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Proofgold Asset
asset id
8f9e416549f1765368bd0443bf274d941adfb5983536cc35695446fa0e075ca6
asset hash
3f5bb1144fd0281b0fff49c3c4491b1e471fb722c2c67a380c1f9c04838f63cd
bday / block
3249
tx
06954..
preasset
doc published by
PrCx1..
Known
46237..
LoopI
:
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
binop_on
x0
x1
⟶
binop_on
x0
x2
⟶
binop_on
x0
x3
⟶
(
∀ x5 .
In
x5
x0
⟶
and
(
x1
x4
x5
=
x5
)
(
x1
x5
x4
=
x5
)
)
⟶
(
∀ x5 .
In
x5
x0
⟶
∀ x6 .
In
x6
x0
⟶
and
(
and
(
and
(
x2
x5
(
x1
x5
x6
)
=
x6
)
(
x1
x5
(
x2
x5
x6
)
=
x6
)
)
(
x3
(
x1
x5
x6
)
x6
=
x5
)
)
(
x1
(
x3
x5
x6
)
x6
=
x5
)
)
⟶
Loop
x0
x1
x2
x3
x4
Known
7c02f..
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Known
22d81..
and4I
:
∀ x0 x1 x2 x3 : ο .
x0
⟶
x1
⟶
x2
⟶
x3
⟶
and
(
and
(
and
x0
x1
)
x2
)
x3
Known
9ae18..
SingE
:
∀ x0 x1 .
In
x1
(
Sing
x0
)
⟶
x1
=
x0
Known
86824..
binop_on_I
:
∀ x0 .
∀ x1 :
ι →
ι → ι
.
(
∀ x2 .
In
x2
x0
⟶
∀ x3 .
In
x3
x0
⟶
In
(
x1
x2
x3
)
x0
)
⟶
binop_on
x0
x1
Known
1f15b..
SingI
:
∀ x0 .
In
x0
(
Sing
x0
)
Theorem
f3ad1..
:
Loop
(
Sing
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
Sing
0
)
(proof)
Known
20499..
Loop_with_defs_I
:
∀ 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
x0
x1
x2
x3
x4
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
x5
x14
x15
=
x2
(
x1
x15
x14
)
(
x1
x14
x15
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x6
x14
x15
x16
=
x2
(
x1
x14
(
x1
x15
x16
)
)
(
x1
(
x1
x14
x15
)
x16
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
and
(
and
(
and
(
and
(
x7
x14
x15
=
x2
x14
(
x1
x15
x14
)
)
(
x10
x14
x15
=
x1
x14
(
x1
x15
(
x2
x14
x4
)
)
)
)
(
x11
x14
x15
=
x1
(
x1
(
x3
x4
x14
)
x15
)
x14
)
)
(
x12
x14
x15
=
x1
(
x2
x14
x15
)
(
x2
(
x2
x14
x4
)
x4
)
)
)
(
x13
x14
x15
=
x1
(
x3
x4
(
x3
x4
x14
)
)
(
x3
x15
x14
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
and
(
x8
x14
x15
x16
=
x2
(
x1
x15
x14
)
(
x1
x15
(
x1
x14
x16
)
)
)
(
x9
x14
x15
x16
=
x3
(
x1
(
x1
x16
x14
)
x15
)
(
x1
x14
x15
)
)
)
⟶
Loop_with_defs
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
Known
1bd08..
and5I
:
∀ x0 x1 x2 x3 x4 : ο .
x0
⟶
x1
⟶
x2
⟶
x3
⟶
x4
⟶
and
(
and
(
and
(
and
x0
x1
)
x2
)
x3
)
x4
Theorem
6e0b2..
:
Loop_with_defs
(
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
)
(proof)
Known
9d0a4..
Loop_with_defs_cex1_I
:
∀ 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
.
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
Loop_with_defs
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
not
(
x5
(
x1
(
x2
(
x8
x15
x16
x14
)
x4
)
x14
)
x17
=
x4
)
⟶
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
Known
8106d..
notI
:
∀ x0 : ο .
(
x0
⟶
False
)
⟶
not
x0
Known
2901c..
EmptyE
:
∀ x0 .
In
x0
0
⟶
False
Theorem
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
)
(proof)
Known
7054a..
Loop_with_defs_cex2_I
:
∀ 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
.
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
Loop_with_defs
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
not
(
x6
x18
(
x1
(
x3
x4
x14
)
(
x9
x15
x16
x14
)
)
x17
=
x4
)
⟶
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
Theorem
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
)
(proof)
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Theorem
05200..
