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Proofgold Asset
asset id
84f0e44e95b67fc7c4d2e79c898f64c55af56b871beddbb67bd182a68a5982e7
asset hash
748ae460d973592be43b7e844ce1f9cf735a93b3a3f750ca534c609661b02074
bday / block
28250
tx
9ceb6..
preasset
doc published by
PrQUS..
Param
SNo
SNo
:
ι
→
ο
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Param
bbc71..
:
ι
→
ι
→
ι
→
ι
→
ι
→
ι
→
ι
→
ι
→
ι
Definition
1eb0a..
:=
λ x0 .
∀ x1 : ο .
(
∀ x2 .
and
(
SNo
x2
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
∀ x15 : ο .
(
∀ x16 .
and
(
SNo
x16
)
(
x0
=
bbc71..
x2
x4
x6
x8
x10
x12
x14
x16
)
⟶
x15
)
⟶
x15
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
⟶
x1
)
⟶
x1
Known
andI
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Theorem
d5242..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
1eb0a..
(
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
)
(proof)
Definition
d4639..
:=
λ x0 :
ι → ι
.
λ x1 .
prim0
(
λ x2 .
and
(
SNo
x2
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x1
=
bbc71..
(
x0
x1
)
x2
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
Known
Eps_i_ex
Eps_i_ex
:
∀ x0 :
ι → ο
.
(
∀ x1 : ο .
(
∀ x2 .
x0
x2
⟶
x1
)
⟶
x1
)
⟶
x0
(
prim0
x0
)
Theorem
26f49..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
d4639..
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x1
=
bbc71..
(
x0
x1
)
(
d4639..
x0
x1
)
x3
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
(proof)
Theorem
fd099..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
∀ x1 .
1eb0a..
x1
⟶
SNo
(
d4639..
x0
x1
)
(proof)
Known
babe8..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
SNo
x14
⟶
SNo
x15
⟶
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
=
bbc71..
x8
x9
x10
x11
x12
x13
x14
x15
⟶
x1
=
x9
Theorem
33b4a..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 x2 x3 x4 x5 x6 x7 x8 .
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
d4639..
x0
(
bbc71..
x1
x2
x3
x4
x5
x6
x7
x8
)
=
x2
(proof)
Definition
50208..
:=
λ x0 x1 :
ι → ι
.
λ x2 .
prim0
(
λ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x3
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
Theorem
9b0e1..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
50208..
x0
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
(
50208..
x0
x1
x2
)
x4
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
(proof)
Theorem
1131a..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
∀ x2 .
1eb0a..
x2
⟶
SNo
(
50208..
x0
x1
x2
)
(proof)
Known
6488e..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
SNo
x14
⟶
SNo
x15
⟶
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
=
bbc71..
x8
x9
x10
x11
x12
x13
x14
x15
⟶
x2
=
x10
Theorem
90339..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
50208..
x0
x1
(
bbc71..
x2
x3
x4
x5
x6
x7
x8
x9
)
=
x4
(proof)
Definition
8d7df..
:=
λ x0 x1 x2 :
ι → ι
.
λ x3 .
prim0
(
λ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x4
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
Theorem
0a376..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
8d7df..
x0
x1
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
(
8d7df..
x0
x1
x2
x3
)
x5
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
(proof)
Theorem
5e734..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
∀ x3 .
1eb0a..
x3
⟶
SNo
(
8d7df..
x0
x1
x2
x3
)
(proof)
Known
ad01a..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
SNo
x14
⟶
SNo
x15
⟶
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
=
bbc71..
x8
x9
x10
x11
x12
x13
x14
x15
⟶
x3
=
x11
Theorem
71a99..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 x4 x5 x6 x7 x8 x9 x10 .
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
8d7df..
x0
x1
x2
(
bbc71..
x3
x4
x5
x6
x7
x8
x9
x10
)
=
x6
(proof)
Definition
41ec1..
:=
λ x0 x1 x2 x3 :
ι → ι
.
λ x4 .
prim0
(
λ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x5
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
Theorem
15adc..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
41ec1..
x0
x1
x2
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
(
41ec1..
x0
x1
x2
x3
x4
)
x6
x8
x10
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
(proof)
Theorem
af528..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
∀ x4 .
1eb0a..
x4
⟶
SNo
(
41ec1..
x0
x1
x2
x3
x4
)
(proof)
Known
f4559..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
SNo
x14
⟶
SNo
x15
⟶
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
=
bbc71..
x8
x9
x10
x11
x12
x13
x14
x15
⟶
x4
=
x12
Theorem
26f65..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 x5 x6 x7 x8 x9 x10 x11 .
