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Proofgold Asset
asset id
49b15ab9b1464d61b51e5020f7fd5d3af3536ef9c7cb0f1285500a3e3ec41a6b
asset hash
94b426d458fa1e37a02c2d9ee1d98d943d5e3df4cb913a811f164ca01a1c267f
bday / block
5026
tx
908cc..
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
311c6..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x10
x16
x17
)
)
=
x12
x15
(
x10
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
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
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
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
⟶
x7
x14
(
x10
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
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
(
x7
x16
(
x12
x17
(
x13
x18
x19
)
)
)
=
x12
x17
(
x13
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ 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
(
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
(
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
(
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
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
81288..
:
(
∀ 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
(
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
⟶
x9
x14
x15
(
x10
x14
(
x7
x15
(
x9
x14
x15
(
x10
x14
(
x7
x15
(
x9
x14
x15
(
x10
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
⟶
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
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
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
(
x12
x16
(
x7
x17
(
x13
x18
x19
)
)
)
=
x7
x17
(
x13
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
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
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
31528..
:
(
∀ 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
(
x7
x14
(
x8
x15
x16
(
x8
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
(
x8
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x7
x16
(
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
(
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
⟶
x7
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
⟶
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
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ 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
(
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
⟶
∀ 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
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
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
(
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
f4c37..
:
(
∀ 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
(
x7
x15
(
x10
x16
x17
)
)
=
x7
x15
(
x10
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
x18
)
)
=
x10
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
(
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
⟶
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
⟶
x7
x14
(
x13
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
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
(
x10
x16
(
x12
x17
(
x13
x18
x19
)
)
)
=
x12
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
⟶
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
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
⟶
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
(
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
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
4a040..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
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
⟶
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
⟶
x10
x14
(
x7
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x10
x18
x19
)
)
)
=
x13
x17
(
x10
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x13
x18
x19
)
)
)
=
x13
x17
(
x13
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
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
(
x12
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
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
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
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
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
9ffca..
:
(
∀ 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
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x8
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
(
x7
x15
(
x12
x16
x17
)
)
=
x7
x15
(
x12
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
x7
x14
(
x10
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x13
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x13
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x13
x18
x19
)
)
)
=
x7
x17
(
x13
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
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
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
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
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
14600..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
(
x9
x14
x15
(
x10
x16
(
x7
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x10
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x10
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x10
x14
(
x9
x15
x16
(
x8
x14
x15
(
x12
x16
(
x10
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
x17
)
)
=
x7
x15
(
x12
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x13
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
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
⟶
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
⟶
x12
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x13
x18
x19
)
)
)
=
x10
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
⟶
x9
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
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
(
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
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
d75e1..
:
(
∀ 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
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x8
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
x17
)
)
=
x7
x15
(
x12
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
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
⟶
x12
x14
(
x13
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
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
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
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
(
x7
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
x17
(
x7
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x13
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
36c30..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x10
x14
(
x7
x15
(
x9
x14
x15
(
x10
x14
(
x7
x15
(
x9
x14
x15
(
x10
x14
(
x7
x15
(
x9
x14
x15
(
x10
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x13
x16
(
x8
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
x17
)
)
=
x10
x15
(
x10
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
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
(
x10
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x13
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x7
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
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
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
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
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
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
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
338a3..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x7
x15
(
x10
x16
x17
)
)
=
x7
x15
(
x10
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
x18
)
)
=
x7
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x7
x15
(
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
⟶
x13
x14
(
x10
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x13
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x13
x16
(
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
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
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
(
x10
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x7
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x9
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
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
(
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
⟶
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
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
ec761..
:
(
∀ 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
(
x10
x15
(
x8
x14
x15
(
x7
x14
(
x10
x15
(
x8
x14
x15
(
x7
x14
(
x10
x15
(
x8
x14
x15
(
x7
x14
(
x10
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x10
x14
(
x10
x15
(
x9
x14
x15
(
x10
x14
(
x10
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
x17
)
)
=
x12
x15
(
x13
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
x18
)
)
=
x7
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
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
(
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
⟶
x7
x14
(
x10
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
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
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
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
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
d045b..
