
)
(
λ x9 :
ι → ι
.
setsum
(
x9
(
x9
(
x5
(
λ x10 :
ι → ι
.
0
)
0
(
λ x10 .
0
)
0
)
)
)
0
)
(
λ x9 .
Inj1
0
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 x7 .
x0
(
λ x9 .
x3
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
x9
)
0
)
0
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
0
)
(
λ x9 .
0
)
=
x3
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
Inj0
x6
)
x4
)
⟶
False
(proof)
Theorem
83068..
:
∀ x0 :
(
ι → ι
)
→
ι →
ι →
ι →
ι →
ι → ι
.
∀ x1 :
(
(
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
)
→
ι →
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x2 :
(
(
ι →
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
(
(
ι →
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι →
ι → ι
)
→ ι
)
→
(
(
ι →
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
(
∀ x4 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x5 :
(
ι →
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x6 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x7 :
ι →
ι →
ι →
ι → ι
.
x3
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι →
ι → ι
.
0
)
(
λ x9 :
ι →
ι →
ι → ι
.
0
)
(
λ x9 .
0
)
=
x7
(
Inj1
(
Inj0
(
x2
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
setsum
0
0
)
)
)
)
0
(
setsum
0
(
x2
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x10 .
x3
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 .
0
)
)
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
0
)
0
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
0
(
x2
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
0
)
)
0
0
)
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
x9
(
x2
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
0
)
)
)
)
)
(
x3
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι →
ι → ι
.
0
)
(
λ x9 :
ι →
ι →
ι → ι
.
0
)
(
λ x9 .
0
)
)
)
⟶
(
∀ x4 :
ι →
ι →
ι →
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x7 :
(
ι →
ι →
ι → ι
)
→ ι
.
x3
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι →
ι → ι
.
x3
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι →
ι → ι
.
x10
(
x3
(
λ x13 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 :
ι →
ι →
ι → ι
.
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
Inj1
0
)
(
λ x13 .
Inj1
0
)
)
(
x0
(
λ x13 .
x12
0
0
0
)
(
x0
(
λ x13 .
0
)
0
0
0
0
0
)
(
Inj0
0
)
(
setsum
0
0
)
(
x10
0
0
0
)
(
x0
(
λ x13 .
0
)
0
0
0
0
0
)
)
(
x9
(
λ x13 .
λ x14 :
ι → ι
.
x3
(
λ x15 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 :
ι →
ι →
ι → ι
.
0
)
(
λ x15 :
ι →
ι →
ι → ι
.
0
)
(
λ x15 .
0
)
)
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
Inj0
(
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
Inj1
0
)
0
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
x10
0
0
0
)
)
)
(
λ x11 .
0
)
)
(
λ x9 :
ι →
ι →
ι → ι
.
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
x2
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x13 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 :
ι →
ι →
ι → ι
.
x1
(
λ x15 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x16 .
0
)
0
(
λ x15 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x13 :
ι →
ι →
ι → ι
.
setsum
0
0
)
(
λ x13 .
x12
0
(
λ x14 :
ι → ι
.
0
)
)
)
(
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
x3
(
λ x15 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 :
ι →
ι →
ι → ι
.
setsum
0
0
)
(
λ x15 :
ι →
ι →
ι → ι
.
Inj0
0
)
(
λ x15 .
x1
(
λ x16 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x17 .
0
)
0
(
λ x16 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
(
setsum
0
(
Inj0
(
x5
0
)
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x2
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
x1
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 .
0
)
0
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
setsum
0
0
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
)
(
λ x11 :
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
x3
(
λ x14 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι →
ι → ι
.
x0
(
λ x16 .
0
)
0
0
0
0
0
)
(
λ x14 :
ι →
ι →
ι → ι
.
x1
(
λ x15 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x16 .
0
)
0
(
λ x15 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x14 .
x0
(
λ x15 .
0
)
0
0
0
0
0
)
)
)
)
(
λ x9 .
0
)
=
setsum
0
(
x0
(
λ x9 .
Inj1
0
)
0
(
setsum
0
(
x5
(
x4
0
0
0
0
)
)
)
(
Inj0
(
x7
(
λ x9 x10 x11 .
setsum
0
0
)
)
)
(
setsum
0
(
setsum
(
x4
0
0
0
0
)
(
x6
0
0
(
λ x9 .
0
)
0
)
)
)
(
x6
(
x2
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
x0
(
λ x12 .
0
)
0
0
0
0
0
)
)
(
Inj0
0
)
(
λ x9 .
x5
0
)
(
x4
(
Inj0
0
)
(
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
0
)
0
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
)
0
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
0
(
x9
(
x5
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
0
)
0
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
(
λ x10 :
ι → ι
.
x2
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
0
)
0
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x11 :
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
0
)
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
x0
(
λ x14 .
x14
)
0
(
x0
(
λ x14 .
setsum
0
0
)
(
x1
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 .
0
)
0
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
x3
(
λ x14 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι →
ι → ι
.
0
)
(
λ x14 :
ι →
ι →
ι → ι
.
0
)
(
λ x14 .
0
)
)
(
x11
0
)
(
x2
(
λ x14 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x14 :
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
0
)
)
x10
)
(
x1
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 .
Inj0
0
)
0
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
x12
(
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 :
ι → ι
.
λ x17 .
0
)
)
)
(
Inj1
(
x11
0
)
)
(
setsum
(
x12
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
λ x16 .
0
)
)
(
x12
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
λ x16 .
0
)
)
)
)
(
x11
(
Inj0
(
x2
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
0
)
)
)
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
=
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
x9
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
Inj0
(
x3
(
λ x14 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι →
ι → ι
.
setsum
0
0
)
(
λ x14 :
ι →
ι →
ι → ι
.
Inj0
0
)
(
λ x14 .
setsum
0
0
)
)
)
)
x7
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
Inj0
(
setsum
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
setsum
0
0
)
(
Inj0
0
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
setsum
(
x0
(
λ x10 .
0
)
0
0
0
0
0
)
0
)
)
)
)
⟶
(
∀ x4 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
(
ι → ι
)
→
ι → ι
.
∀ x7 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
x2
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
x11
(
Inj1
(
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
x1
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 .
0
)
0
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
x3
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
ι →
ι →
ι → ι
.
0
)
(
λ x12 :
ι →
ι →
ι → ι
.
0
)
(
λ x12 .
0
)
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
=
x4
(
x4
(
Inj0
(
setsum
(
x5
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
)
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x2
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x9
(
setsum
0
0
)
)
(
λ x11 :
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x10
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
0
)
(
Inj0
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
Inj0
(
Inj1
0
)
)
=
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
setsum
(
Inj0
0
)
x6
)
(
x3
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι →
ι → ι
.
x3
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι →
ι → ι
.
Inj0
(
x11
(
λ x13 .
λ x14 :
ι → ι
.
0
)
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 .
0
)
)
(
λ x9 :
ι →
ι →
ι → ι
.
setsum
(
x5
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
x0
(
λ x12 .
0
)
0
0
0
0
0
)
(
setsum
0
0
)
(
λ x10 .
x0
(
λ x11 .
0
)
0
0
0
0
0
)
(
x2
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
0
)
)
)
0
)
(
λ x9 .
x7
)
)
(
λ x9 .
x3
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι →
ι → ι
.
0
)
(
λ x10 :
ι →
ι →
ι → ι
.
Inj0
(
x10
x9
x7
(
x3
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 .
0
)
)
)
)
(
λ x10 .
Inj1
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
0
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 :
(
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x7 :
(
ι →
ι →
ι → ι
)
→ ι
.
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
x0
(
λ x11 .
x11
)
(
Inj1
(
x3
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι →
ι → ι
.
Inj0
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
x10
)
(
λ x11 .
x10
)
)
)
(
setsum
x10
0
)
(
Inj0
(
x7
(
λ x11 x12 x13 .
Inj0
0
)
)
)
(
x3
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι →
ι → ι
.
setsum
0
(
Inj1
0
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
Inj1
(
x3
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
ι →
ι →
ι → ι
.
0
)
(
λ x12 :
ι →
ι →
ι → ι
.
0
)
(
λ x12 .
0
)
)
)
(
λ x11 .
0
)
)
(
x7
(
λ x11 x12 x13 .
x0
(
λ x14 .
0
)
(
x0
(
λ x14 .
0
)
0
0
0
0
0
)
(
x1
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 .
0
)
0
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
x3
(
λ x14 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι →
ι → ι
.
0
)
(
λ x14 :
ι →
ι →
ι → ι
.
0
)
(
λ x14 .
0
)
)
x11
0
)
)
)
(
x3
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι →
ι → ι
.
0
)
(
λ x9 :
ι →
ι →
ι → ι
.
x2
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
0
)
0
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x11 :
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
0
)
)
(
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
x12
(
Inj0
0
)
)
)
(
λ x9 .
0
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
x3
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
ι →
ι →
ι → ι
.
x0
(
λ x14 .
x14
)
0
(
x12
(
λ x14 .
λ x15 :
ι → ι
.
0
)
)
x11
0
0
)
(
λ x12 :
ι →
ι →
ι → ι
.
x3
(
λ x13 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 :
ι →
ι →
ι → ι
.
setsum
0
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
x1
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 .
0
)
0
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x13 .
0
)
)
(
λ x12 .
Inj1
(
setsum
0
0
)
)
)
(
x5
(
λ x10 :
ι →
ι → ι
.
x1
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 .
x11
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
λ x15 .
0
)
)
(
x1
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 .
0
)
0
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
)
(
λ x10 .
x1
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 .
setsum
0
0
)
0
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x10 .
0
)
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
0
)
0
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x7
(
λ x11 x12 x13 .
0
)
)
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
=
setsum
(
x3
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι →
ι → ι
.
x3
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι →
ι → ι
.
Inj1
(
x12
0
0
0
)
)
(
λ x11 :
ι →
ι →
ι → ι
.
x2
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj1
0
)
(
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
x3
(
λ x15 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 :
ι →
ι →
ι → ι
.
0
)
(
λ x15 :
ι →
ι →
ι → ι
.
0
)
(
λ x15 .
0
)
)
)
(
λ x11 .
x10
(
x2
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
0
)
)
(
setsum
0
0
)
(
x9
(
λ x12 .
λ x13 :
ι → ι
.
0
)
)
)
)
(
λ x9 :
ι →
ι →
ι → ι
.
x6
(
λ x10 :
ι → ι
.
0
)
0
(
λ x10 .
x7
(
λ x11 x12 x13 .
x11
)
)
(
Inj0
(
x9
0
0
0
)
)
)
(
λ x9 .
Inj0
0
)
)
0
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 x7 .
x0
(
λ x9 .
0
)
0
0
(
setsum
(
setsum
x7
0
)
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
)
0
(
x5
0
)
=
setsum
0
0
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 :
(
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 .
x0
(
λ x9 .
0
)
0
(
x4
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
setsum
0
0
)
(
Inj1
0
)
(
x0
(
λ x9 .
x6
(
λ x10 :
ι → ι
.
setsum
(
x3
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 .
0
)
)
(
x2
(
λ x11 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 :
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
0
)
)
)
x9
(
λ x10 .
x7
)
)
0
(
x2
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj1
(
setsum
0
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
x2
(
λ x12 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
Inj0
0
)
)
)
0
(
x0
(
λ x9 .
0
)
x7
0
(
x3
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι →
ι → ι
.
setsum
0
0
)
(
λ x9 :
ι →
ι →
ι → ι
.
setsum
0
0
)
(
λ x9 .
x6
(
λ x10 :
ι → ι
.
0
)
0
(
λ x10 .
0
)
)
)
(
Inj1
(
setsum
0
0
)
)
(
Inj1
(
x4
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
(
x2
(
λ x9 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x9
x7
(
λ x10 :
ι → ι
.
x7
)
)
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
Inj0
x7
)
)
)
=
x4
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
setsum
(
Inj1
0
)
(
setsum
x7
(
x2
(
λ x10 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
setsum
0
0
)
)
)
)
)
⟶
False
(proof)
Theorem
acb5d..
:
∀ x0 :
(
ι →
(
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x1 :
(
ι →
ι →
(
ι →
ι → ι
)
→ ι
)
→
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x2 :
(
(
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
(
ι →
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x3 :
(
ι → ι
)
→
ι → ι
.
(
∀ x4 x5 x6 .
∀ x7 :
(
ι →
ι → ι
)
→
ι →
ι → ι
.
x3
(
λ x9 .
x2
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
setsum
0
0
)
x9
(
Inj0
x6
)
)
x4
=
setsum
x6
(
Inj0
0
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
x3
(
λ x9 .
x7
(
λ x10 :
(
ι → ι
)
→ ι
.
setsum
(
setsum
(
x0
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
0
)
0
(
λ x11 .
0
)
)
(
Inj1
0
)
)
(
x10
(
λ x11 .
Inj1
0
)
)
)
(
Inj0
(
Inj1
0
)
)
)
(
x0
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
setsum
0
(
setsum
0
(
x0
(
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
0
)
0
(
λ x13 .
0
)
)
)
)
(
x4
(
x0
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
0
)
x5
(
λ x9 .
0
)
)
)
(
λ x9 .
x0
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
x10
)
(
x2
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
x12
0
0
)
(
x0
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
0
)
0
(
λ x10 .
0
)
)
(
x1
(
λ x10 x11 .
λ x12 :
ι →
ι → ι
.
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
)
(
λ x10 .
0
)
)
)
=
Inj1
(
x1
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
x7
(
λ x12 :
(
ι → ι
)
→ ι
.
x12
(
λ x13 .
x2
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 .
λ x16 :
ι →
ι → ι
.
0
)
0
0
)
)
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x10 x11 .
λ x12 :
ι →
ι → ι
.
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
x10
(
λ x11 x12 .
x10
(
λ x13 x14 .
0
)
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
x11
(
x0
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
setsum
(
setsum
0
0
)
x14
)
(
x11
(
x9
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
0
)
)
0
)
(
λ x12 .
x1
(
λ x13 x14 .
λ x15 :
ι →
ι → ι
.
Inj1
0
)
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
x10
)
)
)
(
x11
(
x1
(
λ x12 x13 .
λ x14 :
ι →
ι → ι
.
0
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
Inj1
0
)
)
0
)
)
(
x1
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
x9
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x6
(
x9
(
λ x10 x11 .
x11
)
)
)
)
(
x3
(
λ x9 .
Inj0
x5
)
(
Inj1
(
x1
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
x0
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
0
)
0
(
λ x12 .
0
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
setsum
0
0
)
)
)
)
=
Inj0
(
x0
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
x2
(
λ x13 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 .
λ x15 :
ι →
ι → ι
.
x13
(
λ x16 :
(
ι → ι
)
→
ι → ι
.
λ x17 :
ι → ι
.
0
)
)
(
Inj1
(
setsum
0
0
)
)
x9
)
0
(
λ x9 .
0
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 x7 .
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
Inj0
(
Inj0
(
setsum
(
x9
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
0
)
)
(
Inj1
0
)
)
)
)
(
x3
(
λ x9 .
Inj0
x5
)
x7
)
(
Inj0
(
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
0
)
(
setsum
(
Inj1
0
)
(
x3
(
λ x9 .
0
)
0
)
)
0
)
)
=
x3
(
λ x9 .
x2
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
x10
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
x0
(
λ x15 .
λ x16 :
ι → ι
.
λ x17 .
λ x18 :
ι → ι
.
x3
(
λ x19 .
0
)
0
)
0
(
λ x15 .
x0
(
λ x16 .
λ x17 :
ι → ι
.
λ x18 .
λ x19 :
ι → ι
.
0
)
0
(
λ x16 .
0
)
)
)
)
(
setsum
(
setsum
(
Inj1
0
)
0
)
(
x2
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
(
Inj0
0
)
(
Inj1
0
)
)
)
x5
)
(
setsum
(
x1
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
x7
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
Inj1
(
x1
(
λ x10 x11 .
λ x12 :
ι →
ι → ι
.
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
)
)
(
Inj0
(
setsum
x5
(
x0
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
0
)
0
(
λ x9 .
0
)
)
)
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x1
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
Inj1
(
Inj1
x10
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
setsum
x5
(
setsum
x7
0
)
)
=
x6
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
setsum
0
(
x2
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 .
λ x14 :
ι →
ι → ι
.
x0
(
λ x15 .
λ x16 :
ι → ι
.
λ x17 .
λ x18 :
ι → ι
.
0
)
(
x14
0
0
)
(
λ x15 .
0
)
)
x10
0
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
Inj0
0
)
=
setsum
(
x1
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
x0
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
setsum
(
x3
(
λ x16 .
0
)
0
)
x14
)
(
setsum
0
(
Inj0
0
)
)
(
λ x12 .
0
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
Inj1
x6
)
)
0
)
⟶
(
∀ x4 x5 x6 x7 .
x0
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
0
)
(
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
0
)
(
setsum
(
setsum
0
0
)
x5
)
0
)
(
λ x9 .
x0
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
x11
(
x3
(
λ x14 .
0
)
(
x1
(
λ x14 x15 .
λ x16 :
ι →
ι → ι
.
0
)
(
λ x14 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
)
)
(
x0
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
0
)
0
(
λ x10 .
x0
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
x14
0
)
(
x1
(
λ x11 x12 .
λ x13 :
ι →
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
(
λ x11 .
x3
(
λ x12 .
0
)
0
)
)
)
(
λ x10 .
setsum
x10
x10
)
)
=
x0
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
x11
)
(
Inj0
(
Inj1
0
)
)
(
λ x9 .
setsum
(
x3
(
λ x10 .
setsum
0
0
)
0
)
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
ι → ι
.
x0
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
0
)
0
(
λ x9 .
x2
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
x9
(
x6
(
λ x10 :
ι →
ι → ι
.
x2
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 .
λ x13 :
ι →
ι → ι
.
x10
0
0
)
(
Inj1
0
)
(
x6
(
λ x11 :
ι →
ι → ι
.
0
)
)
)
)
)
=
Inj0
(
x1
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
setsum
(
x0
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
x15
0
)
(
x1
(
λ x12 x13 .
λ x14 :
ι →
ι → ι
.
0
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
(
λ x12 .
x3
(
λ x13 .
0
)
0
)
)
(
x3
(
λ x12 .
0
)
(
x3
(
λ x12 .
0
)
0
)
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
setsum
(
setsum
(
x6
(
λ x10 :
ι →
ι → ι
.
0
)
)
(
x2
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
0
0
)
)
0
)
)
)
⟶
False
(proof)
Theorem
60ce1..
:
∀ x0 :
(
ι →
ι → ι
)
→
(
(
(
ι →
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x1 :
(
ι → ι
)
→
ι →
ι →
ι →
ι →
ι → ι
.
∀ x2 :
(
ι →
(
ι →
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
(
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
)
→
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x3
(
λ x9 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x12 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
x7
(
x9
(
λ x12 x13 .
x2
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
(
λ x14 :
ι → ι
.
x12
)
)
(
x11
(
λ x12 .
x1
(
λ x13 .
0
)
0
0
0
0
0
)
(
x3
(
λ x12 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
(
setsum
0
(
Inj0
0
)
)
x7
)
)
(
x5
0
(
λ x9 :
ι → ι
.
0
)
)
(
x1
(
λ x9 .
x7
)
(
x3
(
λ x9 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
0
(
x1
(
λ x9 .
setsum
0
0
)
0
0
(
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
0
)
)
(
Inj0
0
)
0
)
)
(
Inj0
0
)
(
x0
(
λ x9 x10 .
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x3
(
λ x10 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
x0
(
λ x13 x14 .
0
)
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 .
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 :
ι → ι
.
λ x14 .
0
)
)
0
(
x0
(
λ x10 x11 .
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
setsum
0
(
x6
0
)
)
)
(
x5
(
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x2
(
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
(
λ x11 :
ι → ι
.
0
)
)
(
λ x9 :
ι → ι
.
x9
0
)
)
(
λ x9 :
ι → ι
.
x5
(
Inj0
0
)
(
λ x10 :
ι → ι
.
x9
0
)
)
)
(
Inj1
(
Inj1
(
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
0
)
)
)
)
)
=
Inj1
(
Inj1
0
)
)
⟶
(
∀ x4 :
ι →
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x3
(
λ x9 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x0
(
λ x12 x13 .
0
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
x1
(
λ x15 .
x15
)
0
(
setsum
(
x11
(
λ x15 .
0
)
0
)
(
x3
(
λ x15 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x16 .
λ x17 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
0
0
(
Inj1
(
x13
(
λ x15 .
0
)
0
)
)
)
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
Inj1
(
setsum
(
x4
(
x6
0
)
(
λ x9 .
x6
0
)
)
0
)
)
0
=
x0
(
λ x9 x10 .
x2
(
λ x11 .
λ x12 :
ι →
ι → ι
.
x12
(
x1
(
λ x13 .
Inj0
0
)
0
x9
(
setsum
0
0
)
(
x3
(
λ x13 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x14 .
λ x15 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
0
)
(
Inj0
0
)
)
(
λ x11 :
ι → ι
.
Inj1
(
setsum
(
x2
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
)
0
)
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
x1
(
λ x12 .
0
)
(
setsum
(
Inj0
(
x2
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
)
)
0
)
0
0
0
(
x0
(
λ x12 x13 .
x13
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
x3
(
λ x15 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x16 .
λ x17 :
(
ι → ι
)
→
ι → ι
.
x2
(
λ x18 .
λ x19 :
ι →
ι → ι
.
0
)
(
λ x18 :
ι → ι
.
0
)
)
(
Inj1
0
)
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 .
Inj1
0
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x7
)
(
λ x9 :
ι → ι
.
λ x10 .
Inj1
x7
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x7 :
ι →
ι → ι
.
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x7
x9
(
x7
(
x3
(
λ x11 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
x2
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
(
λ x14 :
ι → ι
.
0
)
)
(
x0
(
λ x11 x12 .
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 :
ι → ι
.
λ x12 .
0
)
)
(
x3
(
λ x11 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
0
)
)
(
λ x9 :
ι → ι
.
setsum
(
x1
(
λ x10 .
x6
(
λ x11 :
ι →
ι → ι
.
0
)
0
)
(
x1
(
λ x10 .
x3
(
λ x11 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
(
x5
0
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
(
Inj1
0
)
(
Inj1
0
)
(
x2
(
λ x10 .
λ x11 :
ι →
ι → ι
.
0
)
(
λ x10 :
ι → ι
.
0
)
)
0
)
(
x5
(
setsum
0
0
)
(
λ x10 :
ι → ι
.
λ x11 .
x10
0
)
)
0
0
(
x6
(
λ x10 :
ι →
ι → ι
.
setsum
0
0
)
0
)
)
(
x1
(
λ x10 .
x9
(
x9
0
)
)
(
x0
(
λ x10 x11 .
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
x2
(
λ x13 .
λ x14 :
ι →
ι → ι
.
0
)
(
λ x13 :
ι → ι
.
0
)
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x10
(
λ x11 .
0
)
)
(
λ x10 :
ι → ι
.
λ x11 .
x11
)
)
(
x5
0
(
λ x10 :
ι → ι
.
λ x11 .
x10
0
)
)
0
0
(
Inj1
0
)
)
)
=
x7
(
x5
0
(
λ x9 :
ι → ι
.
λ x10 .
x6
(
λ x11 :
ι →
ι → ι
.
0
)
(
x3
(
λ x11 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
x13
(
λ x14 .
0
)
0
)
(
Inj0
0
)
(
x1
(
λ x11 .
0
)
0
0
0
0
0
)
)
)
)
(
Inj1
x4
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 x7 .
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
setsum
0
x7
)
(
λ x9 :
ι → ι
.
Inj0
(
x0
(
λ x10 x11 .
Inj0
x10
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
x10
(
λ x13 x14 .
Inj1
0
)
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x7
)
(
λ x10 :
ι → ι
.
λ x11 .
x0
(
λ x12 x13 .
0
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
x2
(
λ x13 .
λ x14 :
ι →
ι → ι
.
0
)
(
λ x13 :
ι → ι
.
0
)
)
(
λ x12 :
ι → ι
.
λ x13 .
x2
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
(
λ x14 :
ι → ι
.
0
)
)
)
)
)
=
x6
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x7 .
x1
(
λ x9 .
x7
)
0
0
(
setsum
(
setsum
(
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x7
)
(
λ x9 :
ι → ι
.
0
)
)
(
Inj1
(
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 .
0
)
(
λ x9 .
0
)
)
)
)
(
x0
(
λ x9 x10 .
x6
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x0
(
λ x13 x14 .
0
)
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 .
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 :
ι → ι
.
λ x14 .
0
)
)
(
λ x11 .
0
)
(
λ x11 .
0
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
x7
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 .
x0
(
λ x11 x12 .
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
x3
(
λ x12 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
(
λ x11 :
ι → ι
.
λ x12 .
Inj0
0
)
)
)
)
(
setsum
(
setsum
(
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x1
(
λ x11 .
0
)
0
0
0
0
0
)
(
λ x9 .
x7
)
(
λ x9 .
x1
(
λ x10 .
0
)
0
0
0
0
0
)
)
(
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 .
x1
(
λ x10 .
0
)
0
0
0
0
0
)
(
λ x9 .
x7
)
)
)
x7
)
0
=
x7
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x9 .
Inj1
0
)
0
(
setsum
(
x3
(
λ x9 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x9
(
λ x12 x13 .
setsum
0
0
)
x10
(
x2
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
)
(
Inj1
0
)
)
(
Inj0
0
)
0
)
0
)
(
Inj0
x4
)
x4
(
Inj0
0
)
=
setsum
x5
0
)
⟶
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x7 .
x0
(
λ x9 x10 .
x6
(
λ x11 x12 .
setsum
0
x9
)
(
λ x11 :
ι → ι
.
x9
)
x7
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 .
setsum
(
Inj0
(
x0
(
λ x11 x12 .
x10
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 .
x2
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
(
λ x14 :
ι → ι
.
0
)
)
(
λ x11 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 :
ι → ι
.
λ x12 .
setsum
0
0
)
)
)
(
x1
(
λ x11 .
0
)
(
Inj0
x10
)
(
x0
(
λ x11 x12 .
setsum
0
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 .
Inj0
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x11 :
ι → ι
.
λ x12 .
x11
0
)
)
(
x6
(
λ x11 x12 .
setsum
0
0
)
(
λ x11 :
ι → ι
.
x0
(
λ x12 x13 .
0
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
Inj0
0
)
)
(
setsum
(
x3
(
λ x11 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
(
x6
(
λ x11 x12 .
0
)
(
λ x11 :
ι → ι
.
0
)
0
)
)
(
x3
(
λ x11 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
(
setsum
0
0
)
0
)
)
)
=
x6
(
λ x9 x10 .
x3
(
λ x11 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
x12
)
(
x6
(
λ x11 x12 .
0
)
(
λ x11 :
ι → ι
.
x10
)
(
x2
(
λ x11 .
λ x12 :
ι →
ι → ι
.
setsum
0
0
)
(
λ x11 :
ι → ι
.
0
)
)
)
(
x3
(
λ x11 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
x0
(
λ x14 x15 .
0
)
(
λ x14 :
(
ι →
ι → ι
)
→ ι
.
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 .
x3
(
λ x17 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
λ x18 .
λ x19 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
(
λ x14 :
(
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x14 :
ι → ι
.
λ x15 .
0
)
)
(
Inj1
x9
)
(
setsum
(
x1
(
λ x11 .
0
)
0
0
0
0
0
)
0
)
)
)
(
λ x9 :
ι → ι
.
x5
)
(
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
x7
)
)
)
⟶
(
∀ x4 .
∀ x5 x6 :
ι → ι
.
∀ x7 .
x0
(
λ x9 x10 .
setsum
(
Inj1
(
setsum
0
0
)
)
x7
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
x9
(
λ x12 x13 .
x0
(
λ x14 x15 .
0
)
(
λ x14 :
(
ι →
ι → ι
)
→ ι
.
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 .
Inj0
(
setsum
0
0
)
)
(
λ x14 :
(
ι → ι
)
→ ι
.
x11
)
(
λ x14 :
ι → ι
.
λ x15 .
0
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x6
x7
)
(
λ x9 :
ι → ι
.
λ x10 .
x7
)
=
setsum
0
(
x0
(
λ x9 x10 .
setsum
(
Inj1
0
)
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
Inj0
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
)
⟶
False
(proof)
Theorem
8ab24..
:
∀ x0 :
(
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x1 :
(
ι → ι
)
→
ι → ι
.
∀ x2 :
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
ι →
ι → ι
)
→
ι → ι
.
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 x7 .
x3
(
λ x9 x10 .
setsum
(
setsum
0
0
)
(
x0
(
λ x11 .
x7
)
(
Inj1
(
Inj1
0
)
)
x7
x10
)
)
0
=
x4
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 x7 .
x3
(
λ x9 x10 .
x3
(
λ x11 x12 .
setsum
x11
0
)
(
Inj0
(
x2
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
x11
(
λ x12 x13 .
0
)
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
x2
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
0
)
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x12
0
)
)
)
)
(
x5
(
setsum
(
x3
(
λ x9 x10 .
0
)
(
x3
(
λ x9 x10 .
0
)
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x10
)
(
λ x9 .
0
)
)
=
x3
(
λ x9 x10 .
Inj1
0
)
(
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
x0
(
λ x11 .
x10
)
0
0
x7
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x1
(
λ x11 .
x0
(
λ x12 .
0
)
(
x9
(
λ x12 .
0
)
0
)
0
x7
)
0
)
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
ι →
(
ι → ι
)
→ ι
.
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x3
(
λ x10 x11 .
Inj1
(
x7
(
x7
0
(
λ x12 .
0
)
)
(
λ x12 .
x11
)
)
)
(
x1
(
λ x10 .
setsum
(
x9
(
λ x11 x12 .
0
)
)
(
x2
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
)
)
(
Inj1
(
x3
(
λ x10 x11 .
0
)
0
)
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
setsum
(
x7
(
x9
(
λ x11 .
x2
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 .
0
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
0
)
)
)
(
λ x11 .
x10
)
)
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
setsum
(
x9
(
λ x11 .
0
)
(
x0
(
λ x11 .
x3
(
λ x12 x13 .
0
)
0
)
(
x0
(
λ x11 .
0
)
0
0
0
)
(
setsum
0
0
)
(
setsum
0
0
)
)
)
0
)
=
x3
(
λ x9 x10 .
Inj1
(
x1
(
λ x11 .
0
)
(
x7
(
setsum
0
0
)
(
λ x11 .
0
)
)
)
)
(
x3
(
λ x9 x10 .
setsum
0
(
setsum
(
Inj1
0
)
(
x3
(
λ x11 x12 .
0
)
0
)
)
)
(
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x10 .
x3
(
λ x11 x12 .
0
)
0
)
(
Inj0
0
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
setsum
(
x7
0
(
λ x10 .
0
)
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x3
(
λ x10 x11 .
0
)
(
setsum
(
setsum
0
(
x1
(
λ x10 .
0
)
0
)
)
(
Inj1
(
x5
0
)
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
setsum
(
x6
(
λ x11 .
λ x12 :
ι → ι
.
x0
(
λ x13 .
0
)
(
x2
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
0
)
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
)
0
0
)
)
(
x9
(
λ x11 .
x7
)
(
Inj0
x7
)
)
)
=
x3
(
λ x9 x10 .
x2
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
x7
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
x11
(
λ x13 .
x12
)
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
)
(
Inj1
(
x5
(
x0
(
λ x9 .
x2
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
0
)
)
(
x6
(
λ x9 .
λ x10 :
ι → ι
.
0
)
)
(
Inj0
0
)
0
)
)
)
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x1
Inj1
(
setsum
0
0
)
=
Inj0
(
x7
(
x0
(
λ x9 .
setsum
0
(
x6
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
)
)
0
0
(
setsum
(
setsum
0
0
)
x5
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x2
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
Inj0
(
setsum
0
0
)
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x2
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
setsum
0
0
)
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
x13
(
λ x15 .
0
)
0
)
)
)
0
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x9 .
setsum
x9
(
Inj1
(
x7
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x7
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
)
(
x1
(
λ x9 .
x6
)
(
setsum
0
(
x7
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
=
Inj1
(
x1
(
λ x9 .
x0
(
λ x10 .
x10
)
x5
(
x2
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
x7
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
setsum
0
0
)
)
(
setsum
0
0
)
)
(
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x5
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
x1
(
λ x11 .
x3
(
λ x12 x13 .
0
)
0
)
(
Inj0
0
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x6
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x0
(
λ x9 .
x9
)
(
x6
x4
)
x4
(
setsum
x4
x4
)
=
setsum
(
x0
(
λ x9 .
x5
(
x2
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
x9
)
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
x10
(
λ x12 .
0
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
0
)
)
(
λ x10 :
ι → ι
.
λ x11 .
x11
)
0
)
(
setsum
0
(
x6
0
)
)
(
x5
(
Inj1
(
Inj1
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x1
(
λ x11 .
x1
(
λ x12 .
0
)
0
)
(
x6
0
)
)
x4
)
(
x1
(
λ x9 .
0
)
(
x7
(
λ x9 :
ι → ι
.
x2
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
0
)
)
(
λ x9 x10 .
x10
)
(
λ x9 .
x5
0
(
λ x10 :
ι → ι
.
λ x11 .
0
)
0
)
)
)
)
(
setsum
(
Inj1
(
x3
(
λ x9 x10 .
0
)
0
)
)
(
x1
(
λ x9 .
setsum
(
x1
(
λ x10 .
0
)
0
)
0
)
(
x3
(
λ x9 x10 .
x2
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
)
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x0
(
λ x9 .
0
)
0
0
(
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x5
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x9
(
λ x11 .
x0
(
λ x12 .
setsum
0
0
)
(
x3
(
λ x12 x13 .
0
)
0
)
(
x2
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 .
0
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
0
)
)
0
)
0
)
)
=
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x9
(
λ x10 x11 .
0
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
x2
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
setsum
(
x0
(
λ x12 .
x11
(
λ x13 x14 .
0
)
)
(
Inj0
0
)
(
x1
(
λ x12 .
0
)
0
)
0
)
x10
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
Inj1
(
x1
(
λ x13 .
x10
)
x10
)
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
Inj0
(
x9
(
λ x11 .
