Search for blocks/addresses/...
Proofgold Address
address
PUTfEGDRaCVf1PG1ZvaWAj5GwJ3BjCNkJL9
total
0
mg
-
conjpub
-
current assets
01458..
/
ee81c..
bday:
2894
doc published by
PrGxv..
Param
0fc90..
:
ι
→
(
ι
→
ι
) →
ι
Param
4ae4a..
:
ι
→
ι
Param
4a7ef..
:
ι
Param
If_i
:
ο
→
ι
→
ι
→
ι
Param
1216a..
:
ι
→
(
ι
→
ο
) →
ι
Definition
ba600..
:=
λ x0 .
λ x1 x2 :
ι → ι
.
λ x3 x4 :
ι → ο
.
0fc90..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
λ x5 .
If_i
(
x5
=
4a7ef..
)
x0
(
If_i
(
x5
=
4ae4a..
4a7ef..
)
(
0fc90..
x0
x1
)
(
If_i
(
x5
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
0fc90..
x0
x2
)
(
If_i
(
x5
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
(
1216a..
x0
x3
)
(
1216a..
x0
x4
)
)
)
)
)
Param
f482f..
:
ι
→
ι
→
ι
Known
7d2e2..
:
∀ x0 x1 x2 x3 x4 .
f482f..
(
0fc90..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
λ x6 .
If_i
(
x6
=
4a7ef..
)
x0
(
If_i
(
x6
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
x4
)
)
)
)
)
4a7ef..
=
x0
Theorem
5c6b7..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 :
ι → ο
.
x0
=
ba600..
x1
x2
x3
x4
x5
⟶
x1
=
f482f..
x0
4a7ef..
(proof)
Theorem
34d3b..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 :
ι → ο
.
x0
=
f482f..
(
ba600..
x0
x1
x2
x3
x4
)
4a7ef..
(proof)
Known
504a8..
:
∀ x0 x1 x2 x3 x4 .
f482f..
(
0fc90..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
λ x6 .
If_i
(
x6
=
4a7ef..
)
x0
(
If_i
(
x6
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
x4
)
)
)
)
)
(
4ae4a..
4a7ef..
)
=
x1
Known
f22ec..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
prim1
x2
x0
⟶
f482f..
(
0fc90..
x0
x1
)
x2
=
x1
x2
Theorem
38231..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 :
ι → ο
.
x0
=
ba600..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x2
x6
=
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
x6
(proof)
Theorem
56c1e..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 :
ι → ο
.
∀ x5 .
prim1
x5
x0
⟶
x1
x5
=
f482f..
(
f482f..
(
ba600..
x0
x1
x2
x3
x4
)
(
4ae4a..
4a7ef..
)
)
x5
(proof)
Known
fb20c..
:
∀ x0 x1 x2 x3 x4 .
f482f..
(
0fc90..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
λ x6 .
If_i
(
x6
=
4a7ef..
)
x0
(
If_i
(
x6
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
x4
)
)
)
)
)
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
=
x2
Theorem
e2c49..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 :
ι → ο
.
x0
=
ba600..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x3
x6
=
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x6
(proof)
Theorem
66d42..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 :
ι → ο
.
∀ x5 .
prim1
x5
x0
⟶
x2
x5
=
f482f..
(
f482f..
(
ba600..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x5
(proof)
Param
decode_p
:
ι
→
ι
→
ο
Known
431f3..
:
∀ x0 x1 x2 x3 x4 .
f482f..
(
0fc90..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
λ x6 .
If_i
(
x6
=
4a7ef..
)
x0
(
If_i
(
x6
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
x4
)
)
)
)
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
=
x3
Known
931fe..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
x0
⟶
decode_p
(
1216a..
x0
x1
)
x2
=
x1
x2
Theorem
7a9a7..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 :
ι → ο
.
x0
=
ba600..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x4
x6
=
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x6
(proof)
Theorem
2a85a..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 :
ι → ο
.
∀ x5 .
prim1
x5
x0
⟶
x3
x5
=
decode_p
(
f482f..
(
ba600..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x5
(proof)
Known
ffdcd..
:
∀ x0 x1 x2 x3 x4 .
f482f..
(
0fc90..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
λ x6 .
If_i
(
x6
=
4a7ef..
)
x0
(
If_i
(
x6
=
4ae4a..
4a7ef..
)
x1
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
x2
(
If_i
(
x6
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
x4
)
)
)
)
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
=
x4
Theorem
5e2e5..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 :
ι → ο
.
x0
=
ba600..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x5
x6
=
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x6
(proof)
Theorem
671d1..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 :
ι → ο
.