:
(
∀ 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
(
x12
x15
(
x9
x14
x15
(
x12
x14
(
x12
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x10
x14
(
x10
x15
(
x9
x14
x15
(
x10
x14
(
x10
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
x17
)
)
=
x12
x15
(
x10
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
x18
)
)
=
x10
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
⟶
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
⟶
x10
x14
(
x7
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
⟶
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
⟶
∀ 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
⟶
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
⟶
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
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
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
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
d112a..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x13
x14
(
x12
x15
(
x9
x14
x15
(
x13
x14
(
x12
x15
(
x9
x14
x15
(
x13
x14
(
x12
x15
(
x9
x14
x15
(
x13
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
x18
)
)
=
x7
x16
(
x10
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
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
(
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
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
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
(
x13
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
x17
x18
(
x13
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
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
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
e5067..
:
(
∀ 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
(
x10
x16
x17
)
)
=
x10
x15
(
x10
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
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
(
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
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
x16
(
x13
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
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
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
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
⟶
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
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x13
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
4dd6e..
:
(
∀ 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
(
x12
x15
(
x8
x14
x15
(
x10
x14
(
x12
x15
(
x8
x14
x15
(
x10
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
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
⟶
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
(
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
⟶
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
⟶
x7
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ 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
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
0b023..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
(
x9
x14
x15
(
x7
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
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
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
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
⟶
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
(
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
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
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
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
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
(
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
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
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
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
6c9ba..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
(
x8
x14
x15
(
x12
x16
(
x12
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x7
x15
(
x13
x16
x17
)
)
=
x7
x15
(
x13
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
x18
)
)
=
x12
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
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
⟶
x13
x14
(
x7
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x13
x18
x19
)
)
)
=
x10
x17
(
x13
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ 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
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
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
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
b2253..
:
(
∀ 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
(
x13
x14
(
x9
x15
x16
(
x8
x14
x15
(
x7
x16
(
x13
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x7
x14
(
x10
x15
(
x9
x14
x15
(
x7
x14
(
x10
x15
(
x9
x14
x15
(
x7
x14
(
x10
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
(
x12
x17
x18
)
)
=
x10
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x13
x18
x19
)
)
)
=
x10
x17
(
x13
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
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
(
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
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
e10c4..
:
(
∀ 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
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x13
x14
(
x7
x15
(
x8
x14
x15
(
x13
x14
(
x7
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x7
x15
(
x7
x16
x17
)
)
=
x7
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
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
(
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
⟶
∀ 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
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x13
x18
x19
)
)
)
=
x10
x17
(
x13
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
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
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
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
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
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
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
46e63..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x13
x16
(
x7
x14
(
x8
x15
x16
(
x9
x14
x15
(
x13
x16
(
x7
x14
(
x8
x15
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ 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
⟶
x7
x14
(
x10
x15
(
x7
x16
x17
)
)
=
x10
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
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
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x10
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
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
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
e7923..
:
(
∀ 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
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x12
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x8
x14
x15
(
x13
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x13
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x13
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x13
x16
(
x9
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x12
x15
(
x10
x16
x17
)
)
=
x12
x15
(
x10
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
x18
)
)
=
x12
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
⟶
x7
x14
(
x10
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
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
(
x12
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x13
x19
x20
)
)
)
=
x9
x17
x18
(
x13
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x7
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
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
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
b4664..
:
(
∀ 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
(
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
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
x18
)
)
=
x10
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x12
x16
(
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
⟶
x10
x14
(
x7
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
⟶
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
(
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
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
08ba1..
:
(
∀ 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
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
⟶
x10
x14
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x7
x18
x19
)
)
)
=
x10
x17
(
x7
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
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
(
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
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
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
⟶
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
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
1f5d7..
:
(
∀ 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
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
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
x18
)
)
=
x12
x16
(
x10
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
(
x13
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x7
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
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
(
x10
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
∀ 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
(
x13
x18
x19
)
)
)
=
x12
x17
(
x13
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
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
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
8c7f3..
:
(
∀ 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
(
x13
x16
(
x13
x17
x18
)
)
=
x13
x16
(
x13
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
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
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
x12
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
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
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
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
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
21636..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x7
x15
(
x12
x16
x17
)
)
=
x7
x15
(
x12
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
x7
x14
(
x13
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x7
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x13
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
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
(
x12
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
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
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
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
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
49d0b..
:
(
∀ 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
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
x18
)
)
=
x10
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
x16
(
x13
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
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
⟶
∀ 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
(
x7
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
x17
(
x7
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
⟶
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
(
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
(
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
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
1a515..
