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Proofgold Asset
asset id
ab1b35d8bce7266561a0ed568c5adca916a4901fe8d2d0adc75b9ea77e61444d
asset hash
5e5de724461df1eb7b75e49894dd1bb81d65562fd5e51faebd226857565137b4
bday / block
4871
tx
1d763..
preasset
doc published by
PrCx1..
Known
bc517..
:
Loop_with_defs_cex1
(
Sing
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
Sing
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
Known
8fd72..
:
Loop_with_defs_cex2
(
Sing
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
Sing
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
9ae18..
SingE
:
∀ x0 x1 .
In
x1
(
Sing
x0
)
⟶
x1
=
x0
Theorem
4ef83..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x13
x15
(
x7
x16
x17
)
)
=
x13
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
x18
)
)
=
x12
x16
(
x13
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x13
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x7
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
383c5..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
x18
)
)
=
x10
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
e8a2e..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
(
x8
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
(
x8
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
(
x8
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x13
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x13
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x13
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x13
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x7
x15
(
x10
x16
x17
)
)
=
x7
x15
(
x10
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
x18
)
)
=
x7
x16
(
x10
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
x16
(
x13
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
189cd..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x10
x14
(
x7
x15
(
x9
x14
x15
(
x10
x14
(
x7
x15
(
x9
x14
x15
(
x10
x14
(
x7
x15
(
x9
x14
x15
(
x10
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
x17
)
)
=
x10
x15
(
x12
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
x18
)
)
=
x7
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
x16
(
x13
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
x16
(
x13
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
x17
(
x7
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x13
x18
x19
)
)
)
=
x7
x17
(
x13
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
513e7..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
x17
)
)
=
x10
x15
(
x7
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
x18
)
)
=
x10
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
x16
(
x13
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x7
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
b743e..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x7
x15
(
x7
x16
x17
)
)
=
x7
x15
(
x7
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
x18
)
)
=
x12
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
a0428..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
x17
)
)
=
x7
x15
(
x10
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
x18
)
)
=
x7
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
x16
(
x13
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
6d2a9..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x10
x14
(
x7
x15
(
x8
x14
x15
(
x10
x14
(
x7
x15
(
x8
x14
x15
(
x10
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x13
x16
x17
)
)
=
x12
x15
(
x13
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
x18
)
)
=
x12
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
x16
(
x13
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
ad010..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
x17
)
)
=
x10
x15
(
x10
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
x17
(
x7
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
345a8..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x13
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
x16
(
x13
x17
(
x7
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
b3e00..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x7
x14
(
x7
x15
(
x8
x14
x15
(
x7
x14
(
x7
x15
(
x8
x14
x15
(
x7
x14
(
x7
x15
(
x8
x14
x15
(
x7
x14
(
x7
x15
(
x8
x14
x15
(
x7
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
x18
)
)
=
x13
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
x17
x18
(
x13
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
1d62d..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x10
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x10
x16
(
x12
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x13
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x13
x16
(
x8
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x13
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x13
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x13
x18
x19
)
)
)
=
x10
x17
(
x13
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
eed3f..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x12
x14
(
x12
x15
(
x9
x14
x15
(
x12
x14
(
x12
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
x18
)
)
=
x7
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x7
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
b77c7..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
x18
)
)
=
x12
x16
(
x10
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x13
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x13
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x13
x18
x19
)
)
)
=
x7
x17
(
x13
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
638e0..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x13
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x13
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x13
x16
(
x9
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
x17
)
)
=
x10
x15
(
x7
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
x18
)
)
=
x12
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
e6d24..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x10
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
x18
)
)
=
x10
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
14578..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x7
x15
(
x12
x16
x17
)
)
=
x7
x15
(
x12
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
x18
)
)
=
x10
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x13
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x13
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
x16
(
x13
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
5cd3c..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x13
x14
(
x7
x15
(
x8
x14
x15
(
x13
x14
(
x7
x15
(
x8
x14
x15
(
x13
x14
(
x7
x15
(
x8
x14
x15
(
x13
x14
(
x7
x15
(
x8
x14
x15
(
x13
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
x17
)
)
=
x10
x15
(
x12
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x7
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
3f6c5..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x10
x15
(
x7
x16
x17
)
)
=
x10
x15
(
x7
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
04802..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
x17
)
)
=
x7
x15
(
x10
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