:
(
∀ 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
(
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
⟶
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
(
x13
x15
(
x10
x16
x17
)
)
=
x13
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
(
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
(
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
(
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
⟶
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
⟶
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
⟶
x7
x14
(
x7
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
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
⟶
∀ 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
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
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
(
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
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
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
(
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
(
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
(
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
⟶
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
(
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
(
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
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
d8f77..
:
(
∀ 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
(
x7
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x12
x16
(
x8
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
(
x13
x15
(
x12
x16
x17
)
)
=
x13
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
(
x12
x17
x18
)
)
=
x7
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
(
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
⟶
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
(
x7
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
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
(
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
⟶
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
⟶
∀ 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
(
x13
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
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
(
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
(
x12
x19
x20
)
)
)
=
x9
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
(
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
(
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
(
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
(
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
(
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
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
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
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
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
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
b87e6..
:
(
∀ 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
⟶
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
(
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
⟶
x10
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
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
⟶
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
⟶
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
⟶
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
⟶
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
⟶
∀ 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
⟶
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
(
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
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
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
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
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
(
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
⟶
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
⟶
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
(
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
(
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
(
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
(
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
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
10075..
:
(
∀ 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
(
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
(
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
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
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
⟶
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
⟶
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
⟶
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
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
(
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
⟶
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
⟶
∀ 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
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
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
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
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
(
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
⟶
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
(
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
(
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
⟶
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
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
f53e4..
:
(
∀ 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
(
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
(
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
⟶
x13
x14
(
x12
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
⟶
x12
x14
(
x7
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
⟶
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
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
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
⟶
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
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
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x13
x18
x19
)
)
)
=
x7
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
(
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
(
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
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
⟶
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
⟶
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
(
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
⟶
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
(
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
⟶
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
⟶
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
e367a..
:
(
∀ 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
(
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
⟶
x12
x14
(
x10
x15
(
x7
x16
x17
)
)
=
x10
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
⟶
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
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
(
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
⟶
x7
x14
(
x12
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
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
(
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
⟶
∀ 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
⟶
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
⟶
∀ 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
⟶
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
⟶
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
⟶
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
(
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
(
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
(
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
(
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
(
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
(
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
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
ab599..
:
(
∀ 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
(
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
⟶
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
⟶
x13
x14
(
x12
x15
(
x10
x16
x17
)
)
=
x12
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
(
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
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
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
⟶
x13
x14
(
x10
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
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
⟶
x10
x14
(
x10
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
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
⟶
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
⟶
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
⟶
∀ 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
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
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
(
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
⟶
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
⟶
∀ 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
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
(
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
(
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
(
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
⟶
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
(
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
(
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
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
30ac6..
:
(
∀ 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
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x7
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x7
x16
(
x12
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
(
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
⟶
x13
x14
(
x10
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
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
⟶
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
(
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
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
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
⟶
x13
x14
(
x10
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
(
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
(
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
(
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
(
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
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
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
⟶
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
(
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
(
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
(
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
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
770dd..
:
(
∀ 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
⟶
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
(
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
⟶
x7
x14
(
x10
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
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
(
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
⟶
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
⟶
x12
x14
(
x10
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
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
(
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
(
x13
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
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
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
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
(
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
⟶
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
(
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
(
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
⟶
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
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
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
(
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
(
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
⟶
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
(
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
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
870df..
:
(
∀ 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
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x12
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
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x9
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
⟶
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
(
x7
x16
(
x13
x17
x18
)
)
=
x7
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
⟶
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
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
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
⟶
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
⟶
x13
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
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
(
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
(
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
⟶
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
⟶
x8
x14
x15
(
x13
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
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
(
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
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
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
(
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
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
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
(
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
⟶
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
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
(
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
(
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
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
3031c..
:
(
∀ 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
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ 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
(
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
⟶
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
⟶
x7
x14
(
x13
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
(
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
⟶
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
(
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
⟶
∀ 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
(
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
⟶
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
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
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
⟶
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
⟶
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
⟶
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
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
(
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
(
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
(
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
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
8b77e..
:
(
∀ 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
(
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
⟶
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x7
x16
(
x8
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
(
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
⟶
x10
x14
(
x7
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
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
(
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
(
x7
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
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
⟶
x13
x14
(
x10
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
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
⟶
x12
x14
(
x10
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
⟶
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
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
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
⟶
∀ 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
(
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
⟶
∀ 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
(
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
(
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
⟶
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
⟶
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
(
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
(
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
⟶
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
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
6ce07..
