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Proofgold Asset
asset id
41b00573fa3deda6a8927c9b5b5cd89ef1cbc7b88dccd95efd21406bb22e39ab
asset hash
b74cb3a68ccf25166d118778cd5ff8f244b213d42038c82bdadbca56a276e84a
bday / block
18372
tx
aa678..
preasset
doc published by
Pr4zB..
Param
and
and
:
ο
→
ο
→
ο
Definition
ChurchNums_3x8_eq
:=
λ x0 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ 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
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
and
(
x0
=
x2
)
(
x1
=
x3
)
Param
not
not
:
ο
→
ο
Definition
ChurchNums_3x8_neq
:=
λ x0 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ 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
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
not
(
ChurchNums_3x8_eq
x0
x1
x2
x3
)
Known
andI
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Theorem
f6916..
:
∀ x0 x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x2 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
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
x0
=
x1
⟶
x2
=
x3
⟶
ChurchNums_3x8_eq
x0
x2
x1
x3
(proof)
Definition
ChurchNums_8_perm_1_2_3_4_5_6_7_0
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 x2 x3 x4 x5 x6 x7 x8 :
(
ι → ι
)
→
ι → ι
.
x0
x2
x3
x4
x5
x6
x7
x8
x1
Definition
ChurchNums_8_perm_2_3_4_5_6_7_0_1
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 x2 x3 x4 x5 x6 x7 x8 :
(
ι → ι
)
→
ι → ι
.
x0
x3
x4
x5
x6
x7
x8
x1
x2
Definition
ChurchNums_8_perm_3_4_5_6_7_0_1_2
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 x2 x3 x4 x5 x6 x7 x8 :
(
ι → ι
)
→
ι → ι
.
x0
x4
x5
x6
x7
x8
x1
x2
x3
Definition
ChurchNums_8_perm_4_5_6_7_0_1_2_3
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 x2 x3 x4 x5 x6 x7 x8 :
(
ι → ι
)
→
ι → ι
.
x0
x5
x6
x7
x8
x1
x2
x3
x4
Definition
ChurchNums_8_perm_5_6_7_0_1_2_3_4
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 x2 x3 x4 x5 x6 x7 x8 :
(
ι → ι
)
→
ι → ι
.
x0
x6
x7
x8
x1
x2
x3
x4
x5
Definition
ChurchNums_8_perm_6_7_0_1_2_3_4_5
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 x2 x3 x4 x5 x6 x7 x8 :
(
ι → ι
)
→
ι → ι
.
x0
x7
x8
x1
x2
x3
x4
x5
x6
Definition
ChurchNums_8_perm_7_0_1_2_3_4_5_6
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 x2 x3 x4 x5 x6 x7 x8 :
(
ι → ι
)
→
ι → ι
.
x0
x8
x1
x2
x3
x4
x5
x6
x7
Theorem
accc4..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8_perm_1_2_3_4_5_6_7_0
(
ChurchNums_8_perm_7_0_1_2_3_4_5_6
x0
)
=
x0
(proof)
Theorem
13c67..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8_perm_2_3_4_5_6_7_0_1
(
ChurchNums_8_perm_6_7_0_1_2_3_4_5
x0
)
=
x0
(proof)
Theorem
d720a..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8_perm_3_4_5_6_7_0_1_2
(
ChurchNums_8_perm_5_6_7_0_1_2_3_4
x0
)
=
x0
(proof)
Theorem
5a925..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8_perm_4_5_6_7_0_1_2_3
(
ChurchNums_8_perm_4_5_6_7_0_1_2_3
x0
)
=
x0
(proof)
Theorem
44285..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8_perm_5_6_7_0_1_2_3_4
(
ChurchNums_8_perm_3_4_5_6_7_0_1_2
x0
)
=
x0
(proof)
Theorem
abce4..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8_perm_6_7_0_1_2_3_4_5
(
ChurchNums_8_perm_2_3_4_5_6_7_0_1
x0
)
=
x0
(proof)
Theorem
06461..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8_perm_7_0_1_2_3_4_5_6
(
ChurchNums_8_perm_1_2_3_4_5_6_7_0
x0
)
=
x0
(proof)
Definition
ChurchNums_8x3_lt1_id_ge1_rot2
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
Definition
ChurchNums_8x3_lt2_id_ge2_rot2
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
Definition
ChurchNums_8x3_to_3_lt3_id_ge3_rot2
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
Definition
ChurchNums_8x3_to_3_lt4_id_ge4_rot2
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
Definition
ChurchNums_8x3_to_3_lt5_id_ge5_rot2
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
Definition
ChurchNums_8x3_to_3_lt6_id_ge6_rot2
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x3
x4
x2
)
(
x1
x3
x4
x2
)
Definition
ChurchNums_8x3_to_3_lt7_id_ge7_rot2
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x3
x4
x2
)
Definition
ChurchNums_8x3_lt1_id_ge1_rot1
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
Definition
ChurchNums_8x3_lt2_id_ge2_rot1
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
Definition
ChurchNums_8x3_lt3_id_ge3_rot1
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
Definition
ChurchNums_8x3_lt4_id_ge4_rot1
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
Definition
ChurchNums_8x3_lt5_id_ge5_rot1
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
Definition
ChurchNums_8x3_lt6_id_ge6_rot1
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x4
x2
x3
)
(
x1
x4
x2
x3
)
Definition
ChurchNums_8x3_lt7_id_ge7_rot1
:=
λ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x0
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x2
x3
x4
)
(
x1
x4
x2
x3
)
Definition
ChurchNum_3ary_proj_p
:=
λ 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
)
→ ο
.
