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Proofgold Asset
asset id
eb883e8c5b7d708ca8e44274f1c9b049bc43782d4c5cd9f3c795660ac02a02dd
asset hash
c95433ac00f65a5f0d534a14a7fd19bbb2f03f5456320323a5d31543a1a0d333
bday / block
19013
tx
7184d..
preasset
doc published by
Pr4zB..
Param
ordsucc
ordsucc
:
ι
→
ι
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
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Known
ordsuccI1
ordsuccI1
:
∀ x0 .
x0
⊆
ordsucc
x0
Known
e0b58..
:
13
∈
16
Theorem
7315d..
:
u13
∈
u17
(proof)
Known
d4076..
:
14
∈
16
Theorem
35e01..
:
u14
∈
u17
(proof)
Known
0e6a7..
:
15
∈
16
Theorem
31b8d..
:
u15
∈
u17
(proof)
Known
ordsuccI2
ordsuccI2
:
∀ x0 .
x0
∈
ordsucc
x0
Theorem
dfaf3..
:
u16
∈
u17
(proof)
Param
ap
ap
:
ι
→
ι
→
ι
Param
lam
Sigma
:
ι
→
(
ι
→
ι
) →
ι
Param
If_i
If_i
:
ο
→
ι
→
ι
→
ι
Known
733b2..
:
u13
=
0
⟶
∀ x0 : ο .
x0
Known
16246..
:
u13
=
u1
⟶
∀ x0 : ο .
x0
Known
40d25..
:
u13
=
u2
⟶
∀ x0 : ο .
x0
Known
19222..
:
u13
=
u3
⟶
∀ x0 : ο .
x0
Known
4d850..
:
u13
=
u4
⟶
∀ x0 : ο .
x0
Known
29333..
:
u13
=
u5
⟶
∀ x0 : ο .
x0
Known
02f5c..
:
u13
=
u6
⟶
∀ x0 : ο .
x0
Known
d9b35..
:
u13
=
u7
⟶
∀ x0 : ο .
x0
Known
0b225..
:
u13
=
u8
⟶
∀ x0 : ο .
x0
Known
3f24c..
:
u13
=
u9
⟶
∀ x0 : ο .
x0
Known
78358..
:
u13
=
u10
⟶
∀ x0 : ο .
x0
Known
bf497..
:
u13
=
u11
⟶
∀ x0 : ο .
x0
Known
ad02f..
:
u13
=
u12
⟶
∀ x0 : ο .
x0
Theorem
af3c7..
:
(
∀ x0 x1 .
∀ x2 :
ι →
ι → ι
.
∀ x3 .
x3
∈
x1
⟶
ap
(
lam
x1
(
λ x5 .
If_i
(
x5
=
x3
)
x0
(
x2
(
ordsucc
x3
)
x5
)
)
)
x3
=
x0
)
⟶
(
∀ x0 x1 .
∀ x2 :
ι →
ι → ι
.
∀ x3 x4 .
(
x4
=
x3
⟶
∀ x5 : ο .
x5
)
⟶
ap
(
lam
x1
(
λ x6 .
If_i
(
x6
=
x3
)
x0
(
x2
(
ordsucc
x3
)
x6
)
)
)
x4
=
ap
(
lam
x1
(
x2
(
ordsucc
x3
)
)
)
x4
)
⟶
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 .
ap
(
lam
17
(
λ x18 .
If_i
(
x18
=
0
)
x0
(
If_i
(
x18
=
1
)
x1
(
If_i
(
x18
=
2
)
x2
(
If_i
(
x18
=
3
)
x3
(
If_i
(
x18
=
4
)
x4
(
If_i
(
x18
=
5
)
x5
(
If_i
(
x18
=
6
)
x6
(
If_i
(
x18
=
7
)
x7
(
If_i
(
x18
=
8
)
x8
(
If_i
(
x18
=
9
)
x9
(
If_i
(
x18
=
10
)
x10
(
If_i
(
x18
=
11
)
x11
(
If_i
(
x18
=
12
)
x12
(
If_i
(
x18
=
13
)
x13
(
If_i
(
x18
=
14
)
x14
(
If_i
(
x18
=
15
)
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u13
=
x13
(proof)
Known
fc551..
:
u14
=
0
⟶
∀ x0 : ο .
x0
Known
ac679..
:
u14
=
u1
⟶
∀ x0 : ο .
x0
Known
0bb18..
:
u14
=
u2
⟶
∀ x0 : ο .
x0
Known
d0fe4..
:
u14
=
u3
⟶
∀ x0 : ο .
x0
Known
ffd62..