:
(
∀ 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
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ 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
(
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
⟶
x12
x14
(
x10
x15
(
x7
x16
x17
)
)
=
x10
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
(
x10
x17
x18
)
)
=
x10
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
(
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
⟶
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
(
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
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
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
⟶
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
(
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
(
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
⟶
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
⟶
∀ 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
⟶
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
(
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
(
x12
x20
x21
)
)
)
)
=
x8
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
⟶
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
(
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
(
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
(
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
984a6..
:
(
∀ 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
(
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
(
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
(
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
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
(
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
⟶
x12
x14
(
x10
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
⟶
x10
x14
(
x12
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
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
(
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
⟶
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
(
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
(
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
⟶
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
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
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
(
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
(
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
(
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
(
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
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
3f597..
:
(
∀ 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
(
x13
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x8
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
⟶
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
⟶
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
(
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
⟶
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
⟶
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
(
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
(
x7
x17
x18
)
)
)
=
x10
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
(
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
⟶
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
⟶
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
⟶
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
⟶
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
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
(
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
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
(
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
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
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
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
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
(
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
⟶
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
a70db..
:
(
∀ 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
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x7
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
⟶
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
⟶
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
⟶
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
⟶
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
⟶
x10
x14
(
x10
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
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
(
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
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
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
⟶
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
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x7
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
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
⟶
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
(
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
(
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
⟶
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
(
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
(
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
⟶
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
(
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
(
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
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
652e5..
:
(
∀ 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
(
x7
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x7
x14
(
x9
x15
x16
(
x8
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
⟶
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
⟶
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
(
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
(
x7
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
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
⟶
x13
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
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
(
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
(
x7
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
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
(
x13
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
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
⟶
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
(
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
⟶
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
⟶
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
(
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
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
(
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
⟶
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
⟶
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
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
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
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
bddf5..
:
(
∀ 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
(
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
⟶
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
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
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
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
⟶
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
⟶
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
⟶
∀ 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
⟶
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
⟶
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
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
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
(
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
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
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
⟶
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
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
(
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
(
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
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
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
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
47db6..
:
(
∀ 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
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ 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
⟶
x12
x14
(
x10
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
(
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
(
x7
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
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
(
x10
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
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
(
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
(
x7
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
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
⟶
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
⟶
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
(
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
(
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
(
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
(
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
⟶
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
(
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
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
6cd78..
:
(
∀ 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
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ 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
(
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
⟶
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
⟶
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
⟶
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
⟶
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
⟶
x13
x14
(
x13
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x13
x18
x19
)
)
)
=
x10
x17
(
x13
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
⟶
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
(
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
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
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
(
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
(
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
⟶
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
⟶
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
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
(
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
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
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
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
76eff..
:
(
∀ 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
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x13
x14
(
x10
x15
(
x8
x14
x15
(
x13
x14
(
x10
x15
(
x8
x14
x15
(
x13
x14
(
x10
x15
(
x8
x14
x15
(
x13
x14
(
x10
x15
(
x8
x14
x15
(
x13
x14
(
x10
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ 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
⟶
x12
x14
(
x7
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
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
⟶
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
⟶
x7
x14
(
x13
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
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
(
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
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
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
⟶
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
⟶
∀ 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
(
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
(
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
(
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
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
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
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
(
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
(
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
(
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
f114e..
:
(
∀ 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
(
x8
x14
x15
(
x10
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ 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
(
x9
x14
x15
(
x12
x14
(
x12
x15
(
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
⟶
x12
x14
(
x7
x15
(
x7
x16
x17
)
)
=
x7
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
(
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
⟶
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
⟶
x12
x14
(
x13
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
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
⟶
x13
x14
(
x7
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
(
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
(
x10
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
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
⟶
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
(
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
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
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
⟶
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
(
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
(
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
(
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
⟶
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
⟶
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
(
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
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
66e5f..
:
(
∀ 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
(
x7
x14
(
x13
x15
(
x9
x14
x15
(
x7
x14
(
x13
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ 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
(
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
(
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
⟶
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
⟶
x12
x14
(
x12
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
(
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
(
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
(
x7
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
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
(
x7
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
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
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
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
(
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
⟶
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
⟶
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
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ 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
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
⟶
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
(
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
⟶
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
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
0b2b0..
:
(
∀ 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
⟶
x7
x14
(
x7
x15
(
x10
x16
x17
)
)
=
x7
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
⟶
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
⟶
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
(
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
(
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
(
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
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
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
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
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
(
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
⟶
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
(
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
(
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
⟶
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
⟶
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
(
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
(
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
(
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
⟶
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
(
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
059f6..
