Step
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1
2
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2
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4
of Lemma
rv-line-circle-0
1. n : ℕ
2. a : ℝ^n
3. b : ℝ^n
4. p : ℝ^n
5. q : ℝ^n
6. p ≠ q
7. d(a;p) ≤ d(a;b)
8. d(a;b) ≤ d(a;q)
9. pp : ℝ^n
10. p - a = pp ∈ ℝ^n
11. qq : ℝ^n
12. q - a = qq ∈ ℝ^n
13. (d(a;p) < d(a;b)) ⇒ (||pp|| < d(a;b))
14. ||pp|| ≤ d(a;b)
15. (d(a;b) < d(a;q)) ⇒ (d(a;b) < ||qq||)
16. d(a;b) ≤ ||qq||
17. pp ≠ qq
18. r0 < ||qq - pp||
19. r0 < ||qq - pp||^2
20. r0 ≤ (((r(2) * pp⋅qq - pp) * r(2) * pp⋅qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - d(a;b)^2))
21. d(a;p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
22. d(a;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
23. quadratic1(||qq - pp||^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2) ∈ [r0, r1]
24. ∀X:ℝ^n. req-vec(n;pp + X;p + X - a)
25. real-vec-be(n;q;p;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp)
26. (d(a;p) < d(a;b))
⇒ (q-p-p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp
∧ ((d(a;b) < d(a;q)) ⇒ q-p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp-p))
27. d(a;p) = d(a;b)
28. p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp ≠ p + quadratic2(d(qq;pp)^2;r(2)
* pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp
⊢ (req-vec(n;p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp;p) ∧ (r0 < pp⋅q - p))
∨ (req-vec(n;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp;p) ∧ (pp⋅q - p < r0))
BY
{ Assert ⌜(r0 < (r(2) * pp⋅qq - pp)) ∨ ((r(2) * pp⋅qq - pp) < r0)⌝⋅ }
1
.....assertion.....
1. n : ℕ
2. a : ℝ^n
3. b : ℝ^n
4. p : ℝ^n
5. q : ℝ^n
6. p ≠ q
7. d(a;p) ≤ d(a;b)
8. d(a;b) ≤ d(a;q)
9. pp : ℝ^n
10. p - a = pp ∈ ℝ^n
11. qq : ℝ^n
12. q - a = qq ∈ ℝ^n
13. (d(a;p) < d(a;b)) ⇒ (||pp|| < d(a;b))
14. ||pp|| ≤ d(a;b)
15. (d(a;b) < d(a;q)) ⇒ (d(a;b) < ||qq||)
16. d(a;b) ≤ ||qq||
17. pp ≠ qq
18. r0 < ||qq - pp||
19. r0 < ||qq - pp||^2
20. r0 ≤ (((r(2) * pp⋅qq - pp) * r(2) * pp⋅qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - d(a;b)^2))
21. d(a;p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
22. d(a;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
23. quadratic1(||qq - pp||^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2) ∈ [r0, r1]
24. ∀X:ℝ^n. req-vec(n;pp + X;p + X - a)
25. real-vec-be(n;q;p;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp)
26. (d(a;p) < d(a;b))
⇒ (q-p-p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp
∧ ((d(a;b) < d(a;q)) ⇒ q-p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp-p))
27. d(a;p) = d(a;b)
28. p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp ≠ p + quadratic2(d(qq;pp)^2;r(2)
* pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp
⊢ (r0 < (r(2) * pp⋅qq - pp)) ∨ ((r(2) * pp⋅qq - pp) < r0)
2
1. n : ℕ
2. a : ℝ^n
3. b : ℝ^n
4. p : ℝ^n
5. q : ℝ^n
6. p ≠ q
7. d(a;p) ≤ d(a;b)
8. d(a;b) ≤ d(a;q)
9. pp : ℝ^n
10. p - a = pp ∈ ℝ^n
11. qq : ℝ^n
12. q - a = qq ∈ ℝ^n
13. (d(a;p) < d(a;b)) ⇒ (||pp|| < d(a;b))
14. ||pp|| ≤ d(a;b)