∀ x5 .
prim1
x5
x0
⟶
x4
x5
=
decode_p
(
f482f..
(
ba600..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x5
(proof)
Definition
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Known
and5I
:
∀ x0 x1 x2 x3 x4 : ο .
x0
⟶
x1
⟶
x2
⟶
x3
⟶
x4
⟶
and
(
and
(
and
(
and
x0
x1
)
x2
)
x3
)
x4
Theorem
730cb..
:
∀ x0 x1 .
∀ x2 x3 x4 x5 :
ι → ι
.
∀ x6 x7 x8 x9 :
ι → ο
.
ba600..
x0
x2
x4
x6
x8
=
ba600..
x1
x3
x5
x7
x9
⟶
and
(
and
(
and
(
and
(
x0
=
x1
)
(
∀ x10 .
prim1
x10
x0
⟶
x2
x10
=
x3
x10
)
)
(
∀ x10 .
prim1
x10
x0
⟶
x4
x10
=
x5
x10
)
)
(
∀ x10 .
prim1
x10
x0
⟶
x6
x10
=
x7
x10
)
)
(
∀ x10 .
prim1
x10
x0
⟶
x8
x10
=
x9
x10
)
(proof)
Param
iff
:
ο
→
ο
→
ο
Known
ee7ef..
:
∀ x0 .
∀ x1 x2 :
ι → ο
.
(
∀ x3 .
prim1
x3
x0
⟶
iff
(
x1
x3
)
(
x2
x3
)
)
⟶
1216a..
x0
x1
=
1216a..
x0
x2
Known
4402a..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
(
∀ x3 .
prim1
x3
x0
⟶
x1
x3
=
x2
x3
)
⟶
0fc90..
x0
x1
=
0fc90..
x0
x2
Theorem
0a986..
:
∀ x0 .
∀ x1 x2 x3 x4 :
ι → ι
.
∀ x5 x6 x7 x8 :
ι → ο
.
(
∀ x9 .
prim1
x9
x0
⟶
x1
x9
=
x2
x9
)
⟶
(
∀ x9 .
prim1
x9
x0
⟶
x3
x9
=
x4
x9
)
⟶
(
∀ x9 .
prim1
x9
x0
⟶
iff
(
x5
x9
)
(
x6
x9
)
)
⟶
(
∀ x9 .
prim1
x9
x0
⟶
iff
(
x7
x9
)
(
x8
x9
)
)
⟶
ba600..
x0
x1
x3
x5
x7
=
ba600..
x0
x2
x4
x6
x8
(proof)
Definition
7aeb9..
:=
λ x0 .
∀ x1 :
ι → ο
.
(
∀ x2 .
∀ x3 :
ι → ι
.
(
∀ x4 .
prim1
x4
x2
⟶
prim1
(
x3
x4
)
x2
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
prim1
x5
x2
⟶
prim1
(
x4
x5
)
x2
)
⟶
∀ x5 x6 :
ι → ο
.
x1
(
ba600..
x2
x3
x4
x5
x6
)
)
⟶
x1
x0
Theorem
d78b8..
:
∀ x0 .
∀ x1 :
ι → ι
.
(
∀ x2 .
prim1
x2
x0
⟶
prim1
(
x1
x2
)
x0
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
prim1
x3
x0
⟶
prim1
(
x2
x3
)
x0
)
⟶
∀ x3 x4 :
ι → ο
.
7aeb9..
(
ba600..
x0
x1
x2
x3
x4
)
(proof)
Theorem
86929..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 :
ι → ο
.
7aeb9..
(
ba600..
x0
x1
x2
x3
x4
)
⟶
∀ x5 .
prim1
x5
x0
⟶
prim1
(
x1
x5
)
x0
(proof)
Theorem
3efb3..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 :
ι → ο
.
7aeb9..
(
ba600..
x0
x1
x2
x3
x4
)
⟶
∀ x5 .
prim1
x5
x0
⟶
prim1
(
x2
x5
)
x0
(proof)
Known
iff_refl
:
∀ x0 : ο .
iff
x0
x0
Theorem
1efa9..
:
∀ x0 .
7aeb9..
x0
⟶
x0
=
ba600..
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
(proof)
Definition
d8d71..
:=
λ x0 .
λ x1 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ο
)
→
(
ι → ο
)
→ ι
.
x1
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
Theorem
b5643..
:
∀ x0 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ο
)
→
(
ι → ο
)
→ ι
.
∀ x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 :
ι → ο
.
