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PUU98..
/
f375c..
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
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Param
69aae..
exp_nat
:
ι
→
ι
→
ι
Known
2931a..
:
∀ x0 x1 x2 x3 .
equip
x0
(
ordsucc
x3
)
⟶
equip
x1
x3
⟶
equip_mod
x0
x1
x2
Known
02169..
add_nat_0R
:
∀ x0 .
add_nat
x0
0
=
x0
Known
9e914..
:
∀ x0 x1 .
nat_p
x1
⟶
add_nat
x0
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x1
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
add_nat
x0
x1
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
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
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
9b388..
mul_nat_1R
:
∀ x0 .
mul_nat
x0
1
=
x0
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
7bb3f..
:
∀ x0 x1 .
nat_p
x1
⟶
add_nat
x0
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x1
)
)
)
)
=
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
add_nat
x0
x1
)
)
)
)
Known
a62e2..
:
∀ x0 .
equip
x0
4
⟶
equip
(
Power
x0
)
16
Known
3fe1c..
add_nat_com
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
add_nat
x0
x1
=
add_nat
x1
x0
Known
0ac77..
:
∀ x0 x1 .
nat_p
x1
⟶
add_nat
x0
(
ordsucc
(
ordsucc
x1
)
)
=
ordsucc
(
ordsucc
(
add_nat
x0
x1
)
)
Known
b2788..
:
∀ x0 .
nat_p
x0
⟶
nat_p
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x0
)
)
)
)
)
)
)
)
Known
5af12..
:
∀ x0 .
add_nat
x0
2
=
ordsucc
(
ordsucc
x0
)
Known
fddcf..
:
69aae..
2
2
=
4
Known
c83a0..
:
∀ x0 .
nat_p
x0
⟶
nat_p
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x0
)
)
)
)
Known
37c59..
exp_nat_1
:
∀ x0 .
69aae..
x0
1
=
x0
Known
74cfe..
:
∀ x0 .
nat_p
x0
⟶
nat_p
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
(
ordsucc
x0
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
Known
a1573..
:
∀ x0 .
equip
x0
2
⟶
equip
(
Power
x0
)
4
Known
2b6ba..
:
nat_p
4
Known
f4890..
exp_nat_S
:
∀ x0 x1 .
nat_p
x1
⟶
69aae..
x0
(
ordsucc
x1
)
=
mul_nat
x0
(
69aae..
x0
x1
)
Known
02b3e..
:
∀ x0 .
equip
x0
3
⟶
equip
(
Power
x0
)
8
Known
c7c31..
nat_1
:
nat_p
1
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
9814f..
:
nat_p
3
Known
08405..
nat_0
:
nat_p
0
Known
36841..
nat_2
:
nat_p
2
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
12817..
:
(
∀ x0 x1 x2 .
equip_mod
(
setsum
(
setsum
(
setprod
(
Power
(
binrep
(
Power
(
Power
0
)
)
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
(
Power
(
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
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
0
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
binrep
(
Power
(
Power
0
)
)
0
)
)
x2
)
)
)
(
setsum
(
setprod
(
binrep
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
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
)
)
(
Power
0
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
Power
(
Power
0
)
)
)
(
setexp
x2
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
)
(
setsum
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
Power
(
Power
0
)
)
)
(
setexp
x2
(
Power
(
Power
0
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
(
setexp
x1
(
Power
(
Power
0
)
)
)
x2
)
)
)
(
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
)
)
(
setexp
x1
(
Power
(
Power
0
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
0
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
x1
(
setexp
x2
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
)
(
setsum
(
setprod
(
Power
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setprod
x1
(
setexp
x2
(
Power
(
Power
0
)
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
Power
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
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
0
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x2
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x2
(
Power
(
Power
0
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
(
setprod
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
x2
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
0
)
(
setexp
x0
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
(
setsum
(
setprod
(
binrep
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
0
)
(
setprod
(
setexp
x0
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x1
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x2
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
0
)
(
setprod
(
setexp
x0
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x1
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
setexp
x2
(
Power
(
Power
0
)
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
(
setprod
(
setexp
x0
(
Power
(
Power
0
)
)
)
(
setexp
x1
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
(
setsum
(
setprod
(
setexp
x0
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x1
(
Power
(
Power
0
)
)
)
(
setexp
x2
(
binrep
(
Power
(
Power
0
)
)
0
)
)
)
)
(
setsum
(
setprod
(
Power
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x0
(
Power
(
Power
0
)
)
)
(
setprod
(
setexp
x1
(
Power
(
Power
0
)
)
)
(
setexp
x2
(
Power
(
Power
0
)
)
)
)
)
)
(
setsum
(
setprod
(
binrep
(
Power
(
binrep
(
Power
(
Power
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)
)
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
0
)
)
(
binrep
(
binrep
(
Power
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
0
)
)
(
Power
(
Power
0
)
)
)
0
)
)
(
binrep
(
Power
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
0
)
)
(
Power
(
Power
0
)
)
)
)
(
binrep
(
binrep
(
binrep
(
binrep
(
Power
(
Power
(
Power
(
Power
0
)
)
)
)
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
0
)
)
(
binrep
(
binrep
(
binrep
(
Power
(
Power
(
Power
(
Power
0
)
)
)
)
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
)
(
binrep
(
binrep
(
binrep
(
Power
(
Power
(
Power
(
Power
0
)
)
)
)
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
0
)
)
(
binrep
(
binrep
(
Power
(
Power
(
Power
(
Power
0
)
)
)
)
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
)
(
binrep
(
binrep
(
Power
(
Power
(
Power
(
Power
0
)
)
)
)
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
)
(
binrep
(
binrep
(
Power
(
Power
(
Power
(
Power
0
)
)
)
)
(
Power
(
Power
0
)
)
)
0
)
)
(
binrep
(
binrep
(
Power
(
Power
(
Power
(
Power
0
)
)
)
)
(
Power
0
)
)
0
)
)
(
Power
(
Power
(
Power
(
Power
0
)
)
)
)
)
(
binrep
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
)
(
binrep
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
0
)
)
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
(
Power
(
Power
0
)
)
)
)
(
binrep
(
Power
(
binrep
(
Power
(
Power
0
)
)
0
)
)
0
)
)
(
binrep
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
0
)
)
(
binrep
(
Power
(
Power
(
Power
0
)
)
)
(
Power
0
)
)
)
(
Power
0
)
)
0
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)