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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
SNo
x0
.
Assume H1:
SNoLe
0
x0
.
Assume H2:
∀ x1 .
x1
∈
SNoS_
(
SNoLev
x0
)
⟶
SNoLe
0
x1
⟶
and
(
and
(
SNo
(
sqrt_SNo_nonneg
x1
)
)
(
SNoLe
0
(
sqrt_SNo_nonneg
x1
)
)
)
(
mul_SNo
(
sqrt_SNo_nonneg
x1
)
(
sqrt_SNo_nonneg
x1
)
=
x1
)
.
Assume H3:
SNoCutP
(
famunion
omega
(
λ x1 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x1
)
0
)
)
(
famunion
omega
(
λ x1 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x1
)
1
)
)
.
Assume H4:
SNoLe
0
(
SNoCut
(
famunion
omega
(
λ x1 .
(
λ x2 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x2
)
0
)
x1
)
)
(
famunion
omega
(
λ x1 .
(
λ x2 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x2
)
1
)
x1
)
)
)
.
Assume H5:
SNoLt
x0
(
mul_SNo
(
SNoCut
(
famunion
omega
(
λ x1 .
(
λ x2 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x2
)
0
)
x1
)
)
(
famunion
omega
(
λ x1 .
(
λ x2 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x2
)
1
)
x1
)
)
)
(
SNoCut
(
famunion
omega
(
λ x1 .
(
λ x2 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x2
)
0
)
x1
)
)
(
famunion
omega
(
λ x1 .
(
λ x2 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x2
)
1
)
x1
)
)
)
)
.
Apply H3 with
False
.
Assume H6:
and
(
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x2
)
0
)
⟶
SNo
x1
)
(
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x2
)
1
)
⟶
SNo
x1
)
.
Apply H6 with
(
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x2
)
0
)
⟶
∀ x2 .
x2
∈
famunion
omega
(
λ x3 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x3
)
1
)
⟶
SNoLt
x1
x2
)
⟶
False
.
Assume H7:
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
(
λ x3 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x3
)
0
)
x2
)
⟶
SNo
x1
.
Assume H8:
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
(
λ x3 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x3
)
1
)
x2
)
⟶
SNo
x1
.
Assume H9:
∀ x1 .
x1
∈
famunion
omega
(
λ x2 .
(
λ x3 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x3
)
0
)
x2
)
⟶
∀ x2 .
x2
∈
famunion
omega
(
λ x3 .
(
λ x4 .
ap
(
SNo_sqrtaux
x0
sqrt_SNo_nonneg
x4
)
1
)
x3
)
⟶
SNoLt
x1
x2
.
Apply SNoCutP_SNoCut_impred with
famunion
omega
(
λ x1 .
(
λ x2 .
ap
(
SNo_sqrtaux
...
...
...
)
0
)
...
)
,
...
,
...
leaving 2 subgoals.
...
...
■