(
∀ x6 :
ι → ι
.
(
∀ x7 .
prim1
x7
x1
⟶
x2
x7
=
x6
x7
)
⟶
∀ x7 :
ι → ι
.
(
∀ x8 .
prim1
x8
x1
⟶
x3
x8
=
x7
x8
)
⟶
∀ x8 :
ι → ο
.
(
∀ x9 .
prim1
x9
x1
⟶
iff
(
x4
x9
)
(
x8
x9
)
)
⟶
∀ x9 :
ι → ο
.
(
∀ x10 .
prim1
x10
x1
⟶
iff
(
x5
x10
)
(
x9
x10
)
)
⟶
x0
x1
x6
x7
x8
x9
=
x0
x1
x2
x3
x4
x5
)
⟶
d8d71..
(
ba600..
x1
x2
x3
x4
x5
)
x0
=
x0
x1
x2
x3
x4
x5
(proof)
Definition
d0331..
:=
λ x0 .
λ x1 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ο
)
→
(
ι → ο
)
→ ο
.
x1
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
Theorem
97885..
:
∀ x0 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ο
)
→
(
ι → ο
)
→ ο
.
∀ x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 :
ι → ο
.
(
∀ x6 :
ι → ι
.
(
∀ x7 .
prim1
x7
x1
⟶
x2
x7
=
x6
x7
)
⟶
∀ x7 :
ι → ι
.
(
∀ x8 .
prim1
x8
x1
⟶
x3
x8
=
x7
x8
)
⟶
∀ x8 :
ι → ο
.
(
∀ x9 .
prim1
x9
x1
⟶
iff
(
x4
x9
)
(
x8
x9
)
)
⟶
∀ x9 :
ι → ο
.
(
∀ x10 .
prim1
x10
x1
⟶
iff
(
x5
x10
)
(
x9
x10
)
)
⟶
x0
x1
x6
x7
x8
x9
=
x0
x1
x2
x3
x4
x5
)
⟶
d0331..
(
ba600..
x1
x2
x3
x4
x5
)
x0
=
x0
x1
x2
x3
x4
x5
(proof)
Definition
7e46e..
:=
λ x0 .
λ x1 x2 :
ι → ι
.
λ x3 :
ι → ο
.
λ x4 .
0fc90..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
λ x5 .
If_i
(
x5
=
4a7ef..
)
x0
(
If_i
(
x5
=
4ae4a..
4a7ef..
)
(
0fc90..
x0
x1
)
(
If_i
(
x5
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
0fc90..
x0
x2
)
(
If_i
(
x5
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
(
1216a..
x0
x3
)
x4
)
)
)
)
Theorem
c9763..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 :
ι → ο
.
∀ x5 .
x0
=
7e46e..
x1
x2
x3
x4
x5
⟶
x1
=
f482f..
x0
4a7ef..
(proof)
Theorem
70120..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 :
ι → ο
.
∀ x4 .
x0
=
f482f..
(
7e46e..
x0
x1
x2
x3
x4
)
4a7ef..
(proof)
Theorem
6071f..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 :
ι → ο
.
∀ x5 .
x0
=
7e46e..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x2
x6
=
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
x6
(proof)
Theorem
b23c6..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 :
ι → ο
.
∀ x4 x5 .
prim1
x5
x0
⟶
x1
x5
=
f482f..
(
f482f..
(
7e46e..
x0
x1
x2
x3
x4
)
(
4ae4a..
4a7ef..
)
)
x5
(proof)
Theorem
7da76..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 :
ι → ο
.
∀ x5 .
x0
=
7e46e..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x3
x6
=
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x6
(proof)
Theorem
9794d..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 :
ι → ο
.
∀ x4 x5 .
prim1
x5
x0
⟶
x2
x5
=
f482f..
(
f482f..
(
7e46e..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x5
(proof)
Theorem
d6e0e..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 :
ι → ο
.
∀ x5 .
x0
=
7e46e..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x4
x6
=
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x6
(proof)
Theorem
6d178..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 :
ι → ο
.
∀ x4 x5 .
prim1
x5
x0
⟶
x3
x5
=
decode_p
(
f482f..
(
7e46e..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x5
(proof)
Theorem
cea9c..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 :
ι → ο
.
∀ x5 .
x0
=
7e46e..
x1
x2
x3
x4
x5
⟶
x5
=
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(proof)
Theorem
ad831..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 :
ι → ο
.
∀ x4 .
x4
=
f482f..
(
7e46e..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(proof)
Theorem
af02e..
:
∀ x0 x1 .
