Step
*
2
2
of Lemma
rv-compass-compass-lemma
1. a : ℝ^2
2. b : ℝ^2
3. c : ℝ^2
4. d : ℝ^2
5. a ≠ c
6. ↓∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) ≤ d(c;d)) ∧ (d(a;q) ≤ d(a;b)))
7. ∃u,v:ℝ^2
(((||u|| = d(a;b)) ∧ (||u - c - a|| = d(c;d)))
∧ ((||v|| = d(a;b)) ∧ (||v - c - a|| = d(c;d)))
∧ (((d(a;b)^2 - d(c;d)^2) + ||c - a||^2^2 < (r(4) * ||c - a||^2 * d(a;b)^2))
⇒ (r2-left(u;c - a;λi.r0) ∧ r2-left(v;λi.r0;c - a))))
8. ∀x,y,z:ℝ^2. (d(x;y) = d(x;z) ⇐⇒ ||z - x|| = d(x;y))
⊢ ∃u,v:{p:ℝ^2| ab=ap ∧ cd=cp}
((↓∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) < d(c;d)) ∧ (d(a;q) < d(a;b))))
⇒ (r2-left(u;c;a) ∧ r2-left(v;a;c)))
BY
{ (ExRepD
THEN (InstConcl [⌜u + a⌝;⌜v + a⌝]⋅
THENA (Auto THEN MemTypeCD THEN Auto THEN Try ((UnfoldTopAb 0 THEN (RWO "14" 0 THENA Auto))))
)
) }
1
1. a : ℝ^2
2. b : ℝ^2
3. c : ℝ^2
4. d : ℝ^2
5. a ≠ c
6. ∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) ≤ d(c;d)) ∧ (d(a;q) ≤ d(a;b)))
7. u : ℝ^2
8. v : ℝ^2
9. ||u|| = d(a;b)
10. ||u - c - a|| = d(c;d)
11. ||v|| = d(a;b)
12. ||v - c - a|| = d(c;d)
13. ((d(a;b)^2 - d(c;d)^2) + ||c - a||^2^2 < (r(4) * ||c - a||^2 * d(a;b)^2))
⇒ (r2-left(u;c - a;λi.r0) ∧ r2-left(v;λi.r0;c - a))
14. ∀x,y,z:ℝ^2. (d(x;y) = d(x;z) ⇐⇒ ||z - x|| = d(x;y))
⊢ ||u + a - a|| = d(a;b)
2
1. a : ℝ^2
2. b : ℝ^2
3. c : ℝ^2
4. d : ℝ^2
5. a ≠ c
6. ∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) ≤ d(c;d)) ∧ (d(a;q) ≤ d(a;b)))
7. u : ℝ^2
8. v : ℝ^2
9. ||u|| = d(a;b)
10. ||u - c - a|| = d(c;d)
11. ||v|| = d(a;b)
12. ||v - c - a|| = d(c;d)
13. ((d(a;b)^2 - d(c;d)^2) + ||c - a||^2^2 < (r(4) * ||c - a||^2 * d(a;b)^2))
⇒ (r2-left(u;c - a;λi.r0) ∧ r2-left(v;λi.r0;c - a))
14. ∀x,y,z:ℝ^2. (d(x;y) = d(x;z) ⇐⇒ ||z - x|| = d(x;y))
15. ab=au + a
⊢ ||u + a - c|| = d(c;d)
3
1. a : ℝ^2
2. b : ℝ^2
3. c : ℝ^2
4. d : ℝ^2
5. a ≠ c
6. ∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) ≤ d(c;d)) ∧ (d(a;q) ≤ d(a;b)))
7. u : ℝ^2
8. v : ℝ^2
9. ||u|| = d(a;b)
10. ||u - c - a|| = d(c;d)
11. ||v|| = d(a;b)
12. ||v - c - a|| = d(c;d)
13. ((d(a;b)^2 - d(c;d)^2) + ||c - a||^2^2 < (r(4) * ||c - a||^2 * d(a;b)^2))
⇒ (r2-left(u;c - a;λi.r0) ∧ r2-left(v;λi.r0;c - a))
14. ∀x,y,z:ℝ^2. (d(x;y) = d(x;z) ⇐⇒ ||z - x|| = d(x;y))
⊢ ||v + a - a|| = d(a;b)
4
1. a : ℝ^2
2. b : ℝ^2
3. c : ℝ^2
4. d : ℝ^2
5. a ≠ c
6. ∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) ≤ d(c;d)) ∧ (d(a;q) ≤ d(a;b)))
7. u : ℝ^2
8. v : ℝ^2
9. ||u|| = d(a;b)
10. ||u - c - a|| = d(c;d)
11. ||v|| = d(a;b)
12. ||v - c - a|| = d(c;d)
13. ((d(a;b)^2 - d(c;d)^2) + ||c - a||^2^2 < (r(4) * ||c - a||^2 * d(a;b)^2))
⇒ (r2-left(u;c - a;λi.r0) ∧ r2-left(v;λi.r0;c - a))
14. ∀x,y,z:ℝ^2. (d(x;y) = d(x;z) ⇐⇒ ||z - x|| = d(x;y))
15. ab=av + a
⊢ ||v + a - c|| = d(c;d)
5
1. a : ℝ^2
2. b : ℝ^2
3. c : ℝ^2
4. d : ℝ^2
5. a ≠ c
6. ↓∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) ≤ d(c;d)) ∧ (d(a;q) ≤ d(a;b)))
7. u : ℝ^2
8. v : ℝ^2
9. ||u|| = d(a;b)
10. ||u - c - a|| = d(c;d)
11. ||v|| = d(a;b)
12. ||v - c - a|| = d(c;d)
13. ((d(a;b)^2 - d(c;d)^2) + ||c - a||^2^2 < (r(4) * ||c - a||^2 * d(a;b)^2))
⇒ (r2-left(u;c - a;λi.r0) ∧ r2-left(v;λi.r0;c - a))
14. ∀x,y,z:ℝ^2. (d(x;y) = d(x;z) ⇐⇒ ||z - x|| = d(x;y))
⊢ (↓∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) < d(c;d)) ∧ (d(a;q) < d(a;b))))
⇒ (r2-left(u + a;c;a) ∧ r2-left(v + a;a;c))
Latex:
Latex:
1. a : \mBbbR{}\^{}2
2. b : \mBbbR{}\^{}2
3. c : \mBbbR{}\^{}2
4. d : \mBbbR{}\^{}2
5. a \mneq{} c
6. \mdownarrow{}\mexists{}p,q:\mBbbR{}\^{}2. (((d(a;b) = d(a;p)) \mwedge{} (d(c;d) = d(c;q))) \mwedge{} (d(c;p) \mleq{} d(c;d)) \mwedge{} (d(a;q) \mleq{} d(a;b)))
7. \mexists{}u,v:\mBbbR{}\^{}2
(((||u|| = d(a;b)) \mwedge{} (||u - c - a|| = d(c;d)))
\mwedge{} ((||v|| = d(a;b)) \mwedge{} (||v - c - a|| = d(c;d)))
\mwedge{} (((d(a;b)\^{}2 - d(c;d)\^{}2) + ||c - a||\^{}2\^{}2 < (r(4) * ||c - a||\^{}2 * d(a;b)\^{}2))
{}\mRightarrow{} (r2-left(u;c - a;\mlambda{}i.r0) \mwedge{} r2-left(v;\mlambda{}i.r0;c - a))))
8. \mforall{}x,y,z:\mBbbR{}\^{}2. (d(x;y) = d(x;z) \mLeftarrow{}{}\mRightarrow{} ||z - x|| = d(x;y))
\mvdash{} \mexists{}u,v:\{p:\mBbbR{}\^{}2| ab=ap \mwedge{} cd=cp\}
((\mdownarrow{}\mexists{}p,q:\mBbbR{}\^{}2. (((d(a;b) = d(a;p)) \mwedge{} (d(c;d) = d(c;q))) \mwedge{} (d(c;p) < d(c;d)) \mwedge{} (d(a;q) < d(a;b))))
{}\mRightarrow{} (r2-left(u;c;a) \mwedge{} r2-left(v;a;c)))
By
Latex:
(ExRepD
THEN (InstConcl [\mkleeneopen{}u + a\mkleeneclose{};\mkleeneopen{}v + a\mkleeneclose{}]\mcdot{}
THENA (Auto
THEN MemTypeCD
THEN Auto
THEN Try ((UnfoldTopAb 0 THEN (RWO "14" 0 THENA Auto))))
)
)
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