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Proofgold Proof
pf
Assume H0:
∀ x0 .
prime_nat
x0
⟶
odd_nat
x0
⟶
∃ x1 .
and
(
x1
∈
setminus
x0
(
Sing
0
)
)
(
∃ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
∃ x5 .
and
(
x5
∈
omega
)
(
mul_SNo
x1
x0
=
add_SNo
(
(
λ x6 .
mul_SNo
x6
x6
)
x2
)
(
add_SNo
(
(
λ x6 .
mul_SNo
x6
x6
)
x3
)
(
add_SNo
(
(
λ x6 .
mul_SNo
x6
x6
)
x4
)
(
(
λ x6 .
mul_SNo
x6
x6
)
x5
)
)
)
)
)
)
)
)
.
Assume H1:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
omega
⟶
∀ x2 .
x2
∈
omega
⟶
∀ x3 .
x3
∈
omega
⟶
∀ x4 .
x4
∈
omega
⟶
mul_SNo
2
x0
=
add_SNo
(
(
λ x5 .
mul_SNo
x5
x5
)
x1
)
(
add_SNo
(
(
λ x5 .
mul_SNo
x5
x5
)
x2
)
(
add_SNo
(
(
λ x5 .
mul_SNo
x5
x5
)
x3
)
(
(
λ x5 .
mul_SNo
x5
x5
)
x4
)
)
)
⟶
∀ x5 : ο .
(
∀ x6 .
x6
∈
omega
⟶
∀ x7 .
x7
∈
omega
⟶
∀ x8 .
x8
∈
omega
⟶
∀ x9 .
x9
∈
omega
⟶
x0
=
add_SNo
(
(
λ x10 .
mul_SNo
x10
x10
)
x6
)
(
add_SNo
(
(
λ x10 .
mul_SNo
x10
x10
)
x7
)
(
add_SNo
(
(
λ x10 .
mul_SNo
x10
x10
)
x8
)
(
(
λ x10 .
mul_SNo
x10
x10
)
x9
)
)
)
⟶
x5
)
⟶
x5
.
Let x0 of type
ι
be given.
Assume H2:
prime_nat
x0
.
Assume H3:
odd_nat
x0
.
Apply H3 with
∃ x1 .
and
(
x1
∈
omega
)
(
∃ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
x0
=
add_SNo
(
mul_SNo
x1
x1
)
(
add_SNo
(
mul_SNo
x2
x2
)
(
add_SNo
(
mul_SNo
x3
x3
)
(
mul_SNo
x4
x4
)
)
)
)
)
)
)
.
Assume H4:
x0
∈
omega
.
Assume H5:
∀ x1 .
x1
∈
omega
⟶
x0
=
mul_nat
2
x1
⟶
∀ x2 : ο .
x2
.
Apply dneg with
∃ x1 .
and
(
x1
∈
omega
)
(
∃ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
x0
=
add_SNo
(
mul_SNo
x1
x1
)
(
add_SNo
(
mul_SNo
x2
x2
)
(
add_SNo
(
mul_SNo
x3
x3
)
(
mul_SNo
x4
x4
)
)
)
)
)
)
)
.
Assume H6:
not
(
∃ x1 .
and
(
x1
∈
omega
)
(
∃ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
x0
=
...
)
)
)
)
)
.
...
■