0
)
x6
)
)
)
⟶
False
(proof)
Theorem
51ded..
:
∀ x0 :
(
ι → ι
)
→
(
(
ι →
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x1 :
(
ι →
(
(
ι →
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
)
→
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x2 :
(
ι →
ι → ι
)
→
ι → ι
.
∀ x3 :
(
(
ι →
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
(
ι →
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
x9
(
x3
(
λ x10 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
x3
(
λ x11 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
setsum
0
0
)
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
x9
0
(
x1
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x11 .
0
)
)
(
λ x11 .
0
)
0
)
)
0
(
λ x10 .
Inj1
x7
)
0
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
Inj0
0
)
=
Inj1
x7
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
(
ι → ι
)
→
ι → ι
.
∀ x7 :
ι → ι
.
x3
(
λ x9 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
0
)
=
x5
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
x2
(
λ x9 x10 .
x10
)
(
x2
(
λ x9 x10 .
0
)
(
x3
(
λ x9 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
setsum
0
x5
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
Inj0
x6
)
)
)
=
x2
(
λ x9 x10 .
setsum
x6
0
)
(
setsum
x6
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 x10 .
x6
(
λ x11 :
ι →
ι → ι
.
0
)
)
(
x0
(
λ x9 .
x3
(
λ x10 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
x6
(
λ x11 :
ι →
ι → ι
.
0
)
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
x0
(
λ x11 .
x2
(
λ x12 x13 .
0
)
0
)
(
λ x11 :
ι →
ι →
ι → ι
.
x0
(
λ x12 .
0
)
(
λ x12 :
ι →
ι →
ι → ι
.
0
)
)
)
)
(
λ x9 :
ι →
ι →
ι → ι
.
0
)
)
=
Inj1
0
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x7 .
x1
(
λ x9 .
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι → ι
.
x0
(
λ x13 .
Inj1
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
x11
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
Inj1
x7
)
(
x2
(
λ x9 x10 .
0
)
(
setsum
(
setsum
(
x3
(
λ x9 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
(
setsum
0
0
)
)
0
)
)
(
λ x9 .
0
)
=
Inj0
(
setsum
(
x3
(
λ x9 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
Inj0
(
x5
0
)
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
Inj1
x7
)
)
(
x2
(
λ x9 x10 .
0
)
(
Inj0
(
x3
(
λ x9 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x6 :
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x1
(
λ x9 .
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι → ι
.
x9
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x10 .
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
Inj1
(
x3
(
λ x10 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
x9
(
λ x11 x12 .
0
)
)
)
)
(
λ x10 .
Inj1
0
)
)
(
x5
(
λ x9 .
λ x10 :
ι → ι
.
x1
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι → ι
.
Inj1
(
Inj0
0
)
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
x0
(
λ x12 .
x0
(
λ x13 .
0
)
(
λ x13 :
ι →
ι →
ι → ι
.
0
)
)
(
λ x12 :
ι →
ι →
ι → ι
.
Inj1
0
)
)
(
x2
(
λ x11 x12 .
0
)
(
setsum
0
0
)
)
(
λ x11 .
0
)
)
(
setsum
(
Inj1
0
)
0
)
)
(
λ x9 .
x6
(
λ x10 .
setsum
0
(
Inj0
(
x3
(
λ x11 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
)
)
(
Inj0
(
x3
(
λ x10 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
x3
(
λ x11 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
)
)
)
=
setsum
x4
(
x3
(
λ x9 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
x7
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
ι →
ι → ι
.
x0
(
λ x9 .
Inj0
0
)
(
λ x9 :
ι →
ι →
ι → ι
.
0
)
=
setsum
(
setsum
x4
(
x0
(
λ x9 .
x9
)
(
λ x9 :
ι →
ι →
ι → ι
.
x2
(
λ x10 x11 .
x11
)
(
x1
(
λ x10 .
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x10 .
0
)
)
)
)
)
x4
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι →
ι → ι
.
∀ x6 x7 .
x0
(
λ x9 .
x6
)
(
λ x9 :
ι →
ι →
ι → ι
.
x3
(
λ x10 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
x2
(
λ x11 x12 .
x1
(
λ x13 .
λ x14 :
(
ι →
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
ι → ι
.
x2
(
λ x17 x18 .
0
)
0
)
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
setsum
0
0
)
(
x9
0
0
0
)
(
λ x13 .
setsum
0
0
)
)
(
x1
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι → ι
.
x13
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x11 .
x11
)
)
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
=
Inj1
(
x3
(
λ x9 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
x2
(
λ x10 x11 .
x0
(
λ x12 .
x3
(
λ x13 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x13 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
(
λ x12 :
ι →
ι →
ι → ι
.
Inj0
0
)
)
0
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
)
⟶
False
(proof)
Theorem
efa1a..
:
∀ x0 :
(
ι →
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
ι → ι
.
∀ x1 :
(
ι →
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x2 :
(
(
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x3 :
(
(
ι → ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
(
∀ x4 :
(
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 x7 .
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x9 .
0
)
(
x0
(
λ x9 .
λ x10 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
Inj1
x12
)
0
)
=
Inj1
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x9 .
x0
(
λ x10 .
λ x11 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
x11
(
x0
(
λ x15 .
λ x16 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x17 :
ι →
ι → ι
.
λ x18 x19 .
0
)
0
)
(
λ x15 .
x15
)
0
)
(
setsum
(
x2
(
λ x10 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
0
)
(
λ x10 x11 .
0
)
)
0
)
)
x7
)
)
⟶
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x5 :
(
ι →
ι →
ι → ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x1
(
λ x11 .
λ x12 :
(
ι → ι
)
→
ι →
ι → ι
.
Inj1
x11
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
Inj0
(
x0
(
λ x12 .
λ x13 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι →
ι → ι
.
λ x15 x16 .
x13
0
(
λ x17 .
0
)
0
)
(
x11
(
λ x12 .
0
)
0
)
)
)
(
x1
(
λ x11 .
λ x12 :
(
ι → ι
)
→
ι →
ι → ι
.
setsum
(
x1
(
λ x13 .
λ x14 :
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
(
x2
(
λ x13 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 x14 .
0
)
)
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x11
(
λ x14 .
0
)
0
)
(
λ x12 .
x2
(
λ x13 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 x14 .
0
)
)
(
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x12 .
0
)
0
)
)
(
setsum
(
Inj1
0
)
(
x7
(
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
)
)
)
)
(
λ x9 .
Inj0
(
Inj1
(
Inj1
(
x2
(
λ x10 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
0
)
(
λ x10 x11 .
0
)
)
)
)
)
(
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x10
(
x6
0
)
)
(
λ x9 x10 .
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x0
(
λ x13 .
λ x14 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι →
ι → ι
.
λ x16 x17 .
x14
0
(
λ x18 .
0
)
0
)
(
setsum
0
0
)
)
(
λ x11 .
x10
)
(
x1
(
λ x11 .
λ x12 :
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x12 .
0
)
0
)
(
setsum
0
0
)
)
)
)
=
setsum
(
x0
(
λ x9 .
λ x10 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x10
(
setsum
0
0
)
(
λ x14 .
x13
)
(
Inj0
(
x10
0
(
λ x14 .
0
)
0
)
)
)
(
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 x10 .
x6
(
x1
(
λ x11 .
λ x12 :
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
)
)
)
0
)
⟶
(
∀ x4 x5 x6 x7 .
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 x10 .
x1
(
λ x11 .
λ x12 :
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
(
setsum
(
x0
(
λ x11 .
λ x12 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 x15 .
setsum
0
0
)
x9
)
(
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x2
(
λ x13 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 x14 .
0
)
)
(
λ x11 .
0
)
0
)
)
)
=
setsum
x7
(
Inj0
0
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
setsum
0
(
setsum
0
(
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x3
(
λ x13 :
ι → ι
.
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x13 .
0
)
0
)
(
λ x11 .
x10
0
)
(
setsum
0
0
)
)
)
)
(
λ x9 x10 .
x9
)
=
Inj1
(
Inj1
(
x0
(
λ x9 .
λ x10 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x3
(
λ x14 :
ι → ι
.
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x1
(
λ x16 .
λ x17 :
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x16 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
(
λ x14 .
x14
)
x12
)
0
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x1
(
λ x9 .
λ x10 :
(
ι → ι
)
→
ι →
ι → ι
.
x2
(
λ x11 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
(
λ x11 x12 .
Inj1
x9
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
Inj1
(
x1
(
λ x10 .
λ x11 :
(
ι → ι
)
→
ι →
ι → ι
.
x0
(
λ x12 .
λ x13 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι →
ι → ι
.
λ x15 x16 .
Inj0
0
)
0
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x11 .
λ x12 :
(
ι → ι
)
→
ι →
ι → ι
.
x11
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
(
x6
(
λ x11 .
0
)
)
)
(
x0
(
λ x10 .
λ x11 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
x0
(
λ x15 .
λ x16 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x17 :
ι →
ι → ι
.
λ x18 x19 .
0
)
0
)
(
Inj1
0
)
)
)
)
(
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 x10 .
x10
)
)
=
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
setsum
0
(
setsum
0
(
Inj0
(
setsum
0
0
)
)
)
)
(
λ x9 x10 .
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x11
(
x0
(
λ x13 .
λ x14 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι →
ι → ι
.
λ x16 x17 .
0
)
(
setsum
0
0
)
)
)
(
λ x11 .
0
)
(
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
setsum
(
x12
(
λ x13 :
ι → ι
.
λ x14 .
0
)
)
(
x12
(
λ x13 :
ι → ι
.
λ x14 .
0
)
)
)
(
λ x11 .
Inj1
0
)
0
)
)
)
⟶
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 .
λ x10 :
(
ι → ι
)
→
ι →
ι → ι
.
x7
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
setsum
(
x3
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x10 .
x10
)
(
x2
(
λ x10 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
x11
0
)
(
λ x10 x11 .
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x12 .
0
)
0
)
)
)
(
Inj1
(
x0
(
λ x10 .
λ x11 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
x2
(
λ x15 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x16 :
ι → ι
.
0
)
(
λ x15 x16 .
0
)
)
(
Inj0
0
)
)
)
)
0
=
Inj0
(
x4
(
setsum
0
x5
)
(
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x0
(
λ x11 .
λ x12 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 x15 .
x1
(
λ x16 .
λ x17 :
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x16 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
(
setsum
0
0
)
)
(
λ x9 x10 .
setsum
(
setsum
0
0
)
x6
)
)
(
λ x9 .
setsum
0
(
x0
(
λ x10 .
λ x11 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
Inj1
0
)
(
Inj0
0
)
)
)
(
x1
(
λ x9 .
λ x10 :
(
ι → ι
)
→
ι →
ι → ι
.
Inj1
(
setsum
0
0
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
Inj1
(
x3
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x10 .
0
)
0
)
)
(
x4
(
Inj0
0
)
(
Inj0
0
)
(
λ x9 .
Inj1
0
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x9 .
0
)
0
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x0
(
λ x9 .
λ x10 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
0
)
(
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x6
)
(
λ x9 x10 .
Inj0
(
x0
(
λ x11 .
λ x12 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 x15 .
x3
(
λ x16 :
ι → ι
.
λ x17 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x16 .
0
)
0
)
(
x1
(
λ x11 .
λ x12 :
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
)
)
)
=
x2
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x6
)
(
λ x9 x10 .
Inj0
(
setsum
(
x0
(
λ x11 .
λ x12 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 x15 .
x1
(
λ x16 .
λ x17 :
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x16 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
x7
)
(
setsum
(
setsum
0
0
)
(
x0
(
λ x11 .
λ x12 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 x15 .
0
)
0
)
)
)
)
)
⟶
(
∀ x4 :
ι →
ι →
ι →
ι → ι
.
∀ x5 .
∀ x6 :
ι →
(
ι →
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
x0
(
λ x9 .
λ x10 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x13
)
(
setsum
0
0
)
=
x6
(
setsum
(
Inj0
(
x4
(
x4
0
0
0
0
)
(
Inj0
0
)
0
0
)
)
0
)
(
λ x9 x10 .
Inj1
(
x0
(
λ x11 .
λ x12 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 x15 .
x15
)
(
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x1
(
λ x13 .
λ x14 :
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
(
λ x11 .
Inj0
0
)
(
x2
(
λ x11 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
(
λ x11 x12 .
0
)
)
)
)
)
)
⟶
False
(proof)
Theorem
35436..
:
∀ x0 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x1 :
(
ι →
ι →
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι →
ι →
ι → ι
)
→ ι
.
∀ x2 :
(
ι →
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
(
∀ x4 x5 .
∀ x6 x7 :
ι → ι
.
x3
(
λ x9 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
x10
(
λ x11 .
x10
(
λ x12 .
x0
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
(
setsum
0
0
)
(
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
0
)
)
)
)
(
λ x9 :
ι → ι
.
0
)
(
x1
(
λ x9 x10 .
λ x11 :
(
ι → ι
)
→ ι
.
Inj1
(
setsum
x9
(
Inj1
0
)
)
)
(
λ x9 x10 x11 x12 .
0
)
)
0
=
x1
(
λ x9 x10 .
λ x11 :
(
ι → ι
)
→ ι
.
x0
(
λ x12 :
(
ι → ι
)
→ ι
.
Inj1
(
Inj0
0
)
)
0
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
)
(
λ x9 x10 x11 x12 .
setsum
x10
(
setsum
0
x12
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
0
)
(
λ x9 :
ι → ι
.
x6
)
(
setsum
0
x6
)
x5
=
x6
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x2
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x2
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
setsum
(
Inj1
x9
)
(
setsum
0
(
x3
(
λ x13 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 :
ι → ι
.
0
)
0
0
)
)
)
(
λ x11 :
ι → ι
.
λ x12 :
ι →
ι → ι
.
0
)
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
x10
0
(
x2
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x11
)
(
λ x11 :
ι → ι
.
λ x12 :
ι →
ι → ι
.
Inj1
(
x11
0
)
)
)
)
=
Inj0
x4
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x2
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
x0
(
λ x11 :
(
ι → ι
)
→ ι
.
x2
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
Inj0
(
Inj1
0
)
)
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
x3
(
λ x14 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x15 :
(
ι → ι
)
→ ι
.
0
)
(
λ x14 :
ι → ι
.
x2
(
λ x15 .
λ x16 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x15 :
ι → ι
.
λ x16 :
ι →
ι → ι
.
0
)
)
(
setsum
0
0
)
(
x13
0
0
)
)
)
(
Inj1
x6
)
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x10
(
x12
(
setsum
0
0
)
)
(
x12
(
x1
(
λ x14 x15 .
λ x16 :
(
ι → ι
)
→ ι
.
0
)
(
λ x14 x15 x16 x17 .
0
)
)
)
)
)
=
x0
(
λ x9 :
(
ι → ι
)
→ ι
.
x1
(
λ x10 x11 .
λ x12 :
(
ι → ι
)
→ ι
.
Inj0
0
)
(
λ x10 x11 x12 x13 .
x11
)
)
x4
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
Inj1
(
x2
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x2
(
λ x14 .
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x2
(
λ x16 .
λ x17 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x16 :
ι → ι
.
λ x17 :
ι →
ι → ι
.
0
)
)
(
λ x14 :
ι → ι
.
λ x15 :
ι →
ι → ι
.
x3
(
λ x16 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x17 :
(
ι → ι
)
→ ι
.
0
)
(
λ x16 :
ι → ι
.
0
)
0
0
)
)
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
setsum
0
(
x1
(
λ x14 x15 .
λ x16 :
(
ι → ι
)
→ ι
.
0
)
(
λ x14 x15 x16 x17 .
0
)
)
)
)
)
)
⟶
(
∀ x4 :
ι →
ι →
ι →
ι → ι
.
∀ x5 :
(
ι →
ι →
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
ι → ι
)
→ ι
.
x1
(
λ x9 x10 .
λ x11 :
(
ι → ι
)
→ ι
.
Inj1
0
)
(
λ x9 x10 x11 x12 .
x10
)
=
x7
(
λ x9 .
x6
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x6 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x7 :
ι → ι
.
x1
(
λ x9 x10 .
λ x11 :
(
ι → ι
)
→ ι
.
0
)
(
λ x9 x10 x11 x12 .
x1
(
λ x13 x14 .
λ x15 :
(
ι → ι
)
→ ι
.
x15
(
λ x16 .
x15
(
λ x17 .
x2
(
λ x18 .
λ x19 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x18 :
ι → ι
.
λ x19 :
ι →
ι → ι
.
0
)
)
)
)
(
λ x13 x14 x15 x16 .
setsum
(
Inj0
(
x3
(
λ x17 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x18 :
(
ι → ι
)
→ ι
.
0
)
(
λ x17 :
ι → ι
.
0
)
0
0
)
)
x13
)
)
=
Inj0
(
setsum
(
x2
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
x7
0
)
)
0
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 x7 .
x0
(
λ x9 :
(
ι → ι
)
→ ι
.
Inj0
(
Inj0
(
setsum
0
(
x3
(
λ x10 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
0
)
0
0
)
)
)
)
(
x2
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
setsum
(
x9
0
)
(
Inj0
0
)
)
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
Inj0
0
)
=
x2
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x3
(
λ x11 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
x9
)
(
λ x11 :
ι → ι
.
0
)
(
setsum
(
setsum
0
(
x1
(
λ x11 x12 .
λ x13 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 x13 x14 .
0
)
)
)
x6
)
(
Inj1
0
)
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
setsum
0
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x9 :
(
ι → ι
)
→ ι
.
x7
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
setsum
0
(
x1
(
λ x9 x10 .
λ x11 :
(
ι → ι
)
→ ι
.
x2
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x13
(
λ x14 :
ι → ι
.
λ x15 .
0
)
(
λ x14 .
0
)
0
)
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
setsum
0
0
)
)
(
λ x9 x10 x11 x12 .
0
)
)
)
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
Inj1
(
x3
(
λ x12 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x13 :
(
ι → ι
)
→ ι
.
x10
(
Inj1
0
)
)
(
λ x12 :
ι → ι
.
x3
(
λ x13 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x14 :
(
ι → ι
)
→ ι
.
Inj0
0
)
(
λ x13 :
ι → ι
.
0
)
0
(
Inj1
0
)
)
(
x2
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
0
)
)
(
x3
(
λ x12 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x13 :
(
ι → ι
)
→ ι
.
x3
(
λ x14 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x15 :
(
ι → ι
)
→ ι
.
0
)
(
λ x14 :
ι → ι
.
0
)
0
0
)
(
λ x12 :
ι → ι
.
Inj0
0
)
(
x2
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
0
)
)
x11
)
)
)
=
x7
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
setsum
(
Inj1
(
x1
(
λ x10 x11 .
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x10 x11 x12 x13 .
0
)
)
)
(
x7
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x11 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
x9
(
λ x13 .
0
)
0
)
(
λ x11 :
ι → ι
.
x7
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
x6
(
x7
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
)
)
⟶
False
(proof)
Theorem
5ce35..
:
∀ x0 :
(
ι → ι
)
→
ι →
ι →
(
ι →
ι → ι
)
→ ι
.
∀ x1 :
(
(
(
ι → ι
)
→ ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x2 :
(
(
ι → ι
)
→ ι
)
→
(
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x3 :
(
ι → ι
)
→
ι → ι
.
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
x3
(
λ x9 .
0
)
(
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x1
(
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x16 :
ι →
ι → ι
.
λ x17 :
ι → ι
.
λ x18 .
Inj0
(
setsum
0
0
)
)
(
setsum
(
Inj1
0
)
(
x1
(
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x16 :
ι →
ι → ι
.
λ x17 :
ι → ι
.
λ x18 .
0
)
0
)
)
)
0
)
=
Inj0
(
Inj1
(
Inj1
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x6 x7 .
x3
(
λ x9 .
setsum
(
x1
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
λ x14 .
setsum
0
0
)
x9
)
(
x0
(
λ x10 .
x2
(
λ x11 :
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x9
)
)
(
x2
(
λ x10 :
ι → ι
.
x9
)
(
λ x10 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x9
)
)
(
x2
(
λ x10 :
ι → ι
.
setsum
0
0
)
(
λ x10 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
)
(
λ x10 x11 .
Inj1
(
Inj0
0
)
)
)
)
(
x0
(
λ x9 .
Inj1
0
)
0
(
Inj1
(
setsum
0
(
x2
(
λ x9 :
ι → ι
.
0
)
(
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
)
)
)
(
λ x9 x10 .
0
)
)
=
x0
(
λ x9 .
x1
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
λ x14 .
x0
(
λ x15 .
Inj1
(
x2
(
λ x16 :
ι → ι
.
0
)
(
λ x16 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
)
)
(
x11
(
λ x15 :
ι → ι
.
λ x16 .
0
)
)
0
(
λ x15 .
setsum
0
)
)
0
)
(
setsum
0
(
x3
(
λ x9 .
x0
(
λ x10 .
setsum
0
0
)
(
x5
0
(
λ x10 :
ι → ι
.
0
)
0
)
(
Inj1
0
)
(
λ x10 x11 .
x7
)
)
0
)
)
(
x0
(
λ x9 .
x0
(
λ x10 .
x2
(
λ x11 :
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x1
(
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x14 :
ι →
ι → ι
.
λ x15 :
ι → ι
.
λ x16 .
0
)
0
)
)
x9
(
x2
(
λ x10 :
ι → ι
.
x2
(
λ x11 :
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
)
(
λ x10 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
Inj1
0
)
)
(
λ x10 x11 .
x11
)
)
x7
(
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x13
)
0
)
(
λ x9 x10 .
x6
)
)
(
λ x9 x10 .
x9
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
x2
(
λ x9 :
ι → ι
.
x7
(
x7
(
setsum
(
x0
(
λ x10 .
0
)
0
0
(
λ x10 x11 .
0
)
)
0
)
)
)
(
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x9
(
λ x10 x11 .
0
)
(
λ x10 .
x7
(
x7
0
)
)
)
=
setsum
(
x3
(
λ x9 .
0
)
(
x7
(
Inj0
(
x3
(
λ x9 .
0
)
0
)
)
)
)
(
x6
(
λ x9 :
ι → ι
.
λ x10 .
x10
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι →
ι → ι
.
x2
(
λ x9 :
ι → ι
.
x7
(
x9
(
Inj0
(
x7
0
0
)
)
)
(
setsum
(
Inj1
0
)
(
setsum
(
Inj1
0
)
0
)
)
)
(
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x1
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
λ x14 .
x11
(
λ x15 :
ι → ι
.
λ x16 .
0
)
)
(
Inj1
x5
)
)
=
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x2
(
λ x14 :
ι → ι
.
x2
(
λ x15 :
ι → ι
.
x14
(
x3
(
λ x16 .
0
)
0
)
)
(
λ x15 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x3
(
λ x16 .
x16
)
0
)
)
(
λ x14 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x1
(
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x17 :
ι →
ι → ι
.
λ x18 :
ι → ι
.
λ x19 .
Inj1
(
setsum
0
0
)
)
(
setsum
(
setsum
0
0
)
0
)
)
)
(
x0
(
λ x9 .
x1
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
λ x14 .
0
)
(
x7
(
x3
(
λ x10 .
0
)
0
)
(
Inj1
0
)
)
)
(
setsum
(
x4
(
setsum
0
0
)
)
(
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
)
(
x2
(
λ x9 :
ι → ι
.
0
)
(
λ x9 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
)
(
λ x9 x10 .
x3
(
λ x11 .
0
)
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 x7 .
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x12
(
x3
(
λ x14 .
x12
(
Inj0
0
)
)
0
)
)
x7
=
x7
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→ ι
.
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
0
=
x4
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
x0
(
λ x9 .
0
)
(
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x11
(
x0
(
λ x14 .
Inj1
0
)
0
(
setsum
0
0
)
(
λ x14 x15 .
x1
(
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x18 :
ι →
ι → ι
.
λ x19 :
ι → ι
.
λ x20 .
0
)
0
)
)
(
x10
(
λ x14 :
ι → ι
.
λ x15 .
x3
(
λ x16 .
0
)
0
)
)
)
(
Inj0
(
setsum
(
x7
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
0
0
)
(
x4
0
)
)
)
)
x5
(
λ x9 x10 .
0
)
=
Inj1
0
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 x6 x7 .
x0
(
λ x9 .
x1
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
λ x14 .
x13
(
x13
(
x13
0
)
)
)
(
Inj0
x9
)
)
(
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x2
(
λ x14 :
ι → ι
.
setsum
(
x3
(
λ x15 .
0
)
0
)
x13
)
(
λ x14 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
)
(
Inj0
0
)
)
x7
(
λ x9 x10 .
x9
)
=
setsum
(
setsum
(
x4
(
setsum
0
x6
)
(
Inj0
0
)
)
x5
)
0
)
⟶
False
(proof)
Theorem
fbca8..
:
∀ x0 :
(
(
ι →
(
ι →
ι → ι
)
→ ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x1 :
(
(
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
∀ x2 :
(
(
(
ι →
ι → ι
)
→ ι
)
→
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
ι →
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 .
x2
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
x2
(
λ x15 :
(
ι →
ι → ι
)
→ ι
.
λ x16 .
λ x17 :
ι →
ι → ι
.
λ x18 x19 .
x17
(
x3
(
λ x20 .
0
)
(
λ x20 .
0
)
)
0
)
(
setsum
0
(
x1
(
λ x15 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
0
)
)
(
λ x15 .
λ x16 :
ι → ι
.
x15
)
)
0
(
λ x10 .
λ x11 :
ι → ι
.
x7
)
)
(
λ x9 .
x9
)
=
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x13
)
(
Inj0
x4
)
(
λ x9 .
λ x10 :
ι → ι
.
x6
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 x7 :
ι → ι
.
x3
(
λ x9 .
setsum
(
x7
(
x7
0
)
)
(
Inj0
0
)
)
(
λ x9 .
0
)
=
setsum
(
x3
(
λ x9 .
x5
(
λ x10 .
0
)
(
λ x10 :
ι → ι
.
λ x11 .
x10
0
)
)
(
λ x9 .
x7
(
x0
(
λ x10 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
(
λ x10 :
ι → ι
.
x10
0
)
)
)
)
(
setsum
0
(
setsum
0
(
setsum
(
x7
0
)
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
ι →
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x2
(
λ x14 :
(
ι →
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
ι →
ι → ι
.
λ x17 x18 .
x17
)
(
x11
0
(
setsum
x13
(
Inj0
0
)
)
)
(
λ x14 .
λ x15 :
ι → ι
.
Inj1
0
)
)
x7
(
λ x9 .
λ x10 :
ι → ι
.
x0
(
λ x11 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
x0
(
λ x14 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x16 .
Inj1
(
setsum
0
0
)
)
(
λ x14 :
ι → ι
.
x12
(
λ x15 :
ι → ι
.
λ x16 .
Inj0
0
)
(
Inj1
0
)
)
)
(
λ x11 :
ι → ι
.
0
)
)
=
Inj1
(
x3
(
λ x9 .
x2
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
x1
(
λ x15 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
Inj1
0
)
0
)
(
x3
(
λ x10 .
x1
(
λ x11 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
0
)
(
λ x10 .
x10
)
)
(
λ x10 .
λ x11 :
ι → ι
.
setsum
x10
(
x1
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
0
)
)
)
(
λ x9 .
Inj1
x5
)
)
)
⟶
(
∀ x4 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x2
(
λ x14 :
(
ι →
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
ι →
ι → ι
.
λ x17 x18 .
0
)
(
setsum
(
setsum
x12
(
x0
(
λ x14 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x16 .
0
)
(
λ x14 :
ι → ι
.
0
)
)
)
0
)
(
λ x14 .
λ x15 :
ι → ι
.
0
)
)
(
x3
(
λ x9 .
x2
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
λ x13 .
x1
(
λ x14 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x14
(
λ x15 .
0
)
(
λ x15 x16 .
0
)
)
)
(
Inj0
0
)
(
λ x10 .
λ x11 :
ι → ι
.
0
)
)
(
λ x9 .
0
)
)
(
λ x9 .
λ x10 :
ι → ι
.
setsum
(
setsum
0
(
x3
(
λ x11 .
setsum
0
0
)
(
λ x11 .
setsum
0
0
)
)
)
0
)
=
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
setsum
(
setsum
x10
(
Inj0
0
)
)
(
x3
(
λ x14 .
0
)
(
λ x14 .
Inj0
(
setsum
0
0
)
)
)
)
(
Inj1
(
setsum
(
x4
(
Inj1
0
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
(
x6
0
)
0
)
0
)
)
(
λ x9 .
λ x10 :
ι → ι
.
x9
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
setsum
0
(
x3
(
λ x10 .
0
)
(
λ x10 .
setsum
(
x1
(
λ x11 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
0
)
(
x1
(
λ x11 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
0
)
)
)
)
0
=
x5
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
ι →
ι → ι
)
→ ι
.
∀ x7 .
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x9
(
λ x10 .
x0
(
λ x11 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
(
λ x11 :
ι → ι
.
0
)
)
(
λ x10 x11 .
setsum
(
Inj0
(
setsum
0
0
)
)
x10
)
)
(
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
setsum
(
x3
(
λ x14 .
0
)
(
λ x14 .
0
)
)
(
x2
(
λ x14 :
(
ι →
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
ι →
ι → ι
.
λ x17 x18 .
0
)
(
setsum
0
0
)
(
λ x14 .
λ x15 :
ι → ι
.
0
)
)
)
(
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
setsum
0
0
)
)
(
λ x9 .
λ x10 :
ι → ι
.
x2
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι →
ι → ι
.
λ x14 x15 .
setsum
0
(
x0
(
λ x16 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x17 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x18 .
0
)
(
λ x16 :
ι → ι
.
0
)
)
)
(
setsum
0
0
)
(
λ x11 .
λ x12 :
ι → ι
.
0
)
)
)
=
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 .
setsum
0
)
(
setsum
x4
(
x6
(
λ x9 x10 .
x0
(
λ x11 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
setsum
0
0
)
(
λ x11 :
ι → ι
.
0
)
)
)
)
(
λ x9 .
λ x10 :
ι → ι
.
Inj1
(
Inj1
(
Inj0
(
x10
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
(
λ x9 :
ι → ι
.
x2
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
setsum
0
(
setsum
0
(
setsum
0
0
)
)
)
(
x9
(
setsum
0
x5
)
)
(
λ x10 .
λ x11 :
ι → ι
.
Inj1
(
x3
(
λ x12 .
0
)
(
λ x12 .
Inj1
0
)
)
)
)
=
setsum
0
(
Inj0
(
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 .
Inj1
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
ι → ι
)
→ ι
.
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 .
x2
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι →
ι → ι
.
λ x15 x16 .
0
)
0
(
λ x12 .
λ x13 :
ι → ι
.
Inj1
0
)
)
(
λ x9 :
ι → ι
.
0
)
=
Inj0
(
Inj1
(
Inj0
x6
)
)
)
⟶
False
(proof)
Theorem
27d8b..
:
∀ x0 :
(
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
ι →
(
ι → ι
)
→ ι
)
→
(
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x1 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x2 :
(
ι → ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι →
(
ι → ι
)
→ ι
.
∀ x3 :
(
ι →
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
(
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x7 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x11 :
ι →
ι → ι
.
x3
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x13 .
setsum
(
x10
(
λ x14 :
ι → ι
.
0
)
)
0
)
)
(
Inj1
x9
)
)
(
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
0
)
=
x1
(
λ x9 :
ι →
ι → ι
.
x3
(
λ x10 .
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x10
)
(
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x11 .
Inj1
(
x10
(
λ x12 .
x10
(
λ x13 .
0
)
(
λ x13 .
0
)
)
(
λ x12 .
x11
)
)
)
)
(
setsum
(
x5
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj0
0
)
(
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
x3
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
0
)
)
)
)
(
Inj0
0
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x7
)
(
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
0
)
=
setsum
(
setsum
x6
0
)
(
setsum
0
(
setsum
x6
(
setsum
(
x0
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
0
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
(
x2
(
λ x9 .
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
(
λ x9 .
0
)
)
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 .
setsum
0
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
x2
(
λ x10 .
0
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
(
x0
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 .
λ x13 :
ι → ι
.
setsum
0
(
Inj1
0
)
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 :
ι → ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x6
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
Inj1
0
)
)
)
(
x0
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 .
λ x13 :
ι → ι
.
setsum
0
0
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 :
ι → ι
.
x2
(
λ x12 .
Inj1
0
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
0
(
x10
(
λ x12 :
ι → ι
.
λ x13 .
0
)
0
0
)
(
λ x12 .
x11
0
)
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
)
(
λ x10 .
setsum
(
x6
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
)
(
x3
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x10
)
(
λ x11 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
0
)
)
)
)
0
(
x2
(
λ x9 .
Inj0
(
Inj1
x7
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
Inj1
x7
)
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x6
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
x11
(
λ x13 .
0
)
0
)
)
(
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
setsum
(
x9
(
λ x11 .
0
)
(
λ x11 .
0
)
)
(
setsum
0
0
)
)
)
x4
(
λ x9 .
Inj1
0
)
)
(
λ x9 .
Inj1
(
x6
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
)
)
=
setsum
0
(
setsum
x5
(
x0
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
x12
x10
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 :
ι → ι
.
x3
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x13 :
ι →
(
ι → ι
)
→ ι
.
λ x14 x15 .
λ x16 :
ι → ι
.
0
)
(
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
)
(
λ x11 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 .
0
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x7
)
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
x2
(
λ x9 .
x9
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
setsum
(
x0
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 .
λ x13 :
ι → ι
.
x10
x11
(
λ x14 .
x13
0
)
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 :
ι → ι
.
x11
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
)
(
Inj1
0
)
)
0
(
setsum
0
(
x7
(
λ x9 x10 :
ι → ι
.
0
)
0
(
λ x9 .
0
)
(
Inj1
(
Inj1
0
)
)
)
)
(
λ x9 .
x0
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 .
λ x13 :
ι → ι
.
x11
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 :
ι → ι
.
x7
(
λ x12 x13 :
ι → ι
.
x13
(
x2
(
λ x14 .
0
)
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
(
λ x14 .
0
)
)
)
(
Inj1
(
x10
(
λ x12 :
ι → ι
.