15. (d(a;b) < d(a;q)) ⇒ (d(a;b) < ||qq||)
16. d(a;b) ≤ ||qq||
17. pp ≠ qq
18. r0 < ||qq - pp||
19. r0 < ||qq - pp||^2
20. r0 ≤ (((r(2) * pp⋅qq - pp) * r(2) * pp⋅qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - d(a;b)^2))
21. d(a;p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
22. d(a;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
23. quadratic1(||qq - pp||^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2) ∈ [r0, r1]
24. ∀X:ℝ^n. req-vec(n;pp + X;p + X - a)
25. real-vec-be(n;q;p;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp)
26. (d(a;p) < d(a;b))
⇒ (q-p-p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp
∧ ((d(a;b) < d(a;q)) ⇒ q-p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp-p))
27. d(a;p) = d(a;b)
28. p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp ≠ p + quadratic2(d(qq;pp)^2;r(2)
* pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp
29. (r0 < (r(2) * pp⋅qq - pp)) ∨ ((r(2) * pp⋅qq - pp) < r0)
⊢ (req-vec(n;p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp;p) ∧ (r0 < pp⋅q - p))
∨ (req-vec(n;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp;p) ∧ (pp⋅q - p < r0))
Latex:
Latex:
1. n : \mBbbN{}
2. a : \mBbbR{}\^{}n
3. b : \mBbbR{}\^{}n
4. p : \mBbbR{}\^{}n
5. q : \mBbbR{}\^{}n
6. p \mneq{} q
7. d(a;p) \mleq{} d(a;b)
8. d(a;b) \mleq{} d(a;q)
9. pp : \mBbbR{}\^{}n
10. p - a = pp
11. qq : \mBbbR{}\^{}n
12. q - a = qq
13. (d(a;p) < d(a;b)) {}\mRightarrow{} (||pp|| < d(a;b))
14. ||pp|| \mleq{} d(a;b)
15. (d(a;b) < d(a;q)) {}\mRightarrow{} (d(a;b) < ||qq||)
16. d(a;b) \mleq{} ||qq||
17. pp \mneq{} qq
18. r0 < ||qq - pp||
19. r0 < ||qq - pp||\^{}2
20. r0 \mleq{} (((r(2) * pp\mcdot{}qq - pp) * r(2) * pp\mcdot{}qq - pp) - r(4) * ||qq - pp||\^{}2 * (||pp||\^{}2 - d(a;b)\^{}2))
21. d(a;p + quadratic1(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp) = d(a;b)
22. d(a;p + quadratic2(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp) = d(a;b)
23. quadratic1(||qq - pp||\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2) \mmember{} [r0, r1]
24. \mforall{}X:\mBbbR{}\^{}n. req-vec(n;pp + X;p + X - a)
25. real-vec-be(n;q;p;p + quadratic2(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp)
26. (d(a;p) < d(a;b))
{}\mRightarrow{} (q-p-p + quadratic2(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp
\mwedge{} ((d(a;b) < d(a;q))
{}\mRightarrow{} q-p + quadratic1(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp-p))
27. d(a;p) = d(a;b)
28. p + quadratic1(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2
- d(a;b)\^{}2)*qq - pp \mneq{} p + quadratic2(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp
\mvdash{} (req-vec(n;p + quadratic1(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp;p)
\mwedge{} (r0 < pp\mcdot{}q - p))
\mvee{} (req-vec(n;p + quadratic2(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp;p)
\mwedge{} (pp\mcdot{}q - p < r0))
By
Latex:
Assert \mkleeneopen{}(r0 < (r(2) * pp\mcdot{}qq - pp)) \mvee{} ((r(2) * pp\mcdot{}qq - pp) < r0)\mkleeneclose{}\mcdot{}
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