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Proofgold Asset
asset id
97e2fda7e170720968710e7c84da407b827d16d8ff0fc19a1526cc3c3845aa4d
asset hash
7c6d16e2ae93622ee4638e7c5592cc365e3d0fd727932956393174d2d7f0ddb5
bday / block
18750
tx
0ee1b..
preasset
doc published by
Pr4zB..
Param
ordsucc
ordsucc
:
ι
→
ι
Definition
ChurchNums_3x8_to_u24
:=
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
.
x0
(
x1
(
λ x2 :
ι → ι
.
λ x3 .
x3
)
(
λ x2 :
ι → ι
.
x2
)
(
λ x2 :
ι → ι
.
λ x3 .
x2
(
x2
x3
)
)
(
λ x2 :
ι → ι
.
λ x3 .
x2
(
x2
(
x2
x3
)
)
)
(
λ x2 :
ι → ι
.
λ x3 .
x2
(
x2
(
x2
(
x2
x3
)
)
)
)
(
λ x2 :
ι → ι
.
λ x3 .
x2
(
x2
(
x2
(
x2
(
x2
x3
)
)
)
)
)
(
λ x2 :
ι → ι
.
λ x3 .
x2
(
x2
(
x2
(
x2
(
x2
(
x2
x3
)
)
)
)
)
)
(
λ x2 :
ι → ι
.
λ x3 .
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
x3
)
)
)
)
)
)
)
)
(
λ x2 :
ι → ι
.
λ x3 .
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x1
(
λ x4 :
ι → ι
.
λ x5 .
x5
)
(
λ x4 :
ι → ι
.
x4
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
x5
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
x5
)
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
(
x4
x5
)
)
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
(
x4
(
x4
x5
)
)
)
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
(
x4
(
x4
(
x4
x5
)
)
)
)
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
(
x4
(
x4
(
x4
(
x4
x5
)
)
)
)
)
)
)
x2
x3
)
)
)
)
)
)
)
)
)
(
λ x2 :
ι → ι
.
λ x3 .
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x2
(
x1
(
λ x4 :
ι → ι
.
λ x5 .
x5
)
(
λ x4 :
ι → ι
.
x4
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
x5
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
x5
)
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
(
x4
x5
)
)
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
(
x4
(
x4
x5
)
)
)
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
(
x4
(
x4
(
x4
x5
)
)
)
)
)
)
(
λ x4 :
ι → ι
.
λ x5 .
x4
(
x4
(
x4
(
x4
(
x4
(
x4
(
x4
x5
)
)
)
)
)
)
)
x2
x3
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
ordsucc
0
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
Definition
u1
:=
1
Definition
u2
:=
ordsucc
u1
Definition
u3
:=
ordsucc
u2
Definition
u4
:=
ordsucc
u3
Definition
u5
:=
ordsucc
u4
Definition
u6
:=
ordsucc
u5
Definition
u7
:=
ordsucc
u6
Definition
u8
:=
ordsucc
u7
Definition
u9
:=
ordsucc
u8
Definition
u10
:=
ordsucc
u9
Definition
u11
:=
ordsucc
u10
Definition
u12
:=
ordsucc
u11
Definition
u13
:=
ordsucc
u12
Definition
u14
:=
ordsucc
u13
Definition
u15
:=
ordsucc
u14
Definition
u16
:=
ordsucc
u15
Definition
u17
:=
ordsucc
u16
Definition
u18
:=
ordsucc
u17
Definition
u19
:=
ordsucc
u18
Definition
u20
:=
ordsucc
u19
Definition
u21
:=
ordsucc
u20
Definition
u22
:=
ordsucc
u21
Definition
u23
:=
ordsucc
u22
Definition
u24
:=
ordsucc
u23
Known
4c607..
:
0
∈
u24
Known
610bf..
:
u1
∈
u24
Known
30da6..
:
u2
∈
u24
Known
3cb84..
:
u3
∈
u24
Known
43f74..
:
u4
∈
u24
Known
8f227..