λ x13 .
0
)
0
0
)
)
(
λ x12 .
x9
)
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x10
(
λ x11 .
setsum
x11
0
)
)
)
=
setsum
(
setsum
(
setsum
(
Inj0
(
x2
(
λ x9 .
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
(
λ x9 .
0
)
)
)
(
x0
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
setsum
0
0
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 :
ι → ι
.
x6
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x5
)
)
)
x6
)
(
Inj0
(
x7
(
λ x9 x10 :
ι → ι
.
setsum
(
setsum
0
0
)
(
x9
0
)
)
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
0
0
)
(
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
Inj0
0
)
)
(
λ x9 .
0
)
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x1
(
λ x9 :
ι →
ι → ι
.
0
)
x7
=
setsum
0
(
setsum
(
setsum
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x9 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x10 .
Inj1
0
)
)
(
setsum
0
0
)
)
(
x5
x4
(
Inj1
x7
)
(
λ x9 .
Inj0
(
x2
(
λ x10 .
0
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
(
λ x10 .
0
)
)
)
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
(
ι →
ι →
ι → ι
)
→ ι
.
x1
(
λ x9 :
ι →
ι → ι
.
x9
0
0
)
0
=
x6
0
(
λ x9 :
ι → ι
.
setsum
(
Inj1
(
x7
(
λ x10 x11 x12 .
x0
(
λ x13 :
ι →
(
ι → ι
)
→ ι
.
λ x14 x15 .
λ x16 :
ι → ι
.
0
)
(
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
(
x6
(
x0
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 .
λ x13 :
ι → ι
.
x3
(
λ x14 .
λ x15 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x14 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x15 .
0
)
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 :
ι → ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x2
(
λ x11 .
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
(
λ x11 .
0
)
)
)
(
λ x10 :
ι → ι
.
0
)
)
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 :
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x0
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
0
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 :
ι → ι
.
x7
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x3
(
λ x10 .
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
(
x9
(
λ x12 .
0
)
)
(
Inj1
0
)
)
(
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x11 .
x1
(
λ x12 :
ι →
ι → ι
.
Inj0
0
)
x7
)
)
=
Inj1
x7
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
(
ι →
ι →
ι → ι
)
→
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x0
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
Inj1
(
setsum
x10
0
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 :
ι → ι
.
setsum
(
x1
(
λ x11 :
ι →
ι → ι
.
0
)
(
x10
(
setsum
0
0
)
)
)
(
x6
(
x2
(
λ x11 .
Inj0
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x12 :
ι →
ι → ι
.
0
)
0
)
0
0
(
λ x11 .
x9
(
λ x12 :
ι → ι
.
λ x13 .
0
)
0
0
)
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x6
0
)
=
Inj0
(
x0
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
x0
(
λ x13 :
ι →
(
ι → ι
)
→ ι
.
λ x14 x15 .
λ x16 :
ι → ι
.
x15
)
(
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x6
(
x0
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 .
λ x13 :
ι → ι
.
Inj1
0
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
λ x11 :
ι → ι
.
Inj1
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
)
⟶
False
(proof)
Theorem
a0efe..
:
∀ x0 :
(
(
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x1 :
(
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x2 :
(
ι → ι
)
→
ι →
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x3 :
(
ι → ι
)
→
(
ι → ι
)
→
(
ι →
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
(
∀ x4 x5 .
∀ x6 :
(
ι →
ι →
ι → ι
)
→ ι
.
∀ x7 :
ι →
ι → ι
.
x3
(
λ x9 .
0
)
(
λ x9 .
0
)
(
λ x9 .
λ x10 :
ι → ι
.
setsum
(
setsum
(
setsum
0
(
x3
(
λ x11 .
0
)
(
λ x11 .
0
)
(
λ x11 .
λ x12 :
ι → ι
.
0
)
(
λ x11 .
0
)
)
)
(
x3
(
λ x11 .
x11
)
(
λ x11 .
x7
0
0
)
(
λ x11 .
λ x12 :
ι → ι
.
setsum
0
0
)
(
λ x11 .
0
)
)
)
(
setsum
0
0
)
)
(
λ x9 .
Inj1
(
Inj1
x9
)
)
=
setsum
(
Inj0
x4
)
(
Inj1
(
x7
(
setsum
(
Inj0
0
)
(
Inj1
0
)
)
x5
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x6 x7 .
x3
(
λ x9 .
x9
)
(
λ x9 .
x7
)
(
λ x9 .
λ x10 :
ι → ι
.
x9
)
(
λ x9 .
Inj0
(
Inj0
(
x3
(
λ x10 .
x6
)
(
λ x10 .
0
)
(
λ x10 .
λ x11 :
ι → ι
.
x3
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x12 .
0
)
)
(
λ x10 .
x7
)
)
)
)
=
setsum
(
x1
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
x9
(
λ x11 :
ι → ι
.
λ x12 .
setsum
0
(
setsum
0
0
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
)
(
Inj0
(
x5
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x10 .
0
)
(
λ x10 .
x0
(
λ x11 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x10 .
λ x11 :
ι → ι
.
x3
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x12 .
0
)
)
(
λ x10 .
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
x2
(
λ x9 .
x1
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
x1
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj1
x9
)
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x11 .
x0
(
λ x12 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
x10
(
λ x13 :
ι → ι
.
0
)
)
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
x10
(
λ x14 :
ι → ι
.
0
)
)
)
(
λ x11 .
Inj1
0
)
(
λ x11 .
λ x12 :
ι → ι
.
setsum
x9
x9
)
(
λ x11 .
Inj1
(
x3
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x12 .
0
)
)
)
)
)
(
x2
(
λ x9 .
setsum
0
(
x2
(
λ x10 .
x2
(
λ x11 .
0
)
0
(
λ x11 x12 :
ι → ι
.
λ x13 .
0
)
0
)
(
setsum
0
0
)
(
λ x10 x11 :
ι → ι
.
λ x12 .
0
)
0
)
)
(
x3
(
λ x9 .
x3
(
λ x10 .
x0
(
λ x11 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x10 .
setsum
0
0
)
(
λ x10 .
λ x11 :
ι → ι
.
0
)
(
λ x10 .
x7
(
λ x11 :
ι → ι
.
0
)
(
λ x11 :
ι → ι
.
0
)
0
0
)
)
(
λ x9 .
x7
(
λ x10 :
ι → ι
.
x3
(
λ x11 .
0
)
(
λ x11 .
0
)
(
λ x11 .
λ x12 :
ι → ι
.
0
)
(
λ x11 .
0
)
)
(
λ x10 :
ι → ι
.
0
)
x6
x6
)
(
λ x9 .
λ x10 :
ι → ι
.
x2
(
λ x11 .
x10
0
)
(
setsum
0
0
)
(
λ x11 x12 :
ι → ι
.
λ x13 .
x0
(
λ x14 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x14 :
ι → ι
.
λ x15 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
setsum
0
0
)
)
(
λ x9 .
x7
(
λ x10 :
ι → ι
.
Inj1
0
)
(
λ x10 :
ι → ι
.
0
)
(
x1
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
0
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
)
(
x2
(
λ x10 .
0
)
0
(
λ x10 x11 :
ι → ι
.
λ x12 .
0
)
0
)
)
)
(
λ x9 x10 :
ι → ι
.
λ x11 .
0
)
x5
)
(
λ x9 x10 :
ι → ι
.
λ x11 .
x11
)
(
Inj1
0
)
=
x2
(
λ x9 .
x1
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
0
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
(
x7
(
λ x11 :
ι → ι
.
0
)
(
λ x11 :
ι → ι
.
x1
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
)
(
x3
(
λ x11 .
0
)
(
λ x11 .
0
)
(
λ x11 .
λ x12 :
ι → ι
.
0
)
(
λ x11 .
0
)
)
(
x1
(
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
)
)
0
)
)
x6
(
λ x9 x10 :
ι → ι
.
λ x11 .
x9
(
Inj1
(
setsum
0
(
setsum
0
0
)
)
)
)
(
x2
(
λ x9 .
x7
(
λ x10 :
ι → ι
.
0
)
(
λ x10 :
ι → ι
.
setsum
(
Inj0
0
)
0
)
0
0
)
(
setsum
x6
(
x7
(
λ x9 :
ι → ι
.
Inj0
0
)
(
λ x9 :
ι → ι
.
x0
(
λ x10 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
)
0
0
)
)
(
λ x9 x10 :
ι → ι
.
λ x11 .
0
)
x4
)
)
⟶
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 x7 .
x2
(
λ x9 .
x3
(
λ x10 .
x7
)
(
λ x10 .
x6
)
(
λ x10 .
λ x11 :
ι → ι
.
setsum
(
setsum
(
x11
0
)
x9
)
(
x1
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
setsum
0
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj1
0
)
)
)
(
λ x10 .
0
)
)
(
Inj0
0
)
(
λ x9 x10 :
ι → ι
.
λ x11 .
Inj0
(
Inj0
(
x3
(
λ x12 .
x0
(
λ x13 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x13 :
ι → ι
.
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x12 .
setsum
0
0
)
(
λ x12 .
λ x13 :
ι → ι
.
x3
(
λ x14 .
0
)
(
λ x14 .
0
)
(
λ x14 .
λ x15 :
ι → ι
.
0
)
(
λ x14 .
0
)
)
(
λ x12 .
0
)
)
)
)
x6
=
Inj1
0
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
Inj0
0
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj1
(
x9
(
λ x10 :
ι → ι
.
0
)
)
)
=
x7
(
setsum
(
x0
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
x0
(
λ x10 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
x2
(
λ x11 .
0
)
0
(
λ x11 x12 :
ι → ι
.
λ x13 .
0
)
0
)
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x12 .
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x11 .
x1
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
)
(
λ x11 .
0
)
(
λ x11 .
λ x12 :
ι → ι
.
x1
(
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x14 .
0
)
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
)
(
λ x11 .
x9
0
)
)
)
x6
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι →
ι → ι
.
∀ x7 .
x1
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
setsum
(
x3
(
λ x11 .
x9
(
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
λ x11 .
Inj1
(
setsum
0
0
)
)
(
λ x11 .
λ x12 :
ι → ι
.
0
)
(
λ x11 .
0
)
)
(
setsum
(
x6
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x13
)
(
λ x11 .
x3
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x12 .
0
)
)
(
x9
(
λ x11 :
ι → ι
.
λ x12 .
0
)
)
0
)
(
x6
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x2
(
λ x14 .
0
)
0
(
λ x14 x15 :
ι → ι
.
λ x16 .
0
)
0
)
(
λ x11 .
Inj0
0
)
x10
0
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
x2
(
λ x12 .
x2
(
λ x13 .
x3
(
λ x14 .
0
)
(
λ x14 .
0
)
(
λ x14 .
λ x15 :
ι → ι
.
0
)
(
λ x14 .
0
)
)
(
setsum
0
0
)
(
λ x13 x14 :
ι → ι
.
λ x15 .
x14
0
)
(
x2
(
λ x13 .
0
)
0
(
λ x13 x14 :
ι → ι
.
λ x15 .
0
)
0
)
)
0
(
λ x12 x13 :
ι → ι
.
λ x14 .
x11
)
(
x1
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
setsum
0
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x11
)
)
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
0
(
x3
(
λ x11 .
0
)
(
λ x11 .
x10
(
λ x12 :
ι → ι
.
0
)
)
(
λ x11 .
λ x12 :
ι → ι
.
Inj0
0
)
(
λ x11 .
x7
)
)
)
)
=
x1
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
Inj0
(
Inj1
(
x3
(
λ x11 .
0
)
(
λ x11 .
x3
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x12 .
0
)
)
(
λ x11 .
λ x12 :
ι → ι
.
0
)
(
λ x11 .
0
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj1
(
x6
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x10
)
(
λ x10 .
x2
(
λ x11 .
Inj0
0
)
x10
(
λ x11 x12 :
ι → ι
.
λ x13 .
0
)
(
x3
(
λ x11 .
0
)
(
λ x11 .
0
)
(
λ x11 .
λ x12 :
ι → ι
.
0
)
(
λ x11 .
0
)
)
)
0
(
x2
(
λ x10 .
x2
(
λ x11 .
0
)
0
(
λ x11 x12 :
ι → ι
.
λ x13 .
0
)
0
)
(
x9
(
λ x10 :
ι → ι
.
0
)
)
(
λ x10 x11 :
ι → ι
.
λ x12 .
x3
(
λ x13 .
0
)
(
λ x13 .
0
)
(
λ x13 .
λ x14 :
ι → ι
.
0
)
(
λ x13 .
0
)
)
(
x3
(
λ x10 .
0
)
(
λ x10 .
0
)
(
λ x10 .
λ x11 :
ι → ι
.
0
)
(
λ x10 .
0
)
)
)
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x0
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
Inj0
(
Inj1
(
Inj1
(
setsum
0
0
)
)
)
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
=
setsum
0
(
Inj0
x5
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x0
(
λ x9 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
x9
(
setsum
0
(
x2
(
λ x10 .
x10
)
0
(
λ x10 x11 :
ι → ι
.
λ x12 .
x1
(
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x14 .
0
)
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
)
(
Inj0
0
)
)
)
(
λ x10 :
ι → ι
.
λ x11 .
setsum
(
x2
(
λ x12 .
setsum
0
0
)
0
(
λ x12 x13 :
ι → ι
.
λ x14 .
Inj1
0
)
(
setsum
0
0
)
)
(
Inj0
(
x0
(
λ x12 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
(
λ x10 .
x7
(
x7
(
Inj0
0
)
)
)
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
x2
(
λ x11 .
x0
(
λ x12 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
x11
)
)
(
x7
0
)
(
λ x11 x12 :
ι → ι
.
λ x13 .
x1
(
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 .
x1
(
λ x16 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x17 .
x0
(
λ x18 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x18 :
ι → ι
.
λ x19 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x16 :
(
(
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x17 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x17 :
ι → ι
.
λ x18 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
Inj0
x13
)
)
(
x1
(
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
0
)
)
)
=
setsum
0
(
setsum
0
0
)
)
⟶
False
(proof)
Theorem
23307..
:
∀ x0 :
(
(
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x1 :
(
(
ι → ι
)
→ ι
)
→
(
ι →
ι →
ι → ι
)
→ ι
.
∀ x2 :
(
ι →
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x3 :
(
ι → ι
)
→
ι → ι
.
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x7 .
x3
(
λ x9 .
x9
)
(
x1
(
λ x9 :
ι → ι
.
0
)
(
λ x9 x10 x11 .
0
)
)
=
x1
(
λ x9 :
ι → ι
.
x5
)
(
λ x9 x10 x11 .
x11
)
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 .
∀ x6 x7 :
ι → ι
.
x3
(
λ x9 .
Inj0
x5
)
(
Inj1
(
x6
(
x1
(
λ x9 :
ι → ι
.
setsum
0
0
)
(
λ x9 x10 x11 .
setsum
0
0
)
)
)
)
=
x4
0
(
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
(
x1
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
ι → ι
.
0
)
(
λ x10 x11 x12 .
0
)
)
(
λ x9 x10 x11 .
x9
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
(
ι → ι
)
→
ι →
ι → ι
.
∀ x7 :
ι → ι
.
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x2
(
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
0
)
x5
=
setsum
(
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
(
setsum
(
x4
x5
)
0
)
)
(
x3
(
λ x9 .
x7
(
x6
(
λ x10 .
setsum
0
0
)
(
x0
(
λ x10 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 :
ι →
ι → ι
.
0
)
0
(
λ x10 .
0
)
)
(
x3
(
λ x10 .
0
)
0
)
)
)
0
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→ ι
.
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
(
setsum
(
Inj1
(
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 .
x9
(
λ x12 .
0
)
)
)
)
x4
)
=
x6
(
λ x9 :
(
ι → ι
)
→ ι
.
setsum
0
(
x6
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
(
x7
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
0
)
)
)
)
x4
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x1
(
λ x9 :
ι → ι
.
x3
(
λ x10 .
x1
(
λ x11 :
ι → ι
.
setsum
(
x0
(
λ x12 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 :
ι →
ι → ι
.
0
)
0
(
λ x12 .
0
)
)
(
setsum
0
0
)
)
(
λ x11 x12 x13 .
x3
(
λ x14 .
x12
)
(
Inj0
0
)
)
)
(
setsum
(
x5
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x10
(
λ x11 .
0
)
0
)
(
λ x10 .
x2
(
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
0
)
(
setsum
0
0
)
)
x7
)
)
(
λ x9 x10 x11 .
setsum
0
(
Inj0
(
Inj0
(
x3
(
λ x12 .
0
)
0
)
)
)
)
=
setsum
x7
0
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x9 :
ι → ι
.
0
)
(
λ x9 x10 x11 .
x2
(
λ x12 .
λ x13 :
ι →
ι → ι
.
setsum
(
x2
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
x10
)
0
)
(
x1
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 x14 .
0
)
)
)
=
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
Inj1
(
x10
(
x1
(
λ x11 :
ι → ι
.
x1
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 x14 .
0
)
)
(
λ x11 x12 x13 .
x2
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
0
)
)
x9
)
)
(
x3
(
λ x9 .
0
)
(
Inj0
(
Inj0
(
x3
(
λ x9 .
0
)
0
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
x1
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 x14 .
Inj0
(
x11
0
(
x11
0
0
)
)
)
)
(
x3
(
λ x9 .
Inj0
x6
)
x6
)
(
λ x9 .
0
)
=
Inj0
(
Inj1
(
setsum
(
x2
(
λ x9 .
λ x10 :
ι →
ι → ι
.
setsum
0
0
)
(
setsum
0
0
)
)
(
x4
(
x3
(
λ x9 .
0
)
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 :
ι → ι
.
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
x0
(
λ x12 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 :
ι →
ι → ι
.
x14
(
x3
(
λ x15 .
x2
(
λ x16 .
λ x17 :
ι →
ι → ι
.
0
)
0
)
(
x1
(
λ x15 :
ι → ι
.
0
)
(
λ x15 x16 x17 .
0
)
)
)
0
)
(
x11
(
x2
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
(
x11
0
0
)
)
0
)
(
λ x12 .
x2
(
λ x13 .
λ x14 :
ι →
ι → ι
.
x13
)
0
)
)
(
Inj0
(
x6
(
x7
0
)
)
)
(
λ x9 .
setsum
0
0
)
=
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
Inj0
0
)
(
Inj0
(
Inj1
(
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
x10
0
)
(
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
0
)
0
(
λ x9 .
0
)
)
(
λ x9 .
x1
(
λ x10 :
ι → ι
.
0
)
(
λ x10 x11 x12 .
0
)
)
)
)
)
(
λ x9 .
Inj0
(
Inj1
(
setsum
(
x6
0
)
(
x7
0
)
)
)
)
)
⟶
False
(proof)
Theorem
c725c..
:
∀ x0 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι →
(
ι →
ι → ι
)
→ ι
.
∀ x1 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι →
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x2 :
(
(
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
)
→
(
ι →
ι →
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x3 :
(
(
ι →
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 x6 x7 .
x3
(
λ x9 :
ι →
(
ι →
ι → ι
)
→ ι
.
setsum
x5
0
)
0
=
setsum
x7
(
Inj0
x7
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x3
(
λ x9 :
ι →
(
ι →
ι → ι
)
→ ι
.
0
)
(
Inj0
0
)
=
x5
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι →
ι →
ι → ι
.
x9
(
λ x11 .
λ x12 :
ι → ι
.
setsum
(
x2
(
λ x13 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι →
ι →
ι → ι
.
x11
)
(
λ x13 x14 .
x2
(
λ x15 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x16 :
ι →
ι →
ι → ι
.
0
)
(
λ x15 x16 .
0
)
)
)
(
x0
(
λ x13 :
ι →
ι → ι
.
Inj1
0
)
0
(
λ x13 x14 .
0
)
)
)
(
x6
(
x0
(
λ x11 :
ι →
ι → ι
.
Inj0
0
)
(
x10
0
0
0
)
(
λ x11 x12 .
x11
)
)
)
(
λ x11 .
0
)
(
setsum
(
x1
(
λ x11 :
ι →
ι → ι
.
0
)
(
λ x11 x12 .
λ x13 :
ι → ι
.
Inj0
0
)
)
(
Inj1
(
x3
(
λ x11 :
ι →
(
ι →
ι → ι
)
→ ι
.
0
)
0
)
)
)
)
(
λ x9 x10 .
setsum
(
x6
x10
)
(
Inj0
(
setsum
x9
0
)
)
)
=
x4
(
x6
(
Inj0
(
x1
(
λ x9 :
ι →
ι → ι
.
x3
(
λ x10 :
ι →
(
ι →
ι → ι
)
→ ι
.
0
)
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
x11
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 :
ι →
(
ι → ι
)
→ ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι →
ι →
ι → ι
.
setsum
(
Inj0
(
x6
(
setsum
0
0
)
(
λ x11 .
setsum
0
0
)
)
)
(
x0
(
λ x11 :
ι →
ι → ι
.
x10
(
setsum
0
0
)
(
x1
(
λ x12 :
ι →
ι → ι
.
0
)
(
λ x12 x13 .
λ x14 :
ι → ι
.
0
)
)
0
)
(
Inj1
(
x0
(
λ x11 :
ι →
ι → ι
.
0
)
0
(
λ x11 x12 .
0
)
)
)
(
λ x11 x12 .
Inj1
(
Inj1
0
)
)
)
)
(
λ x9 x10 .
x10
)
=
x5
(
x3
(
λ x9 :
ι →
(
ι →
ι → ι
)
→ ι
.
x0
(
λ x10 :
ι →
ι → ι
.
x6
(
setsum
0
0
)
(
λ x11 .
x7
(
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
)
)
(
x6
(
x0
(
λ x10 :
ι →
ι → ι
.
0
)
0
(
λ x10 x11 .
0
)
)
(
λ x10 .
0
)
)
(
λ x10 x11 .
x1
(
λ x12 :
ι →
ι → ι
.
0
)
(
λ x12 x13 .
λ x14 :
ι → ι
.
0
)
)
)
x4
)
)
⟶
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
x1
(
λ x9 :
ι →
ι → ι
.
x1
(
λ x10 :
ι →
ι → ι
.
x2
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 x12 .
x1
(
λ x13 :
ι →
ι → ι
.
x12
)
(
λ x13 x14 .
λ x15 :
ι → ι
.
setsum
0
0
)
)
)
(
λ x10 x11 .
λ x12 :
ι → ι
.
setsum
(
x0
(
λ x13 :
ι →
ι → ι
.
0
)
(
setsum
0
0
)
(
λ x13 x14 .
0
)
)
0
)
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
x0
(
λ x12 :
ι →
ι → ι
.
0
)
0
(
λ x12 x13 .
x13
)
)
=
x1
(
λ x9 :
ι →
ι → ι
.
setsum
(
x3
(
λ x10 :
ι →
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x11 :
ι →
ι → ι
.
x9
0
0
)
(
λ x11 x12 .
λ x13 :
ι → ι
.
0
)
)
x6
)
(
setsum
0
0
)
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
x10
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
x1
(
λ x9 :
ι →
ι → ι
.
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
setsum
(
x7
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
x0
(
λ x13 :
ι →
ι → ι
.
x10
)
0
(
λ x13 x14 .
setsum
0
0
)
)
0
(
λ x12 .
setsum
(
Inj1
0
)
x9
)
(
x2
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι →
ι →
ι → ι
.
0
)
(
λ x12 x13 .
x2
(
λ x14 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι →
ι →
ι → ι
.
0
)
(
λ x14 x15 .
0
)
)
)
)
x9
)
=
x5
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
λ x11 .
0
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x0
(
λ x9 :
ι →
ι → ι
.
x7
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
x2
(
λ x13 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι →
ι →
ι → ι
.
setsum
0
0
)
(
λ x13 x14 .
Inj0
(
x2
(
λ x15 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x16 :
ι →
ι →
ι → ι
.
0
)
(
λ x15 x16 .
0
)
)
)
)
(
λ x10 x11 .
x7
(
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x1
(
λ x15 :
ι →
ι → ι
.
x3
(
λ x16 :
ι →
(
ι →
ι → ι
)
→ ι
.
0
)
0
)
(
λ x15 x16 .
λ x17 :
ι → ι
.
Inj0
0
)
)
(
λ x12 x13 .
x0
(
λ x14 :
ι →
ι → ι
.
0
)
(
Inj0
0
)
(
λ x14 x15 .
setsum
0
0
)
)
)
)
x5
(
λ x9 x10 .
x0
(
λ x11 :
ι →
ι → ι
.
Inj1
x9
)
x6
(
λ x11 x12 .
0
)
)
=
Inj1
(
x1
(
λ x9 :
ι →
ι → ι
.
x9
(
x0
(
λ x10 :
ι →
ι → ι
.
x0
(
λ x11 :
ι →
ι → ι
.
0
)
0
(
λ x11 x12 .
0
)
)
(
setsum
0
0
)
(
λ x10 x11 .
x2
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι →
ι →
ι → ι
.
0
)
(
λ x12 x13 .
0
)
)
)
(
x1
(
λ x10 :
ι →
ι → ι
.
x6
)
(
λ x10 x11 .
λ x12 :
ι → ι
.
x11
)
)
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 :
ι →
ι → ι
.
x0
(
λ x9 :
ι →
ι → ι
.
x3
(
λ x10 :
ι →
(
ι →
ι → ι
)
→ ι
.
x6
(
λ x11 .
x10
(
x2
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι →
ι →
ι → ι
.
0
)
(
λ x12 x13 .
0
)
)
(
λ x12 x13 .
0
)
)
)
(
x6
(
λ x10 .
Inj1
(
x1
(
λ x11 :
ι →
ι → ι
.
0
)
(
λ x11 x12 .
λ x13 :
ι → ι
.
0
)
)
)
)
)
(
x6
(
λ x9 .
0
)
)
(
λ x9 x10 .
x9
)
=
x6
(
λ x9 .
x9
)
)
⟶
False
(proof)
Theorem
1af22..
:
∀ x0 :
(
(
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x1 :
(
ι → ι
)
→
ι →
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x2 :
(
ι → ι
)
→
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
ι → ι
.
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x10 .
0
)
(
λ x10 :
ι →
ι → ι
.
Inj0
(
Inj0
(
x2
(
λ x11 .
0
)
(
λ x11 :
ι →
ι → ι
.
0
)
)
)
)
)
(
setsum
(
x2
(
λ x9 .
setsum
x7
x7
)
(
λ x9 :
ι →
ι → ι
.
0
)
)
(
x6
(
λ x9 :
ι →
ι → ι
.
Inj0
0
)
)
)
=
x2
(
λ x9 .
setsum
(
setsum
(
setsum
(
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
(
setsum
0
0
)
)
(
x0
(
λ x10 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
x6
(
λ x11 :
ι →
ι → ι
.
0
)
)
(
λ x10 :
ι → ι
.
λ x11 .
setsum
0
0
)
)
)
(
x0
(
λ x10 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 .
setsum
0
(
setsum
0
0
)
)
)
)
(
λ x9 :
ι →
ι → ι
.
x9
x7
x7
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
(
ι →
ι → ι
)
→ ι
.
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x10 .
0
)
(
x7
(
λ x10 .
setsum
(
x1
(
λ x11 .
0
)
0
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
0
0
0
)
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
setsum
(
x0
(
λ x12 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
x10
(
λ x13 .
0
)
0
)
(
λ x12 :
ι → ι
.
λ x13 .
x11
)
)
0
)
0
(
setsum
(
x9
(
λ x10 :
ι →
ι → ι
.
0
)
)
0
)
0
)
(
x1
(
λ x9 .
Inj1
(
Inj0
(
x0
(
λ x10 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
)
)
x6
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
0
)
(
setsum
0
0
)
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
0
(
setsum
0
0
)
)
0
)
(
Inj0
(
setsum
(
setsum
0
0
)
0
)
)
)
=
Inj1
x6
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 :
ι →
(
ι →
ι → ι
)
→ ι
.
∀ x6 x7 .
x2
(
λ x9 .
0
)
(
λ x9 :
ι →
ι → ι
.
Inj0
(
x5
(
Inj1
(
x2
(
λ x10 .
0
)
(
λ x10 :
ι →
ι → ι
.
0
)
)
)
(
λ x10 x11 .
x9
0
(
x3
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
)
)
)
=
x7
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 x6 x7 .
x2
(
λ x9 .
Inj1
0
)
(
λ x9 :
ι →
ι → ι
.
x0
(
λ x10 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
x10
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
Inj1
0
)
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
=
Inj1
x6
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
x1
(
λ x9 .
0
)
(
x1
(
λ x9 .
Inj0
0
)
0
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
x0
(
λ x11 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
setsum
(
x2
(
λ x12 .
0
)
(
λ x12 :
ι →
ι → ι
.
0
)
)
(
x3
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
)
(
λ x11 :
ι → ι
.
λ x12 .
x0
(
λ x13 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x13 :
ι → ι
.
λ x14 .
x2
(
λ x15 .
0
)
(
λ x15 :
ι →
ι → ι
.
0
)
)
)
)
0
(
Inj1
0
)
(
Inj1
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj1
0
)
(
setsum
0
0
)
)
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
x10
)
(
Inj0
(
Inj1
x5
)
)
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
(
Inj1
0
)
(
x9
(
λ x10 :
ι →
ι → ι
.
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
)
)
(
x1
(
λ x9 .
x1
(
λ x10 .
0
)
(
setsum
0
0
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
setsum
0
0
)
0
x5
x5
)
x4
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
0
)
0
(
x1
(
λ x9 .
x1
(
λ x10 .
0
)
0
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
0
0
0
)
(
x2
(
λ x9 .
0
)
(
λ x9 :
ι →
ι → ι
.
0
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
0
)
0
(
x1
(
λ x9 .
0
)
0
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
0
)
0
0
0
)
(
x7
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
(
λ x9 .
0
)
)
)
0
)
)
(
x6
x5
)
=
setsum
(
x6
(
x2
(
λ x9 .
setsum
0
(
x7
0
(
λ x10 :
ι → ι
.
λ x11 .
0
)
(
λ x10 .
0
)
)
)
(
λ x9 :
ι →
ι → ι
.
setsum
(
x0
(
λ x10 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
(
Inj0
0
)
)
)
)
(
Inj1
(
setsum
x5
(
setsum
0
(
x6
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 .
x5
0
(
λ x10 :
ι → ι
.
λ x11 .
x0
(
λ x12 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
Inj0
x11
)
(
λ x12 :
ι → ι
.
λ x13 .
x10
(
x12
0
)
)
)
(
λ x10 .
Inj0
x9
)
)
0
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
x2
(
λ x11 .
x10
)
(
λ x11 :
ι →
ι → ι
.
x3
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
0
(
x1
(
λ x13 .
0
)
0
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
0
)
0
0
0
)
)
0
)
)
0
x6
(
x7
(
Inj0
(
x0
(
λ x9 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
)
)
=
x5
(
x0
(
λ x9 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
x1
(
λ x10 .
0
)
(
Inj0
(
setsum
0
0
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
x10
(
λ x12 .
setsum
0
0
)
0
)
0
(
setsum
(
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
(
x5
0
(
λ x10 :
ι → ι
.
λ x11 .
0
)
(
λ x10 .
0
)
)
)
(
x5
(
x2
(
λ x10 .
0
)
(
λ x10 :
ι →
ι → ι
.
0
)
)
(
λ x10 :
ι → ι
.
λ x11 .
x0
(
λ x12 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
λ x10 .
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
(
Inj1
x10
)
0
)
(
x2
(
λ x11 .
x2
(
λ x12 .
0
)
(
λ x12 :
ι →
ι → ι
.
setsum
0
0
)
)
(
λ x11 :
ι →
ι → ι
.
x0
(
λ x12 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
x12
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
0
)
)
(
λ x12 :
ι → ι
.
λ x13 .
Inj1
0
)
)
)
)
(
λ x9 .
x5
x6
(
λ x10 :
ι → ι
.
λ x11 .
x9
)
(
λ x10 .
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x12
(
λ x13 :
ι →
ι → ι
.
0
)
)
(
x0
(
λ x12 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x12 :
ι → ι
.
λ x13 .
0
)
)
)
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 x5 :
ι →
ι → ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x0
(
λ x9 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 .
Inj1
0
)
=
x5
(
Inj1
(
setsum
0
0
)
)
(
x5
(
Inj0
(
x7
(
Inj1
0
)
(
λ x9 :
ι → ι
.
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
)
(
λ x9 .
0
)
)
)
(
x4
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x10 .
0
)
(
λ x10 :
ι →
ι → ι
.
0
)
)
0
)
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x6
(
λ x10 .
0
)
)
(
Inj0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x7 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
x0
(
λ x9 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
=
setsum
(
Inj0
0
)
(
x2
(
λ x9 .
x0
(
λ x10 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
Inj0
0
)
)
(
λ x10 :
ι → ι
.
λ x11 .
x7
(
λ x12 :
ι → ι
.
Inj0
0
)
(
x10
0
)
(
x1
(
λ x12 .
0
)
0
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
0
0
0
)
)
)
(
λ x9 :
ι →
ι → ι
.
setsum
(
setsum
(
Inj1
0
)
(
x2
(
λ x10 .
0
)
(
λ x10 :
ι →
ι → ι
.
0
)
)
)
(
Inj0
(
x5
(
λ x10 .
0
)
(
λ x10 .
0
)
)
)
)
)
)
⟶
False
(proof)
Theorem
aa8c7..
:
∀ x0 :
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x1 :
(
(
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
)
→
(
ι →
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x2 :
(
ι → ι
)
→
ι →
ι →
(
ι → ι
)
→
ι →
ι → ι
.
∀ x3 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
(
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x9 :
ι → ι
.
0
)
(
x2
(
λ x9 .
x5
(
x2
(
λ x10 .
setsum
0
0
)
(
x5
0
0
)
(
Inj1
0
)
(
λ x10 .
x1
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x12 x13 x14 .
0
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
)
(
x5
0
0
)
0
)
(
x6
0
)
)
(
x2
(
λ x9 .
x2
(
λ x10 .
x9
)
(
setsum
0
0
)
(
x7
(
λ x10 .
λ x11 :
ι → ι
.
0
)
)
(
λ x10 .