∀ x2 x3 x4 x5 :
ι → ι
.
∀ x6 x7 :
ι → ο
.
∀ x8 x9 .
7e46e..
x0
x2
x4
x6
x8
=
7e46e..
x1
x3
x5
x7
x9
⟶
and
(
and
(
and
(
and
(
x0
=
x1
)
(
∀ x10 .
prim1
x10
x0
⟶
x2
x10
=
x3
x10
)
)
(
∀ x10 .
prim1
x10
x0
⟶
x4
x10
=
x5
x10
)
)
(
∀ x10 .
prim1
x10
x0
⟶
x6
x10
=
x7
x10
)
)
(
x8
=
x9
)
(proof)
Theorem
6d0ba..
:
∀ x0 .
∀ x1 x2 x3 x4 :
ι → ι
.
∀ x5 x6 :
ι → ο
.
∀ x7 .
(
∀ x8 .
prim1
x8
x0
⟶
x1
x8
=
x2
x8
)
⟶
(
∀ x8 .
prim1
x8
x0
⟶
x3
x8
=
x4
x8
)
⟶
(
∀ x8 .
prim1
x8
x0
⟶
iff
(
x5
x8
)
(
x6
x8
)
)
⟶
7e46e..
x0
x1
x3
x5
x7
=
7e46e..
x0
x2
x4
x6
x7
(proof)
Definition
57517..
:=
λ x0 .
∀ x1 :
ι → ο
.
(
∀ x2 .
∀ x3 :
ι → ι
.
(
∀ x4 .
prim1
x4
x2
⟶
prim1
(
x3
x4
)
x2
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
prim1
x5
x2
⟶
prim1
(
x4
x5
)
x2
)
⟶
∀ x5 :
ι → ο
.
∀ x6 .
prim1
x6
x2
⟶
x1
(
7e46e..
x2
x3
x4
x5
x6
)
)
⟶
x1
x0
Theorem
03342..
:
∀ x0 .
∀ x1 :
ι → ι
.
(
∀ x2 .
prim1
x2
x0
⟶
prim1
(
x1
x2
)
x0
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
prim1
x3
x0
⟶
prim1
(
x2
x3
)
x0
)
⟶
∀ x3 :
ι → ο
.
∀ x4 .
prim1
x4
x0
⟶
57517..
(
7e46e..
x0
x1
x2
x3
x4
)
(proof)
Theorem
ddaaf..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 :
ι → ο
.
∀ x4 .
57517..
(
7e46e..
x0
x1
x2
x3
x4
)
⟶
∀ x5 .
prim1
x5
x0
⟶
prim1
(
x1
x5
)
x0
(proof)
Theorem
c310c..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 :
ι → ο
.
∀ x4 .
57517..
(
7e46e..
x0
x1
x2
x3
x4
)
⟶
∀ x5 .
prim1
x5
x0
⟶
prim1
(
x2
x5
)
x0
(proof)
Theorem
75de5..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 :
ι → ο
.
∀ x4 .
57517..
(
7e46e..
x0
x1
x2
x3
x4
)
⟶
prim1
x4
x0
(proof)
Theorem
e3820..
:
∀ x0 .
57517..
x0
⟶
x0
=
7e46e..
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(proof)
Definition
311a4..
:=
λ x0 .
λ x1 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ο
)
→
ι → ι
.
x1
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
Theorem
e5821..
:
∀ x0 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ο
)
→
ι → ι
.
∀ x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 :
ι → ο
.
∀ x5 .
(
∀ x6 :
ι → ι
.
(
∀ x7 .
prim1
x7
x1
⟶
x2
x7
=
x6
x7
)
⟶
∀ x7 :
ι → ι
.
(
∀ x8 .
prim1
x8
x1
⟶
x3
x8
=
x7
x8
)
⟶
∀ x8 :
ι → ο
.
(
∀ x9 .
prim1
x9
x1
⟶
iff
(
x4
x9
)
(
x8
x9
)
)
⟶
x0
x1
x6
x7
x8
x5
=
x0
x1
x2
x3
x4
x5
)
⟶
311a4..
(
7e46e..
x1
x2
x3
x4
x5
)
x0
=
x0
x1
x2
x3
x4
x5
(proof)
Definition
734b8..
:=
λ x0 .
λ x1 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ο
)
→
ι → ο
.
x1
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
Theorem
aa6de..
:
∀ x0 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
(
ι → ο
)
→
ι → ο
.
∀ x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 :
ι → ο
.
∀ x5 .
(
∀ x6 :
ι → ι
.
