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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
nat_p
x0
.
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.
Assume H1:
odd_nat
x1
.
Assume H2:
odd_nat
x2
.
Assume H3:
odd_nat
x3
.
Assume H4:
odd_nat
x4
.
Assume H5:
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
)
)
)
.
Apply H1 with
∀ 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
.
Assume H6:
x1
∈
omega
.
Assume H7:
∀ x5 .
x5
∈
omega
⟶
x1
=
mul_nat
2
x5
⟶
∀ x6 : ο .
x6
.
Apply H2 with
∀ 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
.
Assume H8:
x2
∈
omega
.
Assume H9:
∀ x5 .
x5
∈
omega
⟶
x2
=
mul_nat
2
x5
⟶
∀ x6 : ο .
x6
.
Apply H3 with
∀ 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
.
Assume H10:
x3
∈
omega
.
Assume H11:
∀ x5 .
x5
∈
omega
⟶
x3
=
mul_nat
2
x5
⟶
∀ x6 : ο .
x6
.
Apply H4 with
∀ 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
.
Assume H12:
x4
∈
...
.
...
■