0
)
x9
(
x5
0
0
)
)
(
x3
(
λ x9 :
ι → ι
.
x2
(
λ x10 .
0
)
0
0
(
λ x10 .
0
)
0
0
)
(
x3
(
λ x9 :
ι → ι
.
0
)
0
0
)
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
0
)
)
)
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
Inj1
0
)
)
(
λ x9 .
x5
0
(
setsum
0
0
)
)
(
x3
(
λ x9 :
ι → ι
.
0
)
0
(
setsum
0
0
)
)
(
x3
(
λ x9 :
ι → ι
.
Inj1
0
)
(
x2
(
λ x9 .
0
)
0
0
(
λ x9 .
0
)
0
0
)
(
setsum
0
0
)
)
)
(
x6
0
)
(
λ x9 .
setsum
(
Inj1
(
x3
(
λ x10 :
ι → ι
.
0
)
0
0
)
)
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
0
)
)
)
(
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
Inj0
(
x0
(
λ x13 .
0
)
(
λ x13 .
0
)
(
λ x13 .
0
)
)
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
)
(
setsum
(
x5
(
x6
0
)
(
x0
(
λ x9 .
0
)
(
λ x9 .
0
)
(
λ x9 .
0
)
)
)
0
)
)
(
x3
(
λ x9 :
ι → ι
.
0
)
(
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
0
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
setsum
0
0
)
)
(
x5
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
0
)
)
0
)
)
=
x2
(
λ x9 .
x9
)
(
Inj0
x4
)
(
Inj0
x4
)
(
λ x9 .
x5
0
(
x7
(
λ x10 .
λ x11 :
ι → ι
.
x3
(
λ x12 :
ι → ι
.
x12
0
)
0
(
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x13 x14 x15 .
0
)
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
)
)
)
)
(
Inj0
(
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
0
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
Inj1
0
)
)
)
(
setsum
0
(
x7
(
λ x9 .
λ x10 :
ι → ι
.
x10
0
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x7 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
x3
(
λ x9 :
ι → ι
.
x0
(
λ x10 .
Inj0
(
x1
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x12 x13 x14 .
setsum
0
0
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
)
)
(
λ x10 .
Inj0
(
x0
(
λ x11 .
setsum
0
0
)
(
λ x11 .
0
)
(
λ x11 .
x7
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
0
)
(
λ x12 .
0
)
0
)
)
)
(
λ x10 .
Inj0
(
x9
0
)
)
)
(
Inj0
(
setsum
0
(
Inj0
x5
)
)
)
(
x7
(
λ x9 .
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
x0
(
λ x14 .
setsum
0
0
)
(
λ x14 .
x1
(
λ x15 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x16 x17 x18 .
0
)
(
λ x15 .
λ x16 :
ι →
ι → ι
.
0
)
)
(
λ x14 .
0
)
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
0
)
)
(
λ x9 :
ι → ι
.
Inj1
(
setsum
(
x0
(
λ x10 .
0
)
(
λ x10 .
0
)
(
λ x10 .
0
)
)
(
setsum
0
0
)
)
)
(
λ x9 .
0
)
(
Inj0
0
)
)
=
Inj0
0
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x2
(
λ x9 .
x9
)
(
Inj1
0
)
(
x3
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
Inj1
(
x3
(
λ x12 :
ι → ι
.
0
)
0
0
)
)
)
(
x4
(
x5
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x10 :
ι → ι
.
0
)
0
0
)
(
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
0
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
)
(
λ x9 .
0
)
)
)
(
setsum
(
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
x11
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x3
(
λ x11 :
ι → ι
.
0
)
0
0
)
)
(
x5
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
0
(
λ x9 .
setsum
0
0
)
)
)
)
(
λ x9 .
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
Inj1
(
setsum
0
(
setsum
0
0
)
)
)
)
(
setsum
(
x0
(
λ x9 .
x0
(
λ x10 .
x1
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x12 x13 x14 .
0
)
(
λ x11 .
λ x12 :
ι →
ι → ι
.
0
)
)
(
λ x10 .
Inj0
0
)
(
λ x10 .
0
)
)
(
λ x9 .
0
)
(
λ x9 .
x5
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x7
0
(
λ x11 :
ι → ι
.
λ x12 .
0
)
)
x9
(
λ x10 .
0
)
)
)
(
x6
(
λ x9 :
ι →
ι → ι
.
λ x10 :
ι → ι
.
x2
(
λ x11 .
x3
(
λ x12 :
ι → ι
.
0
)
0
0
)
(
x7
0
(
λ x11 :
ι → ι
.
λ x12 .
0
)
)
(
x3
(
λ x11 :
ι → ι
.
0
)
0
0
)
(
λ x11 .
x2
(
λ x12 .
0
)
0
0
(
λ x12 .
0
)
0
0
)
(
x3
(
λ x11 :
ι → ι
.
0
)
0
0
)
(
setsum
0
0
)
)
)
)
0
=
x3
(
λ x9 :
ι → ι
.
x7
(
x9
(
x3
(
λ x10 :
ι → ι
.
x2
(
λ x11 .
0
)
0
0
(
λ x11 .
0
)
0
0
)
0
(
Inj0
0
)
)
)
(
λ x10 :
ι → ι
.
λ x11 .
x2
(
λ x12 .
0
)
0
0
(
λ x12 .
0
)
0
0
)
)
(
x2
(
λ x9 .
x0
(
λ x10 .
0
)
(
λ x10 .
0
)
(
λ x10 .
x7
(
Inj0
0
)
(
λ x11 :
ι → ι
.
λ x12 .
x0
(
λ x13 .
0
)
(
λ x13 .
0
)
(
λ x13 .
0
)
)
)
)
(
x4
0
)
0
(
λ x9 .
0
)
(
setsum
0
(
setsum
(
x3
(
λ x9 :
ι → ι
.
0
)
0
0
)
(
x0
(
λ x9 .
0
)
(
λ x9 .
0
)
(
λ x9 .
0
)
)
)
)
(
x2
(
λ x9 .
x7
0
(
λ x10 :
ι → ι
.
λ x11 .
x9
)
)
0
(
x2
(
λ x9 .
x9
)
(
x7
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
(
setsum
0
0
)
(
λ x9 .
0
)
(
x4
0
)
0
)
(
λ x9 .
x3
(
λ x10 :
ι → ι
.
x6
(
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
0
)
)
(
x6
(
λ x10 :
ι →
ι → ι
.
λ x11 :
ι → ι
.
0
)
)
(
setsum
0
0
)
)
(
setsum
0
(
x0
(
λ x9 .
0
)
(
λ x9 .
0
)
(
λ x9 .
0
)
)
)
(
Inj0
(
x7
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
)
)
)
(
setsum
(
x5
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
x5
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
0
(
λ x10 .
x10
)
)
(
setsum
0
(
x4
0
)
)
(
λ x9 .
0
)
)
0
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 x6 x7 .
x2
(
λ x9 .
x2
(
λ x10 .
0
)
(
Inj1
0
)
(
x0
(
λ x10 .
setsum
x7
(
setsum
0
0
)
)
(
λ x10 .
0
)
(
λ x10 .
0
)
)
(
λ x10 .
x10
)
x6
0
)
0
x5
(
λ x9 .
x9
)
x5
0
=
x5
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
x11
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
=
x5
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 :
(
ι → ι
)
→ ι
.
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
Inj0
(
x9
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x15 x16 x17 .
Inj0
0
)
(
λ x14 .
λ x15 :
ι →
ι → ι
.
x15
0
0
)
)
(
setsum
(
x0
(
λ x13 .
0
)
(
λ x13 .
0
)
(
λ x13 .
0
)
)
(
x1
(
λ x13 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x14 x15 x16 .
0
)
(
λ x13 .
λ x14 :
ι →
ι → ι
.
0
)
)
)
(
λ x13 .
setsum
x10
0
)
)
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x7
(
λ x11 .
0
)
)
=
x7
(
λ x9 .
x2
(
λ x10 .
x6
0
)
x9
(
setsum
(
x7
(
λ x10 .
0
)
)
(
x7
(
λ x10 .
Inj0
0
)
)
)
(
λ x10 .
x2
(
λ x11 .
0
)
0
(
x6
0
)
(
λ x11 .
x7
(
λ x12 .
0
)
)
(
x3
(
λ x11 :
ι → ι
.
x0
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
0
)
)
(
Inj1
0
)
0
)
x9
)
(
setsum
(
x3
(
λ x10 :
ι → ι
.
x2
(
λ x11 .
0
)
0
0
(
λ x11 .
0
)
0
0
)
0
x5
)
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
x1
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x15 x16 x17 .
0
)
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
x0
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
0
)
)
)
)
(
x6
(
x0
(
λ x10 .
x2
(
λ x11 .
0
)
0
0
(
λ x11 .
0
)
0
0
)
(
λ x10 .
x7
(
λ x11 .
0
)
)
(
λ x10 .
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x5 :
ι →
ι →
ι →
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x0
(
λ x9 .
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
x2
(
λ x14 .
Inj1
(
setsum
0
0
)
)
0
(
setsum
x12
(
x2
(
λ x14 .
0
)
0
0
(
λ x14 .
0
)
0
0
)
)
(
λ x14 .
x14
)
(
setsum
(
x3
(
λ x14 :
ι → ι
.
0
)
0
0
)
(
x1
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x15 x16 x17 .
0
)
(
λ x14 .
λ x15 :
ι →
ι → ι
.
0
)
)
)
(
Inj1
(
Inj1
0
)
)
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
Inj0
(
x2
(
λ x12 .
x9
)
(
setsum
0
0
)
(
x0
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
0
)
)
(
λ x12 .
x12
)
(
setsum
0
0
)
(
x0
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
0
)
)
)
)
)
(
λ x9 .
Inj0
0
)
(
λ x9 .
0
)
=
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 .
setsum
0
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x3
(
λ x11 :
ι → ι
.
Inj1
(
x3
(
λ x12 :
ι → ι
.
0
)
x9
(
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x13 x14 x15 .
0
)
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
)
)
)
(
setsum
(
Inj0
0
)
(
x2
(
λ x11 .
0
)
(
Inj1
0
)
(
x3
(
λ x11 :
ι → ι
.
0
)
0
0
)
(
λ x11 .
0
)
(
Inj0
0
)
(
setsum
0
0
)
)
)
0
)
)
⟶
(
∀ x4 :
ι →
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 :
(
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
x0
(
λ x9 .
x3
(
λ x10 :
ι → ι
.
x0
(
λ x11 .
Inj0
(
Inj1
0
)
)
(
λ x11 .
setsum
(
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x13 x14 x15 .
0
)
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
)
0
)
(
λ x11 .
x0
(
λ x12 .
x1
(
λ x13 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x14 x15 x16 .
0
)
(
λ x13 .
λ x14 :
ι →
ι → ι
.
0
)
)
(
λ x12 .
setsum
0
0
)
(
λ x12 .
0
)
)
)
(
Inj1
(
x7
(
λ x10 :
ι → ι
.
0
)
(
λ x10 x11 .
x0
(
λ x12 .
0
)
(
λ x12 .
0
)
(
λ x12 .
0
)
)
)
)
(
setsum
(
Inj1
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
0
)
)
)
(
x0
(
λ x10 .
Inj1
0
)
(
λ x10 .
x6
0
)
(
λ x10 .
x7
(
λ x11 :
ι → ι
.
0
)
(
λ x11 x12 .
0
)
)
)
)
)
(
λ x9 .
0
)
(
λ x9 .
setsum
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
x0
(
λ x12 .
x12
)
(
λ x12 .
x0
(
λ x13 .
0
)
(
λ x13 .
0
)
(
λ x13 .
0
)
)
(
λ x12 .
x9
)
)
)
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
x11
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
x11
0
(
x7
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 .
0
)
)
)
)
)
=
x3
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
x7
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 .
x0
(
λ x14 .
x13
)
(
λ x14 .
x11
0
0
)
(
λ x14 .
x1
(
λ x15 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x16 x17 x18 .
0
)
(
λ x15 .
λ x16 :
ι →
ι → ι
.
0
)
)
)
)
)
(
setsum
0
(
x6
(
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
x3
(
λ x13 :
ι → ι
.
0
)
0
0
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x6
0
)
)
)
)
x5
)
⟶
False
(proof)
Theorem
7ec4f..
:
∀ x0 :
(
(
ι →
ι →
ι → ι
)
→ ι
)
→
(
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x1 :
(
(
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
∀ x2 :
(
(
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
)
→ ι
)
→
(
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x3 :
(
ι → ι
)
→
(
ι →
ι →
(
ι → ι
)
→ ι
)
→ ι
.
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 .
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
0
)
=
x6
)
⟶
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x3
(
λ x9 .
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
x1
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
Inj0
(
x12
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
x12
(
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 .
0
)
)
)
)
x10
)
=
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
x2
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
0
)
(
λ x10 .
λ x11 x12 :
ι → ι
.
λ x13 .
x11
(
x11
x10
)
)
)
(
setsum
0
(
Inj0
0
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 :
(
ι → ι
)
→
ι → ι
.
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
0
)
(
λ x9 .
λ x10 x11 :
ι → ι
.
λ x12 .
x0
(
λ x13 :
ι →
ι →
ι → ι
.
0
)
(
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x14 :
ι →
ι → ι
.
x2
(
λ x15 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
0
)
(
λ x15 .
λ x16 x17 :
ι → ι
.
λ x18 .
setsum
(
x0
(
λ x19 :
ι →
ι →
ι → ι
.
0
)
(
λ x19 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x20 :
ι →
ι → ι
.
0
)
)
(
setsum
0
0
)
)
)
)
=
x0
(
λ x9 :
ι →
ι →
ι → ι
.
x3
(
λ x10 .
0
)
(
λ x10 x11 .
λ x12 :
ι → ι
.
0
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
x1
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
0
)
)
⟶
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
ι → ι
.
∀ x7 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
Inj0
0
)
(
λ x9 .
λ x10 x11 :
ι → ι
.
λ x12 .
0
)
=
setsum
0
(
x4
(
setsum
(
x4
(
setsum
0
0
)
(
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
0
)
(
λ x9 .
0
)
)
(
x3
(
λ x9 .
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
setsum
0
0
)
)
)
(
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
x0
(
λ x10 :
ι →
ι →
ι → ι
.
x7
(
λ x11 :
ι →
ι → ι
.
0
)
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
x7
(
λ x12 :
ι →
ι → ι
.
0
)
)
)
(
λ x9 .
λ x10 x11 :
ι → ι
.
λ x12 .
0
)
)
(
λ x9 .
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
x2
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
0
)
(
λ x11 .
λ x12 x13 :
ι → ι
.
λ x14 .
0
)
)
(
setsum
0
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
0
)
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
setsum
(
x0
(
λ x10 :
ι →
ι →
ι → ι
.
x10
0
x7
0
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
x3
(
λ x12 .
x10
(
λ x13 :
ι → ι
.
λ x14 .
0
)
)
(
λ x12 x13 .
λ x14 :
ι → ι
.
0
)
)
)
(
Inj0
0
)
)
0
=
Inj1
x4
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x7 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
x1
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
x5
)
(
Inj1
0
)
=
Inj0
(
x3
(
λ x9 .
x7
(
x2
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
x6
(
λ x11 x12 :
ι → ι
.
0
)
(
λ x11 .
0
)
)
(
λ x10 .
λ x11 x12 :
ι → ι
.
λ x13 .
setsum
0
0
)
)
(
Inj1
0
)
(
λ x10 .
x2
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
x10
)
(
λ x11 .
λ x12 x13 :
ι → ι
.
λ x14 .
x12
0
)
)
(
Inj0
0
)
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
x9
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x0
(
λ x9 :
ι →
ι →
ι → ι
.
x6
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
x3
(
λ x11 .
Inj0
(
setsum
x11
(
x7
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
0
)
)
)
(
λ x11 x12 .
λ x13 :
ι → ι
.
setsum
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
(
Inj0
(
x13
0
)
)
)
)
=
x6
)
⟶
(
∀ x4 .
∀ x5 :
(
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x9 :
ι →
ι →
ι → ι
.
setsum
(
Inj1
(
setsum
0
(
x1
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
0
)
)
)
(
Inj1
0
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
setsum
x6
(
Inj0
(
x10
(
x2
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
0
)
(
λ x11 .
λ x12 x13 :
ι → ι
.
λ x14 .
0
)
)
(
x10
0
0
)
)
)
)
=
x7
0
(
λ x9 :
ι → ι
.
x9
0
)
)
⟶
False
(proof)
Theorem
212f6..
:
∀ x0 :
(
(
ι → ι
)
→
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x1 :
(
ι → ι
)
→
ι → ι
.
∀ x2 :
(
ι → ι
)
→
ι →
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 x10 .
x9
)
(
λ x9 .
x1
(
λ x10 .
0
)
x5
)
=
Inj0
(
x3
(
λ x9 x10 .
0
)
(
λ x9 .
x7
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
x3
(
λ x9 x10 .
0
)
(
λ x9 .
x5
x9
(
λ x10 .
x7
(
λ x11 :
(
ι → ι
)
→ ι
.
0
)
)
(
λ x10 .
setsum
0
(
x1
(
λ x11 .
x10
)
0
)
)
(
x5
(
setsum
(
setsum
0
0
)
(
x5
0
(
λ x10 .
0
)
(
λ x10 .
0
)
0
)
)
(
λ x10 .
setsum
x6
(
setsum
0
0
)
)
(
λ x10 .
Inj0
(
x3
(
λ x11 x12 .
0
)
(
λ x11 .
0
)
)
)
(
setsum
x9
0
)
)
)
=
x5
(
x3
(
λ x9 x10 .
0
)
(
λ x9 .
Inj0
0
)
)
(
λ x9 .
x1
(
λ x10 .
x2
(
λ x11 .
0
)
(
setsum
(
x7
(
λ x11 :
(
ι → ι
)
→ ι
.
0
)
)
(
setsum
0
0
)
)
(
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
x0
(
λ x13 :
ι → ι
.
λ x14 .
λ x15 :
ι →
ι → ι
.
λ x16 x17 .
0
)
(
Inj1
0
)
(
x2
(
λ x13 .
0
)
0
(
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 :
ι → ι
.
0
)
)
)
(
λ x11 :
ι → ι
.
0
)
)
0
)
(
λ x9 .
x7
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
)
(
Inj1
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 .
x6
0
)
(
x0
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x1
Inj1
(
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
)
x4
(
setsum
(
Inj1
(
Inj0
0
)
)
(
Inj0
x4
)
)
)
(
λ x9 :
ι →
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
0
)
=
setsum
0
(
setsum
(
setsum
0
(
setsum
0
(
setsum
0
0
)
)
)
(
x0
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
setsum
x13
x13
)
x7
(
x1
(
λ x9 .
x5
)
0
)
)
)
)
⟶
(
∀ x4 x5 :
ι → ι
.
∀ x6 :
ι →
ι → ι
.
∀ x7 .
x2
(
λ x9 .
setsum
(
x5
(
x3
(
λ x10 x11 .
x11
)
(
λ x10 .
Inj0
0
)
)
)
(
setsum
(
x0
(
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
x11
)
0
(
x3
(
λ x10 x11 .
0
)
(
λ x10 .
0
)
)
)
(
x3
(
λ x10 x11 .
0
)
(
λ x10 .
0
)
)
)
)
(
x1
(
λ x9 .
0
)
x7
)
(
λ x9 :
ι →
ι → ι
.
λ x10 :
ι → ι
.
Inj1
(
x1
(
λ x11 .
0
)
(
x3
(
λ x11 x12 .
Inj0
0
)
(
λ x11 .
x2
(
λ x12 .
0
)
0
(
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
)
)
)
)
(
λ x9 :
ι → ι
.
x0
(
λ x10 :
ι → ι
.
λ x11 .
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
x13
)
0
0
)
=
Inj1
x7
)
⟶
(
∀ x4 x5 x6 x7 .
x1
(
λ x9 .
x1
(
λ x10 .
0
)
(
setsum
0
x5
)
)
(
setsum
0
x7
)
=
x1
(
λ x9 .
setsum
(
Inj1
(
Inj1
(
x1
(
λ x10 .
0
)
0
)
)
)
(
x3
(
λ x10 x11 .
x7
)
(
λ x10 .
x9
)
)
)
x6
)
⟶
(
∀ x4 :
ι →
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 .
x9
)
0
=
x7
(
x2
(
λ x9 .
x3
(
λ x10 x11 .
x0
(
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι →
ι → ι
.
λ x15 x16 .
x2
(
λ x17 .
0
)
0
(
λ x17 :
ι →
ι → ι
.
λ x18 :
ι → ι
.
0
)
(
λ x17 :
ι → ι
.
0
)
)
(
x1
(
λ x12 .
0
)
0
)
0
)
(
λ x10 .
x3
(
λ x11 x12 .
0
)
(
λ x11 .
0
)
)
)
0
(
λ x9 :
ι →
ι → ι
.
λ x10 :
ι → ι
.
setsum
x6
(
Inj0
(
setsum
0
0
)
)
)
(
λ x9 :
ι → ι
.
x1
(
λ x10 .
Inj0
0
)
(
Inj0
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 x5 :
ι → ι
.
∀ x6 x7 .
x0
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
Inj0
(
setsum
(
x1
(
λ x14 .
0
)
(
Inj0
0
)
)
x13
)
)
x7
(
Inj1
(
Inj0
0
)
)
=
x7
)
⟶
(
∀ x4 :
ι →
ι →
ι →
ι → ι
.
∀ x5 .
∀ x6 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x12
)
(
setsum
0
x5
)
x5
=
setsum
(
x1
(
λ x9 .
setsum
0
(
x3
(
λ x10 x11 .
Inj0
0
)
(
λ x10 .
x2
(
λ x11 .
0
)
0
(
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
0
)
(
λ x11 :
ι → ι
.
0
)
)
)
)
(
x2
(
λ x9 .
x9
)
(
x4
0
(
x7
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
x0
(
λ x9 :
ι → ι
.
λ x10 .
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
0
)
0
0
)
(
setsum
0
0
)
)
(
λ x9 :
ι →
ι → ι
.
λ x10 :
ι → ι
.
x7
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x12 x13 .
0
)
(
λ x12 .
0
)
)
)
(
λ x9 :
ι → ι
.
x7
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
)
)
)
(
x2
(
λ x9 .
x6
0
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
(
x3
(
λ x9 x10 .
x2
(
λ x11 .
x11
)
(
x7
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
x2
(
λ x13 .
0
)
0
(
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
0
)
(
λ x13 :
ι → ι
.
0
)
)
(
λ x11 :
ι → ι
.
0
)
)
(
λ x9 .
x5
)
)
(
λ x9 :
ι →
ι → ι
.
λ x10 :
ι → ι
.
0
)
(
λ x9 :
ι → ι
.
x3
(
λ x10 x11 .
x0
(
λ x12 :
ι → ι
.
λ x13 .
λ x14 :
ι →
ι → ι
.
λ x15 x16 .
setsum
0
0
)
(
x2
(
λ x12 .
0
)
0
(
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
0
)
(
λ x12 :
ι → ι
.
0
)
)
(
x3
(
λ x12 x13 .
0
)
(
λ x12 .
0
)
)
)
(
setsum
0
)
)
)
)
⟶
False
(proof)
Theorem
e23eb..
:
∀ x0 :
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x1 :
(
(
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
∀ x2 :
(
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x3 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
.
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
∀ x7 .
x3
(
λ x9 .
0
)
(
λ x9 :
ι → ι
.
setsum
0
(
x0
(
λ x10 x11 x12 .
0
)
(
λ x10 .
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
Inj0
0
)
(
x6
(
λ x11 :
(
ι → ι
)
→ ι
.
0
)
0
)
(
x2
(
λ x11 :
ι → ι
.
0
)
(
λ x11 x12 .
0
)
)
(
setsum
0
0
)
)
)
)
0
(
x2
(
λ x9 :
ι → ι
.
x5
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x11 .
x0
(
λ x12 x13 x14 .
0
)
(
λ x12 .
0
)
)
(
λ x11 :
ι → ι
.
Inj0
0
)
0
0
(
setsum
0
0
)
)
)
(
λ x9 x10 .
x0
(
λ x11 x12 x13 .
0
)
(
λ x11 .
0
)
)
)
(
setsum
0
0
)
=
x2
(
λ x9 :
ι → ι
.
x9
(
Inj1
(
setsum
(
Inj0
0
)
(
x9
0
)
)
)
)
(
λ x9 x10 .
Inj0
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x3
(
λ x9 .
x3
(
λ x10 .
0
)
(
λ x10 :
ι → ι
.
x3
(
λ x11 .
Inj1
0
)
(
λ x11 :
ι → ι
.
x11
0
)
(
x6
(
x3
(
λ x11 .
0
)
(
λ x11 :
ι → ι
.
0
)
0
0
0
)
)
(
x10
(
x10
0
)
)
0
)
(
x3
(
λ x10 .
x3
(
λ x11 .
setsum
0
0
)
(
λ x11 :
ι → ι
.
x0
(
λ x12 x13 x14 .
0
)
(
λ x12 .
0
)
)
(
Inj1
0
)
(
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
0
)
(
x2
(
λ x11 :
ι → ι
.
0
)
(
λ x11 x12 .
0
)
)
)
(
λ x10 :
ι → ι
.
0
)
(
Inj1
x5
)
x5
(
Inj0
0
)
)
(
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
x2
(
λ x13 :
ι → ι
.
setsum
0
0
)
(
λ x13 x14 .
x0
(
λ x15 x16 x17 .
0
)
(
λ x15 .
0
)
)
)
x9
(
setsum
x5
0
)
x7
)
(
Inj1
(
Inj0
(
Inj0
0
)
)
)
)
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
setsum
(
x10
(
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
x0
(
λ x16 x17 x18 .
0
)
(
λ x16 .
0
)
)
(
x1
(
λ x13 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x14 .
λ x15 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
0
)
)
(
x1
(
λ x13 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x14 .
λ x15 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
(
Inj1
0
)
)
)
(
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
x5
(
x3
(
λ x10 .
x0
(
λ x11 x12 x13 .
0
)
(
λ x11 .
0
)
)
(
λ x10 :
ι → ι
.
0
)
(
x3
(
λ x10 .
0
)
(
λ x10 :
ι → ι
.
0
)
0
0
0
)
(
setsum
0
0
)
(
x6
0
)
)
(
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
0
(
x9
0
)
(
setsum
0
0
)
)
)
0
x7
)
(
setsum
0
(
x6
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
(
Inj1
0
)
0
0
)
)
)
(
x3
(
λ x9 .
x2
(
λ x10 :
ι → ι
.
x2
(
λ x11 :
ι → ι
.
0
)
(
λ x11 x12 .
x11
)
)
(
λ x10 x11 .
0
)
)
(
λ x9 :
ι → ι
.
0
)
x7
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x9
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
x11
(
λ x15 .
0
)
0
)
(
x11
(
λ x12 .
0
)
0
)
)
0
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
0
)
0
0
0
)
(
x2
(
λ x9 :
ι → ι
.
0
)
(
λ x9 x10 .
0
)
)
x5
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
0
)
)
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
Inj0
0
)
(
x3
(
λ x9 .
0
)
(
λ x9 :
ι → ι
.
0
)
0
0
0
)
x7
0
)
)
0
)
(
x6
0
)
=
x3
(
λ x9 .
Inj0
(
Inj0
x7
)
)
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
x2
(
λ x13 :
ι → ι
.
x1
(
λ x14 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x15 .
λ x16 :
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
0
0
(
Inj1
0
)
)
(
λ x13 x14 .
Inj1
0
)
)
(
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x13 .
Inj0
0
)
(
λ x13 :
ι → ι
.
x12
(
λ x14 .
0
)
0
)
(
setsum
0
0
)
(
x0
(
λ x13 x14 x15 .
0
)
(
λ x13 .
0
)
)
(
x10
(
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
0
)
0
)
)
x7
(
setsum
(
x3
(
λ x10 .
0
)
(
λ x10 :
ι → ι
.
0
)
0
0
0
)
(
x2
(
λ x10 :
ι → ι
.
0
)
(
λ x10 x11 .
0
)
)
)
(
Inj0
(
x2
(
λ x10 :
ι → ι
.
0
)
(
λ x10 x11 .
0
)
)
)
)
(
setsum
0
(
x2
(
λ x10 :
ι → ι
.
x0
(
λ x11 x12 x13 .
0
)
(
λ x11 .
0
)
)
(
λ x10 x11 .
Inj0
0
)
)
)
(
x0
(
λ x10 x11 x12 .
x10
)
(
λ x10 .
x9
0
)
)
)
x7
(
setsum
(
Inj0
x7
)
(
x0
(
λ x9 x10 x11 .
setsum
(
x0
(
λ x12 x13 x14 .
0
)
(
λ x12 .
0
)
)
(
x2
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 .
0
)
)
)
(
λ x9 .
x0
(
λ x10 x11 x12 .
x2
(
λ x13 :
ι → ι
.
0
)
(
λ x13 x14 .
0
)
)
(
λ x10 .
x9
)
)
)
)
(
Inj0
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x7 .
x2
(
λ x9 :
ι → ι
.
x6
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x11 .
Inj0
(
Inj0
0
)
)
(
λ x11 :
ι → ι
.
Inj0
(
Inj1
0
)
)
(
x0
(
λ x11 x12 x13 .
x10
(
λ x14 .
0
)
0
)
(
λ x11 .
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
0
)
)
(
x2
(
λ x11 :
ι → ι
.
setsum
0
0
)
(
λ x11 x12 .
0
)
)
(
setsum
(
x6
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
(
x10
(
λ x11 .
0
)
0
)
)
)
(
x3
(
λ x10 .
0
)
(
λ x10 :
ι → ι
.
x2
(
λ x11 :
ι → ι
.
setsum
0
0
)
(
λ x11 x12 .
x3
(
λ x13 .
0
)
(
λ x13 :
ι → ι
.
0
)
0
0
0
)
)
(
setsum
(
x2
(
λ x10 :
ι → ι
.
0
)
(
λ x10 x11 .
0
)
)
(
x9
0
)
)
(
x2
(
λ x10 :
ι → ι
.
x7
)
(
λ x10 x11 .
0
)
)
(
setsum
(
x0
(
λ x10 x11 x12 .
0
)
(
λ x10 .
0
)
)
(
setsum
0
0
)
)
)
(
x0
(
λ x10 x11 x12 .
x10
)
(
λ x10 .
x0
(
λ x11 x12 x13 .
x13
)
(
λ x11 .
x9
0
)
)
)
)
(
λ x9 x10 .
Inj1
0
)
=
Inj1
(
Inj1
(
x0
(
λ x9 x10 x11 .
x2
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 .
x10
)
)
(
λ x9 .
setsum
0
(
x6
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x2
(
λ x9 :
ι → ι
.
0
)
(
λ x9 x10 .
0
)
=
Inj0
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x2
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 .
x0
(
λ x14 x15 x16 .
x15
)
(
λ x14 .
x12
)
)
)
0
x7
x7
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 x7 .
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
setsum
0
(
setsum
0
(
setsum
0
0
)
)
)
(
x11
(
λ x12 .
x12
)
(
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
x11
(
λ x15 .
0
)
0
)
x10
(
x0
(
λ x12 x13 x14 .
0
)
(
λ x12 .
0
)
)
0
)
)
(
setsum
(
setsum
0
(
x2
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 .
0
)
)
)
(
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
(
x11
(
λ x12 .
0
)
0
)
(
x11
(
λ x12 .
0
)
0
)
(
Inj1
0
)
)
)
(
setsum
(
x2
(
λ x12 :
ι → ι
.
setsum
0
0
)
(
λ x12 x13 .
0
)
)
(
setsum
0
(
x2
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 .
0
)
)
)
)
)
0
0
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
setsum
0
(
Inj0
(
Inj0
0
)
)
)
x6
0
x4
)
=
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
x3
(
λ x13 .
Inj1
(
x11
(
λ x14 .
0
)
0
)
)
(
λ x13 :
ι → ι
.
x2
(
λ x14 :
ι → ι
.
setsum
0
0
)
(
λ x14 x15 .
x2
(
λ x16 :
ι → ι
.
0
)
(
λ x16 x17 .
0
)
)
)
0
(
x11
(
λ x13 .
0
)
(
setsum
0
0
)
)
(
Inj1
(
x3
(
λ x13 .
0
)
(
λ x13 :
ι → ι
.
0
)
0
0
0
)
)
)
(
x11
(
λ x12 .
x11
(
λ x13 .
x1
(
λ x14 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x15 .
λ x16 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
0
)
0
)
x10
)
(
setsum
0
(
setsum
(
x2
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 .
0
)
)
(
setsum
0
0
)
)
)
(
setsum
(
x11
(
λ x12 .
x10
)
(
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
0
)
)
(
x0
(
λ x12 x13 x14 .
setsum
0
0
)
(
λ x12 .
Inj0
0
)
)
)
)
x7
(
x2
(
λ x9 :
ι → ι
.
x0
(
λ x10 x11 x12 .
0
)
(
λ x10 .
0
)
)
(
λ x9 x10 .
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
x12
)
0
x7
(
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
Inj0
0
)
(
x0
(
λ x11 x12 x13 .
0
)
(
λ x11 .
0
)
)
0
x6
)
)
)
x4
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
Inj0
(
setsum
0
(
Inj1
0
)
)
)
(
Inj0
0
)
(
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
x0
(
λ x15 x16 x17 .
Inj0
0
)
(
λ x15 .
x2
(
λ x16 :
ι → ι
.
0
)
(
λ x16 x17 .
0
)
)
)
0
(
Inj0
(
x2
(
λ x12 :
ι → ι
.
0
)
(
λ x12 x13 .
0
)
)
)
(
x3
(
λ x12 .
x1
(
λ x13 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x14 .
λ x15 :
(
ι → ι
)
→
ι → ι
.
0
)
0
0
0
)
(
λ x12 :
ι → ι
.
Inj1
0
)
(
x9
(
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
0
)
0
(
Inj0
0
)
)
)
0
)
0
(
x0
(
λ x9 x10 x11 .
x2
(
λ x12 :
ι → ι
.
x12
(
x3
(
λ x13 .
0
)
(
λ x13 :
ι → ι
.
