Step
*
2
1
2
1
1
1
1
of Lemma
punctured-ball-boundary-retraction
1. n : ℕ
2. p : ℝ^n
3. ||p|| < r1
4. ¬(n = 0 ∈ ℤ)
5. b : ℝ^n
6. ||b|| = r1
7. ∀x:{x:ℝ^n| ||x|| = r1} . x ≠ p
8. λq.p + quadratic1(||q - p||^2;r(2) * p⋅q - p;||p||^2 - r1^2)*q - p:FUN({x:ℝ^n| x ≠ p} ;{x:ℝ^n| ||x|| = r1} )
9. a : ℝ^n
10. ||a|| = r1
11. ||a - p||^2 ≠ r0
12. r0 ≤ (((r(2) * p⋅a - p) * r(2) * p⋅a - p) - r(4) * ||a - p||^2 * (||p||^2 - r1^2))
13. r0 < ||a - p||^2
14. ||p + quadratic1(||a - p||^2;r(2) * p⋅a - p;||p||^2 - r1^2)*a - p|| = r1
⊢ (r(2) * p⋅a - p) = -(||a - p||^2 + (||p||^2 - r1^2))
BY
{ (RWO "real-vec-norm-squared" 0 THENA Auto) }
1
1. n : ℕ
2. p : ℝ^n
3. ||p|| < r1
4. ¬(n = 0 ∈ ℤ)
5. b : ℝ^n
6. ||b|| = r1
7. ∀x:{x:ℝ^n| ||x|| = r1} . x ≠ p
8. λq.p + quadratic1(||q - p||^2;r(2) * p⋅q - p;||p||^2 - r1^2)*q - p:FUN({x:ℝ^n| x ≠ p} ;{x:ℝ^n| ||x|| = r1} )
9. a : ℝ^n
10. ||a|| = r1
11. ||a - p||^2 ≠ r0
12. r0 ≤ (((r(2) * p⋅a - p) * r(2) * p⋅a - p) - r(4) * ||a - p||^2 * (||p||^2 - r1^2))
13. r0 < ||a - p||^2
14. ||p + quadratic1(||a - p||^2;r(2) * p⋅a - p;||p||^2 - r1^2)*a - p|| = r1
⊢ (r(2) * p⋅a - p) = -(a - p⋅a - p + (p⋅p - r1^2))
Latex:
Latex:
1. n : \mBbbN{}
2. p : \mBbbR{}\^{}n
3. ||p|| < r1
4. \mneg{}(n = 0)
5. b : \mBbbR{}\^{}n
6. ||b|| = r1
7. \mforall{}x:\{x:\mBbbR{}\^{}n| ||x|| = r1\} . x \mneq{} p
8. \mlambda{}q.p + quadratic1(||q - p||\^{}2;r(2) * p\mcdot{}q - p;||p||\^{}2 - r1\^{}2)*q - p:FUN(\{x:\mBbbR{}\^{}n| x \mneq{} p\} ;\{x:\mBbbR{}\^{}n| ||\000Cx|| = r1\} )
9. a : \mBbbR{}\^{}n
10. ||a|| = r1
11. ||a - p||\^{}2 \mneq{} r0
12. r0 \mleq{} (((r(2) * p\mcdot{}a - p) * r(2) * p\mcdot{}a - p) - r(4) * ||a - p||\^{}2 * (||p||\^{}2 - r1\^{}2))
13. r0 < ||a - p||\^{}2
14. ||p + quadratic1(||a - p||\^{}2;r(2) * p\mcdot{}a - p;||p||\^{}2 - r1\^{}2)*a - p|| = r1
\mvdash{} (r(2) * p\mcdot{}a - p) = -(||a - p||\^{}2 + (||p||\^{}2 - r1\^{}2))
By
Latex:
(RWO "real-vec-norm-squared" 0 THENA Auto)
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