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Proofgold Asset
asset id
ab8cf4b54a8e68df49f6111eb7a4a87d45f99139f36d6122417dfad4d2983585
asset hash
620f464136d1c23ce668494bf358b93b27ddc782dc7579293fe936043aaaeaea
bday / block
2059
tx
cbccf..
preasset
doc published by
PrGxv..
Definition
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Known
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Theorem
and3I
:
∀ x0 x1 x2 : ο .
x0
⟶
x1
⟶
x2
⟶
and
(
and
x0
x1
)
x2
(proof)
Definition
c4def..
:=
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x0
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x0
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x0
x1
)
)
)
)
)
(
prim1
(
λ x0 .
x0
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
Definition
6b90c..
:=
λ x0 x1 .
prim0
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x2
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x2
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x2
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x2
x3
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x2
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x2
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x2
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x2
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x2
x3
)
(
prim0
x3
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x3
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x2
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x3
x3
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x3
x3
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x2
x3
)
(
prim0
x3
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x2
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x2
x3
)
(
prim0
x3
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x2
)
)
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x3
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x2
)
)
(
prim0
(
prim0
x2
x3
)
(
prim0
x3
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x3
x2
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x2
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x2
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x2
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x3
)
)
(
prim0
(
prim0
x2
x2
)
(
prim0
x3
x2
)
)
)
)
)
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
x3
x2
)
(
prim0
x2
x2
)
)
(
prim0
(
prim0
x3
x3
)
(
prim0
x3
x3
)
)
)
)
)
(
prim1
(
λ x2 .
x2
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
(
prim0
x0
x1
)
Definition
5e331..
:=
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x0
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
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Definition
a3eb9..
:=
λ x0 x1 .
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(
prim0
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(
λ x2 .
prim1
(
λ x3 .
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(
prim0
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prim0
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)
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λ x2 .
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λ x3 .
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(
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prim0
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)
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)
)
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λ x2 .
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λ x3 .
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λ x3 .
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λ x2 .
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λ x3 .
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λ x2 .
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(
prim0
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Definition
bf68c..
:=
λ x0 x1 .
prim0
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
prim0
(
prim0
(
prim0
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x2
)
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prim0
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)
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(
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prim0
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prim0
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)
(
prim0
(
prim1
(
λ x2 .
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(
λ x3 .
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(
prim0
(
prim0
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(
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)
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prim0
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)
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)
(
prim0
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prim1
(
λ x2 .
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(
λ x3 .
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(
prim0
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prim0
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λ x2 .
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λ x3 .
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λ x2 .
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λ x3 .
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λ x2 .
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λ x3 .
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λ x2 .
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λ x3 .
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λ x2 .
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(
prim0
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Definition
c9248..
:=
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
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(
prim0
(
prim0
x1
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)
(
prim0
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)
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(
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prim0
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(
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λ x0 .
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λ x1 .
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(
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λ x0 .
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λ x1 .
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(
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λ x0 .
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Definition
a6e19..
:=
prim0
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λ x0 .
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λ x1 .
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Definition
2fe34..
:=
prim0
(
prim0
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λ x0 .
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λ x1 .
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Definition
3e00e..
:=
λ x0 x1 .
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(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
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(
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prim0
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(
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(
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(
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(
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(
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λ x3 .
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λ x3 .
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λ x2 .
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λ x3 .
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(
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λ x2 .
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λ x3 .
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(
prim0
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Definition
f9341..
:=
λ x0 x1 .
prim0
(
prim0
(
prim1
(
λ x2 .
prim1
(
λ x3 .
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(
prim0
(
prim0
x3
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)
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(
prim0
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prim1
(
λ x2 .
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(
λ x3 .
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(
prim0
(
prim0
x3
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λ x2 .
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(
λ x3 .
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λ x2 .
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λ x3 .
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λ x3 .
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λ x2 .
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λ x3 .
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λ x3 .
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λ x2 .
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)
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(
prim0
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Definition
1fa6d..
