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Proofgold Asset
asset id
27801373371c25d4630a63f1f1a608d26229e13d71739e16d009844634b387a3
asset hash
f10c754a2cba8c67376ebd116ddaf4a04e6ed054044784180e5b3b6322aad0d8
bday / block
3616
tx
85332..
preasset
doc published by
PrCx1..
Known
bc517..
:
Loop_with_defs_cex1
(
Sing
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
Sing
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
Known
8fd72..
:
Loop_with_defs_cex2
(
Sing
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
Sing
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 x2 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
(
λ x0 x1 .
0
)
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
9ae18..
SingE
:
∀ x0 x1 .
In
x1
(
Sing
x0
)
⟶
x1
=
x0
Theorem
1a089..
:
(
∀ 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
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
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
⟶
∀ 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
⟶
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
(
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
⟶
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
⟶
x12
x14
(
x10
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
(
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
⟶
x12
x14
(
x13
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
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
⟶
∀ 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
⟶
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
⟶
∀ 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
⟶
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
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
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
(
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
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
1d6b9..
:
(
∀ 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
(
x13
x15
(
x13
x16
x17
)
)
=
x13
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
(
x12
x16
(
x7
x17
x18
)
)
=
x12
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
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
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x13
x18
x19
)
)
)
=
x10
x17
(
x13
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
(
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
⟶
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
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
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
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
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
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
ce4fc..
:
(
∀ 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
(
x10
x15
(
x10
x16
x17
)
)
=
x10
x15
(
x10
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
x18
)
)
=
x7
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
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
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
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
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
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
⟶
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
⟶
∀ 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
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
(
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
(
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
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
266ed..
:
(
∀ 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
(
x12
x16
x17
)
)
=
x7
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
x18
)
)
=
x12
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
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
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ 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
(
x12
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
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
⟶
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
(
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
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
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
(
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
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
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
(
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
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
b782e..
:
(
∀ 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
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
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
⟶
x8
x14
x15
(
x12
x14
(
x13
x15
(
x8
x14
x15
(
x12
x14
(
x13
x15
(
x8
x14
x15
(
x12
x14
(
x13
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x10
x15
(
x12
x16
x17
)
)
=
x10
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
(
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
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
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
⟶
x10
x14
(
x7
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
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
(
x7
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
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
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
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
(
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
(
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
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
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
(
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
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
1405d..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x12
x15
(
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
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
x18
)
)
=
x10
x16
(
x13
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
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
(
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
⟶
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
x17
(
x7
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
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
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
d3e4c..
:
(
∀ 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
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x13
x16
(
x9
x14
x15
(
x7
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
⟶
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
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
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
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
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
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
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
(
x13
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
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
(
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
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
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
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
cb3f3..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
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
⟶
x12
x14
(
x10
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
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
⟶
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
⟶
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
⟶
∀ 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
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
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
⟶
∀ 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
(
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
(
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
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
bbf56..
:
(
∀ 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
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
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
(
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
⟶
x13
x14
(
x10
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
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
⟶
x12
x14
(
x10
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ 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
⟶
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
(
x13
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
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
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
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
(
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
(
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
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
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
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
2a5a9..
:
(
∀ 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
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x9
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
⟶
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
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
⟶
x10
x14
(
x13
x15
(
x13
x16
x17
)
)
=
x13
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
(
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
(
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
⟶
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
⟶
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
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
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
⟶
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
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
⟶
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
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
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
(
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
(
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
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
cf894..
:
(
∀ 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
(
x13
x15
(
x10
x16
x17
)
)
=
x13
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
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
⟶
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
(
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
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ 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
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
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
(
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
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
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
(
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
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
⟶
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
(
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
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
e658c..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
x17
)
)
=
x10
x15
(
x12
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
x18
)
)
=
x10
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
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
⟶
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
⟶
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
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
(
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
(
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
⟶
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
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
⟶
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
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
9510c..
:
(
∀ 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
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x10
x14
(
x10
x15
(
x8
x14
x15
(
x10
x14
(
x10
x15
(
x8
x14
x15
(
x10
x14
(
x10
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x13
x15
(
x7
x16
x17
)
)
=
x13
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
(
x13
x16
(
x10
x17
x18
)
)
=
x13
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
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
⟶
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
⟶
x13
x14
(
x12
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
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
(
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
(
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
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
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
649a7..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
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
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
x18
)
)
=
x13
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
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
(
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
⟶
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
⟶
x13
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
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
⟶
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
(
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
⟶
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
(
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
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
⟶
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
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
91849..
:
(
∀ 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
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
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
⟶
x7
x14
(
x12
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
(
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
⟶
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
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ 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
(
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
(
x10
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
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
⟶
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
(
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
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
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
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
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
(
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
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
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
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
797f5..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x13
x16
x17
)
)
=
x12
x15
(
x13
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
x18
)
)
=
x7
x16
(
x10
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
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
(
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
⟶
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
(
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
⟶
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
⟶
∀ 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
⟶
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
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
x17
x18
(
x13
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
(
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
(
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
(
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
(
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
(
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
⟶
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
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
7d64d..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
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
(
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
(
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
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
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
⟶
x12
x14
(
x13
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
∀ 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
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
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
⟶
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
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
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
(
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
(
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
(
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
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
42cba..