x1
(
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x2
)
⟶
x1
(
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x3
)
⟶
x1
(
λ x2 x3 x4 :
(
ι → ι
)
→
ι → ι
.
x4
)
⟶
x1
x0
Definition
ChurchNum_8ary_proj_p
:=
λ 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 → 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
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
)
→ ο
.
x1
(
λ x2 x3 x4 x5 x6 x7 x8 x9 :
(
ι → ι
)
→
ι → ι
.
x2
)
⟶
x1
(
λ x2 x3 x4 x5 x6 x7 x8 x9 :
(
ι → ι
)
→
ι → ι
.
x3
)
⟶
x1
(
λ x2 x3 x4 x5 x6 x7 x8 x9 :
(
ι → ι
)
→
ι → ι
.
x4
)
⟶
x1
(
λ x2 x3 x4 x5 x6 x7 x8 x9 :
(
ι → ι
)
→
ι → ι
.
x5
)
⟶
x1
(
λ x2 x3 x4 x5 x6 x7 x8 x9 :
(
ι → ι
)
→
ι → ι
.
x6
)
⟶
x1
(
λ x2 x3 x4 x5 x6 x7 x8 x9 :
(
ι → ι
)
→
ι → ι
.
x7
)
⟶
x1
(
λ x2 x3 x4 x5 x6 x7 x8 x9 :
(
ι → ι
)
→
ι → ι
.
x8
)
⟶
x1
(
λ x2 x3 x4 x5 x6 x7 x8 x9 :
(
ι → ι
)
→
ι → ι
.
x9
)
⟶
x1
x0
Theorem
cef55..
:
ChurchNum_3ary_proj_p
(
λ x0 x1 x2 :
(
ι → ι
)
→
ι → ι
.
x0
)
(proof)
Theorem
18961..
:
ChurchNum_3ary_proj_p
(
λ x0 x1 x2 :
(
ι → ι
)
→
ι → ι
.
x1
)
(proof)
Theorem
a5963..
:
ChurchNum_3ary_proj_p
(
λ x0 x1 x2 :
(
ι → ι
)
→
ι → ι
.
x2
)
(proof)
Theorem
208f3..
:
ChurchNum_8ary_proj_p
(
λ x0 x1 x2 x3 x4 x5 x6 x7 :
(
ι → ι
)
→
ι → ι
.
x0
)
(proof)
Theorem
ef1ab..
:
ChurchNum_8ary_proj_p
(
λ x0 x1 x2 x3 x4 x5 x6 x7 :
(
ι → ι
)
→
ι → ι
.
x1
)
(proof)
Theorem
e5caa..
:
ChurchNum_8ary_proj_p
(
λ x0 x1 x2 x3 x4 x5 x6 x7 :
(
ι → ι
)
→
ι → ι
.
x2
)
(proof)
Theorem
446d8..
:
ChurchNum_8ary_proj_p
(
λ x0 x1 x2 x3 x4 x5 x6 x7 :
(
ι → ι
)
→
ι → ι
.
x3
)
(proof)
Theorem
3a83b..
