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Param
ap
ap
:
ι
→
ι
→
ι
Param
lam
Sigma
:
ι
→
(
ι
→
ι
) →
ι
Param
ordsucc
ordsucc
:
ι
→
ι
Param
If_i
If_i
:
ο
→
ι
→
ι
→
ι
Definition
u17_to_Church17
:=
λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 .
ap
(
lam
17
(
λ x18 .
If_i
(
x18
=
0
)
x1
(
If_i
(
x18
=
1
)
x2
(
If_i
(
x18
=
2
)
x3
(
If_i
(
x18
=
3
)
x4
(
If_i
(
x18
=
4
)
x5
(
If_i
(
x18
=
5
)
x6
(
If_i
(
x18
=
6
)
x7
(
If_i
(
x18
=
7
)
x8
(
If_i
(
x18
=
8
)
x9
(
If_i
(
x18
=
9
)
x10
(
If_i
(
x18
=
10
)
x11
(
If_i
(
x18
=
11
)
x12
(
If_i
(
x18
=
12
)
x13
(
If_i
(
x18
=
13
)
x14
(
If_i
(
x18
=
14
)
x15
(
If_i
(
x18
=
15
)
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
x0
Param
u3
:
ι
Known
2a0b0..
:
(
∀ 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
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u3
=
x3
Known
48efb..
:
∀ 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
Known
d21a1..
:
∀ 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
Theorem
fbcaf..
:
∀ 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
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u3
=
x3
(proof)
Param
u4
:
ι
Known
b09cb..
:
(
∀ 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
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u4
=
x4
Theorem
12883..
:
∀ 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
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u4
=
x4
(proof)
Param
u5
:
ι
Known
497c7..
:
(
∀ 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
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u5
=
x5
Theorem
8021a..
:
∀ 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
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
u5
=
x5
(proof)
Known
aa7c9..
:
∀ x0 :
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι → ι
.
∀ x1 .
∀ x2 :
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι →
ι → ι
.
(
∀ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 .
x0
x1
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
x18
x19
=
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
x18
x19
)
⟶
x0
x1
=
x2
Theorem
1ef08..
:
u17_to_Church17
u3
=
λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 .
x4
(proof)
Theorem
05513..
:
u17_to_Church17
u4
=
λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 .
x5
(proof)
Theorem
22977..
:
u17_to_Church17
u5
=
λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 .
x6
(proof)