:=
prim0
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
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(
prim0
x0
x1
)
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prim0
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prim0
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(
prim0
x0
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(
prim0
(
prim1
(
λ x0 .
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(
λ x1 .
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(
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prim0
x1
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)
(
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)
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(
prim0
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(
prim0
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(
prim0
(
prim1
(
λ x0 .
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(
λ x1 .
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(
prim0
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(
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λ x0 .
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λ x1 .
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(
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λ x0 .
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λ x1 .
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(
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(
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λ x0 .
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λ x1 .
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λ x0 .
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λ x1 .
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λ x0 .
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(
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λ x0 .
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)
)
)
)
)
)
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)
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Definition
3a365..
:=
prim0
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
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)
(
prim0
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)
)
)
)
)
(
prim0
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prim1
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λ x0 .
prim1
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λ x1 .
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(
prim0
(
prim0
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)
(
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(
prim0
(
prim0
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)
(
prim0
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)
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)
)
)
(
prim0
(
prim1
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λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
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(
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(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x0
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x0
x1
)
(
prim0
x1
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x0
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x1
x0
)
(
prim0
x1
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x0
x1
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x1
)
)
(
prim0
(
prim0
x0
x0
)
(
prim0
x0
x0
)
)
)
)
)
(
prim0
(
prim1
(
λ x0 .
prim1
(
λ x1 .
prim0
(
prim0
(
prim0
x1
x0
)
(
prim0
x0
x0
)
)
(
prim0
(
prim0
x1
x1
)
(
prim0
x1
x1
)
)
)
)
)
(
prim1
(
λ x0 .
x0
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
Definition
False
:=
∀ x0 : ο .
x0
Known
92e6a..
:
∀ x0 :
ι → ι
.
∀ x1 x2 .
prim1
x0
=
prim0
x1
x2
⟶
False
Known
50787..
:
∀ x0 x1 x2 x3 .
prim0
x0
x1
=
prim0
x2
x3
⟶
x0
=
x2
Theorem
8b073..
:
∀ x0 x1 x2 .
prim0
(
prim1
(
λ x4 .
prim1
(
λ x5 .
prim0
(
prim0
(
prim0
x5
x4
)
(
prim0
x4
x5
)
)
(
prim0
(
prim0
x5
x5
)
(
prim0
x4
x4
)
)
)
)
)
x0
=
prim0
(
prim0
(
prim1
(
λ x4 .
prim1
(
λ x5 .
prim0
(
prim0
(
prim0
x5
x4
)
(
prim0
x4
x5
)
)
(
prim0
(
prim0
x5
x5
)
(
prim0
x4
x4
)
)
)
)
)
x1
)
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
6641d..
:
∀ x0 x1 x2 .
prim0
(
prim0
(
prim1
(
λ x4 .
prim1
(
λ x5 .
prim0
(
prim0
(
prim0
x5
x4
)
(
prim0
x4
x5
)
)
(
prim0
(
prim0
x5
x5
)
(
prim0
x4
x4
)
)
)
)
)
x1
)
x2
=
prim0
(
prim1
(
λ x4 .
prim1
(
λ x5 .
prim0
(
prim0
(
prim0
x5
x4
)
(
prim0
x4
x5
)
)
(
prim0
(
prim0
x5
x5
)
(
prim0
x4
x4
)
)
)
)
)
x0
⟶
∀ x3 : ο .
x3
(proof)
Theorem
9ec26..
:
∀ x0 x1 .
5e331..
=
a3eb9..
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
59f91..
:
∀ x0 x1 .
5e331..
=
bf68c..
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
Param
236c6..
:
ι
Known
f558c..
:
∀ x0 x1 .
236c6..
=
prim0
x0
x1
⟶
∀ x2 : ο .
x2
Known
93754..
:
∀ x0 x1 x2 x3 .
prim0
x0
x1
=
prim0
x2
x3
⟶
x1
=
x3
Known
db6fe..
:
∀ x0 x1 :
ι → ι
.
∀ x2 .
prim1
x0
=
prim1
x1
⟶
x0
x2
=
x1
x2
Theorem
a8e2e..