:
ChurchNum_8ary_proj_p
(
λ x0 x1 x2 x3 x4 x5 x6 x7 :
(
ι → ι
)
→
ι → ι
.
x4
)
(proof)
Theorem
0ab9f..
:
ChurchNum_8ary_proj_p
(
λ x0 x1 x2 x3 x4 x5 x6 x7 :
(
ι → ι
)
→
ι → ι
.
x5
)
(proof)
Theorem
94187..
:
ChurchNum_8ary_proj_p
(
λ x0 x1 x2 x3 x4 x5 x6 x7 :
(
ι → ι
)
→
ι → ι
.
x6
)
(proof)
Theorem
7734d..
:
ChurchNum_8ary_proj_p
(
λ x0 x1 x2 x3 x4 x5 x6 x7 :
(
ι → ι
)
→
ι → ι
.
x7
)
(proof)
Theorem
081ec..
:
∀ x0 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
(
λ x1 x2 x3 :
(
ι → ι
)
→
ι → ι
.
x0
x2
x3
x1
)
(proof)
Theorem
d5994..
:
∀ x0 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
(
λ x1 x2 x3 :
(
ι → ι
)
→
ι → ι
.
x0
x3
x1
x2
)
(proof)
Theorem
b0f0d..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_lt1_id_ge1_rot2
x0
x1
)
(proof)
Theorem
16e4b..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_lt2_id_ge2_rot2
x0
x1
)
(proof)
Theorem
ec90a..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_to_3_lt3_id_ge3_rot2
x0
x1
)
(proof)
Theorem
24233..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_to_3_lt4_id_ge4_rot2
x0
x1
)
(proof)
Theorem
7b754..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_to_3_lt5_id_ge5_rot2
x0
x1
)
(proof)
Theorem
2f553..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_to_3_lt6_id_ge6_rot2
x0
x1
)
(proof)
Theorem
1aa1c..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_to_3_lt7_id_ge7_rot2
x0
x1
)
(proof)
Theorem
0bfaf..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_lt1_id_ge1_rot1
x0
x1
)
(proof)
Theorem
1d5ae..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_lt2_id_ge2_rot1
x0
x1
)
(proof)
Theorem
f7f04..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_lt3_id_ge3_rot1
x0
x1
)
(proof)
Theorem
b68fb..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_lt4_id_ge4_rot1
x0
x1
)
(proof)
Theorem
a776d..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_lt5_id_ge5_rot1
x0
x1
)
(proof)
Theorem
6c736..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_lt6_id_ge6_rot1
x0
x1
)
(proof)
Theorem
f8e90..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_3ary_proj_p
(
ChurchNums_8x3_lt7_id_ge7_rot1
x0
x1
)
(proof)
Theorem
c2824..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_lt1_id_ge1_rot2
x0
(
ChurchNums_8x3_lt1_id_ge1_rot1
x0
x1
)
=
x1
(proof)
Theorem
0e9d3..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_lt2_id_ge2_rot2
x0
(
ChurchNums_8x3_lt2_id_ge2_rot1
x0
x1
)
=
x1
(proof)
Theorem
92519..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_to_3_lt3_id_ge3_rot2
x0
(
ChurchNums_8x3_lt3_id_ge3_rot1
x0
x1
)
=
x1
(proof)
Theorem
ad374..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_to_3_lt4_id_ge4_rot2
x0
(
ChurchNums_8x3_lt4_id_ge4_rot1
x0
x1
)
=
x1
(proof)
Theorem
a48ad..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_to_3_lt5_id_ge5_rot2
x0
(
ChurchNums_8x3_lt5_id_ge5_rot1
x0
x1
)
=
x1
(proof)
Theorem
fa851..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_to_3_lt6_id_ge6_rot2
x0
(
ChurchNums_8x3_lt6_id_ge6_rot1
x0
x1
)
=
x1
(proof)
Theorem
dfbcd..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_to_3_lt7_id_ge7_rot2
x0
(
ChurchNums_8x3_lt7_id_ge7_rot1
x0
x1
)
=
x1
(proof)
Theorem
b782a..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_lt1_id_ge1_rot1
x0
(
ChurchNums_8x3_lt1_id_ge1_rot2
x0
x1
)
=
x1
(proof)
Theorem
53c9b..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_lt2_id_ge2_rot1
x0
(
ChurchNums_8x3_lt2_id_ge2_rot2
x0
x1
)
=
x1
(proof)
Theorem
28253..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_lt3_id_ge3_rot1
x0
(
ChurchNums_8x3_to_3_lt3_id_ge3_rot2
x0
x1
)
=
x1
(proof)
Theorem
49e33..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_lt4_id_ge4_rot1
x0
(
ChurchNums_8x3_to_3_lt4_id_ge4_rot2
x0
x1
)
=
x1
(proof)
Theorem
b1cc9..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_lt5_id_ge5_rot1