x18
)
)
=
x10
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
14314..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x10
x15
(
x8
x14
x15
(
x12
x14
(
x10
x15
(
x8
x14
x15
(
x12
x14
(
x10
x15
(
x8
x14
x15
(
x12
x14
(
x10
x15
(
x8
x14
x15
(
x12
x14
(
x10
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
a1c01..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x7
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x7
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x7
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
x18
)
)
=
x12
x16
(
x13
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
x16
(
x13
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
34033..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x13
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
x18
)
)
=
x10
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
x17
(
x7
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
fa283..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
x18
)
)
=
x10
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
63d1e..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x7
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
x18
)
)
=
x7
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
x16
(
x13
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
x17
(
x7
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
6befe..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x10
x15
(
x12
x16
x17
)
)
=
x10
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
x18
)
)
=
x10
x16
(
x12
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
x17
(
x12
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
f97ef..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x10
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x7
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x7
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x7
x14
(
x8
x15
x16
(
x9
x14
x15
(
x12
x16
(
x7
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
x17
)
)
=
x12
x15
(
x10
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
x18
)
)
=
x10
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x13
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
f49a6..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x12
x14
(
x13
x15
(
x9
x14
x15
(
x12
x14
(
x13
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
x18
)
)
=
x10
x16
(
x13
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
f371f..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x13
x14
(
x7
x15
(
x9
x14
x15
(
x13
x14
(
x7
x15
(
x9
x14
x15
(
x13
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
x18
)
)
=
x12
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
42118..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
x18
)
)
=
x12
x16
(
x13
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
5cb3f..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
x17
)
)
=
x12
x15
(
x13
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
x18
)
)
=
x7
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
x17
x18
(
x13
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
3b276..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x10
x15
(
x8
x14
x15
(
x12
x14
(
x10
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x10
x15
(
x10
x16
x17
)
)
=
x10
x15
(
x10
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x13
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x7
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
66b44..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x13
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x13
x15
(
x10
x16
x17
)
)
=
x13
x15
(
x10
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
x18
)
)
=
x12
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x13
x18
x19
)
)
)
=
x7
x17
(
x13
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
f7e76..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x12
x14
(
x7
x15
(
x9
x14
x15
(
x12
x14
(
x7
x15
(
x9
x14
x15
(
x12
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
x18
)
)
=
x7
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
9c9b7..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x7
x15
(
x13
x16
x17
)
)
=
x7
x15
(
x13
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
5f53c..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x7
x15
(
x10
x16
x17
)
)
=
x7
x15
(
x10
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
x18
)
)
=
x7
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
x17
(
x7
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
c845d..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x13
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x13
x16
(
x7
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x13
x18
x19
)
)
)
=
x12
x17
(
x13
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x13
x18
x19
)
)
)
=
x12
x17
(
x13
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
dc176..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x13
x14
(
x12
x15
(
x8
x14
x15
(
x13
x14
(
x12
x15
(
x8
x14
x15
(
x13
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x13
x15
(
x12
x16
x17
)
)
=
x13
x15
(
x12
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
x18
)
)
=
x7
x16
(
x10
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x13
x18
x19
)
)
)
=
x13
x17
(
x13
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
0b1da..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
x17
)
)
=
x12
x15
(
x13
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
x18
)
)
=
x10
x16
(
x12
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
550fa..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x7
x15
(
x7
x16
x17
)
)
=
x7
x15
(
x7
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
x18
)
)
=
x12
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x13
x18
x19
)
)
)
=
x13
x17
(
x13
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x7
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
c13d5..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x10
x15
(
x8
x14
x15
(
x12
x14
(
x10
x15
(
x8
x14
x15
(
x12
x14
(
x10
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x7
x15
(
x12
x16
x17
)
)
=
x7
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
x18
)
)
=
x12
x16
(
x13
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x13
x18
x19
)
)
)
=
x12
x17
(
x13
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
b2dd6..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
x17
)
)
=
x10
x15
(
x10
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
x18
)
)
=
x13
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x13
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x13
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x13
x18
x19
)
)
)
=
x12
x17
(
x13
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
04725..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x10
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
x17
)
)
=
x7
x15
(
x12
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
x18
)
)
=
x10
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
x17
(
x7
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
0071a..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x13
x15
(
x7
x16
x17
)
)
=
x13
x15
(
x7
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)