:
u5
∈
u24
Known
469db..
:
u6
∈
u24
Known
bb94c..
:
u7
∈
u24
Known
d2f24..
:
u8
∈
u24
Known
fd0d9..
:
u9
∈
u24
Known
df5ed..
:
u10
∈
u24
Known
05672..
:
u11
∈
u24
Known
15af9..
:
u12
∈
u24
Known
e6799..
:
u13
∈
u24
Known
9522d..
:
u14
∈
u24
Known
fc764..
:
u15
∈
u24
Known
2d7ca..
:
u16
∈
u24
Known
e4fb5..
:
u17
∈
u24
Known
f0252..
:
u18
∈
u24
Known
a4364..
:
u19
∈
u24
Known
9da85..
:
u20
∈
u24
Known
3ee80..
:
u21
∈
u24
Known
9ec16..
:
u22
∈
u24
Known
d7b2c..
:
u23
∈
u24
Theorem
16d18..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
x0
x1
∈
u24
(proof)
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_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
Known
neq_1_0
neq_1_0
:
u1
=
0
⟶
∀ x0 : ο .
x0
Known
neq_2_1
neq_2_1
:
u2
=
u1
⟶
∀ x0 : ο .
x0
Known
neq_3_2
neq_3_2
:
u3
=
u2
⟶
∀ x0 : ο .
x0
Known
neq_4_3
neq_4_3
:
u4
=
u3
⟶
∀ x0 : ο .
x0
Known
neq_5_4
neq_5_4
:
u5
=
u4
⟶
∀ x0 : ο .
x0
Known
neq_6_5
neq_6_5
:
u6
=
u5
⟶
∀ x0 : ο .
x0
Known
neq_7_6
neq_7_6
:
u7
=
u6
⟶
∀ x0 : ο .
x0
Known
neq_8_7
neq_8_7
:
u8
=
u7
⟶
∀ x0 : ο .
x0
Known
neq_9_8
neq_9_8
:
u9
=
u8
⟶
∀ x0 : ο .
x0
Known
4fc31..
:
u10
=
u9
⟶
∀ x0 : ο .
x0
Known
ebfb7..
:
u11
=
u10
⟶
∀ x0 : ο .
x0
Known
ab306..
:
u12
=
u11
⟶
∀ x0 : ο .
x0
Known
ad02f..
:
u13
=
u12
⟶
∀ x0 : ο .
x0
Known
e1947..
:
u14
=
u13
⟶
∀ x0 : ο .
x0
Known
b8e82..
:
u15
=
u14
⟶
∀ x0 : ο .
x0
Known
41073..
:
u16
=
u15
⟶
∀ x0 : ο .
x0
Known
7fbc8..
:
u17
=
u16
⟶
∀ x0 : ο .
x0
Known
82c6a..
:
u18
=
u17
⟶
∀ x0 : ο .
x0
Known
97eb4..
:
u19
=
u18
⟶
∀ x0 : ο .
x0
Known
2615b..
:
u20
=
u19
⟶
∀ x0 : ο .
x0
Known
32e25..
:
u21
=
u20
⟶
∀ x0 : ο .
x0
Known
41315..
:
u22
=
u21
⟶
∀ x0 : ο .
x0
Known
3105f..
:
u23
=
u22
⟶
∀ x0 : ο .
x0
Known
neq_i_sym
neq_i_sym
:
∀ x0 x1 .
(
x0
=
x1
⟶
∀ x2 : ο .
x2
)
⟶
x1
=
x0
⟶
∀ x2 : ο .
x2
Known
c432c..