0
)
0
0
0
)
)
(
λ x12 x13 .
x13
)
)
(
λ x9 .
x6
(
λ x10 .
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
x0
(
λ x14 x15 x16 .
0
)
(
λ x14 .
0
)
)
(
setsum
0
0
)
(
x0
(
λ x11 x12 x13 .
0
)
(
λ x11 .
0
)
)
(
x3
(
λ x11 .
0
)
(
λ x11 :
ι → ι
.
0
)
0
0
0
)
)
)
)
(
Inj1
0
)
=
Inj0
(
setsum
(
x6
(
λ x9 .
x3
(
λ x10 .
x10
)
(
λ x10 :
ι → ι
.
x3
(
λ x11 .
0
)
(
λ x11 :
ι → ι
.
0
)
0
0
0
)
(
x6
(
λ x10 .
0
)
)
(
Inj1
0
)
(
x7
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
)
)
)
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x9 x10 x11 .
setsum
0
x7
)
(
λ x9 .
0
)
=
x5
0
(
x2
(
λ x9 :
ι → ι
.
x6
0
)
(
λ x9 x10 .
0
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x9 x10 x11 .
x7
x10
(
λ x12 :
ι → ι
.
x10
)
)
(
λ x9 .
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
x11
)
(
setsum
x5
(
Inj0
(
x7
0
(
λ x10 :
ι → ι
.
0
)
)
)
)
(
Inj1
(
setsum
0
(
setsum
0
0
)
)
)
(
x7
(
x3
(
λ x10 .
x2
(
λ x11 :
ι → ι
.
0
)
(
λ x11 x12 .
0
)
)
(
λ x10 :
ι → ι
.
Inj1
0
)
(
setsum
0
0
)
x9
(
setsum
0
0
)
)
(
λ x10 :
ι → ι
.
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
(
Inj1
0
)
x6
x6
)
)
)
=
x7
(
setsum
(
setsum
0
(
x4
(
x3
(
λ x9 .
0
)
(
λ x9 :
ι → ι
.
0
)
0
0
0
)
)
)
x6
)
(
λ x9 :
ι → ι
.
x5
)
)
⟶
False
(proof)
Theorem
e2aee..
:
∀ x0 :
(
(
(
ι →
ι →
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x1 :
(
(
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
)
→
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x2 :
(
(
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι →
ι →
ι → ι
)
→
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x3 :
(
ι →
ι →
ι → ι
)
→
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x3
(
λ x9 x10 x11 .
0
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
x1
(
λ x13 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
Inj0
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
x13
(
λ x15 .
x3
(
λ x16 x17 x18 .
x0
(
λ x19 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x20 .
λ x21 :
ι → ι
.
0
)
0
)
(
λ x16 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x17 :
ι →
ι → ι
.
λ x18 x19 .
x18
)
(
λ x16 :
(
ι → ι
)
→
ι → ι
.
λ x17 :
ι → ι
.
x1
(
λ x18 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x19 .
λ x20 :
ι → ι
.
0
)
(
λ x18 :
(
ι → ι
)
→ ι
.
λ x19 .
0
)
(
λ x18 .
0
)
0
)
)
)
(
λ x13 .
x10
(
x0
(
λ x14 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
ι → ι
.
setsum
0
0
)
(
setsum
0
0
)
)
(
x2
(
λ x14 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x14 x15 x16 x17 .
0
)
x11
(
λ x14 :
ι → ι
.
λ x15 .
setsum
0
0
)
x12
(
setsum
0
0
)
)
)
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x3
(
λ x11 x12 x13 .
x2
(
λ x14 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x15 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x16 .
λ x17 :
ι → ι
.
x14
(
λ x18 .
0
)
(
λ x18 x19 .
0
)
)
(
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 .
Inj0
0
)
(
λ x15 .
0
)
(
Inj1
0
)
)
(
λ x14 x15 x16 x17 .
0
)
0
(
λ x14 :
ι → ι
.
λ x15 .
x2
(
λ x16 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x16 x17 x18 x19 .
x16
)
(
x0
(
λ x16 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x17 .
λ x18 :
ι → ι
.
0
)
0
)
(
λ x16 :
ι → ι
.
λ x17 .
0
)
0
(
setsum
0
0
)
)
0
(
setsum
(
Inj1
0
)
x11
)
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
setsum
(
x1
(
λ x15 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x16 .
λ x17 :
ι → ι
.
Inj0
0
)
(
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 .
x15
(
λ x17 .
0
)
)
(
λ x15 .
Inj1
0
)
(
x3
(
λ x15 x16 x17 .
0
)
(
λ x15 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 :
ι →
ι → ι
.
λ x17 x18 .
0
)
(
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 :
ι → ι
.
0
)
)
)
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x11
(
λ x13 .
x3
(
λ x14 x15 x16 .
setsum
0
0
)
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι → ι
.
λ x16 x17 .
0
)
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
Inj0
0
)
)
0
)
)
=
Inj0
(
Inj0
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι →
ι →
ι →
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
x3
(
λ x9 x10 x11 .
setsum
0
(
x2
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x9
)
(
λ x12 x13 x14 x15 .
Inj0
(
x2
(
λ x16 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x16 x17 x18 x19 .
0
)
0
(
λ x16 :
ι → ι
.
λ x17 .
0
)
0
0
)
)
0
(
λ x12 :
ι → ι
.
λ x13 .
x1
(
λ x14 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
Inj0
0
)
(
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 .
0
)
(
λ x14 .
setsum
0
0
)
0
)
(
x0
(
λ x12 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι → ι
.
0
)
0
)
(
x7
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
x0
(
λ x13 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
0
)
(
x9
(
λ x13 :
ι → ι
.
Inj0
(
x3
(
λ x14 x15 x16 .
0
)
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι → ι
.
λ x16 x17 .
0
)
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
0
)
)
)
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
setsum
(
x3
(
λ x11 x12 x13 .
Inj1
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
)
(
x7
(
λ x11 :
(
ι → ι
)
→ ι
.
x0
(
λ x12 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι → ι
.
setsum
0
0
)
(
setsum
0
0
)
)
)
)
=
setsum
0
(
Inj1
0
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x7 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x2
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
Inj1
(
x3
(
λ x10 x11 x12 .
Inj1
(
x2
(
λ x13 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 x15 x16 .
0
)
0
(
λ x13 :
ι → ι
.
λ x14 .
0
)
0
0
)
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x13
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
x0
(
λ x12 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι → ι
.
x2
(
λ x15 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x15 x16 x17 x18 .
0
)
0
(
λ x15 :
ι → ι
.
λ x16 .
0
)
0
0
)
0
)
)
)
(
λ x9 x10 x11 x12 .
x3
(
λ x13 x14 x15 .
x14
)
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 :
ι →
ι → ι
.
λ x15 x16 .
0
)
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
Inj0
(
Inj1
(
x2
(
λ x15 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x15 x16 x17 x18 .
0
)
0
(
λ x15 :
ι → ι
.
λ x16 .
0
)
0
0
)
)
)
)
0
(
λ x9 :
ι → ι
.
λ x10 .
x3
(
λ x11 x12 x13 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
x2
(
λ x15 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x14
)
(
λ x15 x16 x17 x18 .
0
)
(
setsum
0
0
)
(
λ x15 :
ι → ι
.
λ x16 .
x13
)
(
Inj0
0
)
(
Inj1
(
x3
(
λ x15 x16 x17 .
0
)
(
λ x15 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 :
ι →
ι → ι
.
λ x17 x18 .
0
)
(
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 :
ι → ι
.
0
)
)
)
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x2
(
λ x13 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x11
(
λ x14 .
0
)
(
setsum
0
0
)
)
(
λ x13 x14 x15 x16 .
x16
)
x10
(
λ x13 :
ι → ι
.
λ x14 .
setsum
0
(
x0
(
λ x15 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x16 .
λ x17 :
ι → ι
.
0
)
0
)
)
(
Inj0
(
x12
0
)
)
(
x0
(
λ x13 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
setsum
0
0
)
(
x9
0
)
)
)
)
(
x0
(
λ x9 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
0
)
0
)
x5
=
Inj1
(
Inj1
(
x4
0
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x2
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x0
(
λ x10 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι → ι
.
x0
(
λ x13 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
0
)
0
)
(
Inj1
(
x7
(
x2
(
λ x10 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x10 x11 x12 x13 .
0
)
0
(
λ x10 :
ι → ι
.
λ x11 .
0
)
0
0
)
)
)
)
(
λ x9 x10 x11 x12 .
x11
)
0
(
λ x9 :
ι → ι
.
λ x10 .
x2
(
λ x11 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 x13 x14 .
x2
(
λ x15 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
setsum
(
x1
(
λ x16 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x17 .
λ x18 :
ι → ι
.
0
)
(
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 .
0
)
(
λ x16 .
0
)
0
)
(
setsum
0
0
)
)
(
λ x15 x16 x17 x18 .
0
)
0
(
λ x15 :
ι → ι
.
λ x16 .
x15
(
Inj0
0
)
)
(
Inj0
(
x0
(
λ x15 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x16 .
λ x17 :
ι → ι
.
0
)
0
)
)
0
)
x6
(
λ x11 :
ι → ι
.
λ x12 .
setsum
0
(
x2
(
λ x13 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x12
)
(
λ x13 x14 x15 x16 .
x14
)
(
setsum
0
0
)
(
λ x13 :
ι → ι
.
λ x14 .
x11
0
)
x10
(
setsum
0
0
)
)
)
0
0
)
0
(
setsum
(
x0
(
λ x9 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
x2
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x12 x13 x14 x15 .
x2
(
λ x16 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x16 x17 x18 x19 .
0
)
0
(
λ x16 :
ι → ι
.
λ x17 .
0
)
0
0
)
(
x2
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x12 x13 x14 x15 .
0
)
0
(
λ x12 :
ι → ι
.
λ x13 .
0
)
0
0
)
(
λ x12 :
ι → ι
.
λ x13 .
x3
(
λ x14 x15 x16 .
0
)
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι → ι
.
λ x16 x17 .
0
)
(
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
0
)
)
x10
(
x11
0
)
)
0
)
0
)
=
Inj1
x6
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
setsum
(
x2
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x13 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x14 .
λ x15 :
ι → ι
.
x12
(
λ x16 .
0
)
(
λ x16 x17 .
0
)
)
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
0
)
(
λ x13 .
x0
(
λ x14 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
ι → ι
.
0
)
0
)
0
)
(
λ x12 x13 x14 x15 .
x13
)
(
x11
(
setsum
0
0
)
)
(
λ x12 :
ι → ι
.
λ x13 .
0
)
0
x10
)
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
x9
(
λ x11 .
x7
(
x2
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x12 x13 x14 x15 .
0
)
0
(
λ x12 :
ι → ι
.
λ x13 .
x13
)
0
(
x2
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x12 x13 x14 x15 .
0
)
0
(
λ x12 :
ι → ι
.
λ x13 .
0
)
0
0
)
)
)
)
(
λ x9 .
Inj1
(
Inj0
(
x0
(
λ x10 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι → ι
.
x11
)
0
)
)
)
0
=
x5
)
⟶
(
∀ x4 x5 x6 x7 .
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
x11
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
Inj0
(
x3
(
λ x11 x12 .
Inj1
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
x11
(
λ x15 :
ι → ι
.
0
)
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
Inj1
(
x11
(
λ x13 .
0
)
0
)
)
)
)
(
λ x9 .
x5
)
(
x0
(
λ x9 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
setsum
0
(
setsum
0
(
x9
(
λ x12 x13 x14 .
0
)
)
)
)
0
)
=
Inj0
(
setsum
(
x0
(
λ x9 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
x11
(
x9
(
λ x12 x13 x14 .
0
)
)
)
(
setsum
(
x3
(
λ x9 x10 x11 .
0
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
)
(
x0
(
λ x9 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
0
)
0
)
)
)
x6
)
)
⟶
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x5 x6 x7 .
x0
(
λ x9 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
x3
(
λ x12 x13 x14 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
ι →
ι → ι
.
λ x14 x15 .
0
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
x11
(
x2
(
λ x14 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x14 x15 x16 x17 .
x2
(
λ x18 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x18 x19 x20 x21 .
0
)
0
(
λ x18 :
ι → ι
.
λ x19 .
0
)
0
0
)
(
x1
(
λ x14 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
0
)
(
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 .
0
)
(
λ x14 .
0
)
0
)
(
λ x14 :
ι → ι
.
λ x15 .
setsum
0
0
)
0
(
x12
(
λ x14 .
0
)
0
)
)
)
)
0
=
setsum
(
setsum
x7
(
x4
(
λ x9 x10 x11 .
0
)
x7
(
x3
(
λ x9 x10 x11 .
x7
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
x2
(
λ x13 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 x15 x16 .
0
)
0
(
λ x13 :
ι → ι
.
λ x14 .
0
)
0
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
0
)
)
)
)
0
)
⟶
(
∀ x4 :
(
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x0
(
λ x9 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
Inj0
(
x2
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x10
)
(
λ x12 x13 x14 x15 .
setsum
(
x2
(
λ x16 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x16 x17 x18 x19 .
0
)
0
(
λ x16 :
ι → ι
.
λ x17 .
0
)
0
0
)
(
x0
(
λ x16 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x17 .
λ x18 :
ι → ι
.
0
)
0
)
)
x10
(
λ x12 :
ι → ι
.
λ x13 .
x1
(
λ x14 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x15 .
λ x16 :
ι → ι
.
0
)
(
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 .
Inj0
0
)
(
λ x14 .
setsum
0
0
)
(
x2
(
λ x14 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x14 x15 x16 x17 .
0
)
0
(
λ x14 :
ι → ι
.
λ x15 .
0
)
0
0
)
)
(
x1
(
λ x12 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
x2
(
λ x15 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x15 x16 x17 x18 .
0
)
0
(
λ x15 :
ι → ι
.
λ x16 .
0
)
0
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 .
setsum
0
0
)
(
λ x12 .
setsum
0
0
)
0
)
0
)
)
(
Inj1
0
)
=
x7
(
x1
(
λ x9 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
setsum
0
(
Inj1
(
x0
(
λ x12 :
(
ι →
ι →
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι → ι
.
0
)
0
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
Inj0
0
)
(
λ x9 .
0
)
(
x3
(
λ x9 x10 x11 .
Inj0
0
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x2
(
λ x11 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 x13 x14 .
setsum
0
0
)
0
(
λ x11 :
ι → ι
.
λ x12 .
Inj1
0
)
(
x3
(
λ x11 x12 x13 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
)
(
x3
(
λ x11 x12 x13 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι →
ι → ι
.
λ x13 x14 .
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
0
)
)
)
)
)
)
⟶
False
(proof)
Theorem
d392e..
:
∀ x0 :
(
(
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
∀ x1 :
(
ι → ι
)
→
(
(
ι → ι
)
→
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x2 :
(
ι →
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι →
ι → ι
)
→ ι
.
∀ x3 :
(
ι → ι
)
→
ι → ι
.
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x3
(
λ x9 .
0
)
(
Inj0
(
x1
(
setsum
0
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 :
ι → ι
.
x11
(
Inj1
0
)
)
)
)
=
x6
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x7 :
(
ι → ι
)
→ ι
.
x3
(
λ x9 .
0
)
(
x6
(
setsum
(
x6
(
Inj0
0
)
(
λ x9 .
0
)
(
λ x9 .
0
)
0
)
(
x3
(
λ x9 .
setsum
0
0
)
0
)
)
(
λ x9 .
x9
)
(
λ x9 .
0
)
(
x0
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
(
x0
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x2
(
λ x10 .
λ x11 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
0
)
)
(
x1
(
λ x9 .
0
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 :
ι → ι
.
0
)
)
)
)
)
=
x6
(
setsum
(
x5
(
setsum
(
x1
(
λ x9 .
0
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 :
ι → ι
.
0
)
)
0
)
)
(
x3
(
λ x9 .
x3
(
λ x10 .
setsum
0
0
)
(
x5
0
)
)
(
Inj1
(
x0
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
0
)
)
)
)
(
λ x9 .
Inj1
(
x7
(
λ x10 .
x6
(
x6
0
(
λ x11 .
0
)
(
λ x11 .
0
)
0
)
(
λ x11 .
x0
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
0
)
(
λ x11 .
0
)
0
)
)
)
(
λ x9 .
setsum
(
x0
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
Inj1
0
)
(
setsum
0
(
x0
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
0
)
)
)
(
x1
(
λ x10 .
Inj1
(
x7
(
λ x11 .
0
)
)
)
(
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
x1
(
λ x13 .
0
)
(
λ x13 :
ι → ι
.
λ x14 :
ι →
ι → ι
.
λ x15 :
ι → ι
.
x0
(
λ x16 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
0
)
)
)
)
(
x6
0
(
λ x9 .
x9
)
(
λ x9 .
0
)
(
x5
(
x1
(
λ x9 .
x0
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
0
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 :
ι → ι
.
x0
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
0
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
x2
(
λ x9 .
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
0
)
=
Inj0
0
)
⟶
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 .
∀ x6 x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
x2
(
λ x9 .
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
x0
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
(
x2
(
λ x11 .
λ x12 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
x0
(
λ x13 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
(
setsum
0
0
)
)
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 x13 x14 .
x14
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
0
)
=
x0
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
Inj1
0
)
(
setsum
(
x6
(
λ x9 :
(
ι → ι
)
→ ι
.
x1
(
λ x10 .
Inj1
0
)
(
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
x0
(
λ x13 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
0
)
0
)
)
)
0
)
)
⟶
(
∀ x4 .
∀ x5 x6 :
ι →
(
ι → ι
)
→ ι
.
∀ x7 .
x1
(
λ x9 .
x7
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 :
ι → ι
.
Inj0
(
x9
(
x1
(
λ x12 .
Inj1
0
)
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
0
)
)
)
)
=
x7
)
⟶
(
∀ x4 :
(
ι → ι
)
→
ι → ι
.
∀ x5 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x9 .
x5
(
λ x10 x11 .
x10
)
(
λ x10 .
x6
)
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 :
ι → ι
.
0
)
=
x5
(
λ x9 x10 .
setsum
0
(
x3
(
λ x11 .
setsum
0
(
x1
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
0
)
)
)
0
)
)
(
λ x9 .
Inj0
(
x7
(
λ x10 :
ι →
ι → ι
.
x3
(
λ x11 .
x1
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
0
)
)
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x0
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x1
(
λ x10 .
Inj1
(
x3
(
λ x11 .
x7
0
)
(
x1
(
λ x11 .
0
)
(
λ x11 :
ι → ι
.
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
0
)
)
)
)
(
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
x10
0
)
)
(
Inj0
(
x2
(
λ x9 .
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
setsum
(
Inj0
0
)
(
Inj0
0
)
)
)
)
=
setsum
(
Inj1
(
x2
(
λ x9 .
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
x9
)
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 x12 .
x11
)
)
)
0
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x0
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x1
(
λ x10 .
x0
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x1
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
Inj0
0
)
)
0
)
(
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
x3
(
λ x13 .
0
)
0
)
)
0
=
x1
(
λ x9 .
x0
(
λ x10 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x6
)
(
Inj0
x7
)
)
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 :
ι → ι
.
setsum
(
Inj1
(
x0
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x10
0
0
)
0
)
)
x7
)
)
⟶
False
(proof)
Theorem
d256b..
:
∀ x0 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
(
ι →
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x2 :
(
ι → ι
)
→
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x3 :
(
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 .
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x9
(
λ x10 :
ι →
ι → ι
.
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 .
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 .
0
)
(
λ x13 .
0
)
)
(
λ x12 .
x3
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 .
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 .
0
)
(
λ x13 .
0
)
)
)
(
λ x11 .
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
setsum
(
setsum
0
0
)
(
x0
(
λ x12 :
ι →
ι → ι
.
λ x13 .
0
)
0
)
)
(
λ x11 x12 .
x1
(
λ x13 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
x11
)
(
λ x13 .
0
)
)
(
λ x11 .
x7
)
)
)
(
λ x9 .
setsum
(
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj1
0
)
(
λ x10 .
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x6
(
λ x11 .
setsum
0
0
)
(
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 .
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 .
0
)
(
λ x11 .
0
)
)
(
λ x11 .
0
)
)
(
λ x10 x11 .
x10
)
(
λ x10 .
Inj1
(
setsum
0
0
)
)
)
(
Inj1
x7
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x6
(
λ x10 .
Inj1
(
x6
(
λ x11 .
x11
)
(
Inj0
0
)
(
λ x11 .
x0
(
λ x12 :
ι →
ι → ι
.
λ x13 .
0
)
0
)
)
)
(
x9
(
λ x10 .
0
)
)
Inj1
)
(
λ x9 x10 .
x9
)
(
λ x9 .
Inj0
0
)
=
setsum
(
x6
(
λ x9 .
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 .
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x12 x13 .
0
)
(
λ x12 .
0
)
)
(
λ x11 .
x3
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 .
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x12 x13 .
0
)
(
λ x12 .
0
)
)
(
λ x11 :
(
ι → ι
)
→ ι
.
x3
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 .
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x12 x13 .
0
)
(
λ x12 .
0
)
)
(
λ x11 x12 .
x3
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 .
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 .
0
)
(
λ x13 .
0
)
)
(
λ x11 .
setsum
0
0
)
)
(
λ x10 .
setsum
(
Inj1
0
)
(
setsum
0
0
)
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 .
x2
(
λ x12 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
0
)
)
(
λ x11 :
(
ι → ι
)
→ ι
.
x9
)
(
λ x11 x12 .
0
)
(
λ x11 .
x2
(
λ x12 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
0
)
)
)
(
λ x10 x11 .
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 .
x10
)
)
(
λ x10 .
Inj0
x7
)
)
0
(
λ x9 .
Inj1
(
Inj0
0
)
)
)
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj0
0
)
(
λ x9 .
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
Inj0
(
x0
(
λ x11 :
ι →
ι → ι
.
λ x12 .
0
)
0
)
)
(
λ x10 .
0
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
setsum
0
(
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x0
(
λ x11 :
ι →
ι → ι
.
λ x12 .
0
)
0
)
(
λ x10 .
Inj0
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x7
)
(
λ x10 x11 .
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 .
0
)
)
(
λ x10 .
x10
)
)
)
(
λ x9 x10 .
x9
)
(
λ x9 .
Inj0
(
x0
(
λ x10 :
ι →
ι → ι
.
λ x11 .
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 .
0
)
)
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x7 .
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x9
(
λ x10 :
ι →
ι → ι
.
x10
(
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 .
Inj0
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
x11
(
λ x12 .
0
)
)
(
λ x11 x12 .
x0
(
λ x13 :
ι →
ι → ι
.
λ x14 .
0
)
0
)
(
λ x11 .
Inj1
0
)
)
(
x6
0
(
λ x11 :
ι → ι
.
λ x12 .
Inj0
0
)
0
0
)
)
)
(
λ x9 .
x0
(
λ x10 :
ι →
ι → ι
.
λ x11 .
x0
(
λ x12 :
ι →
ι → ι
.
λ x13 .
x13
)
x9
)
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
Inj1
x7
)
(
λ x9 x10 .
x9
)
(
λ x9 .
x7
)
=
setsum
x7
0
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 .
Inj0
(
setsum
0
0
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
Inj1
0
)
=
x6
(
λ x9 .
0
)
(
λ x9 .
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj1
(
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
x10
(
λ x12 :
ι →
ι → ι
.
0
)
)
(
λ x11 .
x0
(
λ x12 :
ι →
ι → ι
.
λ x13 .
0
)
0
)
)
)
(
λ x10 .
Inj0
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
(
λ x10 x11 .
x9
)
(
λ x10 .
setsum
(
x2
(
λ x11 .
x9
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x14
)
)
(
x2
(
λ x11 .
x1
(
λ x12 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x12 .
0
)
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x14
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 x6 :
ι → ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι → ι
.
x2
(
λ x9 .
x7
(
λ x10 .
λ x11 :
ι → ι
.
0
)
(
x6
(
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 .
setsum
0
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
(
λ x10 x11 .
0
)
(
λ x10 .
x0
(
λ x11 :
ι →
ι → ι
.
λ x12 .
0
)
0
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
x11
(
x10
(
λ x13 .
x3
(
λ x14 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x14 .
x14
)
(
λ x14 :
(
ι → ι
)
→ ι
.
Inj1
0
)
(
λ x14 x15 .
x12
)
(
λ x14 .
x2
(
λ x15 .
0
)
(
λ x15 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 :
ι → ι
.
λ x18 .
0
)
)
)
)
)
=
setsum
(
x0
(
λ x9 :
ι →
ι → ι
.
λ x10 .
0
)
0
)
0
)
⟶
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
ι →
ι →
(
ι → ι
)
→ ι
.
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x10 .
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
setsum
(
x0
(
λ x12 :
ι →
ι → ι
.
λ x13 .
0
)
0
)
(
x0
(
λ x12 :
ι →
ι → ι
.
λ x13 .
0
)
0
)
)
(
λ x11 .
setsum
(
x7
0
0
(
λ x12 .
0
)
)
(
x7
0
0
(
λ x12 .
0
)
)
)
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
λ x13 .
x13
)
)
(
λ x9 .
0
)
=
setsum
(
setsum
(
x4
(
x0
(
λ x9 :
ι →
ι → ι
.
λ x10 .
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x11 .
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
0
)
(
λ x11 x12 .
0
)
(
λ x11 .
0
)
)
x5
)
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 .
0
)
)
(
λ x9 .
0
)
)
(
λ x9 .
0
)
)
(
x2
(
λ x9 .
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 .
Inj1
0
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
0
)
)
)
0
)
⟶
(
∀ x4 x5 x6 x7 .
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x9 .
x0
(
λ x10 :
ι →
ι → ι
.
λ x11 .
0
)
(
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x12 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
0
)
)
(
λ x11 .
x2
(
λ x12 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
0
)
)
(
λ x11 :
(
ι → ι
)
→ ι
.
x7
)
(
λ x11 x12 .
x12
)
(
λ x11 .
x7
)
)
(
λ x10 .
Inj0
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
(
λ x10 x11 .
x2
(
λ x12 .
x0
(
λ x13 :
ι →
ι → ι
.
λ x14 .
0
)
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
Inj0
0
)
)
(
λ x10 .
x2
(
λ x11 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
0
)
)
)
)
=
Inj1
x6
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x5 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x6 :
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
(
ι →
ι → ι
)
→ ι
.
x0
(
λ x9 :
ι →
ι → ι
.
λ x10 .
x2
(
λ x11 .
x7
(
λ x12 x13 .
x2
(
λ x14 .
0
)
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 :
ι → ι
.
λ x17 .
x2
(
λ x18 .
0
)
(
λ x18 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x19 :
(
ι → ι
)
→ ι
.
λ x20 :
ι → ι
.
λ x21 .
0
)
)
)
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x14
)
)
(
x2
(
λ x9 .
x0
(
λ x10 :
ι →
ι → ι
.
λ x11 .
x0
(
λ x12 :
ι →
ι → ι
.
λ x13 .
Inj0
0
)
0
)
(
x7
(
λ x10 x11 .
0
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
x10
(
λ x13 .
x11
(
x10
(
λ x14 .
0
)
)
)
)
)
=
setsum
(
x0
(
λ x9 :
ι →
ι → ι
.
λ x10 .
x2
(
λ x11 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x14
)
)
0
)
(
x6
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
x7
(
λ x11 x12 .
x2
(
λ x13 .
0
)
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 :
ι → ι
.
λ x16 .
0
)
)
)
(
λ x10 .
x6
(
λ x11 :
ι → ι
.
0
)
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
ι →
ι → ι
.
∀ x7 :
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
.
x0
(
λ x9 :
ι →
ι → ι
.
λ x10 .
x7
(
setsum
(
x2
(
λ x11 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x14
)
)
(
setsum
x10
(
x6
0
0
)
)
)
(
λ x11 x12 .
Inj0
(
Inj0
(
setsum
0
0
)
)
)
(
λ x11 .
0
)
)
0
=
x7
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x10 .
0
)
)
(
λ x9 .
x7
x9
(
λ x10 x11 .
x11
)
(
λ x10 .
0
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
(
λ x9 x10 .
x7
0
(
λ x11 x12 .
0
)
(
λ x11 .
setsum
x9
x9
)
)
(
λ x9 .
Inj0
(
setsum
(
x7
0
(
λ x10 x11 .
0
)
(
λ x10 .
0
)
)
x5
)
)
)
(
λ x9 x10 .
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x12 .
x0
(
λ x13 :
ι →
ι → ι
.
λ x14 .
x2
(
λ x15 .
0
)
(
λ x15 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 :
ι → ι
.
λ x18 .
0
)
)
(
x3
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
(
λ x13 .
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
(
λ x13 x14 .
0
)
(
λ x13 .
0
)
)
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
x3
(
λ x16 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
0
0
)
(
λ x16 .
x13
(
λ x17 .
0
)
)
(
λ x16 :
(
ι → ι
)
→ ι
.
setsum
0
0
)
(
λ x16 x17 .
0
)
(
λ x16 .
0
)
)
)
(
λ x11 .
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
x2
(
λ x12 .
setsum
0
(
setsum
0
0
)
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
λ x15 .
x2
(
λ x16 .
setsum
0
0
)
(
λ x16 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x17 :
(
ι → ι
)
→ ι
.
λ x18 :
ι → ι
.
λ x19 .
Inj1
0
)
)
)
(
λ x11 x12 .
x11
)
(
λ x11 .
x9
)
)
(
λ x9 .
x2
(
λ x10 .
x10
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
)
)
⟶
False
(proof)
Theorem
96ff9..
:
∀ x0 :
(
ι →
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x1 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x2 :
(
ι → ι
)
→
ι → ι
.
∀ x3 :
(
ι →
ι →
ι → ι
)
→
(
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
(
∀ x4 :
(
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x3
(
λ x9 x10 x11 .
x11
)
(
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
x2
(
λ x10 .
x10
)
x7
)
=
Inj0
0
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x7 .
x3
(
λ x9 x10 x11 .
0
)
(
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
setsum
(
Inj0
(
setsum
0
(
x1
(
λ x10 :
ι →
ι → ι
.
0
)
(
λ x10 .
0
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
)
)
)
(
setsum
0
(
x3
(
λ x10 x11 x12 .
0
)
(
λ x10 :
ι →
(
ι → ι
)
→
ι → ι
.
x3
(
λ x11 x12 x13 .
0
)
(
λ x11 :
ι →
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
)
=
x4
)
⟶
(
∀ x4 x5 x6 x7 .
x2
(
λ x9 .
0
)
(
x0
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
Inj1
(
Inj1
(
x0
(
λ x13 .
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 :
ι → ι
.
0
)
0
)
)
)
(
setsum
0
x7
)
)
=
x0
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
Inj1
(
x12
(
x2
(
λ x13 .
setsum
0
0
)
(
x3
(
λ x13 x14 x15 .
0
)
(
λ x13 :
ι →
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
)
(
setsum
(
x1
(
λ x9 :
ι →
ι → ι
.
0
)
(
λ x9 .
x1
(
λ x10 :
ι →
ι → ι
.
x7
)
(
λ x10 .
x10
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
setsum
0
0
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
x7
)
)
0
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
ι → ι
.
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
(
ι →
ι → ι
)
→ ι
.
x2
(
λ x9 .
x6
(
λ x10 :
ι →
ι → ι
.
x6
(
λ x11 :
ι →
ι → ι
.
0
)
)
)
(
x5
(
λ x9 .
x6
(
λ x10 :
ι →
ι → ι
.
Inj1
(
Inj1
0
)
)
)
(
setsum
(
x1
(
λ x9 :
ι →
ι → ι
.
setsum
0
0
)
(
λ x9 .
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
x9
(
λ x11 .
0
)
0
)
)
(
x4
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
0
)
(
λ x9 x10 .
x3
(
λ x11 x12 x13 .
0
)
(
λ x11 :
ι →
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
)
=
setsum
(
x5
(
λ x9 .
x9
)
(
x2
(
λ x9 .
setsum
0
0
)
0
)
)
(
setsum
(
setsum
(
x6
(
λ x9 :
ι →
ι → ι
.
0
)
)
(
x0
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
0
)
(
setsum
0
0
)
)
)
(
Inj1
(
x1
(
λ x9 :
ι →
ι → ι
.
x7
(
λ x10 x11 .
0
)
)
(
λ x9 .
x7
(
λ x10 x11 .
0
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
Inj1
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
x1
(
λ x9 :
ι →
ι → ι
.
x1
(
λ x10 :
ι →
ι → ι
.
x3
(
λ x11 x12 x13 .
0
)
(
λ x11 :
ι →
(
ι → ι
)
→
ι → ι
.
x2
(
λ x12 .
Inj0
0
)
(
x9
0
0
)
)
)
(
λ x10 .
x10
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
x11
)
)
(
λ x9 .
x0
(
λ x10 .
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
x3
(
λ x14 x15 x16 .
0
)
(
λ x14 :
ι →
(
ι → ι
)
→
ι → ι
.
0
)
)
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
Inj1
)
=
Inj0
0
)
⟶
(
∀ x4 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x5 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x1
(
λ x9 :
ι →
ι → ι
.
x7
)
(
λ x9 .
x5
(
x1
(
λ x10 :
ι →
ι → ι
.
Inj1
(
x1
(
λ x11 :
ι →
ι → ι
.
0
)
(
λ x11 .
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
)
)
(
λ x10 .
x2
(
λ x11 .
setsum
0
0
)
(
setsum
0
0
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
setsum
(
x0
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 :
ι → ι
.
0
)
0
)
0
)
)
(
x0
(
λ x10 .
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
setsum
(
setsum
0
0
)
(
x2
(
λ x14 .
0
)
0
)
)
(
x2
(
λ x10 .
x9
)
x7
)
)
(
λ x10 .
0
)
(
x1
(
λ x10 :
ι →
ι → ι
.
x9
)
(
λ x10 .
x3
(
λ x11 x12 x13 .
x10
)
(
λ x11 :
ι →
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
x11
)
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
setsum
(
Inj1
(
x0
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
x3
(
λ x15 x16 x17 .
0
)
(
λ x15 :
ι →
(
ι → ι
)
→
ι → ι
.
0
)
)
(
x2
(
λ x11 .