:
(
∀ 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
(
x8
x14
x15
(
x13
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x13
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x10
x15
(
x10
x16
x17
)
)
=
x10
x15
(
x10
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
x7
x15
(
x10
x16
(
x10
x17
x18
)
)
)
=
x10
x16
(
x10
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x12
x15
(
x13
x16
(
x12
x17
x18
)
)
)
=
x13
x16
(
x12
x17
(
x13
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x13
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x7
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
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
(
x7
x16
(
x13
x17
(
x13
x18
x19
)
)
)
=
x13
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
(
x10
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x12
x17
(
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
⟶
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
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
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
(
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
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
645af..
:
(
∀ 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
(
x13
x14
(
x7
x15
(
x8
x14
x15
(
x13
x14
(
x7
x15
(
x8
x14
x15
(
x13
x14
(
x7
x15
(
x8
x14
x15
(
x13
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x12
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
x7
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
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
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
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
(
x12
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
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
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
(
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
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
d1bed..
:
(
∀ 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
(
x12
x16
x17
)
)
=
x12
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
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
(
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
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
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
⟶
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
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
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
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
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
⟶
x8
x14
x15
(
x10
x16
(
x8
x17
x18
(
x10
x19
x20
)
)
)
=
x8
x17
x18
(
x10
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
⟶
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
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
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
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
1a5e1..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x10
x15
(
x8
x14
x15
(
x12
x14
(
x10
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x12
x14
(
x10
x15
(
x9
x14
x15
(
x12
x14
(
x10
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x13
x15
(
x12
x16
x17
)
)
=
x13
x15
(
x12
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x10
x16
(
x10
x17
x18
)
)
=
x10
x16
(
x10
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x13
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
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
⟶
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
(
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
⟶
x13
x14
(
x13
x15
(
x12
x16
(
x13
x17
x18
)
)
)
=
x12
x16
(
x13
x17
(
x13
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
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
(
x7
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
x17
(
x12
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
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
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
⟶
x9
x14
x15
(
x7
x16
(
x10
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
d18ed..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x7
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
(
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
(
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
(
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
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
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
(
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
⟶
x10
x14
(
x13
x15
(
x12
x16
(
x12
x17
x18
)
)
)
=
x12
x16
(
x12
x17
(
x10
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x12
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x7
x18
x19
)
)
)
=
x12
x17
(
x7
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x10
x18
x19
)
)
)
=
x10
x17
(
x10
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ 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
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
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
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
(
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
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
929f3..
:
(
∀ 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
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
x17
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x10
x14
(
x12
x15
(
x8
x14
x15
(
x10
x14
(
x12
x15
x16
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x10
x15
(
x12
x16
x17
)
)
=
x10
x15
(
x12
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
x18
)
)
=
x7
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x10
x15
(
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
(
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
(
x7
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x7
x18
x19
)
)
)
=
x7
x17
(
x7
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
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
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x10
x16
(
x9
x17
x18
(
x10
x19
x20
)
)
)
=
x9
x17
x18
(
x10
x19
(
x9
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
x8
x17
x18
(
x7
x19
x20
)
)
)
=
x8
x17
x18
(
x7
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x10
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
197b4..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
(
x9
x14
x15
(
x7
x16
(
x8
x14
x15
(
x7
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x13