:
u14
=
u4
⟶
∀ x0 : ο .
x0
Known
d6c57..
:
u14
=
u5
⟶
∀ x0 : ο .
x0
Known
62d80..
:
u14
=
u6
⟶
∀ x0 : ο .
x0
Known
01bf6..
:
u14
=
u7
⟶
∀ x0 : ο .
x0
Known
4f6ad..
:
u14
=
u8
⟶
∀ x0 : ο .
x0
Known
d7730..
:
u14
=
u9
⟶
∀ x0 : ο .
x0
Known
f5ab5..
:
u14
=
u10
⟶
∀ x0 : ο .
x0
Known
4e1aa..
:
u14
=
u11
⟶
∀ x0 : ο .
x0
Known
ef4da..
:
u14
=
u12
⟶
∀ x0 : ο .
x0
Known
e1947..
:
u14
=
u13
⟶
∀ x0 : ο .
x0
Theorem
d3cf8..
:
(
∀ x0 x1 .
∀ x2 :
ι →
ι → ι
.
∀ x3 .
x3
∈
x1
⟶
ap
(
lam
x1
(
λ x5 .
If_i
(
x5
=
x3
)
x0
(
x2
(
ordsucc
x3
)
x5
)
)
)
x3
=
x0
)
⟶
(
∀ x0 x1 .
∀ x2 :
ι →
ι → ι
.
∀ x3 x4 .
(
x4
=
x3
⟶
∀ x5 : ο .
x5
)
⟶
ap
(
lam
x1
(
λ x6 .
If_i
(
x6
=
x3
)
x0
(
x2
(
ordsucc
x3
)
x6
)
)
)
x4
=
ap
(
lam
x1
(
x2
(
ordsucc
x3
)
)
)
x4
)
⟶
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 .
ap
(
lam
17
(
λ x18 .
If_i
(
x18
=
0
)
x0
(
If_i
(
x18
=
1
)
x1
(
If_i
(
x18
=
2
)
x2
(
If_i
(
x18
=
3
)
x3
(
If_i
(
x18
=
4
)
x4
(
If_i
(
x18
=
5
)
x5
(
If_i
(
x18
=
6
)
x6
(
If_i
(
x18
=
7
)
x7
(
If_i
(
x18
=
8
)
x8
(
If_i
(
x18
=
9
)
x9
(
If_i
(
x18
=
10
)
x10
(
If_i
(
x18
=
11
)
x11
(
If_i
(
x18
=
12
)
x12
(
If_i
(
x18
=
13
)
x13
(
If_i
(
x18
=
14
)
x14
(
If_i
(
x18
=
15
)
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u14
=
x14
(proof)
Known
160ad..
:
u15
=
0
⟶
∀ x0 : ο .
x0
Known
174d1..
:
u15
=
u1
⟶
∀ x0 : ο .
x0
Known
4d715..
:
u15
=
u2
⟶
∀ x0 : ο .
x0
Known
70124..
:
u15
=
u3
⟶
∀ x0 : ο .
x0
Known
4b742..
:
u15
=
u4
⟶
∀ x0 : ο .
x0
Known
24fad..
:
u15
=
u5
⟶
∀ x0 : ο .
x0
Known
f5ac7..
:
u15
=
u6
⟶
∀ x0 : ο .
x0
Known
008b1..
:
u15
=
u7
⟶
∀ x0 : ο .
x0
Known
c0d75..
:
u15
=
u8
⟶
∀ x0 : ο .
x0
Known
3a7bc..
:
u15
=
u9
⟶
∀ x0 : ο .
x0
Known
b7f53..
:
u15
=
u10
⟶
∀ x0 : ο .
x0
Known
9c5db..
:
u15
=
u11
⟶
∀ x0 : ο .
x0
Known
72647..
:
u15
=
u12
⟶
∀ x0 : ο .
x0
Known
4d8d4..
:
u15
=
u13
⟶
∀ x0 : ο .
x0
Known
b8e82..
:
u15
=
u14
⟶
∀ x0 : ο .
x0
Theorem
96595..
:
(
∀ x0 x1 .
∀ x2 :
ι →
ι → ι
.
∀ x3 .
x3
∈
x1
⟶
ap
(
lam
x1
(
λ x5 .
If_i
(
x5
=
x3
)
x0
(
x2
(
ordsucc
x3
)
x5
)
)
)
x3
=
x0
)
⟶
(
∀ x0 x1 .
∀ x2 :
ι →
ι → ι
.
∀ x3 x4 .