0
)
0
)
)
)
(
x3
(
λ x11 x12 x13 .
x0
(
λ x14 .
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 :
ι → ι
.
0
)
x10
)
(
λ x11 :
ι →
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
)
)
=
x5
(
Inj0
(
Inj1
(
x3
(
λ x9 x10 x11 .
setsum
0
0
)
(
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
Inj0
0
)
)
)
)
(
x1
(
λ x9 :
ι →
ι → ι
.
x1
(
λ x10 :
ι →
ι → ι
.
x1
(
λ x11 :
ι →
ι → ι
.
x0
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 :
ι → ι
.
0
)
0
)
(
λ x11 .
x2
(
λ x12 .
0
)
0
)
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
x11
(
λ x13 .
0
)
0
)
)
(
x9
(
setsum
0
0
)
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
x3
(
λ x12 x13 x14 .
x2
(
λ x15 .
0
)
0
)
(
λ x12 :
ι →
(
ι → ι
)
→
ι → ι
.
x12
0
(
λ x13 .
0
)
0
)
)
)
(
λ x9 .
x1
(
λ x10 :
ι →
ι → ι
.
Inj1
(
x2
(
λ x11 .
0
)
0
)
)
(
λ x10 .
x10
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
x7
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
Inj0
(
x0
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
0
)
(
Inj0
0
)
)
)
)
(
λ x9 .
x5
0
x9
(
λ x10 .
x0
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
x12
(
λ x15 :
ι → ι
.
λ x16 .
x16
)
)
0
)
0
)
(
Inj0
(
x2
(
λ x9 .
x6
)
(
setsum
(
x3
(
λ x9 x10 x11 .
0
)
(
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
0
)
)
(
x0
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
0
)
0
)
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι →
ι →
ι →
ι → ι
.
∀ x7 .
x0
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
Inj1
(
x1
(
λ x13 :
ι →
ι → ι
.
x10
(
λ x14 :
ι → ι
.
λ x15 .
Inj0
0
)
)
(
λ x13 .
0
)
(
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
x14
)
)
)
0
=
Inj0
x4
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
ι → ι
.
x0
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
0
)
(
setsum
(
x4
(
x1
(
λ x9 :
ι →
ι → ι
.
x2
(
λ x10 .
0
)
0
)
(
λ x9 .
x1
(
λ x10 :
ι →
ι → ι
.
0
)
(
λ x10 .
0
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 .
0
)
)
)
0
)
=
x5
)
⟶
False
(proof)
Theorem
1b320..
:
∀ x0 :
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
ι →
ι → ι
)
→
ι → ι
.
∀ x2 :
(
ι →
(
(
ι →
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x3 :
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι → ι
)
→
ι → ι
.
(
∀ x4 .
∀ x5 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι →
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
x3
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 x11 .
x9
(
λ x12 :
ι → ι
.
0
)
)
0
=
x7
x4
(
λ x9 :
ι → ι
.
0
)
x6
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 :
(
ι →
ι →
ι → ι
)
→ ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x3
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 x11 .
setsum
(
x1
(
λ x12 .
x2
(
λ x13 .
λ x14 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x15 .
0
)
(
λ x13 :
ι → ι
.
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
Inj1
0
)
)
(
x7
(
λ x12 .
λ x13 :
ι → ι
.
x10
)
(
λ x12 .
x11
)
(
λ x12 .
0
)
(
x7
(
λ x12 .
λ x13 :
ι → ι
.
0
)
(
λ x12 .
0
)
(
λ x12 .
0
)
0
)
)
)
(
x7
(
λ x12 .
λ x13 :
ι → ι
.
x1
(
λ x14 x15 .
x0
(
λ x16 .
0
)
0
)
0
)
(
λ x12 .
0
)
(
λ x12 .
x1
(
λ x13 x14 .
0
)
(
x1
(
λ x13 x14 .
0
)
0
)
)
(
Inj0
(
x0
(
λ x12 .
0
)
0
)
)
)
)
0
=
Inj0
0
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
x2
(
λ x9 .
λ x10 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x11 .
0
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
x10
(
λ x12 .
x1
(
λ x13 x14 .
x14
)
(
x10
(
λ x13 .
x13
)
0
)
)
(
x1
(
λ x12 x13 .
setsum
(
x1
(
λ x14 x15 .
0
)
0
)
(
x0
(
λ x14 .
0
)
0
)
)
0
)
)
0
=
Inj0
(
Inj1
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x7 .
x2
(
λ x9 .
λ x10 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x11 .
x1
(
λ x12 x13 .
setsum
(
setsum
0
x13
)
(
x10
(
λ x14 x15 .
0
)
(
Inj1
0
)
)
)
(
x0
(
λ x12 .
x2
(
λ x13 .
λ x14 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x15 .
0
)
(
λ x13 :
ι → ι
.
λ x14 :
(
ι → ι
)
→
ι → ι
.
λ x15 :
ι → ι
.
setsum
0
0
)
(
Inj1
0
)
)
(
x1
(
λ x12 x13 .
0
)
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
setsum
0
0
)
x7
=
Inj0
0
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x5 .
∀ x6 :
ι →
(
ι → ι
)
→ ι
.
∀ x7 .
x1
(
λ x9 x10 .
Inj0
(
x0
(
λ x11 .
x10
)
(
x3
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
0
)
x7
)
)
)
0
=
x4
(
λ x9 .
Inj0
(
x3
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 x12 .
x0
(
λ x13 .
0
)
(
setsum
0
0
)
)
(
x0
(
λ x10 .
setsum
0
0
)
(
x3
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 x12 .
0
)
0
)
)
)
)
(
λ x9 :
ι → ι
.
0
)
(
Inj0
0
)
(
Inj0
0
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x5 :
ι →
ι → ι
.
∀ x6 .
∀ x7 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x9 x10 .
x2
(
λ x11 .
λ x12 :
(
ι →
ι → ι
)
→
ι → ι
.
λ x13 .
x13
)
(
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 :
ι → ι
.
Inj1
(
x11
0
)
)
(
Inj1
(
x0
(
λ x11 .
x0
(
λ x12 .
0
)
0
)
(
setsum
0
0
)
)
)
)
0
=
Inj0
(
Inj0
x6
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x9 .
setsum
x5
(
x6
x7
)
)
0
=
x6
(
setsum
(
x6
x7
)
x7
)
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
x0
(
λ x9 .
x6
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 x12 .
x3
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 x15 .
x12
)
0
)
)
(
Inj1
(
x1
(
λ x9 x10 .
Inj1
(
x6
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 x13 .
0
)
)
)
0
)
)
=
setsum
(
x7
0
)
(
x3
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 x11 .
x11
)
(
x0
(
λ x9 .
x9
)
0
)
)
)
⟶
False
(proof)
Theorem
4acef..
:
∀ x0 :
(
ι → ι
)
→
(
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
)
→
(
ι →
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x1 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x2 :
(
(
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι →
(
ι →
ι → ι
)
→ ι
.
∀ x3 :
(
(
ι → ι
)
→ ι
)
→
(
ι →
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→ ι
.
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x3
(
λ x9 :
ι → ι
.
setsum
(
x0
(
λ x10 .
setsum
0
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι → ι
.
0
)
(
λ x10 x11 x12 .
x11
)
(
λ x10 :
ι → ι
.
λ x11 .
x7
(
λ x12 .
Inj1
0
)
(
λ x12 x13 .
x2
(
λ x14 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
0
)
(
λ x14 .
0
)
0
(
λ x14 x15 .
0
)
)
)
)
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
Inj0
(
x2
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
x1
(
λ x14 x15 .
x15
)
(
λ x14 :
ι → ι
.
0
)
)
(
λ x13 .
x3
(
λ x14 :
ι → ι
.
0
)
(
λ x14 .
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 x17 .
x1
(
λ x18 x19 .
0
)
(
λ x18 :
ι → ι
.
0
)
)
)
0
(
λ x13 .
setsum
0
)
)
)
=
x6
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
x3
(
λ x9 :
ι → ι
.
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
Inj1
)
=
x6
(
λ x9 :
ι → ι
.
λ x10 .
x9
(
x3
(
λ x11 :
ι → ι
.
x10
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 x14 .
Inj0
(
x2
(
λ x15 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
0
)
(
λ x15 .
0
)
0
(
λ x15 x16 .
0
)
)
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
⟶
(
∀ x4 x5 x6 x7 .
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
Inj1
(
x2
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
x0
(
λ x11 .
x0
(
λ x12 .
0
)
(
λ x12 :
ι →
(
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
ι → ι
.
0
)
(
λ x12 x13 x14 .
0
)
(
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
x11
0
(
λ x14 .
0
)
)
(
λ x11 x12 x13 .
x3
(
λ x14 :
ι → ι
.
0
)
(
λ x14 .
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 x17 .
0
)
)
(
λ x11 :
ι → ι
.
λ x12 .
x1
(
λ x13 x14 .
0
)
(
λ x13 :
ι → ι
.
0
)
)
)
(
λ x10 .
x9
(
λ x11 :
ι →
ι → ι
.
0
)
x7
)
x7
(
λ x10 .
x9
(
λ x11 :
ι →
ι → ι
.
0
)
)
)
)
(
λ x9 .
x9
)
x4
(
λ x9 x10 .
x10
)
=
Inj1
(
setsum
(
x3
(
λ x9 :
ι → ι
.
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
x10
(
λ x13 .
x11
)
)
)
x7
)
)
⟶
(
∀ x4 :
ι →
ι →
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
x5
(
Inj1
(
x0
(
λ x10 .
setsum
0
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι → ι
.
x12
0
)
(
λ x10 x11 x12 .
x1
(
λ x13 x14 .
0
)
(
λ x13 :
ι → ι
.
0
)
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
)
)
(
λ x9 .
x5
(
Inj1
(
x6
(
λ x10 .
x1
(
λ x11 x12 .
0
)
(
λ x11 :
ι → ι
.
0
)
)
)
)
)
(
Inj1
(
setsum
(
x1
(
λ x9 x10 .
setsum
0
0
)
(
λ x9 :
ι → ι
.
0
)
)
0
)
)
(
λ x9 x10 .
setsum
(
x2
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
x11
(
λ x12 :
ι →
ι → ι
.
Inj1
0
)
(
x3
(
λ x12 :
ι → ι
.
0
)
(
λ x12 .
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 x15 .
0
)
)
)
(
λ x11 .
0
)
(
x0
(
λ x11 .
x11
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
Inj0
0
)
(
λ x11 x12 x13 .
x0
(
λ x14 .
0
)
(
λ x14 :
ι →
(
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
ι → ι
.
0
)
(
λ x14 x15 x16 .
0
)
(
λ x14 :
ι → ι
.
λ x15 .
0
)
)
(
λ x11 :
ι → ι
.
λ x12 .
x1
(
λ x13 x14 .
0
)
(
λ x13 :
ι → ι
.
0
)
)
)
(
λ x11 x12 .
x1
(
λ x13 x14 .
setsum
0
0
)
(
λ x13 :
ι → ι
.
0
)
)
)
0
)
=
setsum
(
x3
(
λ x9 :
ι → ι
.
Inj1
(
Inj1
0
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
setsum
0
(
Inj0
(
Inj1
0
)
)
)
)
x7
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι →
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x1
(
λ x9 x10 .
0
)
(
λ x9 :
ι → ι
.
x1
(
λ x10 x11 .
Inj1
0
)
(
λ x10 :
ι → ι
.
0
)
)
=
x1
(
λ x9 x10 .
x0
(
λ x11 .
x1
(
λ x12 x13 .
x12
)
(
λ x12 :
ι → ι
.
x10
)
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
ι → ι
.
x12
)
(
λ x11 x12 x13 .
x3
(
λ x14 :
ι → ι
.
setsum
(
setsum
0
0
)
(
Inj1
0
)
)
(
λ x14 .
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 x17 .
x1
(
λ x18 x19 .
x17
)
(
λ x18 :
ι → ι
.
Inj1
0
)
)
)
(
λ x11 :
ι → ι
.
λ x12 .
setsum
(
x11
0
)
(
x0
(
λ x13 .
0
)
(
λ x13 :
ι →
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
setsum
0
0
)
(
λ x13 x14 x15 .
Inj1
0
)
(
λ x13 :
ι → ι
.
λ x14 .
x11
0
)
)
)
)
(
λ x9 :
ι → ι
.
x0
(
λ x10 .
x7
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
ι → ι
.
setsum
0
0
)
(
λ x10 x11 x12 .
Inj0
(
x2
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
x3
(
λ x14 :
ι → ι
.
0
)
(
λ x14 .
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 x17 .
0
)
)
(
λ x13 .
x13
)
(
setsum
0
0
)
(
λ x13 x14 .
Inj1
0
)
)
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
x1
(
λ x9 x10 .
0
)
(
λ x9 :
ι → ι
.
x9
(
x9
(
setsum
(
x9
0
)
x5
)
)
)
=
x5
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x0
(
λ x9 .
x6
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
x2
(
λ x12 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
setsum
(
x2
(
λ x13 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
x13
(
λ x14 :
ι →
ι → ι
.
0
)
0
)
(
λ x13 .
x11
0
)
0
(
λ x13 x14 .
setsum
0
0
)
)
(
x0
(
λ x13 .
0
)
(
λ x13 :
ι →
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
ι → ι
.
0
)
(
λ x13 x14 x15 .
x15
)
(
λ x13 :
ι → ι
.
λ x14 .
x14
)
)
)
x11
0
(
λ x12 x13 .
x0
(
λ x14 .
x3
(
λ x15 :
ι → ι
.
x1
(
λ x16 x17 .
0
)
(
λ x16 :
ι → ι
.
0
)
)
(
λ x15 .
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 x18 .
x3
(
λ x19 :
ι → ι
.
0
)
(
λ x19 .
λ x20 :
(
ι → ι
)
→ ι
.
λ x21 x22 .
0
)
)
)
(
λ x14 :
ι →
(
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
ι → ι
.
x16
0
)
(
λ x14 x15 x16 .
setsum
(
Inj0
0
)
0
)
(
λ x14 :
ι → ι
.
λ x15 .
x1
(
λ x16 x17 .
x3
(
λ x18 :
ι → ι
.
0
)
(
λ x18 .
λ x19 :
(
ι → ι
)
→ ι
.
λ x20 x21 .
0
)
)
(
λ x16 :
ι → ι
.
x15
)
)
)
)
(
λ x9 x10 x11 .
x9
)
(
λ x9 :
ι → ι
.
λ x10 .
x3
(
λ x11 :
ι → ι
.
x10
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 x14 .
0
)
)
=
Inj0
(
Inj0
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
x1
(
λ x10 x11 .
0
)
(
λ x10 :
ι → ι
.
0
)
)
(
λ x9 .
x5
)
0
(
λ x9 x10 .
x1
(
λ x11 x12 .
x12
)
(
λ x11 :
ι → ι
.
Inj1
0
)
)
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x0
(
λ x9 .
x6
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
ι → ι
.
x1
(
λ x12 x13 .
0
)
(
λ x12 :
ι → ι
.
Inj0
(
setsum
(
x3
(
λ x13 :
ι → ι
.
0
)
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 x16 .
0
)
)
(
x9
0
(
λ x13 .
0
)
)
)
)
)
(
λ x9 x10 x11 .
x7
)
(
λ x9 :
ι → ι
.
λ x10 .
x1
(
λ x11 x12 .
Inj1
x10
)
(
λ x11 :
ι → ι
.
Inj1
(
Inj0
0
)
)
)
=
x6
)
⟶
False
(proof)
Theorem
fc3b7..
:
∀ x0 :
(
ι → ι
)
→
(
ι →
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x1 :
(
(
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι →
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x2 :
(
ι → ι
)
→
ι → ι
.
∀ x3 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
(
∀ x4 .
∀ x5 :
ι →
(
ι →
ι → ι
)
→ ι
.
∀ x6 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x7 :
(
ι → ι
)
→ ι
.
x3
(
λ x9 .
x0
(
λ x10 .
Inj1
x9
)
(
λ x10 x11 .
λ x12 :
ι → ι
.
x12
(
x2
(
λ x13 .
setsum
0
0
)
(
Inj1
0
)
)
)
)
(
λ x9 x10 .
x2
(
λ x11 .
0
)
(
setsum
(
x6
(
λ x11 .
x1
(
λ x12 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 x14 .
λ x15 :
ι → ι
.
0
)
0
)
(
λ x11 x12 .
0
)
)
(
x3
(
λ x11 .
0
)
(
λ x11 x12 .
setsum
0
0
)
(
setsum
0
0
)
(
λ x11 :
ι → ι
.
x11
0
)
)
)
)
(
Inj0
(
setsum
(
setsum
(
x6
(
λ x9 .
0
)
(
λ x9 x10 .
0
)
)
0
)
(
x0
(
λ x9 .
setsum
0
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
x10
)
)
)
)
(
λ x9 :
ι → ι
.
0
)
=
setsum
0
0
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x3
(
λ x9 .
x9
)
(
λ x9 x10 .
Inj1
0
)
0
(
λ x9 :
ι → ι
.
setsum
(
Inj0
(
Inj1
0
)
)
(
Inj1
(
setsum
(
Inj0
0
)
(
x9
0
)
)
)
)
=
setsum
0
(
setsum
(
x2
(
λ x9 .
x2
(
λ x10 .
setsum
0
0
)
(
Inj1
0
)
)
(
x5
(
x3
(
λ x9 .
0
)
(
λ x9 x10 .
0
)
0
(
λ x9 :
ι → ι
.
0
)
)
)
)
(
x0
(
λ x9 .
x9
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
x2
(
λ x12 .
0
)
(
x1
(
λ x12 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 x14 .
λ x15 :
ι → ι
.
0
)
0
)
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 .
0
)
0
=
setsum
(
setsum
x4
(
setsum
x7
x7
)
)
(
Inj0
(
x0
(
λ x9 .
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
x9
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x2
(
x1
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
Inj1
0
)
)
0
=
x1
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
x2
(
λ x13 .
0
)
(
x1
(
λ x13 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 x15 .
λ x16 :
ι → ι
.
Inj1
(
x0
(
λ x17 .
0
)
(
λ x17 x18 .
λ x19 :
ι → ι
.
0
)
)
)
(
setsum
0
x10
)
)
)
(
x4
(
Inj1
(
Inj0
0
)
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 x6 x7 .
x1
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
setsum
(
setsum
0
(
x2
(
λ x13 .
0
)
(
x12
0
)
)
)
(
x1
(
λ x13 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 x15 .
λ x16 :
ι → ι
.
Inj1
(
x0
(
λ x17 .
0
)
(
λ x17 x18 .
λ x19 :
ι → ι
.
0
)
)
)
0
)
)
(
x2
(
λ x9 .
setsum
(
Inj1
(
x2
(
λ x10 .
0
)
0
)
)
0
)
0
)
=
setsum
(
Inj0
0
)
0
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
(
(
ι →
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 :
ι → ι
.
x1
(
λ x9 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 x11 .
λ x12 :
ι → ι
.
0
)
(
setsum
(
Inj0
(
x6
(
λ x9 :
ι →
ι → ι
.
x2
(
λ x10 .
0
)
0
)
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 x12 .
λ x13 :
ι → ι
.
0
)
0
)
)
)
(
x3
(
λ x9 .
x0
(
λ x10 .
x7
0
)
(
λ x10 x11 .
λ x12 :
ι → ι
.
x2
(
λ x13 .
0
)
0
)
)
(
λ x9 x10 .
setsum
(
x1
(
λ x11 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 x13 .
λ x14 :
ι → ι
.
0
)
0
)
0
)
0
(
λ x9 :
ι → ι
.
0
)
)
)
=
setsum
(
x3
(
λ x9 .
x5
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
(
λ x10 :
ι → ι
.
0
)
)
(
λ x9 x10 .
setsum
x9
0
)
(
Inj0
0
)
(
λ x9 :
ι → ι
.
0
)
)
(
x0
(
λ x9 .
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
0
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x0
(
λ x9 .
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
0
)
=
x6
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 x7 .
x0
(
λ x9 .
0
)
(
λ x9 x10 .
λ x11 :
ι → ι
.
setsum
(
setsum
x9
0
)
0
)
=
x4
)
⟶
False
(proof)
Theorem
f4c43..
:
∀ x0 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x1 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x2 :
(
(
ι →
(
ι →
ι → ι
)
→ ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x3 :
(
(
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
x0
(
λ x10 .
Inj0
0
)
x6
(
λ x10 .
x2
(
λ x11 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
setsum
0
(
setsum
0
0
)
)
(
x0
(
λ x11 .
setsum
0
0
)
(
Inj0
0
)
(
λ x11 .
setsum
0
0
)
(
λ x11 :
ι → ι
.
λ x12 .
x10
)
)
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
(
λ x9 .
0
)
(
setsum
(
x0
(
λ x9 .
Inj1
x7
)
x5
(
λ x9 .
x7
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
0
)
=
Inj1
0
)
⟶
(
∀ x4 :
ι →
(
ι →
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 x7 .
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
x6
)
(
x2
(
λ x9 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
0
)
)
(
x2
(
λ x9 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
x10
(
λ x13 :
ι → ι
.
setsum
(
x13
0
)
0
)
)
(
Inj1
(
Inj1
(
x4
0
(
λ x9 x10 .
0
)
)
)
)
)
=
x6
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x6 :
ι →
ι → ι
.
∀ x7 .
x2
(
λ x9 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
0
)
(
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
x12
)
(
Inj0
(
x0
(
λ x9 .
x7
)
(
setsum
0
0
)
(
λ x9 .
setsum
0
0
)
(
λ x9 :
ι → ι
.
λ x10 .
x6
0
0
)
)
)
)
=
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
Inj0
(
x9
(
λ x13 .
x0
(
λ x14 .
0
)
x10
(
λ x14 .
x2
(
λ x15 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x16 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x17 :
ι → ι
.
λ x18 .
0
)
0
)
(
λ x14 :
ι → ι
.
λ x15 .
Inj0
0
)
)
)
)
x4
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι →
ι → ι
.
x2
(
λ x9 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
setsum
(
setsum
(
setsum
(
Inj1
0
)
(
x10
(
λ x13 :
ι → ι
.
0
)
)
)
(
x0
(
λ x13 .
Inj1
0
)
(
x1
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 .
0
)
0
)
(
λ x13 .
Inj1
0
)
(
λ x13 :
ι → ι
.
λ x14 .
x13
0
)
)
)
0
)
(
Inj0
(
setsum
(
x0
(
λ x9 .
x0
(
λ x10 .
0
)
0
(
λ x10 .
0
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
0
(
λ x9 .
Inj0
0
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
0
)
)
=
x7
(
x5
(
λ x9 .
x0
(
λ x10 .
0
)
(
Inj1
(
Inj1
0
)
)
(
λ x10 .
x1
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
setsum
0
0
)
0
)
(
λ x10 :
ι → ι
.
λ x11 .
x3
(
λ x12 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
x12
(
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
λ x15 .
0
)
)
(
λ x12 .
x1
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 .
0
)
0
)
0
)
)
(
λ x9 x10 .
x2
(
λ x11 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
Inj0
(
x11
0
(
λ x15 x16 .
0
)
)
)
0
)
)
(
x0
(
λ x9 .
x1
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 .
x12
(
λ x14 .
x1
(
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 .
λ x17 :
(
ι → ι
)
→
ι → ι
.
λ x18 .
0
)
0
)
x11
)
x6
)
(
x5
(
λ x9 .
0
)
(
λ x9 x10 .
x1
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
x11
(
λ x15 .
0
)
)
(
Inj0
0
)
)
)
(
λ x9 .
Inj0
(
x2
(
λ x10 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι → ι
.
λ x13 .
Inj0
0
)
(
x1
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x6
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
(
ι →
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
x1
(
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 .
x0
(
λ x17 .
Inj1
(
setsum
0
0
)
)
(
setsum
(
x3
(
λ x17 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x17 .
0
)
0
)
x16
)
(
λ x17 .
setsum
x17
(
x3
(
λ x18 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x18 .
0
)
0
)
)
(
λ x17 :
ι → ι
.
λ x18 .
x17
(
x1
(
λ x19 :
(
ι → ι
)
→ ι
.
λ x20 .
λ x21 :
(
ι → ι
)
→
ι → ι
.
λ x22 .
0
)
0
)
)
)
x10
)
0
=
Inj1
(
setsum
(
x5
(
x4
(
λ x9 .
x6
0
0
0
)
)
(
λ x9 :
ι → ι
.
x9
(
x1
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
0
)
)
)
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
(
ι → ι
)
→
ι → ι
.
∀ x7 :
ι → ι
.
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
0
)
0
=
x7
(
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
x11
(
λ x13 .
x0
(
λ x14 .
x0
(
λ x15 .
0
)
0
(
λ x15 .
0
)
(
λ x15 :
ι → ι
.
λ x16 .
0
)
)
(
Inj1
0
)
(
λ x14 .
x1
(
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 .
λ x17 :
(
ι → ι
)
→
ι → ι
.
λ x18 .
0
)
0
)
(
λ x14 :
ι → ι
.
λ x15 .
0
)
)
0
)
(
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
x12
)
(
x1
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
Inj0
0
)
x4
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
x0
(
λ x9 .
x5
)
(
Inj1
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
x2
(
λ x10 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 :
ι → ι
.
λ x13 .
Inj0
0
)
0
)
(
λ x9 .
Inj0
(
setsum
0
0
)
)
x4
)
)
(
λ x9 .
x6
0
)
(
λ x9 :
ι → ι
.
λ x10 .
0
)
=
x5
)
⟶
(
∀ x4 .
∀ x5 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 x7 .
x0
(
λ x9 .
x9
)
0
(
λ x9 .
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
x2
(
λ x11 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 .
x13
(
x11
0
(
λ x15 x16 .
0
)
)
)
x7
)
(
λ x10 .
setsum
(
x1
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 .
x2
(
λ x15 :
ι →
(
ι →
ι → ι
)
→ ι
.
λ x16 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x17 :
ι → ι
.
λ x18 .
0
)
0
)
0
)
0
)
0
)
(
λ x9 :
ι → ι
.
λ x10 .
setsum
0
(
setsum
(
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
x0
(
λ x12 .
0
)
0
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
λ x11 .
setsum
0
0
)
x7
)
x7
)
)
=
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x10 .
0
)
0
)
(
λ x9 .
setsum
(
setsum
(
x1
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
λ x13 .
0
)
(
x5
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
)
(
x0
(
λ x10 .
0
)
0
(
λ x10 .
x9
)
(
λ x10 :
ι → ι
.
λ x11 .
x0
(
λ x12 .
0
)
0
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
λ x13 .
0
)
)
)
)
(
x3
(
λ x10 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
setsum
(
x0
(
λ x11 .
0
)
0
(
λ x11 .
0
)
(
λ x11 :
ι → ι
.
λ x12 .
0
)
)
0
)
(
λ x10 .
0
)
0
)
)
(
x3
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
Inj0
x6
)
(
λ x9 .
x7
)
(
x0
(
λ x9 .
0
)
(
x0
(
λ x9 .
setsum
0
0
)
x4
(
λ x9 .
x9
)
(
λ x9 :
ι → ι
.
λ x10 .
x3
(
λ x11 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x11 .
0
)
0
)
)
(
λ x9 .
x0
(
λ x10 .
Inj0
0
)
0
(
λ x10 .
Inj0
0
)
(
λ x10 :
ι → ι
.
λ x11 .
setsum
0
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x6
)
)
)
)
⟶
False
(proof)
Theorem
99c9f..
:
∀ x0 :
(
(
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→ ι
)
→
(
ι →
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
∀ x1 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x2 :
(
ι → ι
)
→
(
(
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x3 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
ι → ι
.
(
∀ x4 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x5 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x3
(
λ x9 :
ι →
ι → ι
.
λ x10 .
Inj0
(
Inj1
x10
)
)
(
x1
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
ι → ι
.
0
)
(
x5
(
λ x10 x11 .
setsum
0
0
)
(
λ x10 :
ι → ι
.
λ x11 .
x1
(
λ x12 :
ι → ι
.
0
)
0
)
)
)
(
Inj0
(
x4
(
λ x9 :
ι → ι
.
λ x10 .
x10
)
(
λ x9 :
ι → ι
.
λ x10 .
x9
0
)
(
λ x9 .
x2
(
λ x10 .
0
)
(
λ x10 :
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
0
(
λ x10 x11 .
0
)
(
λ x10 .
0
)
0
)
)
)
)
=
setsum
(
x3
(
λ x9 :
ι →
ι → ι
.
λ x10 .
x10
)
(
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
setsum
0
(
x9
0
(
λ x10 x11 .
0
)
0
0
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
setsum
(
x0
(
λ x12 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x12 .
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
0
)
)
(
Inj0
0
)
)
)
)
0
)
⟶
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 :
ι →
ι → ι
.
λ x10 .
0
)
0
=
x4
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x7 :
(
ι → ι
)
→
(
ι → ι
)
→
ι →
ι → ι
.
x2
(
λ x9 .
Inj0
0
)
(
λ x9 :
ι →
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x10
)
(
setsum
(
setsum
(
x5
0
)
(
x4
(
λ x9 :
ι → ι
.
λ x10 .
Inj0
0
)
)
)
(
x3
(
λ x9 :
ι →
ι → ι
.
λ x10 .
x7
(
λ x11 .
Inj0
0
)
(
λ x11 .
setsum
0
0
)
(
x6
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
(
λ x11 :
ι → ι
.
0
)
0
)
(
x0
(
λ x11 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 .
0
)
)
)
(
setsum
(
setsum
0
0
)
0
)
)
)
(
λ x9 x10 .
x10
)
Inj0
0
=
Inj0
(
x4
(
λ x9 :
ι → ι
.
λ x10 .
x1
(
λ x11 :
ι → ι
.
x3
(
λ x12 :
ι →
ι → ι
.
λ x13 .
0
)
x10
)
x10
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 .
Inj1
x7
)
(
λ x9 :
ι →
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
(
setsum
0
0
)
(
λ x9 x10 .
x1
(
λ x11 :
ι → ι
.
Inj1
(
setsum
(
x2
(
λ x12 .
0
)
(
λ x12 :
ι →
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
0
)
0
(
λ x12 x13 .
0
)
(
λ x12 .
0
)
0
)
(
x1
(
λ x12 :
ι → ι
.
0
)
0
)
)
)
x7
)
(
λ x9 .
Inj1
(
Inj1
(
setsum
0
(
x6
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 .
0
)
)
)
)
)
(
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
x0
(
λ x10 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
setsum
(
x1
(
λ x13 :
ι → ι
.
0
)
0
)
0
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
setsum
(
setsum
x11
x7
)
(
Inj0
(
Inj1
0
)
)
)
)
=
x1
(
λ x9 :
ι → ι
.
x0
(
λ x10 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
x10
(
x6
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
λ x13 .
x11
(
λ x14 .
0
)
)
)
(
λ x11 x12 .
x9
(
x3
(
λ x13 :
ι →
ι → ι
.
λ x14 .
0
)
0
)
)
0
(
x6
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
λ x13 .
x3
(
λ x14 :
ι →
ι → ι
.
λ x15 .
0
)
0
)
)
)
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
setsum
0
(
x0
(
λ x13 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
x1
(
λ x14 :
ι → ι
.
0
)
0
)
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 .
0
)
)
)
)
(
Inj1
(
x1
(
λ x9 :
ι → ι
.
x0
(
λ x10 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x10 .
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
x12
)
)
x5
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 x6 .
∀ x7 :
ι →
ι →
ι → ι
.
x1
(
λ x9 :
ι → ι
.
0
)
(
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
Inj1
(
Inj0
(
x9
0
(
λ x10 x11 .
0
)
0
0
)
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
)
=
setsum
(
setsum
x6
0
)
0
)
⟶
(
∀ x4 :
(
ι →
ι →
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x1
(
λ x9 :
ι → ι
.
setsum
x5
(
x1
(
λ x10 :
ι → ι
.
0
)
(
setsum
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
(
setsum
(
setsum
x5
x5
)
(
x4
(
λ x9 x10 x11 .
setsum
(
Inj1
0
)
x10
)
)
)
=
setsum
(
x1
(
λ x9 :
ι → ι
.
x6
(
λ x10 .
x2
(
λ x11 .
0
)
(
λ x11 :
ι →
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
x14
)
x7
(
λ x11 x12 .
x10
)
(
λ x11 .
x3
(
λ x12 :
ι →
ι → ι
.
λ x13 .
0
)
0
)
x7
)
(
x2
(
λ x10 .
x2
(
λ x11 .
0
)
(
λ x11 :
ι →
ι → ι
.
λ x12 .
λ x13 :
ι → ι
.
λ x14 .
0
)
0
(
λ x11 x12 .
0
)
(
λ x11 .
0
)
0
)
(
λ x10 :
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
0
)
(
x3
(
λ x10 :
ι →
ι → ι
.
λ x11 .
0
)
0
)
(
λ x10 x11 .
Inj1
0
)
(
λ x10 .
Inj1
0
)
0
)
)
(
x1
(
λ x9 :
ι → ι
.
0
)
(
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
Inj0
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
x1
(
λ x12 :
ι → ι
.
0
)
0
)
)
)
)
0
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
=
setsum
(
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
x7
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
setsum
(
x1
(
λ x12 :
ι → ι
.
x10
(
λ x13 .
0
)
)
(
setsum
0
0
)
)
(
x1
(
λ x12 :
ι → ι
.
x3
(
λ x13 :
ι →
ι → ι
.
λ x14 .
0
)
0
)
0
)
)
)
(
setsum
(
x3
(
λ x9 :
ι →
ι → ι
.
λ x10 .
x1
(
λ x11 :
ι → ι
.
setsum
0
0
)
0
)
(
x2
(
λ x9 .
0
)
(
λ x9 :
ι →
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
x10
)
0
(
λ x9 x10 .
x3
(
λ x11 :
ι →
ι → ι
.
λ x12 .
0
)
0
)
(
λ x9 .
setsum
0
0
)
(
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
)
)
)
(
Inj0
(
x2
(
λ x9 .
0
)
(
λ x9 :
ι →
ι → ι
.
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
setsum
0
0
)
0
(
λ x9 x10 .
x0
(
λ x11 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 .