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
41ec1..
x0
x1
x2
x3
(
bbc71..
x4
x5
x6
x7
x8
x9
x10
x11
)
=
x8
(proof)
Definition
28f5a..
:=
λ x0 x1 x2 x3 x4 :
ι → ι
.
λ x5 .
prim0
(
λ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x6
x8
x10
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
)
Theorem
baa4b..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
⟶
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
28f5a..
x0
x1
x2
x3
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
(
28f5a..
x0
x1
x2
x3
x4
x5
)
x7
x9
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
(proof)
Theorem
69bbd..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
⟶
∀ x5 .
1eb0a..
x5
⟶
SNo
(
28f5a..
x0
x1
x2
x3
x4
x5
)
(proof)
Known
f56fc..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
SNo
x14
⟶
SNo
x15
⟶
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
=
bbc71..
x8
x9
x10
x11
x12
x13
x14
x15
⟶
x5
=
x13
Theorem
62fb0..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
⟶
(
∀ x5 .
1eb0a..
x5
⟶
SNo
(
x4
x5
)
)
⟶
∀ x5 x6 x7 x8 x9 x10 x11 x12 .
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
28f5a..
x0
x1
x2
x3
x4
(
bbc71..
x5
x6
x7
x8
x9
x10
x11
x12
)
=
x10
(proof)
Definition
717b4..
:=
λ x0 x1 x2 x3 x4 x5 :
ι → ι
.
λ x6 .
prim0
(
λ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
x6
=
bbc71..
(
x0
x6
)
(
x1
x6
)
(
x2
x6
)
(
x3
x6
)
(
x4
x6
)
(
x5
x6
)
x7
x9
)
⟶
x8
)
⟶
x8
)
)
Theorem
42518..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
⟶
(
∀ x5 .
1eb0a..
x5
⟶
SNo
(
x4
x5
)
)
⟶
∀ x5 :
ι → ι
.
(
∀ x6 .
1eb0a..
x6
⟶
and
(
SNo
(
x5
x6
)
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
x6
=
bbc71..
(
x0
x6
)
(
x1
x6
)
(
x2
x6
)
(
x3
x6
)
(
x4
x6
)
(
x5
x6
)
x8
x10
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
)
⟶
∀ x6 .
1eb0a..
x6
⟶
and
(
SNo
(
717b4..
x0
x1
x2
x3
x4
x5
x6
)
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
x6
=
bbc71..
(
x0
x6
)
(
x1
x6
)
(
x2
x6
)
(
x3
x6
)
(
x4
x6
)
(
x5
x6
)
(
717b4..
x0
x1
x2
x3
x4
x5
x6
)
x8
)
⟶
x7
)
⟶
x7
)
(proof)
Theorem
9599d..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
⟶
(
∀ x5 .
1eb0a..
x5
⟶
SNo
(
x4
x5
)
)
⟶
∀ x5 :
ι → ι
.
(
∀ x6 .
1eb0a..
x6
⟶
and
(
SNo
(
x5
x6
)
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
x6
=
bbc71..
(
x0
x6
)
(
x1
x6
)
(
x2
x6
)
(
x3
x6
)
(
x4
x6
)
(
x5
x6
)
x8
x10
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
)
⟶
∀ x6 .
1eb0a..
x6
⟶
SNo
(
717b4..
x0
x1
x2
x3
x4
x5
x6
)
(proof)
Known
4a66b..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
SNo
x14
⟶
SNo
x15
⟶
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
=
bbc71..
x8
x9
x10
x11
x12
x13
x14
x15
⟶
x6
=
x14
Theorem
c7d23..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
⟶
(
∀ x5 .
1eb0a..
x5
⟶
SNo
(
x4
x5
)
)
⟶
∀ x5 :
ι → ι
.
(
∀ x6 .
1eb0a..
x6
⟶
and
(
SNo
(
x5
x6
)
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
x6
=
bbc71..
(
x0
x6
)
(
x1
x6
)
(
x2
x6
)
(
x3
x6
)
(
x4
x6
)
(
x5
x6
)
x8
x10
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
)
⟶
(
∀ x6 .
1eb0a..
x6
⟶
SNo
(
x5
x6
)
)
⟶
∀ x6 x7 x8 x9 x10 x11 x12 x13 .
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
717b4..
x0
x1
x2
x3
x4
x5
(
bbc71..
x6
x7
x8
x9
x10
x11
x12
x13
)
=
x12
(proof)
Definition
053de..