:
(
∀ 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
(
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
⟶
x9
x14
x15
(
x10
x14
(
x12
x15
(
x9
x14
x15
(
x10
x14
(
x12
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
(
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
(
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
⟶
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
⟶
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
(
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
⟶
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
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
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
⟶
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
(
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
⟶
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
(
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
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
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
(
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
⟶
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
⟶
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
⟶
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
⟶
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
(
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
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
4a27b..
:
(
∀ 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
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
(
x9
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
⟶
x7
x14
(
x10
x15
(
x7
x16
x17
)
)
=
x10
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
(
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
(
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
⟶
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
⟶
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
⟶
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
(
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
(
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
⟶
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
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
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
(
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
⟶
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
(
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
(
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
(
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
⟶
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
(
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
(
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
⟶
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
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
3d069..
:
(
∀ 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
(
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
⟶
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
(
x13
x17
x18
)
)
=
x7
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
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
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
⟶
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
⟶
x12
x14
(
x12
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
(
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
⟶
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
⟶
∀ 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
⟶
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
⟶
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
(
x7
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
x17
x18
(
x13
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
(
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
(
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
(
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
⟶
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
(
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
⟶
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
(
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
⟶
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
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
ca3ae..
:
(
∀ 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
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x8
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x7
x14
(
x7
x15
(
x9
x14
x15
(
x7
x14
(
x7
x15
(
x9
x14
x15
(
x7
x14
(
x7
x15
(
x9
x14
x15
(
x7
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
(
x10
x16
x17
)
)
=
x12
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
⟶
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
⟶
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
⟶
x12
x14
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
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
⟶
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
(
x13
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
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
(
x12
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
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
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
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
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
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
⟶
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
⟶
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
(
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
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
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
(
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
(
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
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
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
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
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
(
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
(
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
51392..
:
(
∀ 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
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
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
(
x7
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x7
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
⟶
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
(
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
(
x7
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
x7
x14
(
x7
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
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
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
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
(
x10
x17
x18
)
)
)
=
x12
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
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
⟶
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
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
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
(
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
⟶
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
(
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
⟶
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
(
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
(
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
(
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
(
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
⟶
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
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
c847b..
:
(
∀ 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
(
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
⟶
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
⟶
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
(
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
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
(
x13
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
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
(
x10
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
⟶
x13
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
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
⟶
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
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
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
(
x7
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
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
⟶
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
(
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
(
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
(
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
(
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
(
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
(
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
⟶
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
⟶
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
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
4bb04..
:
(
∀ 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
(
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
⟶
x8
x14
x15
(
x7
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x8
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
⟶
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
(
x12
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
⟶
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
⟶
x10
x14
(
x7
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
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
(
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
(
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
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
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
⟶
x9
x14
x15
(
x10
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
x17
x18
(
x13
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
(
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
⟶
∀ 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
⟶
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
(
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
(
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
(
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
(
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
(
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
(
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
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
94d84..
:
(
∀ 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
(
x7
x15
(
x10
x16
x17
)
)
=
x7
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
(
x10
x16
(
x13
x17
x18
)
)
=
x10
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
⟶
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
⟶
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
⟶
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
⟶
x7
x14
(
x10
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
⟶
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
⟶
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
(
x13
x17
(
x13
x18
x19
)
)
)
=
x13
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
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
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
(
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
(
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
(
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
(
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
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
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
(
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
⟶
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
(
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
(
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
⟶
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
b508f..
:
(
∀ 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
(
x10
x14
(
x10
x15
(
x9
x14
x15
(
x10
x14
(
x10
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ 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
(
x9
x14
x15
(
x12
x14
(
x12
x15
(
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
⟶
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
(
x10
x17
x18
)
)
=
x10
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
⟶
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
⟶
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
⟶
x7
x14
(
x13
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
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
(
x7
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
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
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
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
(
x13
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
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
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
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
(
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
⟶
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
(
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
⟶
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
⟶
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
(
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
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
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
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
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
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
bcc6d..
:
(
∀ 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
(
x13
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x13
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x13
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x13
x16
(
x12
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
(
x13
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x13
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x13
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x13
x16
(
x12
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
⟶
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
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
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
⟶
x10
x14
(
x13
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
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
⟶
x12
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
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
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
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
⟶
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
⟶
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
⟶
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
(
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
⟶
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
⟶
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
⟶
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
⟶
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
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
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
(
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
(
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
(
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
(
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
4227b..
:
(
∀ 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
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x13
x15
(
x12
x16
x17
)
)
=
x13
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
(
x12
x17
x18
)
)
=
x7
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
(
x7
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
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
(
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
⟶
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
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
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
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x13
x18
x19
)
)
)
=
x10
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
⟶
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
(
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
⟶
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
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
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
(
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
⟶
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
(
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
(
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
⟶
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
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
46669..