0
)
)
(
λ x9 .
0
)
(
Inj1
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι →
ι → ι
.
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
x0
(
λ x12 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x12 .
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 .
0
)
)
=
x0
(
λ x9 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
Inj0
(
x7
(
x2
(
λ x10 .
0
)
(
λ x10 :
ι →
ι → ι
.
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
x13
)
0
(
λ x10 x11 .
x2
(
λ x12 .
0
)
(
λ x12 :
ι →
ι → ι
.
λ x13 .
λ x14 :
ι → ι
.
λ x15 .
0
)
0
(
λ x12 x13 .
0
)
(
λ x12 .
0
)
0
)
(
λ x10 .
x0
(
λ x11 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 .
0
)
)
0
)
(
Inj0
0
)
)
)
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
Inj1
(
x10
(
λ x12 .
0
)
)
)
)
⟶
False
(proof)
Theorem
b241d..
:
∀ x0 :
(
ι → ι
)
→
(
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x1 :
(
ι → ι
)
→
(
(
ι →
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x2 :
(
(
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x3 :
(
(
ι → ι
)
→ ι
)
→
(
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
(
∀ x4 :
ι → ι
.
∀ x5 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
x3
(
λ x9 :
ι → ι
.
x7
(
x7
(
x5
(
λ x10 .
x1
(
λ x11 .
0
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
(
x6
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
0
0
0
)
(
λ x10 .
x1
(
λ x11 .
0
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
)
(
x6
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
setsum
0
0
)
0
(
x6
(
λ x10 .
λ x11 :
ι → ι
.
λ x12 .
0
)
0
0
0
)
(
setsum
0
0
)
)
)
(
setsum
(
setsum
(
x5
(
λ x10 .
0
)
0
(
λ x10 .
0
)
)
0
)
0
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
0
)
=
x7
(
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
setsum
(
Inj0
(
x7
0
0
)
)
(
Inj1
(
x7
0
0
)
)
)
(
λ x9 .
0
)
)
(
x3
(
λ x9 :
ι → ι
.
x3
(
λ x10 :
ι → ι
.
x2
(
λ x11 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x2
(
λ x12 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x12 .
0
)
)
(
λ x11 .
x11
)
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x11
(
x2
(
λ x14 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x14 .
0
)
)
(
Inj1
0
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 x7 .
x3
(
λ x9 :
ι → ι
.
x9
(
Inj1
(
x0
(
λ x10 .
x1
(
λ x11 .
0
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
Inj1
0
)
(
x3
(
λ x10 :
ι → ι
.
0
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
0
)
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x10 :
ι →
ι → ι
.
λ x11 x12 .
x1
(
λ x13 .
x12
)
(
λ x13 :
ι →
(
ι → ι
)
→ ι
.
x3
(
λ x14 :
ι → ι
.
x14
(
x1
(
λ x15 .
0
)
(
λ x15 :
ι →
(
ι → ι
)
→ ι
.
0
)
)
)
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι → ι
.
λ x16 x17 .
x15
0
0
)
)
)
=
Inj1
(
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x7
)
(
λ x9 .
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 x7 .
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x6
)
(
λ x9 .
setsum
0
x6
)
=
Inj0
0
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x2
(
λ x9 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x1
(
λ x10 .
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
x9
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 .
setsum
x12
(
x3
(
λ x13 :
ι → ι
.
0
)
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 :
ι →
ι → ι
.
λ x15 x16 .
0
)
)
)
(
λ x11 .
0
)
(
λ x11 .
0
)
(
x7
(
λ x11 :
ι →
ι → ι
.
x0
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
0
)
0
)
)
)
)
(
λ x9 .
0
)
=
setsum
(
x6
(
λ x9 .
0
)
)
x4
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
x1
(
λ x9 .
Inj1
x5
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
x7
)
=
Inj0
x6
)
⟶
(
∀ x4 :
ι →
(
ι → ι
)
→ ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x6 .
∀ x7 :
ι →
ι →
(
ι → ι
)
→
ι → ι
.
x1
Inj0
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
setsum
(
x7
0
(
x5
0
(
λ x10 :
ι → ι
.
Inj0
0
)
(
Inj0
0
)
)
(
λ x10 .
x9
(
x7
0
0
(
λ x11 .
0
)
0
)
(
λ x11 .
x9
0
(
λ x12 .
0
)
)
)
(
x3
(
λ x10 :
ι → ι
.
setsum
0
0
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 x13 .
x0
(
λ x14 .
0
)
(
λ x14 :
ι → ι
.
λ x15 :
(
ι → ι
)
→
ι → ι
.
λ x16 :
ι → ι
.
0
)
0
)
)
)
0
)
=
x6
)
⟶
(
∀ x4 :
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 .
∀ x7 :
ι → ι
.
x0
(
λ x9 .
setsum
0
x6
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
x10
(
λ x12 .
setsum
0
0
)
(
x0
(
λ x12 .
0
)
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
Inj0
(
Inj1
0
)
)
(
Inj1
0
)
)
)
0
=
setsum
(
setsum
0
0
)
(
Inj0
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x6 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
x0
(
λ x9 .
x7
)
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
λ x11 :
ι → ι
.
Inj0
(
x0
(
λ x12 .
Inj1
(
x9
0
)
)
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
ι → ι
.
λ x14 :
ι → ι
.
x0
(
λ x15 .
x14
0
)
(
λ x15 :
ι → ι
.
λ x16 :
(
ι → ι
)
→
ι → ι
.
λ x17 :
ι → ι
.
x17
0
)
(
setsum
0
0
)
)
(
setsum
(
Inj1
0
)
(
x10
(
λ x12 .
0
)
0
)
)
)
)
(
setsum
(
x6
(
λ x9 .
λ x10 :
ι → ι
.
x9
)
)
0
)
=
Inj1
(
Inj1
(
x5
x4
(
λ x9 .
0
)
(
λ x9 .
x0
(
λ x10 .
0
)
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
x12
0
)
0
)
)
)
)
⟶
False
(proof)
Theorem
47e4a..
:
∀ x0 :
(
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
(
ι →
ι → ι
)
→
ι → ι
)
→
(
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x2 :
(
ι → ι
)
→
(
ι →
ι →
ι → ι
)
→
(
ι →
ι → ι
)
→
ι →
ι → ι
.
∀ x3 :
(
(
ι → ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
(
∀ x4 :
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
x7
(
λ x12 :
ι → ι
.
λ x13 x14 .
Inj1
0
)
)
(
Inj0
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
x2
(
λ x12 .
x11
0
0
)
(
λ x12 x13 x14 .
0
)
(
λ x12 x13 .
x10
(
λ x14 :
ι → ι
.
0
)
)
(
x10
(
λ x12 :
ι → ι
.
0
)
)
(
Inj1
0
)
)
(
x4
0
(
λ x9 .
x1
(
λ x10 :
ι → ι
.
λ x11 :
ι →
ι → ι
.
λ x12 .
0
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
)
(
x5
(
λ x9 .
0
)
)
)
)
)
=
x7
(
λ x9 :
ι → ι
.
λ x10 x11 .
Inj1
0
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι →
ι → ι
.
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
0
)
(
setsum
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
x11
(
setsum
0
0
)
(
x7
0
0
)
)
x5
)
(
x7
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
0
)
x5
)
(
Inj1
(
x7
0
0
)
)
)
)
=
x6
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x5 x6 x7 .
x2
(
λ x9 .
0
)
(
λ x9 x10 x11 .
x7
)
(
λ x9 x10 .
setsum
(
Inj0
x10
)
(
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 :
ι →
ι → ι
.
x13
0
0
)
(
x2
(
λ x11 .
x0
(
λ x12 .
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι →
ι → ι
.
λ x15 :
ι → ι
.
λ x16 .
0
)
0
0
0
)
(
λ x11 x12 x13 .
0
)
(
λ x11 x12 .
x3
(
λ x13 :
ι → ι
.
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι → ι
.
0
)
0
)
(
setsum
0
0
)
(
setsum
0
0
)
)
)
)
0
x5
=
setsum
x6
x7
)
⟶
(
∀ x4 x5 x6 x7 .
x2
(
λ x9 .
x7
)
(
λ x9 x10 x11 .
x2
(
λ x12 .
x0
(
λ x13 .
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 :
ι →
ι → ι
.
λ x16 :
ι → ι
.
λ x17 .
0
)
(
Inj1
x11
)
x12
(
Inj1
x11
)
)
(
λ x12 x13 x14 .
Inj1
x12
)
(
λ x12 x13 .
x3
(
λ x14 :
ι → ι
.
λ x15 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x16 :
ι →
ι → ι
.
x16
0
(
Inj0
0
)
)
x10
)
x9
(
setsum
0
(
Inj0
x11
)
)
)
(
λ x9 x10 .
Inj1
(
setsum
(
x0
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
λ x15 .
setsum
0
0
)
x6
(
setsum
0
0
)
(
Inj0
0
)
)
(
x0
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
λ x15 .
setsum
0
0
)
0
(
Inj0
0
)
0
)
)
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
x9
(
x2
(
λ x12 .
x2
(
λ x13 .
0
)
(
λ x13 x14 x15 .
0
)
(
λ x13 x14 .
0
)
0
0
)
(
λ x12 x13 x14 .
0
)
(
λ x12 x13 .
x12
)
(
setsum
0
0
)
(
x11
0
0
)
)
)
(
setsum
0
x4
)
)
(
Inj1
(
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
Inj0
(
Inj0
0
)
)
0
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
x0
(
λ x12 .
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι →
ι → ι
.
λ x15 :
ι → ι
.
λ x16 .
0
)
0
0
0
)
x5
)
(
Inj0
(
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
0
0
0
)
)
)
)
=
Inj1
x4
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 x7 .
x1
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 .
setsum
(
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 :
ι →
ι → ι
.
x3
(
λ x15 :
ι → ι
.
λ x16 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x17 :
ι →
ι → ι
.
0
)
(
x0
(
λ x15 .
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 :
ι →
ι → ι
.
λ x18 :
ι → ι
.
λ x19 .
0
)
0
0
0
)
)
(
Inj1
(
x9
0
)
)
)
0
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x5
(
λ x10 .
λ x11 :
ι → ι
.
0
)
)
=
x5
(
λ x9 .
λ x10 :
ι → ι
.
x0
(
λ x11 .
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
λ x15 .
setsum
(
setsum
(
setsum
0
0
)
(
x12
(
λ x16 .
0
)
)
)
0
)
0
0
(
x10
0
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x5 :
ι →
ι →
ι →
ι → ι
.
∀ x6 x7 .
x1
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 .
Inj0
(
x1
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 .
x13
0
x11
)
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x10
(
x3
(
λ x13 :
ι → ι
.
λ x14 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x15 :
ι →
ι → ι
.
0
)
0
)
(
setsum
0
0
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
setsum
0
(
Inj1
x7
)
)
=
setsum
(
x2
(
λ x9 .
0
)
(
λ x9 x10 x11 .
0
)
(
λ x9 x10 .
Inj1
x7
)
x6
(
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x2
(
λ x14 .
x13
)
(
λ x14 x15 x16 .
Inj1
0
)
(
λ x14 x15 .
Inj1
0
)
0
0
)
(
x1
(
λ x9 :
ι → ι
.
λ x10 :
ι →
ι → ι
.
λ x11 .
0
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x2
(
λ x10 .
0
)
(
λ x10 x11 x12 .
0
)
(
λ x10 x11 .
0
)
0
0
)
)
(
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x13
)
x6
(
Inj1
0
)
(
Inj1
0
)
)
(
Inj1
x6
)
)
)
0
)
⟶
(
∀ x4 .
∀ x5 :
(
ι →
ι → ι
)
→ ι
.
∀ x6 x7 .
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
Inj1
0
)
(
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
x7
x6
(
Inj1
(
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
(
x2
(
λ x9 .
0
)
(
λ x9 x10 x11 .
0
)
(
λ x9 x10 .
0
)
0
0
)
(
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
0
0
0
)
(
x2
(
λ x9 .
0
)
(
λ x9 x10 x11 .
0
)
(
λ x9 x10 .
0
)
0
0
)
)
)
)
x4
0
=
x4
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x6 :
ι →
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
∀ x7 .
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x13
)
0
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
0
)
0
)
0
=
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
Inj0
0
)
(
setsum
(
Inj0
0
)
(
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
x12
(
Inj0
0
)
)
(
x2
(
λ x9 .
0
)
(
λ x9 x10 x11 .
setsum
0
0
)
(
λ x9 x10 .
Inj1
0
)
0
(
x4
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
0
)
)
)
(
x4
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
x1
(
λ x11 :
ι → ι
.
λ x12 :
ι →
ι → ι
.
λ x13 .
0
)
(
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
)
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
x1
(
λ x12 :
ι → ι
.
λ x13 :
ι →
ι → ι
.
λ x14 .
0
)
(
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
)
(
x0
(
λ x9 .
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
0
0
0
)
)
)
)
)
⟶
False
(proof)
Theorem
ba301..
:
∀ x0 :
(
ι →
ι →
(
ι →
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x1 :
(
(
ι → ι
)
→ ι
)
→
ι →
(
ι →
(
ι → ι
)
→ ι
)
→ ι
.
∀ x2 :
(
ι → ι
)
→
(
ι →
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
(
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
(
∀ x4 .
∀ x5 :
ι →
ι →
ι → ι
.
∀ x6 x7 .
x3
(
λ x9 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
0
)
(
x1
(
λ x9 :
ι → ι
.
0
)
(
Inj0
(
x3
(
λ x9 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
x1
(
λ x10 :
ι → ι
.
0
)
0
(
λ x10 .
λ x11 :
ι → ι
.
0
)
)
0
)
)
(
λ x9 .
λ x10 :
ι → ι
.
Inj0
0
)
)
=
setsum
(
Inj1
(
Inj1
0
)
)
(
x5
x6
(
Inj1
(
setsum
(
x1
(
λ x9 :
ι → ι
.
0
)
0
(
λ x9 .
λ x10 :
ι → ι
.
0
)
)
(
setsum
0
0
)
)
)
(
x3
(
λ x9 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
setsum
x7
(
x2
(
λ x10 .
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
0
)
)
)
(
Inj0
(
x0
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
0
)
(
λ x9 x10 .
0
)
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x3
(
λ x9 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
x2
(
λ x10 .
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
setsum
(
x11
0
0
)
(
setsum
(
x11
0
0
)
(
x9
(
λ x12 x13 .
0
)
0
0
0
)
)
)
)
x7
=
Inj0
(
Inj1
0
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x5 x6 x7 .
x2
(
λ x9 .
x9
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
0
)
=
x7
)
⟶
(
∀ x4 :
(
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 :
(
ι →
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 .
setsum
(
setsum
(
x0
(
λ x10 x11 .
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
x12
0
0
)
(
λ x10 x11 .
0
)
)
0
)
(
x2
(
λ x10 .
x7
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
Inj0
x10
)
)
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x0
(
λ x11 x12 .
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
x14
(
Inj0
(
x1
(
λ x15 :
ι → ι
.
0
)
0
(
λ x15 .
λ x16 :
ι → ι
.
0
)
)
)
)
(
λ x11 x12 .
x9
)
)
=
x0
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
x3
(
λ x13 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
x2
(
λ x14 .
setsum
(
x0
(
λ x15 x16 .
λ x17 :
ι →
ι → ι
.
λ x18 :
ι → ι
.
0
)
(
λ x15 x16 .
0
)
)
(
x0
(
λ x15 x16 .
λ x17 :
ι →
ι → ι
.
λ x18 :
ι → ι
.
0
)
(
λ x15 x16 .
0
)
)
)
(
λ x14 .
λ x15 :
ι →
ι → ι
.
x14
)
)
(
x12
x10
)
)
(
λ x9 x10 .
setsum
(
setsum
(
x0
(
λ x11 x12 .
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
x11
)
(
λ x11 x12 .
setsum
0
0
)
)
(
Inj1
(
setsum
0
0
)
)
)
(
x3
(
λ x11 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
0
)
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
x1
(
λ x9 :
ι → ι
.
0
)
0
(
λ x9 .
λ x10 :
ι → ι
.
x6
(
x10
0
)
)
=
x6
(
x6
(
x0
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
setsum
(
setsum
0
0
)
(
Inj1
0
)
)
(
λ x9 x10 .
x7
(
λ x11 .
λ x12 :
ι → ι
.
λ x13 .
Inj1
0
)
(
setsum
0
0
)
(
λ x11 .
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
(
ι →
ι → ι
)
→
ι → ι
.
x1
(
λ x9 :
ι → ι
.
0
)
0
(
λ x9 .
λ x10 :
ι → ι
.
Inj1
(
x0
(
λ x11 x12 .
λ x13 :
ι →
ι → ι
.
λ x14 :
ι → ι
.
x14
(
x0
(
λ x15 x16 .
λ x17 :
ι →
ι → ι
.
λ x18 :
ι → ι
.
0
)
(
λ x15 x16 .
0
)
)
)
(
λ x11 x12 .
Inj0
(
x3
(
λ x13 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
0
)
0
)
)
)
)
=
setsum
0
(
Inj1
(
x5
0
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 .
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
x0
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
setsum
0
(
Inj1
(
Inj1
(
x11
0
0
)
)
)
)
(
λ x9 x10 .
0
)
=
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
ι →
ι →
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x7 :
ι →
(
ι →
ι → ι
)
→
ι → ι
.
x0
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
x12
(
Inj1
(
x11
x10
(
x12
0
)
)
)
)
(
λ x9 x10 .
0
)
=
setsum
(
x4
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
x3
(
λ x12 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
x0
(
λ x13 x14 .
λ x15 :
ι →
ι → ι
.
λ x16 :
ι → ι
.
0
)
(
λ x13 x14 .
Inj0
0
)
)
0
)
)
(
setsum
(
x3
(
λ x9 :
(
ι →
ι → ι
)
→
ι →
ι →
ι → ι
.
x2
(
λ x10 .
x9
(
λ x11 x12 .
0
)
0
0
0
)
(
λ x10 .
λ x11 :
ι →
ι → ι
.
x2
(
λ x12 .
0
)
(
λ x12 .
λ x13 :
ι →
ι → ι
.
0
)
)
)
(
x2
(
λ x9 .
x0
(
λ x10 x11 .
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
0
)
(
λ x10 x11 .
0
)
)
(
λ x9 .
λ x10 :
ι →
ι → ι
.
x7
0
(
λ x11 x12 .
0
)
0
)
)
)
(
Inj0
(
x0
(
λ x9 x10 .
λ x11 :
ι →
ι → ι
.
λ x12 :
ι → ι
.
x12
0
)
(
λ x9 x10 .
0
)
)
)
)
)
⟶
False
(proof)
Theorem
27c51..
:
∀ x0 :
(
ι → ι
)
→
ι →
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ 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
.
∀ x2 :
(
(
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x3 :
(
(
ι → ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
(
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 :
(
ι → ι
)
→ ι
.
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
(
Inj1
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x0
(
λ x12 .
x12
)
(
setsum
0
0
)
(
Inj0
0
)
(
λ x12 :
ι → ι
.
λ x13 .
setsum
0
0
)
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x0
(
λ x11 .
0
)
0
0
(
λ x11 :
ι → ι
.
λ x12 .
0
)
)
(
x6
0
)
(
setsum
0
0
)
)
)
)
(
x6
0
)
=
x6
(
Inj1
0
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
ι →
ι → ι
.
∀ x6 x7 .
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
setsum
0
(
Inj0
0
)
)
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
Inj1
(
setsum
0
0
)
)
(
setsum
(
Inj0
0
)
x4
)
0
)
)
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
=
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
setsum
(
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x12
(
x13
(
λ x14 :
ι → ι
.
λ x15 .
0
)
0
)
)
(
setsum
(
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
x7
)
x7
)
(
Inj1
(
x1
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 x15 x16 .
x15
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
0
)
)
)
)
(
setsum
(
x5
0
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
Inj1
0
)
x4
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
(
setsum
(
Inj0
0
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
)
0
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 x15 x16 .
setsum
0
(
setsum
0
0
)
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
0
)
)
(
Inj0
(
x1
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
setsum
(
setsum
0
0
)
(
x3
(
λ x14 :
ι → ι
.
λ x15 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
0
)
)
)
=
x1
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
setsum
(
Inj1
(
setsum
0
(
Inj1
0
)
)
)
(
Inj0
(
Inj0
0
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
Inj0
(
x10
(
λ x12 .
setsum
(
x1
(
λ x13 :
ι → ι
.
λ x14 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x15 x16 x17 .
0
)
(
λ x13 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x14 :
(
ι → ι
)
→ ι
.
λ x15 :
ι → ι
.
0
)
)
(
x3
(
λ x13 :
ι → ι
.
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
)
)
)
⟶
(
∀ x4 :
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
)
→
ι → ι
.
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x9
(
λ x12 :
ι →
ι → ι
.
λ x13 :
ι → ι
.
λ x14 .
setsum
(
x1
(
λ x15 :
ι → ι
.
λ x16 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x17 x18 x19 .
0
)
(
λ x15 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x16 :
(
ι → ι
)
→ ι
.
λ x17 :
ι → ι
.
Inj0
0
)
)
(
Inj1
(
setsum
0
0
)
)
)
(
λ x12 :
ι → ι
.
λ x13 .
0
)
)
(
x5
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x10
(
x0
(
λ x11 .
0
)
(
x7
(
λ x11 :
(
ι → ι
)
→
ι → ι
.
λ x12 :
ι → ι
.
λ x13 .
0
)
0
)
(
x1
(
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x13 x14 x15 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
0
)
)
(
λ x11 :
ι → ι
.
λ x12 .
setsum
0
0
)
)
)
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
(
x4
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
(
λ x9 :
ι → ι
.
λ x10 .
setsum
0
0
)
)
)
)
=
x5
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
Inj0
(
setsum
(
setsum
(
setsum
0
0
)
0
)
0
)
)
(
setsum
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
(
setsum
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
0
)
)
0
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
x1
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
x11
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
0
)
=
x5
(
Inj0
(
x5
(
x5
0
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 x13 x14 .
0
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
0
)
)
)
(
λ x9 :
ι → ι
.
x6
(
λ x10 .
x2
(
λ x11 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
)
)
)
(
λ x9 :
ι → ι
.
setsum
(
setsum
(
x0
(
λ x10 .
x2
(
λ x11 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
0
0
(
λ x10 :
ι → ι
.
λ x11 .
x10
0
)
)
(
x2
(
λ x10 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x11 .
λ x12 :
(
ι → ι
)
→
ι → ι
.
x12
(
λ x13 .
0
)
0
)
(
Inj1
0
)
)
)
(
x1
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 x13 x14 .
x3
(
λ x15 :
ι → ι
.
λ x16 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x13
)
(
x2
(
λ x15 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x16 .
λ x17 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
(
setsum
0
0
)
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
x0
(
λ x13 .
x0
(
λ x14 .
0
)
0
0
(
λ x14 :
ι → ι
.
λ x15 .
0
)
)
0
(
x12
0
)
(
λ x13 :
ι → ι
.
λ x14 .
x2
(
λ x15 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x16 .
λ x17 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
ι →
ι →
ι → ι
.
∀ x6 .
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
.
x1
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
0
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
x9
(
λ x12 :
ι → ι
.
Inj0
(
x10
(
λ x13 .
0
)
)
)
0
(
x1
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 x15 x16 .
0
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
x14
(
x13
(
λ x15 .
0
)
)
)
)
)
=
setsum
(
setsum
(
x1
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
x1
(
λ x14 :
ι → ι
.
λ x15 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x16 x17 x18 .
setsum
0
0
)
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 :
ι → ι
.
0
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
x0
(
λ x12 .
x3
(
λ x13 :
ι → ι
.
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
(
x7
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
0
)
(
x7
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
0
)
(
λ x12 :
ι → ι
.
λ x13 .
x10
(
λ x14 .
0
)
)
)
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
Inj0
(
x1
(
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x13 x14 x15 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
0
)
)
)
(
Inj0
(
Inj0
0
)
)
(
setsum
x6
0
)
)
)
(
setsum
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x7
(
λ x11 :
(
ι → ι
)
→ ι
.
x10
(
λ x12 :
ι → ι
.
λ x13 .
0
)
0
)
(
setsum
0
0
)
)
(
setsum
(
Inj0
0
)
(
setsum
0
0
)
)
x6
)
(
Inj1
(
x5
(
x0
(
λ x9 .
0
)
0
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
)
0
(
x5
0
0
0
0
)
x6
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
∀ x7 .
x0
(
λ x9 .
x9
)
(
Inj0
0
)
x7
(
λ x9 :
ι → ι
.
λ x10 .
x1
(
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x13 x14 x15 .
Inj0
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
x10
)
)
=
Inj0
(
x0
(
λ x9 .
0
)
(
x0
(
λ x9 .
Inj1
(
x3
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
0
(
x0
(
λ x9 .
x9
)
(
Inj1
0
)
0
(
λ x9 :
ι → ι
.
λ x10 .
x3
(
λ x11 :
ι → ι
.
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x2
(
λ x11 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
0
)
)
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x10
(
λ x11 :
ι → ι
.
λ x12 .
0
)
(
x2
(
λ x11 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x12 .
λ x13 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
)
0
(
setsum
(
x3
(
λ x9 :
ι → ι
.
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
(
Inj0
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x1
(
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x13 x14 x15 .
setsum
x14
(
x3
(
λ x16 :
ι → ι
.
λ x17 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
setsum
(
x1
(
λ x14 :
ι → ι
.
λ x15 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x16 x17 x18 .
0
)
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x15 :
(
ι → ι
)
→ ι
.
λ x16 :
ι → ι
.
0
)
)
(
x2
(
λ x14 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι →
ι → ι
)
→ ι
.
∀ x6 x7 .
x0
(
λ x9 .
0
)
(
x0
(
λ x9 .
x3
(
λ x10 :
ι → ι
.
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
x7
)
x7
(
x0
(
λ x9 .
0
)
0
0
(
λ x9 :
ι → ι
.
λ x10 .
Inj1
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
Inj1
(
Inj1
0
)
)
)
(
setsum
(
x1
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
setsum
x11
(
Inj1
0
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
x11
0
)
(
x2
(
λ x12 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x13 .
λ x14 :
(
ι → ι
)
→
ι → ι
.
0
)
0
)
0
)
)
(
setsum
(
Inj0
(
Inj1
0
)
)
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x11
(
λ x12 .
0
)
0
)
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x1
(
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x13 x14 x15 .
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
x2
(
λ x14 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x15 .
λ x16 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x17 :
ι → ι
.
λ x18 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x19 x20 x21 .
Inj0
0
)
(
λ x17 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x18 :
(
ι → ι
)
→ ι
.
λ x19 :
ι → ι
.
x19
0
)
)
0
)
)
=
setsum
(
x0
(
λ x9 .
Inj0
(
x0
(
λ x10 .
0
)
(
x1
(
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x12 x13 x14 .
0
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 :
ι → ι
.
0
)
)
0
(
λ x10 :
ι → ι
.
λ x11 .
x3
(
λ x12 :
ι → ι
.
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
)
(
x1
(
λ x9 :
ι → ι
.
λ x10 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x11 x12 x13 .
0
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
x10
(
λ x12 .
0
)
)
)
x6
(
λ x9 :
ι → ι
.
λ x10 .
x9
0
)
)
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
λ x10 .
λ x11 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x12 :
ι → ι
.
λ x13 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x14 x15 x16 .
x1
(
λ x17 :
ι → ι
.
λ x18 :
(
ι → ι
)
→
(
ι → ι
)
→ ι
.
λ x19 x20 x21 .
setsum
0
0
)
(
λ x17 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x18 :
(
ι → ι
)
→ ι
.
λ x19 :
ι → ι
.
x3
(
λ x20 :
ι → ι
.
λ x21 :
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
0
)
0
0
)
)
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
λ x13 :
(
ι → ι
)
→ ι
.
λ x14 :
ι → ι
.
0
)
)
x4
)
)
⟶
False
(proof)
Theorem
b633b..
:
∀ x0 :
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι →
ι →
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x2 :
(
(
(
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
)
→
(
ι →
ι →
ι →
ι → ι
)
→
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x3 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι →
ι →
ι → ι
)
→ ι
.
(
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 x7 .
x3
(
λ x9 .
0
)
(
λ x9 :
ι → ι
.
λ x10 x11 x12 .
x1
(
λ x13 :
ι →
ι →
ι → ι
.
λ x14 x15 :
ι → ι
.
setsum
0
0
)
(
λ x13 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x14 x15 .
x15
)
)
=
x1
(
λ x9 :
ι →
ι →
ι → ι
.
λ x10 x11 :
ι → ι
.
x9
(
x11
(
Inj1
(
Inj1
0
)
)
)
(
x10
(
x0
(
λ x12 .
0
)
(
setsum
0
0
)
)
)
0
)
(
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 x11 .
Inj1
0
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x3
Inj0
(
λ x9 :
ι → ι
.
λ x10 x11 x12 .
x12
)
=
setsum
0
(
x4
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
setsum
(
setsum
(
x1
(
λ x12 :
ι →
ι →
ι → ι
.
λ x13 x14 :
ι → ι
.
0
)
(
λ x12 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x13 x14 .
0
)
)
x11
)
(
x10
(
x2
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
λ x13 :
ι → ι
.
λ x14 :
(
ι → ι
)
→ ι
.
0
)
(
λ x12 x13 x14 x15 .
0
)
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
)
(
λ x9 :
ι → ι
.
x5
0
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 :
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
Inj0
(
setsum
0
0
)
)
(
λ x9 x10 x11 x12 .
Inj1
0
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
x1
(
λ x10 :
ι →
ι →
ι → ι
.
λ x11 x12 :
ι → ι
.
Inj0
(
Inj0
(
x12
0
)
)
)
(
λ x10 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x11 x12 .
Inj1
(
x0
(
λ x13 .
0
)
(
x1
(
λ x13 :
ι →
ι →
ι → ι
.
λ x14 x15 :
ι → ι
.
0
)
(
λ x13 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x14 x15 .
0
)
)
)
)
)
=
Inj0
0
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
∀ x6 x7 .
x2
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
λ x10 :
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
x7
)
(
λ x9 x10 x11 x12 .
x12
)
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
0
)
=
x7
)
⟶
(
∀ x4 .
∀ x5 :
(
(
ι →
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
(
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 :
(
ι → ι
)
→ ι
.
x1
(
λ x9 :
ι →
ι →
ι → ι
.
λ x10 x11 :
ι → ι
.
x11
0
)
(
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 x11 .
x11
)
=
x5
(
λ x9 :
ι →
ι → ι
.
λ x10 :
ι → ι
.
λ x11 .
x10
(
x10
(
x0
(
λ x12 .
x0
(
λ x13 .
0
)
0
)
0
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 :
ι →
ι →
ι → ι
.
λ x10 x11 :
ι → ι
.
0
)
(
λ x9 :
ι →
(
ι → ι
)
→
ι → ι
.
λ x10 x11 .
setsum
x11
(
x9
x11
(
λ x12 .
setsum
(
setsum
0
0
)
0
)
(
x0
(
λ x12 .
0
)
x10
)
)
)
=
setsum
0
(
x3
(
λ x9 .
0
)
(
λ x9 :
ι → ι
.
λ x10 x11 x12 .
Inj0
(
x9
x11
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι →
ι →
ι → ι
.
∀ x7 :
ι →
(
ι →
ι → ι
)
→
ι →
ι → ι
.
x0
(
λ x9 .
0
)
0
=
x4
)
⟶
(
∀ x4 .
∀ x5 :
ι → ι
.
∀ x6 :
ι →
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x7 :
ι → ι
.
x0
(
λ x9 .
x5
(
x0
(
λ x10 .
x3
(
λ x11 .
Inj0
0
)
(
λ x11 :
ι → ι
.
λ x12 x13 x14 .
x11
0
)
)
(
x0
(
λ x10 .
x9
)
(
x0
(
λ x10 .
0
)
0
)
)
)
)
0
=
Inj1
(
x3
(
λ x9 .
x2
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
λ x11 :
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
0
)
(
λ x10 x11 x12 x13 .
x13
)
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
Inj1
(
Inj0
0
)
)
)
(
λ x9 :
ι → ι
.
λ x10 x11 x12 .
x9
(
x0
(
λ x13 .
Inj1
0
)
(
Inj1
0
)
)
)
)
)
⟶
False
(proof)
Theorem
c0927..
:
∀ x0 :
(
(
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x1 :
(
(
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x2 :
(
ι →
ι → ι
)
→
ι → ι
.
∀ x3 :
(
(
ι →
ι →
ι → ι
)
→ ι
)
→
(
(
ι →
(
ι → ι
)
→ ι
)
→ ι
)
→
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 :
ι →
ι →
ι → ι
.
setsum
(
x9
x7
(
x2
(
λ x10 x11 .
0
)
(
Inj1
0
)
)
0
)
0
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
x7
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x0
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
x7
)
(
Inj0
(
x2
(
λ x10 x11 .
x0
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
(
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 .
0
)
0
0
)
)
)
)
=
Inj1
(
x3
(
λ x9 :
ι →
ι →
ι → ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 .
x11
(
λ x13 :
ι → ι
.
x3
(
λ x14 :
ι →
ι →
ι → ι
.
0
)
(
λ x14 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x14 :
(
ι → ι
)
→ ι
.
0
)
)
)
(
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 .
setsum
0
0
)
(
x3
(
λ x10 :
ι →
ι →
ι → ι
.
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
)
0
)
(
Inj0
0
)
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x9
(
λ x10 .
x10
)
)
)
)
⟶
(
∀ x4 :
ι →
ι →
ι →
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 .
x3
(
λ x9 :
ι →
ι →
ι → ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 .
x3
(
λ x13 :
ι →
ι →
ι → ι
.
x11
(
λ x14 :
ι → ι
.
x11
(
λ x15 :
ι → ι
.
0
)
)
)
(
λ x13 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
x3
(
λ x14 :
ι →
ι →
ι → ι
.
x14
0
0
0
)
(
λ x14 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x14 :
(
ι → ι
)
→ ι
.
Inj0
0
)
)
)
0
x7
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
setsum
(
Inj1
(
setsum
(
x2
(
λ x10 x11 .
