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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Let x2 of type
ι
be given.
Let x3 of type
ι
be given.
Let x4 of type
ι
be given.
Let x5 of type
ι
be given.
Assume H0:
SNo
x1
.
Assume H1:
∀ x6 .
x6
∈
SNoS_
(
SNoLev
x0
)
⟶
mul_SNo
x6
x1
=
mul_SNo
x1
x6
.
Assume H2:
∀ x6 .
x6
∈
SNoS_
(
SNoLev
x1
)
⟶
mul_SNo
x0
x6
=
mul_SNo
x6
x0
.
Assume H3:
∀ x6 .
x6
∈
SNoS_
(
SNoLev
x0
)
⟶
∀ x7 .
x7
∈
SNoS_
(
SNoLev
x1
)
⟶
mul_SNo
x6
x7
=
mul_SNo
x7
x6
.
Assume H4:
∀ x6 .
x6
∈
x3
⟶
∀ x7 : ο .
(
∀ x8 .
x8
∈
SNoL
x0
⟶
∀ x9 .
x9
∈
SNoR
x1
⟶
x6
=
add_SNo
(
mul_SNo
x8
x1
)
(
add_SNo
(
mul_SNo
x0
x9
)
(
minus_SNo
(
mul_SNo
x8
x9
)
)
)
⟶
x7
)
⟶
(
∀ x8 .
x8
∈
SNoR
x0
⟶
∀ x9 .
x9
∈
SNoL
x1
⟶
x6
=
add_SNo
(
mul_SNo
x8
x1
)
(
add_SNo
(
mul_SNo
x0
x9
)
(
minus_SNo
(
mul_SNo
x8
x9
)
)
)
⟶
x7
)
⟶
x7
.
Assume H5:
∀ x6 .
x6
∈
SNoL
x0
⟶
∀ x7 .
x7
∈
SNoR
x1
⟶
add_SNo
(
mul_SNo
x6
x1
)
(
add_SNo
(
mul_SNo
x0
x7
)
(
minus_SNo
(
mul_SNo
x6
x7
)
)
)
∈
x3
.
Assume H6:
∀ x6 .
x6
∈
SNoR
x0
⟶
∀ x7 .
x7
∈
SNoL
x1
⟶
add_SNo
(
mul_SNo
x6
x1
)
(
add_SNo
(
mul_SNo
x0
x7
)
(
minus_SNo
(
mul_SNo
x6
x7
)
)
)
∈
x3
.
Assume H7:
∀ x6 .
x6
∈
x5
⟶
∀ x7 : ο .
(
∀ x8 .
x8
∈
SNoL
x1
⟶
∀ x9 .
x9
∈
SNoR
x0
⟶
x6
=
add_SNo
(
mul_SNo
x8
x0
)
(
add_SNo
(
mul_SNo
x1
x9
)
(
minus_SNo
(
mul_SNo
x8
x9
)
)
)
⟶
x7
)
⟶
(
∀ x8 .
x8
∈
SNoR
x1
⟶
∀ x9 .
x9
∈
SNoL
x0
⟶
x6
=
add_SNo
(
mul_SNo
x8
x0
)
(
add_SNo
(
mul_SNo
x1
x9
)
(
minus_SNo
(
mul_SNo
x8
x9
)
)
)
⟶
x7
)
⟶
x7
.
Assume H8:
∀ x6 .
x6
∈
SNoL
x1
⟶
∀ x7 .
x7
∈
SNoR
x0
⟶
add_SNo
(
mul_SNo
x6
x0
)
(
add_SNo
(
mul_SNo
x1
x7
)
(
minus_SNo
(
mul_SNo
x6
x7
)
)
)
∈
x5
.
Assume H9:
∀ x6 .
x6
∈
SNoR
x1
⟶
∀ x7 .
x7
∈
SNoL
x0
⟶
add_SNo
(
mul_SNo
x6
x0
)
(
add_SNo
(
mul_SNo
x1
x7
)
(
minus_SNo
(
mul_SNo
x6
x7
)
)
)
∈
x5
.
Assume H10:
x2
=
x4
.
Assume H11:
...
.
...
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