(
x4
=
x3
⟶
∀ x5 : ο .
x5
)
⟶
ap
(
lam
x1
(
λ x6 .
If_i
(
x6
=
x3
)
x0
(
x2
(
ordsucc
x3
)
x6
)
)
)
x4
=
ap
(
lam
x1
(
x2
(
ordsucc
x3
)
)
)
x4
)
⟶
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 .
ap
(
lam
17
(
λ x18 .
If_i
(
x18
=
0
)
x0
(
If_i
(
x18
=
1
)
x1
(
If_i
(
x18
=
2
)
x2
(
If_i
(
x18
=
3
)
x3
(
If_i
(
x18
=
4
)
x4
(
If_i
(
x18
=
5
)
x5
(
If_i
(
x18
=
6
)
x6
(
If_i
(
x18
=
7
)
x7
(
If_i
(
x18
=
8
)
x8
(
If_i
(
x18
=
9
)
x9
(
If_i
(
x18
=
10
)
x10
(
If_i
(
x18
=
11
)
x11
(
If_i
(
x18
=
12
)
x12
(
If_i
(
x18
=
13
)
x13
(
If_i
(
x18
=
14
)
x14
(
If_i
(
x18
=
15
)
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u15
=
x15
(proof)
Known
86ae3..
:
u16
=
0
⟶
∀ x0 : ο .
x0
Known
ab690..
:
u16
=
u1
⟶
∀ x0 : ο .
x0
Known
296ac..
:
u16
=
u2
⟶
∀ x0 : ο .
x0
Known
ca5c3..
:
u16
=
u3
⟶
∀ x0 : ο .
x0
Known
7b2eb..
:
u16
=
u4
⟶
∀ x0 : ο .
x0
Known
35bff..
:
u16
=
u5
⟶
∀ x0 : ο .
x0
Known
3bd28..
:
u16
=
u6
⟶
∀ x0 : ο .
x0
Known
d3a2f..
:
u16
=
u7
⟶
∀ x0 : ο .
x0
Known
6c306..
:
u16
=
u8
⟶
∀ x0 : ο .
x0
Known
78b49..
:
u16
=
u9
⟶
∀ x0 : ο .
x0
Known
6879f..
:
u16
=
u10
⟶
∀ x0 : ο .
x0
Known
22184..
:
u16
=
u11
⟶
∀ x0 : ο .
x0
Known
fa664..
:
u16
=
u12
⟶
∀ x0 : ο .
x0
Known
4326e..
:
u16
=
u13
⟶
∀ x0 : ο .
x0
Known
71c5e..
:
u16
=
u14
⟶
∀ x0 : ο .
x0
Known
41073..
:
u16
=
u15
⟶
∀ x0 : ο .
x0
Known
beta
beta
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
x0
⟶
ap
(
lam
x0
x1
)
x2
=
x1
x2
Theorem
8a676..
:
(
∀ x0 x1 .
∀ x2 :
ι →
ι → ι
.
∀ x3 .
x3
∈
x1
⟶
ap
(
lam
x1
(
λ x5 .
If_i
(
x5
=
x3
)
x0
(
x2
(
ordsucc
x3
)
x5
)
)
)
x3
=
x0
)
⟶
(
∀ x0 x1 .
∀ x2 :
ι →
ι → ι
.
∀ x3 x4 .
(
x4
=
x3
⟶
∀ x5 : ο .
x5
)
⟶
ap
(
lam
x1
(
λ x6 .
If_i
(
x6
=
x3
)
x0
(
x2
(
ordsucc
x3
)
x6
)
)
)
x4
=
ap
(
lam
x1
(
x2
(
ordsucc
x3
)
)
)
x4
)
⟶
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 .
ap
(
lam
17
(
λ x18 .
If_i
(
x18
=
0
)
x0
(
If_i
(
x18
=
1
)
x1
(
If_i
(
x18
=
2
)
x2
(
If_i
(
x18
=
3
)
x3
(
If_i
(
x18
=
4
)
x4
(
If_i
(
x18
=
5
)
x5
(
If_i
(
x18
=
6
)
x6
(
If_i
(
x18
=
7
)
x7
(
If_i
(
x18
=
8
)
x8
(
If_i
(
x18
=
9
)
x9
(
If_i
(
x18
=
10
)
x10
(
If_i
(
x18
=
11
)
x11
(
If_i
(
x18
=
12
)
x12
(
If_i
(
x18
=
13
)
x13
(
If_i
(
x18
=
14
)
x14
(
If_i
(
x18
=
15
)
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u16
=
x16
(proof)