:=
λ x0 x1 x2 x3 x4 x5 x6 :
ι → ι
.
λ x7 .
prim0
(
λ x8 .
and
(
SNo
x8
)
(
x7
=
bbc71..
(
x0
x7
)
(
x1
x7
)
(
x2
x7
)
(
x3
x7
)
(
x4
x7
)
(
x5
x7
)
(
x6
x7
)
x8
)
)
Theorem
0b166..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
⟶
(
∀ x5 .
1eb0a..
x5
⟶
SNo
(
x4
x5
)
)
⟶
∀ x5 :
ι → ι
.
(
∀ x6 .
1eb0a..
x6
⟶
and
(
SNo
(
x5
x6
)
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
x6
=
bbc71..
(
x0
x6
)
(
x1
x6
)
(
x2
x6
)
(
x3
x6
)
(
x4
x6
)
(
x5
x6
)
x8
x10
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
)
⟶
(
∀ x6 .
1eb0a..
x6
⟶
SNo
(
x5
x6
)
)
⟶
∀ x6 :
ι → ι
.
(
∀ x7 .
1eb0a..
x7
⟶
and
(
SNo
(
x6
x7
)
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
x7
=
bbc71..
(
x0
x7
)
(
x1
x7
)
(
x2
x7
)
(
x3
x7
)
(
x4
x7
)
(
x5
x7
)
(
x6
x7
)
x9
)
⟶
x8
)
⟶
x8
)
)
⟶
∀ x7 .
1eb0a..
x7
⟶
and
(
SNo
(
053de..
x0
x1
x2
x3
x4
x5
x6
x7
)
)
(
x7
=
bbc71..
(
x0
x7
)
(
x1
x7
)
(
x2
x7
)
(
x3
x7
)
(
x4
x7
)
(
x5
x7
)
(
x6
x7
)
(
053de..
x0
x1
x2
x3
x4
x5
x6
x7
)
)
(proof)
Theorem
0c70a..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
⟶
(
∀ x5 .
1eb0a..
x5
⟶
SNo
(
x4
x5
)
)
⟶
∀ x5 :
ι → ι
.
(
∀ x6 .
1eb0a..
x6
⟶
and
(
SNo
(
x5
x6
)
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
x6
=
bbc71..
(
x0
x6
)
(
x1
x6
)
(
x2
x6
)
(
x3
x6
)
(
x4
x6
)
(
x5
x6
)
x8
x10
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
)
⟶
(
∀ x6 .
1eb0a..
x6
⟶
SNo
(
x5
x6
)
)
⟶
∀ x6 :
ι → ι
.
(
∀ x7 .
1eb0a..
x7
⟶
and
(
SNo
(
x6
x7
)
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
x7
=
bbc71..
(
x0
x7
)
(
x1
x7
)
(
x2
x7
)
(
x3
x7
)
(
x4
x7
)
(
x5
x7
)
(
x6
x7
)
x9
)
⟶
x8
)
⟶
x8
)
)
⟶
∀ x7 .
1eb0a..
x7
⟶
SNo
(
053de..
x0
x1
x2
x3
x4
x5
x6
x7
)
(proof)
Known
7c302..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
SNo
x14
⟶
SNo
x15
⟶
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
=
bbc71..
x8
x9
x10
x11
x12
x13
x14
x15
⟶
x7
=
x15
Theorem
48feb..
:
∀ x0 :
ι → ι
.
(
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x1
=
bbc71..
(
x0
x1
)
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
⟶
(
∀ x1 .
1eb0a..
x1
⟶
SNo
(
x0
x1
)
)
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
1eb0a..
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x2
=
bbc71..
(
x0
x2
)
(
x1
x2
)
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
)
⟶
(
∀ x2 .
1eb0a..
x2
⟶
SNo
(
x1
x2
)
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
1eb0a..
x3
⟶
and
(
SNo
(
x2
x3
)
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
x3
=
bbc71..
(
x0
x3
)
(
x1
x3
)
(
x2
x3
)
x5
x7
x9
x11
x13
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
)
⟶
(
∀ x3 .
1eb0a..
x3
⟶
SNo
(
x2
x3
)
)
⟶
∀ x3 :
ι → ι
.
(
∀ x4 .
1eb0a..
x4
⟶
and
(
SNo
(
x3
x4
)
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
x4
=
bbc71..
(
x0
x4
)
(
x1
x4
)
(
x2
x4
)
(
x3
x4
)
x6
x8
x10
x12
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
)
⟶
(
∀ x4 .