:
(
∀ 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
(
x8
x14
x15
(
x13
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x13
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x13
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x13
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
(
x8
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
(
x8
x14
x15
(
x10
x16
(
x13
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x10
x15
(
x12
x16
x17
)
)
=
x10
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
(
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
⟶
x7
x14
(
x13
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
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
⟶
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
⟶
x7
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
⟶
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
⟶
∀ 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
(
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
⟶
∀ 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
(
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
(
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
(
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
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
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
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
54c97..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x7
x14
(
x12
x15
(
x8
x14
x15
(
x7
x14
(
x12
x15
(
x8
x14
x15
(
x7
x14
(
x12
x15
(
x8
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
(
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
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
x18
)
)
=
x13
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
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
⟶
x13
x14
(
x10
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
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
⟶
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
⟶
x10
x14
(
x13
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
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
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
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
(
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
(
x13
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
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
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
e616f..
:
(
∀ 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
(
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
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
⟶
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
(
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
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x13
x18
x19
)
)
)
=
x13
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
(
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
(
x7
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
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
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
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
(
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
(
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
(
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
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
92c6b..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x13
x15
(
x8
x14
x15
(
x12
x14
(
x13
x15
(
x8
x14
x15
(
x12
x14
(
x13
x15
(
x8
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
(
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
⟶
x13
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
(
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
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ 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
(
x10
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
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
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
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
(
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
(
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
(
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
(
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
⟶
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
(
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
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
683ea..
:
(
∀ 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
(
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
(
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
⟶
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
⟶
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
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x13
x18
x19
)
)
)
=
x12
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
⟶
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
(
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
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
⟶
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
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
(
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
⟶
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
(
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
(
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
⟶
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
f743f..
:
(
∀ 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
(
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
⟶
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x7
x15
(
x12
x16
x17
)
)
=
x7
x15
(
x12
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
x16
(
x13
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
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
(
x12
x18
x19
)
)
)
=
x10
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
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
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
(
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
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
d2523..
:
(
∀ 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
(
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
⟶
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
⟶
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
(
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
(
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
⟶
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
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
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
⟶
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
(
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
⟶
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
(
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
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
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
(
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
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
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
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
bbe7d..
:
(
∀ 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
(
x10
x15
(
x12
x16
x17
)
)
=
x10
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
(
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
(
x7
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
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
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
⟶
∀ 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
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
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
(
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
(
x13
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
2bad4..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x7
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x13
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x13
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x7
x15
(
x13
x16
x17
)
)
=
x7
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
⟶
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
(
x7
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
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
⟶
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
⟶
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
(
x7
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
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
(
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
(
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
⟶
∀ 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
(
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
⟶
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
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
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
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
0fdf3..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x13
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
(
x9
x14
x15
(
x7
x16
(
x9
x14
x15
(
x13
x16
(
x9
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
(
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
(
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
(
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
(
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
⟶
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
⟶
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
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x13
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
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
⟶
∀ 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
(
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
(
x13
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
(
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
(
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
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
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
984ba..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x9
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
⟶
x13
x14
(
x7
x15
(
x7
x16
x17
)
)
=
x7
x15
(
x7
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
x18
)
)
=
x7
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
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
(
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
⟶
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
⟶
∀ 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
(
x13
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
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
(
x7
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
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
⟶
∀ 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
(
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
(
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
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
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
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
(
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
(
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
48512..
:
(
∀ 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
(
x10
x14
(
x7
x15
(
x8
x14
x15
(
x10
x14
(
x7
x15
(
x8
x14
x15
(
x10
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
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
(
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
⟶
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
(
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
⟶
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
⟶
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
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
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
⟶
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
⟶
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
(
x10
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
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
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
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
21303..
:
(
∀ 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
(
x8
x15
x16
(
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
(
x9
x14
x15
(
x7
x16
(
x12
x14
(
x8
x15
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
⟶
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
(
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
⟶
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
⟶
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
(
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
(
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
⟶
∀ 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
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ 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
⟶
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
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
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
(
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
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
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
(
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
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
a0603..
:
(
∀ 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
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
x18
)
)
=
x13
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
⟶
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
(
x13
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
x12
x14
(
x12
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
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
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ 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
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
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
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
(
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
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
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
⟶
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
724ca..
:
(
∀ 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
(
x10
x15
(
x10
x16
x17
)
)
=
x10
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
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
x18
)
)
=
x10
x16
(
x13
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
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
⟶
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
⟶
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
(
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
⟶
∀ 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
⟶
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
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
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
⟶
∀ 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
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
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
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
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
(
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
(
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
(
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
(
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
d6c61..
:
(
∀ 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
(
x9
x14
x15
(
x12
x16
(
x9
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
⟶
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
(
x12
x16
(
x7
x17
x18
)
)
=
x12
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
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
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
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
⟶
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
(
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
⟶
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
⟶
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
⟶
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
(
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
(
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
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
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
(
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
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
73a7e..
:
(
∀ 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
(
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
(
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
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
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
⟶
x10
x14
(
x12
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
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
(
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
(
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
(
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
(
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
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
afb2f..
:
(
∀ 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
(
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
(
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
⟶
x12
x14
(
x13
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
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
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
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
⟶
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
(
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
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
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
(
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
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
⟶
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
⟶
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
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
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
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
f7c49..
:
(
∀ 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
(
x10
x15
(
x7
x16
x17
)
)
=
x10
x15
(
x7
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
x18
)
)
=
x13
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
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
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
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
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
x16
(
x13
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
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
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
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
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
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
(
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
(
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
(
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
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
31826..
:
(
∀ 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
(
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
⟶
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
⟶
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
⟶
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
⟶
x13
x14
(
x13
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
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
⟶
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
(
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
⟶
∀ 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
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
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
⟶
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
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
52254..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
x17
)
)
=
x10
x15
(
x10
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
x13
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x13
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
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
⟶
x7
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
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
⟶
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
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
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
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)