:
u23
=
0
⟶
∀ x0 : ο .
x0
Theorem
8f0b2..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt7_id_ge7_rot2
x1
x0
)
(
ChurchNums_8_perm_1_2_3_4_5_6_7_0
x1
)
=
ChurchNums_3x8_to_u24
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
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_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
Known
neq_2_0
neq_2_0
:
u2
=
0
⟶
∀ x0 : ο .
x0
Known
neq_3_1
neq_3_1
:
u3
=
u1
⟶
∀ x0 : ο .
x0
Known
neq_4_2
neq_4_2
:
u4
=
u2
⟶
∀ x0 : ο .
x0
Known
neq_5_3
neq_5_3
:
u5
=
u3
⟶
∀ x0 : ο .
x0
Known
neq_6_4
neq_6_4
:
u6
=
u4
⟶
∀ x0 : ο .
x0
Known
neq_7_5
neq_7_5
:
u7
=
u5
⟶
∀ x0 : ο .
x0
Known
neq_8_6
neq_8_6
:
u8
=
u6
⟶
∀ x0 : ο .
x0
Known
neq_9_7
neq_9_7
:
u9
=
u7
⟶
∀ x0 : ο .
x0
Known
96175..
:
u10
=
u8
⟶
∀ x0 : ο .
x0
Known
4f03f..
:
u11
=
u9
⟶
∀ x0 : ο .
x0
Known
6c583..
:
u12
=
u10
⟶
∀ x0 : ο .
x0
Known
bf497..
:
u13
=
u11
⟶
∀ x0 : ο .
x0
Known
ef4da..
:
u14
=
u12
⟶
∀ x0 : ο .
x0
Known
4d8d4..
:
u15
=
u13
⟶
∀ x0 : ο .
x0
Known
71c5e..
:
u16
=
u14
⟶
∀ x0 : ο .
x0
Known
ac12b..
:
u17
=
u15
⟶
∀ x0 : ο .
x0
Known
0eaf4..
:
u18
=
u16
⟶
∀ x0 : ο .
x0
Known
3c054..
:
u19
=
u17
⟶
∀ x0 : ο .
x0
Known
75fad..
:
u20
=
u18
⟶
∀ x0 : ο .
x0
Known
44711..
:
u21
=
u19
⟶
∀ x0 : ο .
x0
Known
c8ac0..
:
u22
=
u20
⟶
∀ x0 : ο .
x0
Known
1a616..
:
u23
=
u21
⟶
∀ x0 : ο .
x0
Known
e8714..
:
u22
=
0
⟶
∀ x0 : ο .
x0
Known
13d86..
:
u23
=
u1
⟶
∀ x0 : ο .
x0
Theorem
7bf94..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt6_id_ge6_rot2
x1
x0
)
(
ChurchNums_8_perm_2_3_4_5_6_7_0_1
x1
)
=
ChurchNums_3x8_to_u24
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
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_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
Known
neq_3_0
neq_3_0
:
u3
=
0
⟶
∀ x0 : ο .
x0
Known
neq_4_1
neq_4_1
:
u4
=
u1
⟶
∀ x0 : ο .
x0
Known
neq_5_2
neq_5_2
:
u5
=
u2
⟶
∀ x0 : ο .
x0
Known
neq_6_3
neq_6_3
:
u6
=
u3
⟶
∀ x0 : ο .
x0
Known
neq_7_4
neq_7_4
:
u7
=
u4
⟶
∀ x0 : ο .
x0
Known
neq_8_5
neq_8_5
:
u8
=
u5
⟶
∀ x0 : ο .
x0
Known
neq_9_6
neq_9_6
:
u9
=
u6
⟶
∀ x0 : ο .
x0
Known
7d7a8..
:
u10
=
u7
⟶
∀ x0 : ο .
x0
Known
b3a20..
:
u11
=
u8
⟶
∀ x0 : ο .
x0
Known
22885..
:
u12
=
u9
⟶
∀ x0 : ο .
x0
Known
78358..
:
u13
=
u10
⟶
∀ x0 : ο .
x0
Known
4e1aa..
:
u14
=
u11
⟶
∀ x0 : ο .
x0
Known
72647..
:
u15
=
u12
⟶
∀ x0 : ο .
x0
Known
4326e..
:
u16
=
u13
⟶
∀ x0 : ο .
x0
Known
82608..