x14
(
x10
x15
(
x9
x14
x15
(
x13
x14
(
x10
x15
(
x9
x14
x15
(
x13
x14
(
x10
x15
(
x9
x14
x15
(
x13
x14
(
x10
x15
(
x9
x14
x15
(
x13
x14
(
x10
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x10
x15
(
x7
x16
x17
)
)
=
x10
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
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
⟶
x10
x14
(
x7
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
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
(
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
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x7
x18
x19
)
)
)
=
x13
x17
(
x7
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x10
x18
x19
)
)
)
=
x12
x17
(
x10
x18
(
x9
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
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
(
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
(
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
(
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
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x13
x20
x21
)
)
)
)
=
x8
x18
x19
(
x13
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x10
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
ae201..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x9
x14
x15
(
x7
x14
(
x13
x15
(
x9
x14
x15
(
x7
x14
(
x13
x15
(
x9
x14
x15
(
x7
x14
(
x13
x15
(
x9
x14
x15
(
x7
x14
(
x13
x15
(
x9
x14
x15
(
x7
x14
(
x13
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x10
x15
(
x7
x16
x17
)
)
=
x10
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
x18
)
)
=
x12
x16
(
x13
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x10
x15
(
x7
x16
(
x10
x17
x18
)
)
)
=
x7
x16
(
x10
x17
(
x10
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
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
(
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
⟶
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
⟶
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
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x13
x16
(
x13
x17
(
x12
x18
x19
)
)
)
=
x13
x17
(
x12
x18
(
x8
x14
x15
(
x13
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
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
⟶
∀ 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
⟶
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
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
(
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
(
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
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
9a396..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
(
x8
x14
x15
(
x12
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
(
x9
x14
x15
(
x10
x16
(
x12
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x12
x15
(
x13
x16
x17
)
)
=
x12
x15
(
x13
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x10
x16
(
x7
x17
x18
)
)
=
x10
x16
(
x7
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
x7
x14
(
x12
x15
(
x13
x16
(
x10
x17
x18
)
)
)
=
x13
x16
(
x10
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x13
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
x16
(
x13
x17
(
x7
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
(
x10
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
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
(
x7
x17
(
x10
x18
x19
)
)
)
=
x7
x17
(
x10
x18
(
x8
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x7
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x7
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x10
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x7
x16
(
x13
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
52a4a..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
(
x9
x14
x15
(
x12
x16
(
x8
x14
x15
(
x12
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
(
x8
x14
x15
(
x12
x14
(
x7
x15
x16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
x17
)
)
=
x12
x15
(
x7
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x13
x16
(
x13
x17
x18
)
)
=
x13
x16
(
x13
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
x12
x14
(
x13
x15
(
x10
x16
(
x12
x17
x18
)
)
)
=
x10
x16
(
x12
x17
(
x12
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x10
x16
(
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
⟶
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
⟶
x10
x14
(
x7
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
∀ 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
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
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
(
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
(
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
(
x7
x16
(
x13
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x7
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x12
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
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
(
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
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x8
x14
x15
(
x13
x16
(
x12
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
89e69..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
(
x9
x14
x15
(
x7
x16
(
x10
x14
(
x9
x15
x16
x17
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
(
x9
x14
x15
(
x12
x16
(
x9
x14
x15
(
x10
x16
x17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
=
x17
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x7
x14
(
x12
x15
(
x13
x16
x17
)
)
=
x12
x15
(
x13
x16
(
x7
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x9
x14
x15
(
x10
x16
(
x7
x17
x18
)
)
=
x10
x16
(
x7
x17
(
x9
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
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
⟶
x12
x14
(
x12
x15
(
x7
x16
(
x12
x17
x18
)
)
)
=
x7
x16
(
x12
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x12
x14
(
x12
x15
(
x10
x16
(
x13
x17
x18
)
)
)
=
x10
x16
(
x13
x17
(
x12
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
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
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x10
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x10
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
x17
x18
(
x13
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ 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
(
x13
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x10
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x13
x20
x21
)
)
)
)
=
x9
x18
x19
(
x13