x0
(
ChurchNums_8x3_to_3_lt5_id_ge5_rot2
x0
x1
)
=
x1
(proof)
Theorem
8922d..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_lt6_id_ge6_rot1
x0
(
ChurchNums_8x3_to_3_lt6_id_ge6_rot2
x0
x1
)
=
x1
(proof)
Theorem
e57bb..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
∀ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNums_8x3_lt7_id_ge7_rot1
x0
(
ChurchNums_8x3_to_3_lt7_id_ge7_rot2
x0
x1
)
=
x1
(proof)
Theorem
4ac5f..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
(
ChurchNums_8_perm_1_2_3_4_5_6_7_0
x0
)
(proof)
Theorem
c5de4..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
(
ChurchNums_8_perm_2_3_4_5_6_7_0_1
x0
)
(proof)
Theorem
eaaf4..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
(
ChurchNums_8_perm_3_4_5_6_7_0_1_2
x0
)
(proof)
Theorem
dac10..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
(
ChurchNums_8_perm_4_5_6_7_0_1_2_3
x0
)
(proof)
Theorem
eed30..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
(
ChurchNums_8_perm_5_6_7_0_1_2_3_4
x0
)
(proof)
Theorem
a5ec6..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
(
ChurchNums_8_perm_6_7_0_1_2_3_4_5
x0
)
(proof)
Theorem
e1672..
:
∀ 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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_8ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
(
ChurchNums_8_perm_7_0_1_2_3_4_5_6
x0
)
(proof)
Definition
TwoRamseyGraph_4_5_24_ChurchNums_3x8
:=
λ x0 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x2 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ 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
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
λ x4 .
x0
(
x1
(
x2
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
)
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
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)
(
λ x5 :
ι → ι
.
x5
)
)
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
)
)
(
x2
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
)
(
x3
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
)
(
x3
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
λ x6 .
x6
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
(
λ x5 :
ι → ι
.
x5
)
)
)
)
(
λ x5 .
x4
)
Theorem
f60cd..
:
∀ x0 x1 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x2 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
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_3ary_proj_p
x1
⟶
ChurchNum_8ary_proj_p
x2
⟶
ChurchNum_8ary_proj_p
x3
⟶
TwoRamseyGraph_4_5_24_ChurchNums_3x8
x0
x2
x1
x3
=
TwoRamseyGraph_4_5_24_ChurchNums_3x8
x1
x3
x0
x2
(proof)
Param
ordsucc
ordsucc
:
ι
→
ι
Definition
TwoRamseyGraph_4_5_24
:=
λ x0 x1 .
∀ x2 x3 :
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
∀ x4 x5 :
(
(
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 → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x2
⟶
ChurchNum_3ary_proj_p
x3
⟶
ChurchNum_8ary_proj_p
x4
⟶
ChurchNum_8ary_proj_p
x5
⟶
x0
=
x2
(
λ x7 :
ι → ι
.
λ x8 .
x8
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
ordsucc
(
x4
(
λ x7 :
ι → ι
.
λ x8 .
x8
)
(
λ x7 :
ι → ι
.
x7
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
x8
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
x8
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
)
)
ordsucc
0
)
⟶
x1
=
x3
(
λ x7 :
ι → ι
.
λ x8 .
x8
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
ordsucc
(
x5
(
λ x7 :
ι → ι
.
λ x8 .
x8
)
(
λ x7 :
ι → ι
.
x7
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
x8
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
x8
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
)
(
λ x7 :
ι → ι
.
λ x8 .
x7
(
x7
(
x7
(
x7
(
x7
(
x7
(
x7
x8
)
)
)
)
)
)
)
ordsucc
0
)
⟶
TwoRamseyGraph_4_5_24_ChurchNums_3x8
x2
x4
x3
x5
=
λ x7 x8 .
x7
Theorem
9303b..
:
∀ x0 x1 .
TwoRamseyGraph_4_5_24
x0
x1
⟶
TwoRamseyGraph_4_5_24
x1
x0
(proof)