1eb0a..
x4
⟶
SNo
(
x3
x4
)
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
1eb0a..
x5
⟶
and
(
SNo
(
x4
x5
)
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
x5
=
bbc71..
(
x0
x5
)
(
x1
x5
)
(
x2
x5
)
(
x3
x5
)
(
x4
x5
)
x7
x9
x11
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
)
⟶
(
∀ x5 .
1eb0a..
x5
⟶
SNo
(
x4
x5
)
)
⟶
∀ x5 :
ι → ι
.
(
∀ x6 .
1eb0a..
x6
⟶
and
(
SNo
(
x5
x6
)
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
x6
=
bbc71..
(
x0
x6
)
(
x1
x6
)
(
x2
x6
)
(
x3
x6
)
(
x4
x6
)
(
x5
x6
)
x8
x10
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
)
⟶
(
∀ x6 .
1eb0a..
x6
⟶
SNo
(
x5
x6
)
)
⟶
∀ x6 :
ι → ι
.
(
∀ x7 .
1eb0a..
x7
⟶
and
(
SNo
(
x6
x7
)
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
x7
=
bbc71..
(
x0
x7
)
(
x1
x7
)
(
x2
x7
)
(
x3
x7
)
(
x4
x7
)
(
x5
x7
)
(
x6
x7
)
x9
)
⟶
x8
)
⟶
x8
)
)
⟶
(
∀ x7 .
1eb0a..
x7
⟶
SNo
(
x6
x7
)
)
⟶
∀ x7 x8 x9 x10 x11 x12 x13 x14 .
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
SNo
x14
⟶
053de..
x0
x1
x2
x3
x4
x5
x6
(
bbc71..
x7
x8
x9
x10
x11
x12
x13
x14
)
=
x14
(proof)
Definition
8dd2c..
:=
λ x0 .
prim0
(
λ x1 .
and
(
SNo
x1
)
(
∀ x2 : ο .
(
∀ x3 .
and
(
SNo
x3
)
(
∀ x4 : ο .
(
∀ x5 .
and
(
SNo
x5
)
(
∀ x6 : ο .
(
∀ x7 .
and
(
SNo
x7
)
(
∀ x8 : ο .
(
∀ x9 .
and
(
SNo
x9
)
(
∀ x10 : ο .
(
∀ x11 .
and
(
SNo
x11
)
(
∀ x12 : ο .
(
∀ x13 .
and
(
SNo
x13
)
(
∀ x14 : ο .
(
∀ x15 .
and
(
SNo
x15
)
(
x0
=
bbc71..
x1
x3
x5
x7
x9
x11
x13
x15
)
⟶
x14
)
⟶
x14
)
⟶
x12
)
⟶
x12
)
⟶
x10
)
⟶
x10
)
⟶
x8
)
⟶
x8
)
⟶
x6
)
⟶
x6
)
⟶
x4
)
⟶
x4
)
⟶
x2
)
⟶
x2
)
)
Theorem
340c0..
:
∀ x0 .
1eb0a..
x0
⟶
and
(
SNo
(
8dd2c..
x0
)
)
(
∀ x1 : ο .
(
∀ x2 .
and
(
SNo
x2
)
(
∀ x3 : ο .
(
∀ x4 .
and
(
SNo
x4
)
(
∀ x5 : ο .
(
∀ x6 .
and
(
SNo
x6
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
SNo
x8
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
SNo
x10
)
(
∀ x11 : ο .
(
∀ x12 .
and
(
SNo
x12
)
(
∀ x13 : ο .
(
∀ x14 .
and
(
SNo
x14
)
(
x0
=
bbc71..
(
8dd2c..
x0
)
x2
x4
x6
x8
x10
x12
x14
)
⟶
x13
)
⟶
x13
)
⟶
x11
)
⟶
x11
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
⟶
x1
)
⟶
x1
)
(proof)
Theorem
95571..
:
∀ x0 .
1eb0a..
x0
⟶
SNo
(
8dd2c..
x0
)
(proof)
Known
cca09..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
SNo
x8
⟶
SNo
x9
⟶
SNo
x10
⟶
SNo
x11
⟶
SNo
x12
⟶
SNo
x13
⟶
SNo
x14
⟶
SNo
x15
⟶
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
=
bbc71..
x8
x9
x10
x11
x12
x13
x14
x15
⟶
x0
=
x8
Theorem
212d5..
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNo
x6
⟶
SNo
x7
⟶
8dd2c..
(
bbc71..
x0
x1
x2
x3
x4
x5
x6
x7
)
=
x0
(proof)