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
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
(
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
74501..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x13
x14
(
x12
x15
(
x10
x16
x17
)
)
=
x12
x15
(
x10
x16
(
x13
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x13
x16
(
x12
x17
x18
)
)
=
x13
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x7
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x13
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x13
x14
(
x13
x15
(
x12
x16
(
x7
x17
x18
)
)
)
=
x12
x16
(
x7
x17
(
x13
x14
(
x13
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
x7
x14
(
x10
x15
(
x12
x16
(
x10
x17
x18
)
)
)
=
x12
x16
(
x10
x17
(
x7
x14
(
x10
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x12
x18
x19
)
)
)
=
x10
x17
(
x12
x18
(
x9
x14
x15
(
x10
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x7
x16
(
x12
x17
(
x12
x18
x19
)
)
)
=
x12
x17
(
x12
x18
(
x8
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x9
x14
x15
(
x12
x16
(
x8
x17
x18
(
x13
x19
x20
)
)
)
=
x8
x17
x18
(
x13
x19
(
x9
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x12
x16
(
x9
x17
x18
(
x12
x19
x20
)
)
)
=
x9
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x13
x16
(
x13
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x13
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
⟶
x8
x14
x15
(
x7
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x13
x17
(
x9
x18
x19
(
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
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x7
x20
x21
)
)
)
)
=
x9
x18
x19
(
x7
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
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
(
x13
x17
(
x8
x18
x19
(
x7
x20
x21
)
)
)
)
=
x8
x18
x19
(
x7
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
b269b..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex1
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x9
x14
x15
(
x12
x16
(
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
⟶
x9
x14
x15
(
x12
x14
(
x12
x15
(
x9
x14
x15
(
x12
x14
(
x12
x15
(
x9
x14
x15
(
x12
x14
(
x12
x15
(
x9
x14
x15
(
x12
x14
(
x12
x15
x16
)
)
)
)
)
)
)
)
)
)
)
=
x16
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x12
x14
(
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
(
x7
x16
(
x13
x17
x18
)
)
=
x7
x16
(
x13
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x7
x15
(
x10
x16
(
x7
x17
x18
)
)
)
=
x10
x16
(
x7
x17
(
x10
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x7
x15
(
x13
x16
(
x13
x17
x18
)
)
)
=
x13
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
⟶
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
⟶
x7
x14
(
x7
x15
(
x13
x16
(
x7
x17
x18
)
)
)
=
x13
x16
(
x7
x17
(
x7
x14
(
x7
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x10
x14
(
x12
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
x16
(
x13
x17
(
x10
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x8
x14
x15
(
x12
x16
(
x7
x17
(
x12
x18
x19
)
)
)
=
x7
x17
(
x12
x18
(
x8
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
x9
x14
x15
(
x7
x16
(
x13
x17
(
x13
x18
x19
)
)
)
=
x13
x17
(
x13
x18
(
x9
x14
x15
(
x7
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
x13
x16
(
x9
x17
x18
(
x7
x19
x20
)
)
)
=
x9
x17
x18
(
x7
x19
(
x8
x14
x15
(
x13
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
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
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
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
(
x10
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x10
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x10
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
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
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
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
(
x12
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x12
x16
(
x7
x17
x21
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)
Theorem
9ef6a..
:
(
∀ x0 .
∀ x1 x2 x3 :
ι →
ι → ι
.
∀ x4 .
∀ x5 :
ι →
ι → ι
.
∀ x6 :
ι →
ι →
ι → ι
.
∀ x7 :
ι →
ι → ι
.
∀ x8 x9 :
ι →
ι →
ι → ι
.
∀ x10 x11 x12 x13 :
ι →
ι → ι
.
Loop_with_defs_cex2
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
x10
x14
(
x7
x15
(
x10
x16
x17
)
)
=
x7
x15
(
x10
x16
(
x10
x14
x17
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
x18
)
)
=
x12
x16
(
x12
x17
(
x8
x14
x15
x18
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
x7
x14
(
x12
x15
(
x7
x16
(
x7
x17
x18
)
)
)
=
x7
x16
(
x7
x17
(
x7
x14
(
x12
x15
x18
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
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
⟶
x10
x14
(
x7
x15
(
x7
x16
(
x13
x17
x18
)
)
)
=
x7
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
⟶
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
(
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
⟶
∀ 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
⟶
x9
x14
x15
(
x12
x16
(
x12
x17
(
x13
x18
x19
)
)
)
=
x12
x17
(
x13
x18
(
x9
x14
x15
(
x12
x16
x19
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
x8
x14
x15
(
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
(
x12
x16
(
x8
x17
x18
(
x12
x19
x20
)
)
)
=
x8
x17
x18
(
x12
x19
(
x8
x14
x15
(
x12
x16
x20
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x10
x16
(
x13
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x9
x14
x15
(
x10
x16
(
x13
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x8
x18
x19
(
x10
x20
x21
)
)
)
)
=
x8
x18
x19
(
x10
x20
(
x8
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x9
x14
x15
(
x7
x16
(
x12
x17
(
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
(
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
(
x7
x17
(
x8
x18
x19
(
x12
x20
x21
)
)
)
)
=
x8
x18
x19
(
x12
x20
(
x8
x14
x15
(
x7
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x12
x16
(
x12
x17
(
x9
x18
x19
(
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
(
x12
x17
(
x9
x18
x19
(
x12
x20
x21
)
)
)
)
=
x9
x18
x19
(
x12
x20
(
x9
x14
x15
(
x12
x16
(
x12
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
x8
x14
x15
(
x13
x16
(
x7
x17
(
x9
x18
x19
(
x10
x20
x21
)
)
)
)
=
x9
x18
x19
(
x10
x20
(
x8
x14
x15
(
x13
x16
(
x7
x17
x21
)
)
)
)
)
⟶
(
∀ x14 .
In
x14
x0
⟶
∀ x15 .
In
x15
x0
⟶
∀ x16 .
In
x16
x0
⟶
∀ x17 .
In
x17
x0
⟶
∀ x18 .
In
x18
x0
⟶
∀ x19 .
In
x19
x0
⟶
∀ x20 .
In
x20
x0
⟶
∀ x21 .
In
x21
x0
⟶
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
)
)
)
)
)
⟶
False
)
⟶
∀ x0 : ο .
x0
(proof)