:
u17
=
u14
⟶
∀ x0 : ο .
x0
Known
dfba1..
:
u18
=
u15
⟶
∀ x0 : ο .
x0
Known
0384c..
:
u19
=
u16
⟶
∀ x0 : ο .
x0
Known
9ce5b..
:
u20
=
u17
⟶
∀ x0 : ο .
x0
Known
80a82..
:
u21
=
u18
⟶
∀ x0 : ο .
x0
Known
b0147..
:
u22
=
u19
⟶
∀ x0 : ο .
x0
Known
94779..
:
u23
=
u20
⟶
∀ x0 : ο .
x0
Known
1158c..
:
u21
=
0
⟶
∀ x0 : ο .
x0
Known
9e7b1..
:
u22
=
u1
⟶
∀ x0 : ο .
x0
Known
60a3a..
:
u23
=
u2
⟶
∀ x0 : ο .
x0
Theorem
a9b55..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt5_id_ge5_rot2
x1
x0
)
(
ChurchNums_8_perm_3_4_5_6_7_0_1_2
x1
)
=
ChurchNums_3x8_to_u24
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
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_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
Known
neq_4_0
neq_4_0
:
u4
=
0
⟶
∀ x0 : ο .
x0
Known
neq_5_1
neq_5_1
:
u5
=
u1
⟶
∀ x0 : ο .
x0
Known
neq_6_2
neq_6_2
:
u6
=
u2
⟶
∀ x0 : ο .
x0
Known
neq_7_3
neq_7_3
:
u7
=
u3
⟶
∀ x0 : ο .
x0
Known
neq_8_4
neq_8_4
:
u8
=
u4
⟶
∀ x0 : ο .
x0
Known
neq_9_5
neq_9_5
:
u9
=
u5
⟶
∀ x0 : ο .
x0
Known
d0401..
:
u10
=
u6
⟶
∀ x0 : ο .
x0
Known
4abfa..
:
u11
=
u7
⟶
∀ x0 : ο .
x0
Known
a6a6c..
:
u12
=
u8
⟶
∀ x0 : ο .
x0
Known
3f24c..
:
u13
=
u9
⟶
∀ x0 : ο .
x0
Known
f5ab5..
:
u14
=
u10
⟶
∀ x0 : ο .
x0
Known
9c5db..
:
u15
=
u11
⟶
∀ x0 : ο .
x0
Known
fa664..
:
u16
=
u12
⟶
∀ x0 : ο .
x0
Known
30174..
:
u17
=
u13
⟶
∀ x0 : ο .
x0
Known
d92fd..
:
u18
=
u14
⟶
∀ x0 : ο .
x0
Known
38ccc..
:
u19
=
u15
⟶
∀ x0 : ο .
x0
Known
996e8..
:
u20
=
u16
⟶
∀ x0 : ο .
x0
Known
b821e..
:
u21
=
u17
⟶
∀ x0 : ο .
x0
Known
7957c..
:
u22
=
u18
⟶
∀ x0 : ο .
x0
Known
ad532..
:
u23
=
u19
⟶
∀ x0 : ο .
x0
Known
4552b..
:
u20
=
0
⟶
∀ x0 : ο .
x0
Known
db0cd..
:
u21
=
u1
⟶
∀ x0 : ο .
x0
Known
af720..
:
u22
=
u2
⟶
∀ x0 : ο .
x0
Known
3d5c1..
:
u23
=
u3
⟶
∀ x0 : ο .
x0
Theorem
d6534..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt4_id_ge4_rot2
x1
x0
)
(
ChurchNums_8_perm_4_5_6_7_0_1_2_3
x1
)
=
ChurchNums_3x8_to_u24
x0
x1
⟶
∀ x2 : ο .
x2
(proof)
Known
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
)
Known
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
)
Theorem
471b2..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt6_id_ge6_rot2
x1
x0
)
(
ChurchNums_8_perm_2_3_4_5_6_7_0_1
x1
)
=
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt7_id_ge7_rot2
x1
x0
)
(
ChurchNums_8_perm_1_2_3_4_5_6_7_0
x1
)
⟶
∀ x2 : ο .
