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Proofgold Asset
asset id
1758672292e79f84c7cfb09ed13d73cc820ddbb5f283fb2c9b7dd71fa0b9467b
asset hash
3821c9d2025d68633aff98d0d4ec760dbc01474c6caf9959765660f71b7171ef
bday / block
22396
tx
9d49f..
preasset
doc published by
PrBTh..
Known
30edc..
equip_tra
:
∀ x0 x1 x2 .
equip
x0
x1
⟶
equip
x1
x2
⟶
equip
x0
x2
Known
637fd..
equip_sym
:
∀ x0 x1 .
equip
x0
x1
⟶
equip
x1
x0
Theorem
86c40..
:
∀ x0 x1 x2 .
equip
x1
x0
⟶
equip
x2
x0
⟶
equip
x1
x2
(proof)
Known
82600..
:
∀ x0 .
equip
x0
0
⟶
x0
=
0
Known
d0de4..
Empty_eq
:
∀ x0 .
(
∀ x1 .
nIn
x1
x0
)
⟶
x0
=
0
Known
9d2e6..
nIn_I2
:
∀ x0 x1 .
(
In
x0
x1
⟶
False
)
⟶
nIn
x0
x1
Known
2901c..
EmptyE
:
∀ x0 .
In
x0
0
⟶
False
Known
681fa..
proj0_setprod
:
∀ x0 x1 x2 .
In
x2
(
setprod
x0
x1
)
⟶
In
(
proj0
x2
)
x0
Known
54d4b..
equip_ref
:
∀ x0 .
equip
x0
x0
Theorem
12ef8..
:
∀ x0 x1 .
equip
x0
0
⟶
equip
(
setprod
x0
x1
)
0
(proof)
Known
1cfb6..
proj1_setprod
:
∀ x0 x1 x2 .
In
x2
(
setprod
x0
x1
)
⟶
In
(
proj1
x2
)
x1
Theorem
9f38c..
:
∀ x0 x1 .
equip
x1
0
⟶
equip
(
setprod
x0
x1
)
0
(proof)
Known
c9b7c..
equipE_impred
:
∀ x0 x1 .
equip
x0
x1
⟶
∀ x2 : ο .
(
∀ x3 :
ι → ι
.
bij
x0
x1
x3
⟶
x2
)
⟶
x2
Known
80a11..
bijE_impred
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
bij
x0
x1
x2
⟶
∀ x3 : ο .
(
inj
x0
x1
x2
⟶
(
∀ x4 .
In
x4
x1
⟶
∀ x5 : ο .
(
∀ x6 .
and
(
In
x6
x0
)
(
x2
x6
=
x4
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
Known
e6daf..
injE_impred
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
inj
x0
x1
x2
⟶
∀ x3 : ο .
(
(
∀ x4 .
In
x4
x0
⟶
In
(
x2
x4
)
x1
)
⟶
(
∀ x4 .
In
x4
x0
⟶
∀ x5 .
In
x5
x0
⟶
x2
x4
=
x2
x5
⟶
x4
=
x5
)
⟶
x3
)
⟶
x3
Known
f2c5c..
equipI
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
bij
x0
x1
x2
⟶
equip
x0
x1
Known
e5c63..
bijI
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
inj
x0
x1
x2
⟶
(
∀ x3 .
In
x3
x1
⟶
∀ x4 : ο .
(
∀ x5 .
and
(
In
x5
x0
)
(
x2
x5
=
x3
)
⟶
x4
)
⟶
x4
)
⟶
bij
x0
x1
x2
Known
1796e..
injI
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
(
∀ x3 .
In
x3
x0
⟶
In
(
x2
x3
)
x1
)
⟶
(
∀ x3 .
In
x3
x0
⟶
∀ x4 .
In
x4
x0
⟶
x2
x3
=
x2
x4
⟶
x3
=
x4
)
⟶
inj
x0
x1
x2
Known
c1504..
tuple_setprod_eta
:
∀ x0 x1 x2 .
In
x2
(
setprod
x0
x1
)
⟶
lam
2
(
λ x4 .
If_i
(
x4
=
0
)
(
ap
x2
0
)
(
ap
x2
1
)
)
=
x2
Known
e51a8..
cases_1
:
∀ x0 .
In
x0
1
⟶
∀ x1 :
ι → ο
.
x1
0
⟶
x1
x0
Known
fe28a..
ap0_setprod
:
∀ x0 x1 x2 .
In
x2
(
setprod
x0
x1
)
⟶
In
(
ap
x2
0
)
x0
Known
3342b..
proj1_ap_1
:
∀ x0 .
proj1
x0
=
ap
x0
1
Known
andE
andE
:
∀ x0 x1 : ο .
and
x0
x1
⟶
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Known
7c02f..
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Known
07808..
pair_setprod
:
∀ x0 x1 x2 .
In
x2
x0
⟶
∀ x3 .
In
x3
x1
⟶
In
(
setsum
x2
x3
)
(
setprod
x0
x1
)
Known
0978b..
In_0_1
:
In
0
1
Known
b8fac..
proj1_pair_eq
:
∀ x0 x1 .
proj1
(
setsum
x0
x1
)
=
x1
Theorem
fb646..
:
∀ x0 x1 .
equip
x0
x1
⟶
equip
(
setprod
1
x0
)
x1
(proof)
Known
82f37..
proj0_ap_0
:
∀ x0 .
proj0
x0
=
ap
x0
0
Known
6789e..
ap1_setprod
:
∀ x0 x1 x2 .