(
∀ x7 .
prim1
x7
x1
⟶
x2
x7
=
x6
x7
)
⟶
∀ x7 :
ι → ι
.
(
∀ x8 .
prim1
x8
x1
⟶
x3
x8
=
x7
x8
)
⟶
∀ x8 :
ι → ο
.
(
∀ x9 .
prim1
x9
x1
⟶
iff
(
x4
x9
)
(
x8
x9
)
)
⟶
x0
x1
x6
x7
x8
x5
=
x0
x1
x2
x3
x4
x5
)
⟶
734b8..
(
7e46e..
x1
x2
x3
x4
x5
)
x0
=
x0
x1
x2
x3
x4
x5
(proof)
Definition
264ee..
:=
λ x0 .
λ x1 x2 :
ι → ι
.
λ x3 x4 .
0fc90..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
λ x5 .
If_i
(
x5
=
4a7ef..
)
x0
(
If_i
(
x5
=
4ae4a..
4a7ef..
)
(
0fc90..
x0
x1
)
(
If_i
(
x5
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
0fc90..
x0
x2
)
(
If_i
(
x5
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x3
x4
)
)
)
)
Theorem
b50ff..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 .
x0
=
264ee..
x1
x2
x3
x4
x5
⟶
x1
=
f482f..
x0
4a7ef..
(proof)
Theorem
af877..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 .
x0
=
f482f..
(
264ee..
x0
x1
x2
x3
x4
)
4a7ef..
(proof)
Theorem
9e7c7..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 .
x0
=
264ee..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x2
x6
=
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
x6
(proof)
Theorem
8892a..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 x5 .
prim1
x5
x0
⟶
x1
x5
=
f482f..
(
f482f..
(
264ee..
x0
x1
x2
x3
x4
)
(
4ae4a..
4a7ef..
)
)
x5
(proof)
Theorem
b7d3c..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 .
x0
=
264ee..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x3
x6
=
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x6
(proof)
Theorem
e5a18..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 x5 .
prim1
x5
x0
⟶
x2
x5
=
f482f..
(
f482f..
(
264ee..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x5
(proof)
Theorem
bd572..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 .
x0
=
264ee..
x1
x2
x3
x4
x5
⟶
x4
=
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
(proof)
Theorem
46c2b..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 .
x3
=
f482f..
(
264ee..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
(proof)
Theorem
67806..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 .
x0
=
264ee..
x1
x2
x3
x4
x5
⟶
x5
=
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(proof)
Theorem
4eb22..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 .
x4
=
f482f..
(
264ee..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(proof)
Theorem
e3dba..
:
∀ x0 x1 .
∀ x2 x3 x4 x5 :
ι → ι
.
∀ x6 x7 x8 x9 .
264ee..
x0
x2
x4
x6
x8
=
264ee..
x1
x3
x5
x7
x9
⟶
and
(
and
(
and
(
and
(
x0
=
x1
)
(
∀ x10 .
prim1
x10
x0
⟶
x2
x10
=
x3
x10
)
)
(
∀ x10 .
prim1
x10
x0
⟶
x4
x10
=
x5
x10
)
)
(
x6
=
x7
)
)
(
x8
=
x9
)
(proof)
Theorem
d989f..
:
∀ x0 .
∀ x1 x2 x3 x4 :
ι → ι
.
∀ x5 x6 .
(
∀ x7 .
prim1
x7
x0
⟶
x1
x7
=
x2
x7
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
x3
x7
=
x4
x7
)
⟶
264ee..
x0
x1
x3
x5
x6
=
264ee..
x0
x2
x4
x5
x6
(proof)
Definition
14e81..
:=
λ x0 .
∀ x1 :
ι → ο
.
(
∀ x2 .
∀ x3 :
ι → ι
.
(
∀ x4 .
prim1
x4
x2
⟶
prim1
(
x3
x4
)
x2
)
⟶
∀ x4 :
ι → ι
.
(
∀ x5 .
prim1
x5
x2
⟶
prim1
(
x4
x5
)
x2
)
⟶
∀ x5 .
prim1
x5
x2
⟶
∀ x6 .
prim1
x6
x2
⟶
x1
(
264ee..
x2
x3
x4
x5
x6
)
)
⟶
x1
x0
Theorem
eac13..
:
∀ x0 .
∀ x1 :
ι → ι
.
(
∀ x2 .
prim1
x2
x0
⟶
prim1
(
x1
x2
)
x0
)
⟶
∀ x2 :
ι → ι
.