x2
(proof)
Known
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
)
Known
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
)
Theorem
373fc..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt5_id_ge5_rot2
x1
x0
)
(
ChurchNums_8_perm_3_4_5_6_7_0_1_2
x1
)
=
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt6_id_ge6_rot2
x1
x0
)
(
ChurchNums_8_perm_2_3_4_5_6_7_0_1
x1
)
⟶
∀ x2 : ο .
x2
(proof)
Known
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
)
Known
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
)
Theorem
987aa..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt4_id_ge4_rot2
x1
x0
)
(
ChurchNums_8_perm_4_5_6_7_0_1_2_3
x1
)
=
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt5_id_ge5_rot2
x1
x0
)
(
ChurchNums_8_perm_3_4_5_6_7_0_1_2
x1
)
⟶
∀ x2 : ο .
x2
(proof)
Theorem
08b75..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt5_id_ge5_rot2
x1
x0
)
(
ChurchNums_8_perm_3_4_5_6_7_0_1_2
x1
)
=
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt7_id_ge7_rot2
x1
x0
)
(
ChurchNums_8_perm_1_2_3_4_5_6_7_0
x1
)
⟶
∀ x2 : ο .
x2
(proof)
Theorem
290ad..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt4_id_ge4_rot2
x1
x0
)
(
ChurchNums_8_perm_4_5_6_7_0_1_2_3
x1
)
=
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt6_id_ge6_rot2
x1
x0
)
(
ChurchNums_8_perm_2_3_4_5_6_7_0_1
x1
)
⟶
∀ x2 : ο .
x2
(proof)
Theorem
83ebc..
:
∀ 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 → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
(
ι → ι
)
→
ι → ι
)
→
(
ι → ι
)
→
ι → ι
.
ChurchNum_3ary_proj_p
x0
⟶
ChurchNum_8ary_proj_p
x1
⟶
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt4_id_ge4_rot2
x1
x0
)
(
ChurchNums_8_perm_4_5_6_7_0_1_2_3
x1
)
=
ChurchNums_3x8_to_u24
(
ChurchNums_8x3_to_3_lt7_id_ge7_rot2
x1
x0
)
(
ChurchNums_8_perm_1_2_3_4_5_6_7_0
x1
)
⟶
∀ x2 : ο .
x2
(proof)
Param
nat_p
nat_p
:
ι
→
ο
Definition
False
False
:=
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Definition
inj
inj
:=
λ x0 x1 .
λ x2 :
ι → ι
.
and
(
∀ x3 .
x3
∈
x0
⟶
x2
x3
∈
x1
)
(
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
x2
x3
=
x2
x4
⟶
x3
=
x4
)
Definition
atleastp
atleastp
:=
λ x0 x1 .
∀ x2 : ο .
(
∀ x3 :
ι → ι
.
inj
x0
x1
x3
⟶
x2
)
⟶
x2
Known
PigeonHole_nat
PigeonHole_nat
:
∀ x0 .
nat_p
x0
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
x2
∈
ordsucc
x0
⟶
x1
x2
∈
x0
)
⟶
not
(
∀ x2 .
x2
∈
ordsucc
x0
⟶
∀ x3 .
x3
∈
ordsucc
x0
⟶
x1
x2
=
x1
x3
⟶
x2
=
x3
)
Theorem
4fb58..
Pigeonhole_not_atleastp_ordsucc
:
∀ x0 .
nat_p
x0
⟶
not
(
atleastp
(
ordsucc
x0
)
x0
)
(proof)
Definition
bij
bij
:=
λ x0 x1 .
λ x2 :
ι → ι
.
and
(
and
(
∀ x3 .
x3
∈
x0
⟶
x2
x3
∈
x1
)
(
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
x2
x3
=
x2
x4
⟶
x3
=
x4
)
)
(
∀ x3 .
x3
∈
x1
⟶
∀ x4 : ο .