In
x2
(
setprod
x0
x1
)
⟶
In
(
ap
x2
1
)
x1
Known
d6e1a..
proj0_pair_eq
:
∀ x0 x1 .
proj0
(
setsum
x0
x1
)
=
x0
Theorem
25080..
:
∀ x0 x1 .
equip
x0
x1
⟶
equip
(
setprod
x0
1
)
x1
(proof)
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Param
69aae..
exp_nat
:
ι
→
ι
→
ι
Known
3fe1c..
add_nat_com
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
add_nat
x0
x1
=
add_nat
x1
x0
Known
a2e43..
:
∀ x0 .
nat_p
x0
⟶
nat_p
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x0
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
Known
63495..
:
∀ x0 .
add_nat
x0
1
=
ordsucc
x0
Known
7bb3f..
:
∀ x0 x1 .
nat_p
x1
⟶
add_nat
x0
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x1
)
)
)
)
=
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
add_nat
x0
x1
)
)
)
)
Known
2b177..
:
∀ x0 x1 .
nat_p
x1
⟶
add_nat
x0
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x1
)
)
)
)
)
)
)
)
=
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
add_nat
x0
x1
)
)
)
)
)
)
)
)
Known
5af12..
:
∀ x0 .
add_nat
x0
2
=
ordsucc
(
ordsucc
x0
)
Known
0ac77..
:
∀ x0 x1 .
nat_p
x1
⟶
add_nat
x0
(
ordsucc
(
ordsucc
x1
)
)
=
ordsucc
(
ordsucc
(
add_nat
x0
x1
)
)
Known
9b388..
mul_nat_1R
:
∀ x0 .
mul_nat
x0
1
=
x0
Known
a62e2..
:
∀ x0 .
equip
x0
4
⟶
equip
(
Power
x0
)
16
Known
d6dd4..
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
∀ x2 x3 .
equip
x2
x0
⟶
equip
x3
x1
⟶
equip
(
binrep
x2
x3
)
(
add_nat
x0
(
69aae..
2
x1
)
)
Known
37c59..
exp_nat_1
:
∀ x0 .
69aae..
x0
1
=
x0
Known
b2788..
:
∀ x0 .
nat_p
x0
⟶
nat_p
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x0
)
)
)
)
)
)
)
)
Known
fddcf..
:
69aae..
2
2
=
4
Known
02b3e..
:
∀ x0 .
equip
x0
3
⟶
equip
(
Power
x0
)
8
Known
02169..
add_nat_0R
:
∀ x0 .
add_nat
x0
0
=
x0
Known
f4890..
exp_nat_S
:
∀ x0 x1 .
nat_p
x1
⟶
69aae..
x0
(
ordsucc
x1
)
=
mul_nat
x0
(
69aae..
x0
x1
)
Known
a1573..
:
∀ x0 .
equip
x0
2
⟶
equip
(
Power
x0
)
4
Known
2b6ba..
:
nat_p
4
Known
c83a0..
:
∀ x0 .
nat_p
x0
⟶
nat_p
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x0
)
)
)
)
Known
fad70..
mul_nat_0R
:
∀ x0 .
mul_nat
x0
0
=
0
Known
94de7..
exp_nat_0
:
∀ x0 .
69aae..
x0
0
=
1
Known
e6b7d..
:
∀ x0 .
equip
x0
1
⟶
equip
(
Power
x0
)
2
Known
224e2..
:
∀ x0 .
equip
x0
0
⟶
equip
(
Power
x0
)
1
Known
36841..
nat_2
:
nat_p
2
Known
9814f..
:
nat_p
3
Known
08405..
nat_0
:
nat_p
0
Known
c7c31..
nat_1
:
nat_p
1
Known
4c6ff..
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
∀ x2 x3 .
equip
x2
x0
⟶
equip
x3
x1
⟶
equip
(
setexp
x2
x3
)
(
69aae..
x0
x1
)
Known
912b9..
equip_setprod_mul_nat_2
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
∀ x2 x3 .
equip
x2
x0
⟶
equip
x3
x1
⟶
equip
(
setprod
x2
x3
)
(
mul_nat
x0
x1
)
Known
56707..
equip_setsum_add_nat_2
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
∀ x2 x3 .
equip
x2
x0
⟶
equip
x3
x1
⟶
equip
(
setsum
x2
x3
)
(
add_nat
x0
x1
)
Theorem
b8103..
:
(
∀ x0 x1 x2 .
equip
(
setsum
(
setsum
(
setprod
(
binrep
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x2
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x2
(
Power
(
Power
0
)
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
0
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
binrep
(
Power
(
Power
0
)
)
0
)
)
x2
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x1
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
Power
(
Power
0
)
)
)
(
setexp
x2
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
Power
(
Power
0
)
)
)
(
setexp
x2
(
Power
(
Power
0
)
)
)
)
)
)
(
setsum
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
Power
(
Power
0
)
)
)
x2
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x1
(
Power
(
Power
0
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
binrep
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
x1
(
setexp
x2
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
x1
(
setexp
x2
(
Power
(
Power
0
)
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
x1
x2
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
x1
)
)
(
setsum
(
setprod
(
Power
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x2
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
(
setsum
(
setprod
(
binrep
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
0
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x2
(
Power
(
Power
0
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
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⟶
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⟶
∀ x0 : ο .
x0
(proof)