:
∀ x0 x1 x2 x3 .
a3eb9..
x0
x1
=
bf68c..
x2
x3
⟶
∀ x4 : ο .
x4
(proof)
Known
128d8..
:
∀ x0 x1 x2 x3 .
prim0
x0
x1
=
prim0
x2
x3
⟶
∀ x4 : ο .
(
x0
=
x2
⟶
x1
=
x3
⟶
x4
)
⟶
x4
Theorem
5e750..
:
∀ x0 x1 x2 x3 .
a3eb9..
x0
x1
=
a3eb9..
x2
x3
⟶
and
(
x0
=
x2
)
(
x1
=
x3
)
(proof)
Theorem
2f86f..
:
∀ x0 x1 x2 x3 .
bf68c..
x0
x1
=
bf68c..
x2
x3
⟶
and
(
x0
=
x2
)
(
x1
=
x3
)
(proof)
Theorem
a3634..
:
∀ x0 x1 .
c4def..
=
6b90c..
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
Known
0286c..
:
∀ x0 x1 .
prim0
x0
x1
=
236c6..
⟶
False
Theorem
5e60e..
:
c4def..
=
c9248..
⟶
∀ x0 : ο .
x0
(proof)
Theorem
dc5ae..
:
∀ x0 x1 .
6b90c..
x0
x1
=
c9248..
⟶
∀ x2 : ο .
x2
(proof)
Theorem
924dd..
:
∀ x0 .
c4def..
=
a6e19..
x0
⟶
∀ x1 : ο .
x1
(proof)
Theorem
a9278..
:
∀ x0 x1 x2 .
6b90c..
x0
x1
=
a6e19..
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
84bad..
:
∀ x0 .
c9248..
=
a6e19..
x0
⟶
∀ x1 : ο .
x1
(proof)
Theorem
0cc7e..
:
∀ x0 .
c4def..
=
2fe34..
x0
⟶
∀ x1 : ο .
x1
(proof)
Theorem
dabcf..
:
∀ x0 x1 x2 .
6b90c..
x0
x1
=
2fe34..
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
1832c..
:
∀ x0 .
c9248..
=
2fe34..
x0
⟶
∀ x1 : ο .
x1
(proof)
Theorem
7241c..
:
∀ x0 x1 .
a6e19..
x0
=
2fe34..
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
91964..
:
∀ x0 x1 .
c4def..
=
3e00e..
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
07fb9..
:
∀ x0 x1 x2 x3 .
6b90c..
x0
x1
=
3e00e..
x2
x3
⟶
∀ x4 : ο .
x4
(proof)
Theorem
77b59..
:
∀ x0 x1 .
c9248..
=
3e00e..
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
62a09..
:
∀ x0 x1 x2 .
a6e19..
x0
=
3e00e..
x1
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
a0ec1..
:
∀ x0 x1 x2 .
2fe34..
x0
=
3e00e..
x1
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
a14cf..
:
∀ x0 x1 .
c4def..
=
f9341..
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
72b8e..
:
∀ x0 x1 x2 x3 .
6b90c..
x0
x1
=
f9341..
x2
x3
⟶
∀ x4 : ο .
x4
(proof)
Theorem
b728e..
:
∀ x0 x1 .
c9248..
=
f9341..
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
cec81..
:
∀ x0 x1 x2 .
a6e19..
x0
=
f9341..
x1
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
becdb..
:
∀ x0 x1 x2 .
2fe34..
x0
=
f9341..
x1
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
a1a1b..
:
∀ x0 x1 x2 x3 .
3e00e..
x0
x1
=
f9341..
x2
x3
⟶
∀ x4 : ο .
x4
(proof)
Theorem
b8b08..
:
∀ x0 .
c4def..
=
1fa6d..
x0
⟶
∀ x1 : ο .
x1
(proof)
Theorem
0a32f..
:
∀ x0 x1 x2 .
6b90c..
x0
x1
=
1fa6d..
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
0482c..
:
∀ x0 .
c9248..