:
(
∀ 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
⟶
x12
x14
(
x7
x15
(
x12
x16
x17
)
)
=
x7
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
(
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
⟶
x13
x14
(
x13
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
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
(
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
⟶
x13
x14
(
x12
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
(
x10
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
⟶
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
⟶
∀ 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
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
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
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
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
(
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
⟶
∀ 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
(
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
(
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
(
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
(
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
⟶
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
(
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
(
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
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
eee62..
:
(
∀ 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
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x13
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
⟶
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
(
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
⟶
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
(
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
(
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
⟶
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
(
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
(
x7
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
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
⟶
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
⟶
∀ 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
(
x10
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
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
(
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
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
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
⟶
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
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
(
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
(
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
(
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
eaf6d..
:
(
∀ 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
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
⟶
x12
x14
(
x12
x15
(
x13
x16
x17
)
)
=
x12
x15
(
x13
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
⟶
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
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
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
(
x7
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
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
(
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
⟶
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
(
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
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
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
(
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
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
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
⟶
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
⟶
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
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
(
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
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
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
(
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
(
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
⟶
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
adfa1..
:
(
∀ 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
(
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
⟶
x9
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
(
x9
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
⟶
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
(
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
⟶
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
(
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
(
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
⟶
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
⟶
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
⟶
∀ 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
⟶
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
⟶
∀ 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
(
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
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
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
⟶
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
(
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
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
(
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
⟶
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
(
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
⟶
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
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
bca70..
:
(
∀ 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
⟶
x13
x14
(
x12
x15
(
x13
x16
x17
)
)
=
x12
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
(
x7
x16
(
x12
x17
x18
)
)
=
x7
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
(
x13
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
⟶
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
⟶
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
⟶
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
(
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
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
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
(
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
⟶
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
⟶
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
⟶
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
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
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
(
x12
x20
x21
)
)
)
)
=
x9
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
⟶
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
(
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
(
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
(
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
(
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
e49a8..
:
(
∀ 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
(
x12
x16
x17
)
)
=
x7
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
(
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
(
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
(
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
⟶
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
⟶
x12
x14
(
x10
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
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
⟶
x12
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
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
⟶
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
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
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
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
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
(
x7
x19
x20
)
)
)
=
x9
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
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
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
(
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
⟶
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
(
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
⟶
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
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
(
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
⟶
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
(
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
0fb60..
:
(
∀ 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
(
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
(
x13
x16
(
x13
x17
x18
)
)
=
x13
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
⟶
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
⟶
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
⟶
x10
x14
(
x7
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
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
⟶
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
⟶
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
⟶
∀ 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
(
x10
x16
(
x12
x17
(
x13
x18
x19
)
)
)
=
x12
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
⟶
∀ 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
⟶
x8
x14
x15
(
x13
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
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
⟶
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
(
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
⟶
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
(
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
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
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
⟶
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
⟶
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
(
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
b47eb..
:
(
∀ 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
⟶
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
⟶
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
(
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
⟶
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
(
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
(
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
⟶
x13
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
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
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
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
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
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
(
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
⟶
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
(
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
(
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
(
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
(
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
⟶
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
(
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
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
(
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
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
4d875..
:
(
∀ 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
⟶
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
(
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
⟶
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
⟶
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
⟶
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
(
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
(
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
⟶
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
⟶
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
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
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
(
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
⟶
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
(
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
(
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
(
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
⟶
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
(
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
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
fee22..
:
(
∀ 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
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x13
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
(
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
(
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
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
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
⟶
x12
x14
(
x12
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
(
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
(
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
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
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
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
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
⟶
∀ 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
(
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
(
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
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
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
(
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
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
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
(
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
(
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
⟶
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
⟶
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
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
c8c4e..
:
(
∀ 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
(
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
⟶
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
(
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
(
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
(
x10
x17
x18
)
)
)
=
x12
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
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
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
⟶
x7
x14
(
x13
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
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
(
x12
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
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
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
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
⟶
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
⟶
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
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
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
(
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
(
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
(
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
(
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
(
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
(
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
⟶
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
f687f..
:
(
∀ 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
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x7
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
⟶
x7
x14
(
x13
x15
(
x12
x16
x17
)
)
=
x13
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
(
x13
x16
(
x13
x17
x18
)
)
=
x13
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
⟶
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
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
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
(
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
(
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
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
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
⟶
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
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
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
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
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
⟶
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
⟶
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
⟶
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
(
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
⟶
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
(
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
(
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
⟶
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
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)