(
∀ x3 .
prim1
x3
x0
⟶
prim1
(
x2
x3
)
x0
)
⟶
∀ x3 .
prim1
x3
x0
⟶
∀ x4 .
prim1
x4
x0
⟶
14e81..
(
264ee..
x0
x1
x2
x3
x4
)
(proof)
Theorem
7833c..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 .
14e81..
(
264ee..
x0
x1
x2
x3
x4
)
⟶
∀ x5 .
prim1
x5
x0
⟶
prim1
(
x1
x5
)
x0
(proof)
Theorem
29f0c..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 .
14e81..
(
264ee..
x0
x1
x2
x3
x4
)
⟶
∀ x5 .
prim1
x5
x0
⟶
prim1
(
x2
x5
)
x0
(proof)
Theorem
61077..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 .
14e81..
(
264ee..
x0
x1
x2
x3
x4
)
⟶
prim1
x3
x0
(proof)
Theorem
a6148..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 .
14e81..
(
264ee..
x0
x1
x2
x3
x4
)
⟶
prim1
x4
x0
(proof)
Theorem
f870f..
:
∀ x0 .
14e81..
x0
⟶
x0
=
264ee..
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(proof)
Definition
3be00..
:=
λ x0 .
λ x1 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι →
ι → ι
.
x1
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
Theorem
0c40e..
:
∀ x0 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι →
ι → ι
.
∀ x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 .
(
∀ x6 :
ι → ι
.
(
∀ x7 .
prim1
x7
x1
⟶
x2
x7
=
x6
x7
)
⟶
∀ x7 :
ι → ι
.
(
∀ x8 .
prim1
x8
x1
⟶
x3
x8
=
x7
x8
)
⟶
x0
x1
x6
x7
x4
x5
=
x0
x1
x2
x3
x4
x5
)
⟶
3be00..
(
264ee..
x1
x2
x3
x4
x5
)
x0
=
x0
x1
x2
x3
x4
x5
(proof)
Definition
882cc..
:=
λ x0 .
λ x1 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι →
ι → ο
.
x1
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
f482f..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
Theorem
08db2..
:
∀ x0 :
ι →
(
ι → ι
)
→
(
ι → ι
)
→
ι →
ι → ο
.
∀ x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 .
(
∀ x6 :
ι → ι
.
(
∀ x7 .
prim1
x7
x1
⟶
x2
x7
=
x6
x7
)
⟶
∀ x7 :
ι → ι
.
(
∀ x8 .
prim1
x8
x1
⟶
x3
x8
=
x7
x8
)
⟶
x0
x1
x6
x7
x4
x5
=
x0
x1
x2
x3
x4
x5
)
⟶
882cc..
(
264ee..
x1
x2
x3
x4
x5
)
x0
=
x0
x1
x2
x3
x4
x5
(proof)
Param
d2155..
:
ι
→
(
ι
→
ι
→
ο
) →
ι
Definition
ae02b..
:=
λ x0 .
λ x1 :
ι → ι
.
λ x2 :
ι →
ι → ο
.
λ x3 x4 :
ι → ο
.
0fc90..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
λ x5 .
If_i
(
x5
=
4a7ef..
)
x0
(
If_i
(
x5
=
4ae4a..
4a7ef..
)
(
0fc90..
x0
x1
)
(
If_i
(
x5
=
4ae4a..
(
4ae4a..
4a7ef..
)
)
(
d2155..
x0
x2
)
(
If_i
(
x5
=
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
(
1216a..
x0
x3
)
(
1216a..
x0
x4
)
)
)
)
)
Theorem
cc896..
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
∀ x3 :
ι →
ι → ο
.
∀ x4 x5 :
ι → ο
.
x0
=
ae02b..
x1
x2
x3
x4
x5
⟶
x1
=
f482f..
x0
4a7ef..
(proof)
Theorem
99e8a..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 :
ι →
ι → ο
.
∀ x3 x4 :
ι → ο
.
x0
=
f482f..
(
ae02b..
x0
x1
x2
x3
x4
)
4a7ef..
(proof)
Theorem
cf832..
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
∀ x3 :
ι →
ι → ο
.
∀ x4 x5 :
ι → ο
.
x0
=
ae02b..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x2
x6
=
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
x6
(proof)
Theorem
1e440..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 :
ι →
ι → ο
.
∀ x3 x4 :
ι → ο
.
∀ x5 .
prim1
x5
x0
⟶
x1
x5
=
f482f..
(
f482f..
(
ae02b..
x0
x1
x2
x3
x4
)
(
4ae4a..
4a7ef..