=
1fa6d..
x0
⟶
∀ x1 : ο .
x1
(proof)
Theorem
4bdb9..
:
∀ x0 x1 .
a6e19..
x0
=
1fa6d..
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
a5def..
:
∀ x0 x1 .
2fe34..
x0
=
1fa6d..
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
54459..
:
∀ x0 x1 x2 .
3e00e..
x0
x1
=
1fa6d..
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
38968..
:
∀ x0 x1 x2 .
f9341..
x0
x1
=
1fa6d..
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
f9749..
:
∀ x0 .
c4def..
=
3a365..
x0
⟶
∀ x1 : ο .
x1
(proof)
Theorem
ff84b..
:
∀ x0 x1 x2 .
6b90c..
x0
x1
=
3a365..
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
36b24..
:
∀ x0 .
c9248..
=
3a365..
x0
⟶
∀ x1 : ο .
x1
(proof)
Theorem
62476..
:
∀ x0 x1 .
a6e19..
x0
=
3a365..
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
cdd8a..
:
∀ x0 x1 .
2fe34..
x0
=
3a365..
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
fb77a..
:
∀ x0 x1 x2 .
3e00e..
x0
x1
=
3a365..
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
6774e..
:
∀ x0 x1 x2 .
f9341..
x0
x1
=
3a365..
x2
⟶
∀ x3 : ο .
x3
(proof)
Theorem
9f429..
:
∀ x0 x1 .
1fa6d..
x0
=
3a365..
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
ffc37..
:
∀ x0 x1 x2 x3 .
6b90c..
x0
x1
=
6b90c..
x2
x3
⟶
and
(
x0
=
x2
)
(
x1
=
x3
)
(proof)
Theorem
ffefa..
:
∀ x0 x1 x2 x3 .
6b90c..
x0
x1
=
6b90c..
x2
x3
⟶
x0
=
x2
(proof)
Theorem
af84e..
:
∀ x0 x1 x2 x3 .
6b90c..
x0
x1
=
6b90c..
x2
x3
⟶
x1
=
x3
(proof)
Theorem
84af1..
:
∀ x0 x1 .
a6e19..
x0
=
a6e19..
x1
⟶
x0
=
x1
(proof)
Theorem
2eb6f..
:
∀ x0 x1 .
2fe34..
x0
=
2fe34..
x1
⟶
x0
=
x1
(proof)
Theorem
42e43..
:
∀ x0 x1 x2 x3 .
3e00e..
x0
x1
=
3e00e..
x2
x3
⟶
and
(
x0
=
x2
)
(
x1
=
x3
)
(proof)
Theorem
ee7eb..
:
∀ x0 x1 x2 x3 .
3e00e..
x0
x1
=
3e00e..
x2
x3
⟶
x0
=
x2
(proof)
Theorem
67376..
:
∀ x0 x1 x2 x3 .
3e00e..
x0
x1
=
3e00e..
x2
x3
⟶
x1
=
x3
(proof)
Theorem
063ac..
:
∀ x0 x1 x2 x3 .
f9341..
x0
x1
=
f9341..
x2
x3
⟶
and
(
x0
=
x2
)
(
x1
=
x3
)
(proof)
Theorem
9efcd..
:
∀ x0 x1 x2 x3 .
f9341..
x0
x1
=
f9341..
x2
x3
⟶
x0
=
x2
(proof)
Theorem
2b512..
:
∀ x0 x1 x2 x3 .
f9341..
x0
x1
=
f9341..
x2
x3
⟶
x1
=
x3
(proof)
Theorem
a1c68..
:
∀ x0 x1 .
1fa6d..
x0
=
1fa6d..
x1
⟶
x0
=
x1
(proof)
Theorem
f684d..
:
∀ x0 x1 .
3a365..
x0
=
3a365..
x1
⟶
x0
=
x1
(proof)
Definition
74e69..
:=
λ x0 .
∀ x1 :
ι → ο
.
x1
5e331..