0
)
0
)
(
x0
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
)
)
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
=
Inj0
(
x4
0
(
setsum
(
x4
(
x5
0
)
x7
(
x0
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
(
x3
(
λ x9 :
ι →
ι →
ι → ι
.
0
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
)
(
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
0
)
0
(
setsum
0
0
)
)
)
0
(
Inj1
0
)
)
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x5 x6 x7 .
x2
(
λ x9 x10 .
x9
)
(
Inj0
(
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
x2
(
λ x12 x13 .
0
)
(
setsum
0
0
)
)
(
setsum
x7
x5
)
x7
)
)
=
x4
(
λ x9 :
ι → ι
.
x5
)
(
Inj1
(
x4
(
λ x9 :
ι → ι
.
setsum
x5
0
)
(
Inj0
(
x3
(
λ x9 :
ι →
ι →
ι → ι
.
0
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
)
)
(
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
0
)
(
setsum
0
0
)
(
x2
(
λ x9 x10 .
0
)
0
)
)
)
)
(
x3
(
λ x9 :
ι →
ι →
ι → ι
.
Inj0
(
x9
0
(
setsum
0
0
)
(
x9
0
0
0
)
)
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
Inj1
(
x9
(
x2
(
λ x10 x11 .
0
)
0
)
(
λ x10 .
0
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
setsum
(
Inj1
0
)
)
0
(
setsum
0
x7
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x6 :
(
ι → ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 x10 .
x2
(
λ x11 x12 .
0
)
0
)
(
x3
(
λ x9 :
ι →
ι →
ι → ι
.
x1
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x11 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x12 .
x9
(
x11
(
λ x13 :
ι → ι
.
0
)
)
(
x3
(
λ x13 :
ι →
ι →
ι → ι
.
0
)
(
λ x13 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x13 :
(
ι → ι
)
→ ι
.
0
)
)
0
)
(
x5
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x10 x11 .
x9
0
0
0
)
)
(
Inj0
(
x6
(
λ x10 .
0
)
)
)
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
x3
(
λ x10 :
ι →
ι →
ι → ι
.
x3
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
x0
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
x0
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x9
0
(
λ x11 .
x9
0
(
λ x12 .
0
)
)
)
)
(
λ x9 :
(
ι → ι
)
→ ι
.
setsum
(
x0
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
setsum
0
0
)
(
x0
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
)
0
)
)
=
Inj0
(
Inj1
0
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
x3
(
λ x12 :
ι →
ι →
ι → ι
.
x12
0
x11
(
setsum
0
(
x10
(
λ x13 :
ι → ι
.
0
)
)
)
)
(
λ x12 :
ι →
(
ι → ι
)
→ ι
.
setsum
(
Inj0
(
Inj0
0
)
)
(
setsum
0
x11
)
)
(
λ x12 :
(
ι → ι
)
→ ι
.
x10
(
λ x13 :
ι → ι
.
Inj0
0
)
)
)
0
0
=
x3
(
λ x9 :
ι →
ι →
ι → ι
.
setsum
(
x3
(
λ x10 :
ι →
ι →
ι → ι
.
x0
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
x0
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
Inj1
(
x2
(
λ x11 x12 .
0
)
0
)
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
)
(
Inj0
(
x9
(
setsum
0
0
)
(
x0
(
λ x10 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
(
setsum
0
0
)
)
)
)
(
λ x9 :
ι →
(
ι → ι
)
→ ι
.
x2
(
λ x10 x11 .
setsum
(
x3
(
λ x12 :
ι →
ι →
ι → ι
.
0
)
(
λ x12 :
ι →
(
ι → ι
)
→ ι
.
Inj0
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
x11
)
)
(
x1
(
λ x12 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x13 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x14 .
Inj1
0
)
(
Inj0
0
)
(
x0
(
λ x12 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
)
)
0
)
(
λ x9 :
(
ι → ι
)
→ ι
.
x3
(
λ x10 :
ι →
ι →
ι → ι
.
Inj0
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x10 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x11 .
0
)
0
0
=
Inj0
0
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι →
ι → ι
.
∀ x7 :
ι → ι
.
x0
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
x3
(
λ x10 :
ι →
ι →
ι → ι
.
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
x10
(
x7
(
x10
0
(
λ x11 .
0
)
)
)
(
λ x11 .
x2
(
λ x12 x13 .
Inj0
0
)
(
x3
(
λ x12 :
ι →
ι →
ι → ι
.
0
)
(
λ x12 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x12 :
(
ι → ι
)
→ ι
.
0
)
)
)
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x6
(
x6
(
setsum
0
0
)
(
Inj0
0
)
)
(
setsum
(
Inj0
0
)
(
Inj0
0
)
)
)
)
0
=
Inj0
(
Inj1
(
Inj0
(
x2
(
λ x9 x10 .
x9
)
0
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x9 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
Inj1
(
x5
(
x3
(
λ x10 :
ι →
ι →
ι → ι
.
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
x1
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
λ x12 :
(
(
ι → ι
)
→ ι
)
→ ι
.
λ x13 .
0
)
0
0
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x10
(
λ x11 .
0
)
)
)
(
λ x10 .
x0
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
(
x0
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
)
(
λ x10 .
x6
(
x0
(
λ x11 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
0
)
0
)
)
(
x3
(
λ x10 :
ι →
ι →
ι → ι
.
0
)
(
λ x10 :
ι →
(
ι → ι
)
→ ι
.
x3
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
(
λ x11 :
ι →
(
ι → ι
)
→ ι
.
0
)
(
λ x11 :
(
ι → ι
)
→ ι
.
0
)
)
(
λ x10 :
(
ι → ι
)
→ ι
.
x7
)
)
)
)
0
=
x6
(
setsum
(
x2
(
λ x9 x10 .
0
)
(
setsum
(
x6
0
)
(
Inj0
0
)
)
)
(
Inj0
(
Inj1
(
Inj0
0
)
)
)
)
)
⟶
False
(proof)
Theorem
73e8d..
:
∀ x0 :
(
(
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
(
(
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
∀ x1 :
(
(
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
)
→
(
ι → ι
)
→
ι →
ι → ι
.
∀ x2 :
(
ι → ι
)
→
ι → ι
.
∀ x3 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
(
∀ x4 x5 .
∀ x6 :
ι →
ι → ι
.
∀ x7 .
x3
(
λ x9 :
ι → ι
.
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x10 .
0
)
0
x7
)
x7
=
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
Inj0
(
x3
(
λ x10 :
ι → ι
.
x6
(
x3
(
λ x11 :
ι → ι
.
0
)
0
)
0
)
0
)
)
(
λ x9 .
Inj1
(
setsum
x7
0
)
)
(
setsum
(
Inj1
0
)
(
Inj0
(
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
Inj0
0
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
Inj1
0
)
)
)
)
(
setsum
0
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x9
(
λ x10 .
λ x11 :
ι → ι
.
0
)
(
λ x10 :
ι → ι
.
λ x11 .
0
)
)
(
x2
(
λ x9 .
x9
)
)
(
Inj1
(
Inj1
0
)
)
(
setsum
(
x6
0
0
)
x5
)
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x3
(
λ x9 :
ι → ι
.
Inj1
(
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x10
(
λ x11 .
λ x12 :
ι → ι
.
x2
(
λ x13 .
0
)
0
)
(
λ x11 :
ι → ι
.
λ x12 .
setsum
0
0
)
)
(
λ x10 .
x10
)
0
(
setsum
0
(
x9
0
)
)
)
)
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
setsum
x5
0
)
(
λ x9 .
x5
)
0
(
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
Inj1
(
x3
(
λ x10 :
ι → ι
.
0
)
0
)
)
(
λ x9 .
x6
)
(
setsum
0
0
)
(
Inj0
x5
)
)
)
=
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x6
)
(
λ x9 .
x0
(
λ x10 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x11 .
x10
(
λ x12 :
ι →
ι → ι
.
λ x13 .
x1
(
λ x14 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x14 .
0
)
0
0
)
(
λ x12 x13 .
0
)
)
0
x9
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
x9
)
)
(
setsum
0
(
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x5
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
x6
)
⟶
(
∀ x4 .
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
(
ι → ι
)
→
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
∀ x7 .
x2
(
λ x9 .
0
)
0
=
x4
)
⟶
(
∀ x4 :
(
(
ι → ι
)
→
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x5 .
∀ x6 :
(
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x7 .
x2
(
λ x9 .
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
x9
)
(
λ x10 .
Inj0
(
x3
(
λ x11 :
ι → ι
.
0
)
(
x2
(
λ x11 .
0
)
0
)
)
)
(
x6
(
λ x10 :
ι → ι
.
Inj1
(
x3
(
λ x11 :
ι → ι
.
0
)
0
)
)
0
(
λ x10 .
0
)
(
setsum
0
x7
)
)
(
Inj0
(
x2
(
λ x10 .
x7
)
(
x6
(
λ x10 :
ι → ι
.
0
)
0
(
λ x10 .
0
)
0
)
)
)
)
x5
=
x5
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x6 x7 .
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x9 .
x2
(
λ x10 .
x0
(
λ x11 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
Inj1
(
Inj0
0
)
)
)
0
)
0
x7
=
Inj0
(
Inj1
(
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
x2
(
λ x10 .
x1
(
λ x11 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x11 .
0
)
0
0
)
(
x2
(
λ x10 .
0
)
0
)
)
)
)
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→ ι
.
x1
(
λ x9 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x9 .
0
)
0
(
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
setsum
(
Inj1
(
setsum
0
0
)
)
(
x0
(
λ x10 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x9
(
λ x11 :
ι →
ι → ι
.
λ x12 .
0
)
(
λ x11 x12 .
0
)
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
x11
(
λ x12 .
0
)
0
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
x0
(
λ x11 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
x7
(
λ x12 :
ι →
ι → ι
.
λ x13 .
x13
)
)
(
λ x11 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x12 :
(
ι → ι
)
→
ι → ι
.
x3
(
λ x13 :
ι → ι
.
x0
(
λ x14 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x14 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x15 :
(
ι → ι
)
→
ι → ι
.
0
)
)
0
)
)
)
=
setsum
(
setsum
0
(
x6
(
x6
(
x2
(
λ x9 .
0
)
0
)
)
)
)
(
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
setsum
0
(
Inj1
(
x9
(
λ x10 :
ι →
ι → ι
.
λ x11 .
0
)
(
λ x10 x11 .
0
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x5 :
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x6 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
Inj1
(
setsum
(
setsum
0
(
x0
(
λ x10 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x10 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x11 :
(
ι → ι
)
→
ι → ι
.
0
)
)
)
(
Inj0
0
)
)
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
=
x5
(
λ x9 :
ι → ι
.
x9
(
x7
(
λ x10 .
λ x11 :
ι → ι
.
setsum
0
(
Inj0
0
)
)
(
x2
(
λ x10 .
setsum
0
0
)
0
)
(
x3
(
λ x10 :
ι → ι
.
x2
(
λ x11 .
0
)
0
)
0
)
)
)
(
Inj1
(
x4
(
λ x9 :
(
ι → ι
)
→ ι
.
Inj0
(
x1
(
λ x10 :
(
ι →
(
ι → ι
)
→ ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
0
)
(
λ x10 .
0
)
0
0
)
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
ι →
(
ι →
ι → ι
)
→
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x9 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
0
)
(
λ x9 :
(
(
ι → ι
)
→ ι
)
→
(
ι → ι
)
→ ι
.
λ x10 :
(
ι → ι
)
→
ι → ι
.
0
)
=
Inj1
x7
)
⟶
False
(proof)
Theorem
1cf83..
:
∀ x0 :
(
(
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x1 :
(
ι → ι
)
→
ι → ι
.
∀ x2 :
(
(
ι → ι
)
→ ι
)
→
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→ ι
.
∀ x3 :
(
ι → ι
)
→
(
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι → ι
)
→ ι
.
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
(
(
(
ι → ι
)
→
ι → ι
)
→ ι
)
→ ι
.
x3
(
λ x9 .
Inj1
(
x3
(
λ x10 .
x3
(
λ x11 .
x7
(
λ x12 :
(
ι → ι
)
→
ι → ι
.
0
)
)
(
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 .
Inj1
0
)
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
x2
(
λ x13 :
ι → ι
.
0
)
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
x2
(
λ x12 :
ι → ι
.
x2
(
λ x13 :
ι → ι
.
x10
(
λ x14 .
Inj0
0
)
)
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
x11
)
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
=
x2
(
λ x9 :
ι → ι
.
x3
(
λ x10 .
Inj0
(
x2
(
λ x11 :
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
x0
(
λ x13 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
Inj0
(
Inj0
0
)
)
(
x2
(
λ x13 :
ι → ι
.
0
)
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
Inj0
0
)
)
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x7
(
λ x10 :
(
ι → ι
)
→
ι → ι
.
Inj0
(
setsum
(
Inj0
0
)
(
x2
(
λ x11 :
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x3
(
λ x9 .
x0
(
λ x10 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
setsum
x9
0
)
x9
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
x1
(
λ x12 .
0
)
(
setsum
(
Inj0
x11
)
0
)
)
=
setsum
(
x3
(
λ x9 .
Inj0
(
x3
(
λ x10 .
x1
(
λ x11 .
0
)
0
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
x11
(
λ x13 .
0
)
)
)
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
x0
(
λ x11 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
x10
(
λ x12 .
Inj0
0
)
)
)
)
(
Inj1
(
x4
(
Inj1
x7
)
)
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x2
(
λ x9 :
ι → ι
.
x5
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
=
setsum
(
x3
(
λ x9 .
0
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
x1
(
λ x12 .
x10
(
λ x13 .
x11
)
)
(
Inj1
x11
)
)
)
(
setsum
(
x6
0
)
(
x3
(
λ x9 .
x5
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
)
)
)
⟶
(
∀ x4 :
(
ι →
ι → ι
)
→ ι
.
∀ x5 :
(
(
ι → ι
)
→ ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 :
(
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
∀ x7 .
x2
(
λ x9 :
ι → ι
.
Inj1
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
Inj0
(
setsum
(
x3
(
λ x10 .
x3
(
λ x11 .
0
)
(
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 .
0
)
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
x2
(
λ x13 :
ι → ι
.
0
)
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
)
(
x6
(
λ x10 .
0
)
(
λ x10 :
ι → ι
.
x10
0
)
)
)
)
=
Inj1
0
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 .
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→ ι
.
x1
(
λ x9 .
x9
)
0
=
setsum
0
0
)
⟶
(
∀ x4 :
(
(
ι →
ι → ι
)
→
ι → ι
)
→
(
ι →
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x6 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→ ι
.
∀ x7 .
x1
(
λ x9 .
0
)
(
x6
(
λ x9 .
Inj1
(
x1
(
λ x10 .
x6
(
λ x11 .
0
)
0
(
λ x11 .
0
)
)
0
)
)
(
setsum
x7
(
x2
(
λ x9 :
ι → ι
.
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
)
)
(
λ x9 .
0
)
)
=
Inj1
x7
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 .
x0
(
λ x9 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
x1
(
λ x10 .
setsum
(
Inj0
0
)
0
)
0
)
(
x0
(
λ x9 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
x6
(
setsum
0
(
x3
(
λ x10 .
0
)
(
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 .
0
)
)
)
)
0
)
=
x1
(
λ x9 .
x2
(
λ x10 :
ι → ι
.
x9
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x11 .
0
)
(
setsum
(
x3
(
λ x11 .
0
)
(
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 .
0
)
)
0
)
)
)
(
setsum
(
setsum
(
Inj1
0
)
x7
)
x5
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x5 :
ι → ι
.
∀ x6 .
∀ x7 :
ι →
ι →
ι → ι
.
x0
(
λ x9 :
(
ι → ι
)
→
ι →
(
ι → ι
)
→
ι → ι
.
Inj1
x6
)
(
Inj0
(
Inj0
(
x7
0
(
setsum
0
0
)
(
x3
(
λ x9 .
0
)
(
λ x9 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 .
0
)
)
)
)
)
=
x5
(
x1
(
λ x9 .
x9
)
(
x4
(
λ x9 .
setsum
0
0
)
(
λ x9 .
Inj0
(
x5
0
)
)
(
λ x9 .
0
)
(
Inj0
(
setsum
0
0
)
)
)
)
)
⟶
False
(proof)
Theorem
e46a9..
:
∀ x0 :
(
(
ι →
(
ι → ι
)
→
ι →
ι → ι
)
→ ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
ι →
ι → ι
)
→
ι → ι
.
∀ x2 :
(
(
ι →
ι →
ι → ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x3 :
(
ι →
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ι
)
→
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→
ι →
(
(
ι → ι
)
→
ι → ι
)
→
ι →
ι → ι
.
(
∀ x4 .
∀ x5 :
(
(
ι →
ι → ι
)
→ ι
)
→
ι → ι
.
∀ x6 x7 .
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x9
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x5
(
λ x10 :
ι →
ι → ι
.
x6
)
(
setsum
(
x0
(
λ x10 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
Inj0
0
)
(
λ x10 .
x10
)
(
setsum
0
0
)
)
(
setsum
(
x0
(
λ x10 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x10 .
0
)
0
)
(
Inj1
0
)
)
)
)
(
x2
(
λ x9 :
ι →
ι →
ι → ι
.
0
)
(
setsum
(
x1
(
λ x9 x10 .
x1
(
λ x11 x12 .
0
)
0
)
(
Inj1
0
)
)
(
x0
(
λ x9 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
setsum
0
0
)
(
λ x9 .
x9
)
(
x0
(
λ x9 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x9 .
0
)
0
)
)
)
0
)
(
λ x9 :
ι → ι
.
λ x10 .
x1
(
λ x11 x12 .
x10
)
0
)
(
x1
(
λ x9 x10 .
0
)
(
x2
(
λ x9 :
ι →
ι →
ι → ι
.
x5
(
λ x10 :
ι →
ι → ι
.
Inj1
0
)
(
x3
(
λ x10 .
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x10 :
ι → ι
.
λ x11 .
0
)
0
0
)
)
(
Inj1
(
Inj0
0
)
)
0
)
)
(
x0
(
λ x9 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
x6
)
(
λ x9 .
Inj1
(
x3
(
λ x10 .
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x0
(
λ x12 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x12 .
0
)
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
x3
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x11 :
ι → ι
.
λ x12 .
0
)
0
0
)
x7
(
λ x10 :
ι → ι
.
λ x11 .
0
)
(
x1
(
λ x10 x11 .
0
)
0
)
x9
)
)
(
setsum
(
setsum
0
x4
)
0
)
)
=
x0
(
λ x9 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
x6
)
(
λ x9 .
x2
(
λ x10 :
ι →
ι →
ι → ι
.
Inj1
(
x1
(
λ x11 x12 .
0
)
(
setsum
0
0
)
)
)
0
(
x1
(
λ x10 x11 .
x3
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x0
(
λ x14 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x14 .
0
)
0
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
x12
(
λ x13 x14 .
0
)
)
x9
(
λ x12 :
ι → ι
.
λ x13 .
x13
)
0
(
Inj1
0
)
)
(
x2
(
λ x10 :
ι →
ι →
ι → ι
.
x0
(
λ x11 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 .
0
)
0
)
(
setsum
0
0
)
(
x0
(
λ x10 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x10 .
0
)
0
)
)
)
)
(
Inj0
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x2
(
λ x11 :
ι →
ι →
ι → ι
.
x0
(
λ x12 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x12 .
0
)
0
)
(
x0
(
λ x11 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 .
0
)
0
)
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
Inj1
0
)
(
x1
(
λ x9 x10 .
x2
(
λ x11 :
ι →
ι →
ι → ι
.
0
)
0
0
)
0
)
(
λ x9 :
ι → ι
.
λ x10 .
x9
(
Inj0
0
)
)
(
Inj1
x4
)
(
Inj1
0
)
)
)
)
⟶
(
∀ x4 .
∀ x5 :
(
(
(
ι → ι
)
→ ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x7
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x10 x11 .
x2
(
λ x12 :
ι →
ι →
ι → ι
.
setsum
x10
x11
)
(
Inj0
(
Inj0
0
)
)
(
x2
(
λ x12 :
ι →
ι →
ι → ι
.
x2
(
λ x13 :
ι →
ι →
ι → ι
.
0
)
0
0
)
0
(
x3
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x12 :
ι → ι
.
λ x13 .
0
)
0
0
)
)
)
(
Inj1
(
x3
(
λ x10 .
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x10 :
ι → ι
.
λ x11 .
0
)
x7
(
x2
(
λ x10 :
ι →
ι →
ι → ι
.
0
)
0
0
)
)
)
)
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
x1
(
λ x10 x11 .
x2
(
λ x12 :
ι →
ι →
ι → ι
.
x3
(
λ x13 .
λ x14 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x13 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x13 :
ι → ι
.
λ x14 .
0
)
0
0
)
(
x9
(
λ x12 x13 .
0
)
)
(
x0
(
λ x12 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x12 .
0
)
0
)
)
(
x5
(
λ x10 :
(
ι → ι
)
→ ι
.
x6
0
)
x7
(
Inj0
0
)
)
)
0
(
λ x9 :
ι → ι
.
λ x10 .
Inj1
0
)
0
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
x1
(
λ x9 x10 .
0
)
(
Inj1
0
)
)
(
λ x9 :
ι → ι
.
λ x10 .
setsum
(
x9
0
)
(
x9
0
)
)
0
(
x1
(
λ x9 x10 .
x6
0
)
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
0
0
)
)
)
)
(
λ x9 :
ι → ι
.
λ x10 .
x2
(
λ x11 :
ι →
ι →
ι → ι
.
x11
0
0
0
)
(
x3
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
setsum
0
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
x3
(
λ x12 .
λ x13 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x2
(
λ x14 :
ι →
ι →
ι → ι
.
0
)
0
0
)
(
λ x12 :
(
ι →
ι → ι
)
→ ι
.
x12
(
λ x13 x14 .
0
)
)
0
(
λ x12 :
ι → ι
.
λ x13 .
Inj0
0
)
(
x1
(
λ x12 x13 .
0
)
0
)
0
)
x7
(
λ x11 :
ι → ι
.
λ x12 .
0
)
0
0
)
x7
)
(
setsum
(
x2
(
λ x9 :
ι →
ι →
ι → ι
.
0
)
0
0
)
0
)
(
x1
(
λ x9 x10 .
x7
)
(
x6
x4
)
)
=
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
Inj1
(
x6
0
)
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
setsum
(
x1
(
λ x10 x11 .
x0
(
λ x12 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
Inj0
0
)
(
λ x12 .
0
)
x10
)
(
x1
(
λ x10 x11 .
x9
(
λ x12 x13 .
0
)
)
(
x5
(
λ x10 :
(
ι → ι
)
→ ι
.
0
)
0
0
)
)
)
0
)
(
Inj0
x7
)
(
λ x9 :
ι → ι
.
λ x10 .
setsum
(
setsum
0
(
Inj0
(
x3
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x11 :
ι → ι
.
λ x12 .
0
)
0
0
)
)
)
(
Inj0
(
x3
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x0
(
λ x13 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x13 .
0
)
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
Inj0
0
)
(
Inj1
0
)
(
λ x11 :
ι → ι
.
λ x12 .
setsum
0
0
)
(
Inj0
0
)
(
x1
(
λ x11 x12 .
0
)
0
)
)
)
)
(
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
x4
(
setsum
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x9
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
(
Inj0
0
)
(
λ x9 :
ι → ι
.
λ x10 .
Inj0
0
)
x4
(
x1
(
λ x9 x10 .
0
)
0
)
)
(
x2
(
λ x9 :
ι →
ι →
ι → ι
.
x0
(
λ x10 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x10 .
0
)
0
)
(
setsum
0
0
)
(
x5
(
λ x9 :
(
ι → ι
)
→ ι
.
0
)
0
0
)
)
)
)
(
setsum
(
x1
(
λ x9 x10 .
0
)
0
)
x4
)
)
⟶
(
∀ x4 x5 x6 x7 .
x2
(
λ x9 :
ι →
ι →
ι → ι
.
0
)
(
Inj0
(
setsum
(
x2
(
λ x9 :
ι →
ι →
ι → ι
.
Inj0
0
)
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
0
0
)
x4
)
(
Inj0
(
x3
(
λ x9 .
λ x10 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x9 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x9 :
ι → ι
.
λ x10 .
0
)
0
0
)
)
)
)
x6
=
x6
)
⟶
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
x2
(
λ x9 :
ι →
ι →
ι → ι
.
Inj0
0
)
(
Inj0
(
x2
(
λ x9 :
ι →
ι →
ι → ι
.
x0
(
λ x10 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
x10
0
(
λ x11 .
0
)
0
0
)
(
λ x10 .
x0
(
λ x11 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x11 .
0
)
0
)
(
x1
(
λ x10 x11 .
0
)
0
)
)
0
0
)
)
0
=
x5
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
ι →
ι → ι
.
∀ x6 x7 :
ι → ι
.
x1
(
λ x9 x10 .
0
)
(
x4
0
)
=
Inj1
(
x4
0
)
)
⟶
(
∀ x4 x5 x6 .
∀ x7 :
ι → ι
.
x1
(
λ x9 x10 .
Inj0
(
x7
x6
)
)
x4
=
x4
)
⟶
(
∀ x4 x5 x6 x7 .
x0
(
λ x9 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
setsum
x7
(
x1
(
λ x10 x11 .
setsum
(
x1
(
λ x12 x13 .
0
)
0
)
0
)
x7
)
)
(
λ x9 .
x6
)
0
=
setsum
x4
0
)
⟶
(
∀ x4 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x5 .
∀ x6 :
(
(
ι →
ι → ι
)
→
ι →
ι → ι
)
→ ι
.
∀ x7 :
ι → ι
.
x0
(
λ x9 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
setsum
x5
(
setsum
(
x9
0
(
λ x10 .
x3
(
λ x11 .
λ x12 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x11 :
(
ι →
ι → ι
)
→ ι
.
0
)
0
(
λ x11 :
ι → ι
.
λ x12 .
0
)
0
0
)
(
Inj1
0
)
0
)
(
x3
(
λ x10 .
λ x11 :
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
0
)
(
λ x10 :
(
ι →
ι → ι
)
→ ι
.
x9
0
(
λ x11 .
0
)
0
0
)
0
(
λ x10 :
ι → ι
.
λ x11 .
setsum
0
0
)
(
x9
0
(
λ x10 .
0
)
0
0
)
(
Inj0
0
)
)
)
)
(
λ x9 .
x9
)
(
Inj0
(
x4
0
(
λ x9 .
x7
(
x7
0
)
)
(
λ x9 .
Inj1
0
)
(
setsum
(
Inj0
0
)
(
Inj0
0
)
)
)
)
=
setsum
(
Inj0
(
setsum
(
setsum
(
Inj1
0
)
(
Inj1
0
)
)
0
)
)
(
setsum
(
setsum
(
setsum
(
x0
(
λ x9 :
ι →
(
ι → ι
)
→
ι →
ι → ι
.
0
)
(
λ x9 .
0
)
0
)
(
Inj1
0
)
)
0
)
0
)
)
⟶
False
(proof)
Theorem
88f1b..
:
∀ x0 x1 :
(
ι → ι
)
→
ι → ι
.
∀ x2 :
(
(
(
(
ι →
ι → ι
)
→ ι
)
→ ι
)
→ ι
)
→
ι →
ι → ι
.
∀ x3 :
(
ι → ι
)
→
ι →
ι → ι
.
(
∀ x4 x5 .
∀ x6 :
ι → ι
.
∀ x7 :
ι →
ι →
ι → ι
.
x3
(
λ x9 .
x7
(
Inj1
0
)
x9
(
x7
(
x6
(
setsum
0
0
)
)
0
0
)
)
(
Inj0
(
Inj0
(
setsum
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
0
)
x4
)
)
)
(
x6
(
x0
(
λ x9 .
0
)
(
x6
(
setsum
0
0
)
)
)
)
=
x6
(
x7
0
x4
(
Inj0
(
Inj1
0
)
)
)
)
⟶
(
∀ x4 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→
ι → ι
.
∀ x5 :
(
ι →
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 :
ι → ι
.
∀ x7 .
x3
(
λ x9 .
x9
)
0
(
Inj0
0
)
=
x5
(
λ x9 .
λ x10 :
ι → ι
.
λ x11 .
x9
)
)
⟶
(
∀ x4 :
(
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→ ι
)
→ ι
.
∀ x5 :
ι →
(
ι → ι
)
→
ι → ι
.
∀ x6 x7 .
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
0
)
0
x6
=
setsum
(
x0
(
λ x9 .
setsum
x9
(
setsum
0
(
Inj0
0
)
)
)
x7
)
(
x4
(
λ x9 :
(
ι → ι
)
→
ι → ι
.
λ x10 :
ι → ι
.
x0
(
λ x11 .
0
)
0
)
)
)
⟶
(
∀ x4 :
ι →
ι →
(
ι → ι
)
→ ι
.
∀ x5 :
(
ι → ι
)
→ ι
.
∀ x6 :
(
(
ι → ι
)
→ ι
)
→
(
ι →
ι → ι
)
→
ι → ι
.
∀ x7 :
(
(
(
ι → ι
)
→ ι
)
→
ι →
ι → ι
)
→
(
(
ι → ι
)
→ ι
)
→
ι → ι
.
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj1
(
x6
(
λ x10 :
ι → ι
.
0
)
(
λ x10 x11 .
x1
(
λ x12 .
x1
(
λ x13 .
0
)
0
)
x10
)
(
x9
(
λ x10 :
ι →
ι → ι
.
x10
0
0
)
)
)
)
(
x1
(
λ x9 .
x1
(
λ x10 .
x7
(
λ x11 :
(
ι → ι
)
→ ι
.
λ x12 x13 .
Inj1
0
)
(
λ x11 :
ι → ι
.
0
)
0
)
(
x3
(
λ x10 .
setsum
0
0
)
(
x7
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
0
)
(
λ x10 :
ι → ι
.
0
)
0
)
x9
)
)
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj1
(
Inj1
0
)
)
(
x1
(
λ x9 .
x6
(
λ x10 :
ι → ι
.
0
)
(
λ x10 x11 .
0
)
0
)
(
x4
0
0
(
λ x9 .
0
)
)
)
0
)
)
(
setsum
0
(
x3
(
λ x9 .
x5
(
λ x10 .
Inj1
0
)
)
(
x3
(
λ x9 .
x3
(
λ x10 .
0
)
0
0
)
(
setsum
0
0
)
(
x4
0
0
(
λ x9 .
0
)
)
)
(
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 .
0
)
(
λ x9 :
ι → ι
.
x3
(
λ x10 .
0
)
0
0
)
(
x3
(
λ x9 .
0
)
0
0
)
)
)
)
=
x1
(
λ x9 .
x0
(
λ x10 .
x10
)
(
x5
(
λ x10 .
x2
(
λ x11 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
x7
(
λ x12 :
(
ι → ι
)
→ ι
.
λ x13 x14 .
0
)
(
λ x12 :
ι → ι
.
0
)
0
)
(
Inj0
0
)
(
x3
(
λ x11 .
0
)
0
0
)
)
)
)
(
x7
(
λ x9 :
(
ι → ι
)
→ ι
.
λ x10 x11 .
x11
)
(
λ x9 :
ι → ι
.
x1
(
λ x10 .
0
)
(
x7
(
λ x10 :
(
ι → ι
)
→ ι
.
λ x11 x12 .
x1
(
λ x13 .
0
)
0
)
(
λ x10 :
ι → ι
.
setsum
0
0
)
(
x6
(
λ x10 :
ι → ι
.
0
)
(
λ x10 x11 .
0
)
0
)
)
)
(
x6
(
λ x9 :
ι → ι
.
x0
(
λ x10 .
Inj1
0
)
(
x1
(
λ x10 .
0
)
0
)
)
(
λ x9 x10 .
x0
(
λ x11 .
x1
(
λ x12 .
0
)
0
)
0
)
0
)
)
)
⟶
(
∀ x4 :
(
ι → ι
)
→ ι
.
∀ x5 x6 x7 .
x1
(
λ x9 .
x5
)
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
0
(
x0
(
λ x10 .
x10
)
(
x0
(
λ x10 .
0
)
0
)
)
)
(
Inj0
x7
)
(
x1
(
λ x9 .
setsum
(
setsum
0
0
)
x6
)
(
setsum
(
x1
(
λ x9 .
0
)
0
)
0
)
)
)
=
Inj1
(
setsum
(
setsum
(
setsum
x5
(
Inj0
0
)
)
(
x2
(
λ x9 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
setsum
0
0
)
0
0
)
)
(
setsum
(
x1
(
λ x9 .
setsum
0
0
)
x5
)
(
setsum
x6
x5
)
)
)
)
⟶
(
∀ x4 x5 x6 x7 .
x1
(
λ x9 .
x0
(
λ x10 .
x9
)
(
Inj1
(
x2
(
λ x10 :
(
(
ι →
ι → ι
)
→ ι
)
→ ι
.
Inj1
0
)
x5
(
x3
(
λ x10 .
0
)
0
0
)
)
)
)
(
x1
(
λ x9 .
x3
(
λ x10 .
x7
)
(
setsum
x6
x9
)
(
setsum
(
Inj0
0
)
(
Inj0
0
)
)
)
(
Inj0
0
)
)
=
Inj1
0
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 :
ι →
(
(
ι → ι
)
→
ι → ι
)
→ ι
.
∀ x6 .
∀ x7 :
ι →
ι →
ι → ι
.
x0
(
λ x9 .
0
)
x6
=
Inj1
(
x3
(
λ x9 .
0
)
(
setsum
0
(
Inj0
(
Inj1
0
)
)
)
(
setsum
(
x1
(
λ x9 .
0
)
(
Inj1
0
)
)
(
setsum
(
x1
(
λ x9 .
0
)
0
)
(
setsum
0
0
)
)
)
)
)
⟶
(
∀ x4 :
ι → ι
.
∀ x5 x6 x7 .
x0
(
λ x9 .
setsum
x9
0
)
(
setsum
(
x0
(
λ x9 .
x0
(
λ x10 .
x9
)
(
x3
(
λ x10 .
0
)
0
0
)
)
0
)
x6
)
=
x6
)
⟶
False
(proof)