)
)
x5
(proof)
Param
2b2e3..
:
ι
→
ι
→
ι
→
ο
Known
67416..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
∀ x2 .
prim1
x2
x0
⟶
∀ x3 .
prim1
x3
x0
⟶
2b2e3..
(
d2155..
x0
x1
)
x2
x3
=
x1
x2
x3
Theorem
e670a..
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
∀ x3 :
ι →
ι → ο
.
∀ x4 x5 :
ι → ο
.
x0
=
ae02b..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
∀ x7 .
prim1
x7
x1
⟶
x3
x6
x7
=
2b2e3..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x6
x7
(proof)
Theorem
df2ad..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 :
ι →
ι → ο
.
∀ x3 x4 :
ι → ο
.
∀ x5 .
prim1
x5
x0
⟶
∀ x6 .
prim1
x6
x0
⟶
x2
x5
x6
=
2b2e3..
(
f482f..
(
ae02b..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
x5
x6
(proof)
Theorem
293e1..
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
∀ x3 :
ι →
ι → ο
.
∀ x4 x5 :
ι → ο
.
x0
=
ae02b..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x4
x6
=
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x6
(proof)
Theorem
2473b..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 :
ι →
ι → ο
.
∀ x3 x4 :
ι → ο
.
∀ x5 .
prim1
x5
x0
⟶
x3
x5
=
decode_p
(
f482f..
(
ae02b..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
x5
(proof)
Theorem
5549c..
:
∀ x0 x1 .
∀ x2 :
ι → ι
.
∀ x3 :
ι →
ι → ο
.
∀ x4 x5 :
ι → ο
.
x0
=
ae02b..
x1
x2
x3
x4
x5
⟶
∀ x6 .
prim1
x6
x1
⟶
x5
x6
=
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x6
(proof)
Theorem
d04af..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 :
ι →
ι → ο
.
∀ x3 x4 :
ι → ο
.
∀ x5 .
prim1
x5
x0
⟶
x4
x5
=
decode_p
(
f482f..
(
ae02b..
x0
x1
x2
x3
x4
)
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
x5
(proof)
Theorem
3bc26..
:
∀ x0 x1 .
∀ x2 x3 :
ι → ι
.
∀ x4 x5 :
ι →
ι → ο
.
∀ x6 x7 x8 x9 :
ι → ο
.
ae02b..
x0
x2
x4
x6
x8
=
ae02b..
x1
x3
x5
x7
x9
⟶
and
(
and
(
and
(
and
(
x0
=
x1
)
(
∀ x10 .
prim1
x10
x0
⟶
x2
x10
=
x3
x10
)
)
(
∀ x10 .
prim1
x10
x0
⟶
∀ x11 .
prim1
x11
x0
⟶
x4
x10
x11
=
x5
x10
x11
)
)
(
∀ x10 .
prim1
x10
x0
⟶
x6
x10
=
x7
x10
)
)
(
∀ x10 .
prim1
x10
x0
⟶
x8
x10
=
x9
x10
)
(proof)
Known
62ef7..
:
∀ x0 .
∀ x1 x2 :
ι →
ι → ο
.
(
∀ x3 .
prim1
x3
x0
⟶
∀ x4 .
prim1
x4
x0
⟶
iff
(
x1
x3
x4
)
(
x2
x3
x4
)
)
⟶
d2155..
x0
x1
=
d2155..
x0
x2
Theorem
0a4ae..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
∀ x3 x4 :
ι →
ι → ο
.
∀ x5 x6 x7 x8 :
ι → ο
.
(
∀ x9 .
prim1
x9
x0
⟶
x1
x9
=
x2
x9
)
⟶
(
∀ x9 .
prim1
x9
x0
⟶
∀ x10 .
prim1
x10
x0
⟶
iff
(
x3
x9
x10
)
(
x4
x9
x10
)
)
⟶
(
∀ x9 .
prim1
x9
x0
⟶
iff
(
x5
x9
)
(
x6
x9
)
)
⟶
(
∀ x9 .
prim1
x9
x0
⟶
iff
(
x7
x9
)
(
x8
x9
)
)
⟶
ae02b..
x0
x1
x3
x5
x7
=
ae02b..
x0
x2
x4
x6
x8
(proof)
Definition
a4680..
:=
λ x0 .
∀ x1 :
ι → ο
.
(
∀ x2 .
∀ x3 :
ι → ι
.
(
∀ x4 .
prim1
x4
x2
⟶
prim1
(
x3
x4
)
x2
)
⟶
∀ x4 :
ι →
ι → ο
.