⟶
(
∀ x2 x3 .
x1
x2
⟶
x1
x3
⟶
x1
(
a3eb9..
x2
x3
)
)
⟶
(
∀ x2 x3 .
x1
x2
⟶
x1
x3
⟶
x1
(
bf68c..
x2
x3
)
)
⟶
x1
x0
Theorem
e5778..
:
74e69..
5e331..
(proof)
Theorem
c4252..
:
∀ x0 x1 .
74e69..
x0
⟶
74e69..
x1
⟶
74e69..
(
a3eb9..
x0
x1
)
(proof)
Theorem
fbf8c..
:
∀ x0 x1 .
74e69..
x0
⟶
74e69..
x1
⟶
74e69..
(
bf68c..
x0
x1
)
(proof)
Theorem
facf7..
:
∀ x0 :
ι → ο
.
x0
5e331..
⟶
(
∀ x1 x2 .
74e69..
x1
⟶
x0
x1
⟶
74e69..
x2
⟶
x0
x2
⟶
x0
(
a3eb9..
x1
x2
)
)
⟶
(
∀ x1 x2 .
74e69..
x1
⟶
x0
x1
⟶
74e69..
x2
⟶
x0
x2
⟶
x0
(
bf68c..
x1
x2
)
)
⟶
∀ x1 .
74e69..
x1
⟶
x0
x1
(proof)
Known
FalseE
:
False
⟶
∀ x0 : ο .
x0
Theorem
75262..
:
∀ x0 x1 .
74e69..
(
a3eb9..
x0
x1
)
⟶
and
(
74e69..
x0
)
(
74e69..
x1
)
(proof)
Theorem
826e0..
:
∀ x0 x1 .
74e69..
(
bf68c..
x0
x1
)
⟶
and
(
74e69..
x0
)
(
74e69..
x1
)
(proof)
Definition
e05e6..
:=
λ x0 .
∀ x1 :
ι → ο
.
x1
c4def..
⟶
(
∀ x2 x3 .
x1
x2
⟶
x1
x3
⟶
x1
(
6b90c..
x2
x3
)
)
⟶
x1
c9248..
⟶
(
∀ x2 .
x1
x2
⟶
x1
(
a6e19..
x2
)
)
⟶
(
∀ x2 .
x1
x2
⟶
x1
(
2fe34..
x2
)
)
⟶
(
∀ x2 x3 .
x1
x2
⟶
x1
x3
⟶
x1
(
3e00e..
x2
x3
)
)
⟶
(
∀ x2 x3 .
x1
x2
⟶
x1
x3
⟶
x1
(
f9341..
x2
x3
)
)
⟶
(
∀ x2 .
x1
x2
⟶
x1
(
1fa6d..
x2
)
)
⟶
(
∀ x2 .
x1
x2
⟶
x1
(
3a365..
x2
)
)
⟶
x1
x0
Theorem
b24e2..
:
e05e6..
c4def..
(proof)
Theorem
05df6..
:
∀ x0 x1 .
e05e6..
x0
⟶
e05e6..
x1
⟶
e05e6..
(
6b90c..
x0
x1
)
(proof)
Theorem
9dc64..
:
e05e6..
c9248..
(proof)
Theorem
82f14..
:
∀ x0 .
e05e6..
x0
⟶
e05e6..
(
a6e19..
x0
)
(proof)
Theorem
6d5ea..
:
∀ x0 .
e05e6..
x0
⟶
e05e6..
(
2fe34..
x0
)
(proof)
Theorem
1a774..
:
∀ x0 x1 .
e05e6..
x0
⟶
e05e6..
x1
⟶
e05e6..
(
3e00e..
x0
x1
)
(proof)
Theorem
6891d..
:
∀ x0 x1 .
e05e6..
x0
⟶
e05e6..
x1
⟶
e05e6..
(
f9341..
x0
x1
)
(proof)
Theorem
96df8..
:
∀ x0 .
e05e6..
x0
⟶
e05e6..
(
1fa6d..
x0
)
(proof)
Theorem
b6c01..
:
∀ x0 .
e05e6..
x0
⟶
e05e6..
(
3a365..
x0
)
(proof)