(
∀ x5 .
and
(
x5
∈
x0
)
(
x2
x5
=
x3
)
⟶
x4
)
⟶
x4
)
Definition
equip
equip
:=
λ x0 x1 .
∀ x2 : ο .
(
∀ x3 :
ι → ι
.
bij
x0
x1
x3
⟶
x2
)
⟶
x2
Param
mul_nat
mul_nat
:
ι
→
ι
→
ι
Param
lam
Sigma
:
ι
→
(
ι
→
ι
) →
ι
Definition
setprod
setprod
:=
λ x0 x1 .
lam
x0
(
λ x2 .
x1
)
Known
nat_ind
nat_ind
:
∀ x0 :
ι → ο
.
x0
0
⟶
(
∀ x1 .
nat_p
x1
⟶
x0
x1
⟶
x0
(
ordsucc
x1
)
)
⟶
∀ x1 .
nat_p
x1
⟶
x0
x1
Known
mul_nat_0R
mul_nat_0R
:
∀ x0 .
mul_nat
x0
0
=
0
Known
equip_ref
equip_ref
:
∀ x0 .
equip
x0
x0
Definition
nIn
nIn
:=
λ x0 x1 .
not
(
x0
∈
x1
)
Known
Empty_eq
Empty_eq
:
∀ x0 .
(
∀ x1 .
nIn
x1
x0
)
⟶
x0
=
0
Param
ap
ap
:
ι
→
ι
→
ι
Known
EmptyE
EmptyE
:
∀ x0 .
nIn
x0
0
Known
ap1_Sigma
ap1_Sigma
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
lam
x0
x1
⟶
ap
x2
1
∈
x1
(
ap
x2
0
)
Param
add_nat
add_nat
:
ι
→
ι
→
ι
Known
mul_nat_SR
mul_nat_SR
:
∀ x0 x1 .
nat_p
x1
⟶
mul_nat
x0
(
ordsucc
x1
)
=
add_nat
x0
(
mul_nat
x0
x1
)
Param
setsum
setsum
:
ι
→
ι
→
ι
Known
equip_tra
equip_tra
:
∀ x0 x1 x2 .
equip
x0
x1
⟶
equip
x1
x2
⟶
equip
x0
x2
Known
c88e0..
:
∀ x0 x1 x2 x3 .
nat_p
x0
⟶
nat_p
x1
⟶
equip
x0
x2
⟶
equip
x1
x3
⟶
equip
(
add_nat
x0
x1
)
(
setsum
x2
x3
)
Known
mul_nat_p
mul_nat_p
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
nat_p
(
mul_nat
x0
x1
)
Param
combine_funcs
combine_funcs
:
ι
→
ι
→
(
ι
→
ι
) →
(
ι
→
ι
) →
ι
→
ι
Param
If_i
If_i
:
ο
→
ι
→
ι
→
ι
Param
Inj0
Inj0
:
ι
→
ι
Param
Inj1
Inj1
:
ι
→
ι
Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Known
and3I
and3I
:
∀ x0 x1 x2 : ο .
x0
⟶
x1
⟶
x2
⟶
and
(
and
x0
x1
)
x2
Known
f4c7c..
:
∀ x0 x1 .
∀ x2 :
ι → ο
.
(
∀ x3 .
x3
∈
x0
⟶
x2
(
Inj0
x3
)
)
⟶
(
∀ x3 .
x3
∈
x1
⟶
x2
(
Inj1
x3
)
)
⟶
∀ x3 .
x3
∈
setsum
x0
x1
⟶
x2
x3
Known
tuple_2_setprod
tuple_2_setprod
:
∀ x0 x1 x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x1
⟶
lam
2
(
λ x4 .
If_i
(
x4
=
0
)
x2
x3
)
∈
setprod
x0
x1
Known
ordsuccI2
ordsuccI2
:
∀ x0 .
x0
∈
ordsucc
x0
Known
tuple_2_0_eq
tuple_2_0_eq
:
∀ x0 x1 .
ap
(
lam
2
(
λ x3 .