∀ x5 x6 :
ι → ο
.
x1
(
ae02b..
x2
x3
x4
x5
x6
)
)
⟶
x1
x0
Theorem
58929..
:
∀ x0 .
∀ x1 :
ι → ι
.
(
∀ x2 .
prim1
x2
x0
⟶
prim1
(
x1
x2
)
x0
)
⟶
∀ x2 :
ι →
ι → ο
.
∀ x3 x4 :
ι → ο
.
a4680..
(
ae02b..
x0
x1
x2
x3
x4
)
(proof)
Theorem
0a953..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 :
ι →
ι → ο
.
∀ x3 x4 :
ι → ο
.
a4680..
(
ae02b..
x0
x1
x2
x3
x4
)
⟶
∀ x5 .
prim1
x5
x0
⟶
prim1
(
x1
x5
)
x0
(proof)
Theorem
e88d1..
:
∀ x0 .
a4680..
x0
⟶
x0
=
ae02b..
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
2b2e3..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
(proof)
Definition
8d403..
:=
λ x0 .
λ x1 :
ι →
(
ι → ι
)
→
(
ι →
ι → ο
)
→
(
ι → ο
)
→
(
ι → ο
)
→ ι
.
x1
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
2b2e3..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
Theorem
11ab3..
:
∀ x0 :
ι →
(
ι → ι
)
→
(
ι →
ι → ο
)
→
(
ι → ο
)
→
(
ι → ο
)
→ ι
.
∀ x1 .
∀ x2 :
ι → ι
.
∀ x3 :
ι →
ι → ο
.
∀ x4 x5 :
ι → ο
.
(
∀ x6 :
ι → ι
.
(
∀ x7 .
prim1
x7
x1
⟶
x2
x7
=
x6
x7
)
⟶
∀ x7 :
ι →
ι → ο
.
(
∀ x8 .
prim1
x8
x1
⟶
∀ x9 .
prim1
x9
x1
⟶
iff
(
x3
x8
x9
)
(
x7
x8
x9
)
)
⟶
∀ x8 :
ι → ο
.
(
∀ x9 .
prim1
x9
x1
⟶
iff
(
x4
x9
)
(
x8
x9
)
)
⟶
∀ x9 :
ι → ο
.
(
∀ x10 .
prim1
x10
x1
⟶
iff
(
x5
x10
)
(
x9
x10
)
)
⟶
x0
x1
x6
x7
x8
x9
=
x0
x1
x2
x3
x4
x5
)
⟶
8d403..
(
ae02b..
x1
x2
x3
x4
x5
)
x0
=
x0
x1
x2
x3
x4
x5
(proof)
Definition
6d899..
:=
λ x0 .
λ x1 :
ι →
(
ι → ι
)
→
(
ι →
ι → ο
)
→
(
ι → ο
)
→
(
ι → ο
)
→ ο
.
x1
(
f482f..
x0
4a7ef..
)
(
f482f..
(
f482f..
x0
(
4ae4a..
4a7ef..
)
)
)
(
2b2e3..
(
f482f..
x0
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
(
decode_p
(
f482f..
x0
(
4ae4a..
(
4ae4a..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
)
)
)
)
Theorem
77bfd..
:
∀ x0 :
ι →
(
ι → ι
)
→
(
ι →
ι → ο
)
→
(
ι → ο
)
→
(
ι → ο
)
→ ο
.
∀ x1 .
∀ x2 :
ι → ι
.
∀ x3 :
ι →
ι → ο
.
∀ x4 x5 :
ι → ο
.
(
∀ x6 :
ι → ι
.
(
∀ x7 .
prim1
x7
x1
⟶
x2
x7
=
x6
x7
)
⟶
∀ x7 :
ι →
ι → ο
.
(
∀ x8 .
prim1
x8
x1
⟶
∀ x9 .
prim1
x9
x1
⟶
iff
(
x3
x8
x9
)
(
x7
x8
x9
)
)
⟶
∀ x8 :
ι → ο
.
(
∀ x9 .
prim1
x9
x1
⟶
iff
(
x4
x9
)
(
x8
x9
)
)
⟶
∀ x9 :
ι → ο
.
(
∀ x10 .
prim1
x10
x1
⟶
iff
(
x5
x10
)
(
x9
x10
)
)
⟶
x0
x1
x6
x7
x8
x9
=
x0
x1
x2
x3
x4
x5
)
⟶
6d899..
(
ae02b..
x1
x2
x3
x4
x5
)
x0
=
x0
x1
x2
x3
x4
x5
(proof)
previous assets