If_i
(
x3
=
0
)
x0
x1
)
)
0
=
x0
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
In_irref
In_irref
:
∀ x0 .
nIn
x0
x0
Known
tuple_2_1_eq
tuple_2_1_eq
:
∀ x0 x1 .
ap
(
lam
2
(
λ x3 .
If_i
(
x3
=
0
)
x0
x1
)
)
1
=
x1
Definition
or
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Known
ordsuccE
ordsuccE
:
∀ x0 x1 .
x1
∈
ordsucc
x0
⟶
or
(
x1
∈
x0
)
(
x1
=
x0
)
Known
andI
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Known
Inj1_setsum
Inj1_setsum
:
∀ x0 x1 x2 .
x2
∈
x1
⟶
Inj1
x2
∈
setsum
x0
x1
Known
tuple_Sigma_eta
tuple_Sigma_eta
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
lam
x0
x1
⟶
lam
2
(
λ x4 .
If_i
(
x4
=
0
)
(
ap
x2
0
)
(
ap
x2
1
)
)
=
x2
Known
Inj0_setsum
Inj0_setsum
:
∀ x0 x1 x2 .
x2
∈
x0
⟶
Inj0
x2
∈
setsum
x0
x1
Known
ap0_Sigma
ap0_Sigma
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
lam
x0
x1
⟶
ap
x2
0
∈
x0
Known
setprod_mon
setprod_mon
:
∀ x0 x1 .
x0
⊆
x1
⟶
∀ x2 x3 .
x2
⊆
x3
⟶
setprod
x0
x2
⊆
setprod
x1
x3
Known
Subq_ref
Subq_ref
:
∀ x0 .
x0
⊆
x0
Known
ordsuccI1
ordsuccI1
:
∀ x0 .
x0
⊆
ordsucc
x0
Known
combine_funcs_eq2
combine_funcs_eq2
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 .
combine_funcs
x0
x1
x2
x3
(
Inj1
x4
)
=
x3
x4
Known
combine_funcs_eq1
combine_funcs_eq1
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 .
combine_funcs
x0
x1
x2
x3
(
Inj0
x4
)
=
x2
x4
Theorem
a57cb..
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
equip
(
mul_nat
x0
x1
)
(
setprod
x0
x1
)
(proof)
Definition
u25
:=
ordsucc
u24
Known
nat_4
nat_4
:
nat_p
4
Known
nat_3
nat_3
:
nat_p
3
Known
nat_2
nat_2
:
nat_p
2
Known
nat_1
nat_1
:
nat_p
1
Known
mul_nat_1R
mul_nat_1R
:
∀ x0 .
mul_nat
x0
1
=
x0
Param
ChurchNum_p
:
(
(
ι
→
ι
) →
ι
→
ι
) →
ο
Known
39d66..
:
∀ x0 :
(
ι → ι
)
→
ι → ι
.
ChurchNum_p
x0
⟶
∀ x1 :
(
ι → ι
)
→
ι → ι
.
ChurchNum_p
x1
⟶
x0
ordsucc
(
x1
ordsucc
0
)
=
add_nat
(
x0
ordsucc
0
)
(
x1
ordsucc
0
)
Known
c9952..
:
ChurchNum_p
(
λ x0 :
ι → ι
.
λ x1 .
x0
(
x0
(
x0
(
x0
(
x0
x1
)
)
)
)
)
Known
0a45b..
:
ChurchNum_p
(
λ x0 :
ι → ι
.
λ x1 .
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
x1
)
)
)
)
)
)
)
)
)
)
Known
466b2..
:
ChurchNum_p
(
λ x0 :
ι → ι
.
λ x1 .
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
x1
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
Known
4e557..
:
ChurchNum_p
(
λ x0 :
ι → ι
.
λ x1 .
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
(
x0
x1
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
Theorem
c12f7..
:
mul_nat
u5
u5
=
u25
(proof)