Step
*
1
1
1
1
2
1
1
2
1
1
3
1
1
1
1
1
1
1
of Lemma
real-path-comp-exists
1. f : [r0, r1] ⟶ℝ
2. g : [r0, r1] ⟶ℝ
3. ∀x,y:{x:ℝ| x ∈ [r0, r1]} . ((x = y) ⇒ (f(x) = f(y)))
4. ∀x,y:{x:ℝ| x ∈ [r0, r1]} . ((x = y) ⇒ (g(x) = g(y)))
5. f(r1) = g(r0)
6. ∀t:{x:ℝ| x ∈ [r0, (r1/r(2))]} . (rmin(t;(r1/r(2))) = t)
7. ∀t:{x:ℝ| x ∈ [r0, (r1/r(2))]} . (r(2) * t ∈ [r0, r1])
8. ∀t:{x:ℝ| x ∈ [r0, (r1/r(2))]} . (f(r(2) * rmin(t;(r1/r(2)))) = f(r(2) * t))
9. ∀t:{x:ℝ| x ∈ [(r1/r(2)), r1]} . (rmax(t;(r1/r(2))) = t)
10. ∀t:{x:ℝ| x ∈ [(r1/r(2)), r1]} . ((r(2) * t) - r1 ∈ [r0, r1])
11. ∀t:{x:ℝ| x ∈ [(r1/r(2)), r1]} . (g((r(2) * rmax(t;(r1/r(2)))) - r1) = g((r(2) * t) - r1))
12. ∀t:{x:ℝ| x ∈ [r0, r1]} . (f(r(2) * rmin(t;(r1/r(2)))) ∈ ℝ)
13. ∀t:{x:ℝ| x ∈ [r0, r1]} . (g((r(2) * rmax(t;(r1/r(2)))) - r1) ∈ ℝ)
14. ∀t:{x:ℝ| x ∈ [r0, r1]} . ∀n:ℕ+.
∃m:ℕ+
((|t - (r1/r(2))| ≤ (r1/r(m)))
⇒ (|f(r(2) * rmin(t;(r1/r(2)))) - g((r(2) * rmax(t;(r1/r(2)))) - r1)| ≤ (r1/r(n))))
15. ∀t:{x:ℝ| x ∈ [r0, r1]} . ∀n:ℕ+.
∃x:ℝ
(((t < ((r1/r(2)) + (r1/r(n)))) ∧ (x = f(r(2) * rmin(t;(r1/r(2))))))
∨ ((((r1/r(2)) - (r1/r(n))) < t) ∧ (x = g((r(2) * rmax(t;(r1/r(2)))) - r1))))
16. h : t:{x:ℝ| x ∈ [r0, r1]} ⟶ n:ℕ+ ⟶ ℝ
17. ∀t:{x:ℝ| x ∈ [r0, r1]} . ∀n:ℕ+.
(((t < ((r1/r(2)) + (r1/r(n)))) ∧ ((h t n) = f(r(2) * rmin(t;(r1/r(2))))))
∨ ((((r1/r(2)) - (r1/r(n))) < t) ∧ ((h t n) = g((r(2) * rmax(t;(r1/r(2)))) - r1))))
18. c : ∀t:{x:ℝ| x ∈ [r0, r1]} . cauchy(n.h t (n + 1))
19. ∀t:{x:ℝ| x ∈ [r0, r1]} . lim n→∞.h t (n + 1) = cauchy-limit(n.h t (n + 1);c t)
20. {x:ℝ| x ∈ [r0, (r1/r(2))]} ⊆r {x:ℝ| x ∈ [r0, r1]}
21. {x:ℝ| x ∈ [(r1/r(2)), r1]} ⊆r {x:ℝ| x ∈ [r0, r1]}
22. ∀t:{x:ℝ| x ∈ [r0, (r1/r(2))]} . (λt.cauchy-limit(n.h t (n + 1);c t)(t) = f(r(2) * t))
23. ∀t:{x:ℝ| x ∈ [(r1/r(2)), r1]} . (λt.cauchy-limit(n.h t (n + 1);c t)(t) = g((r(2) * t) - r1))
24. x : {x:ℝ| x ∈ [r0, r1]}
25. y : {x:ℝ| x ∈ [r0, r1]}
26. x = y
27. lim n→∞.h x (n + 1) = cauchy-limit(n.h x (n + 1);c x)
28. k : ℕ+
29. ∃N:ℕ [(∀n:ℕ. ((N ≤ n) ⇒ (|(h y (n + 1)) - cauchy-limit(n.h y (n + 1);c y)| ≤ (r1/r(2 * k)))))]
30. m : ℕ+
31. (|x - (r1/r(2))| ≤ (r1/r(m)))
⇒ (|f(r(2) * rmin(x;(r1/r(2)))) - g((r(2) * rmax(x;(r1/r(2)))) - r1)| ≤ (r1/r(2 * k)))
32. n : ℕ
33. m ≤ n
⊢ |(h x (n + 1)) - h y (n + 1)| ≤ (r1/r(2 * k))
BY
{ ((InstHyp [⌜x⌝;⌜n + 1⌝] 17 ⋅ THENA Auto) THEN (InstHyp [⌜y⌝;⌜n + 1⌝] 17 ⋅ THENA Auto)) }
1
1. f : [r0, r1] ⟶ℝ
2. g : [r0, r1] ⟶ℝ
3. ∀x,y:{x:ℝ| x ∈ [r0, r1]} . ((x = y) ⇒ (f(x) = f(y)))
4. ∀x,y:{x:ℝ| x ∈ [r0, r1]} . ((x = y) ⇒ (g(x) = g(y)))
5. f(r1) = g(r0)
6. ∀t:{x:ℝ| x ∈ [r0, (r1/r(2))]} . (rmin(t;(r1/r(2))) = t)
7. ∀t:{x:ℝ| x ∈ [r0, (r1/r(2))]} . (r(2) * t ∈ [r0, r1])
8. ∀t:{x:ℝ| x ∈ [r0, (r1/r(2))]} . (f(r(2) * rmin(t;(r1/r(2)))) = f(r(2) * t))
9. ∀t:{x:ℝ| x ∈ [(r1/r(2)), r1]} . (rmax(t;(r1/r(2))) = t)
10. ∀t:{x:ℝ| x ∈ [(r1/r(2)), r1]} . ((r(2) * t) - r1 ∈ [r0, r1])
11. ∀t:{x:ℝ| x ∈ [(r1/r(2)), r1]} . (g((r(2) * rmax(t;(r1/r(2)))) - r1) = g((r(2) * t) - r1))
12. ∀t:{x:ℝ| x ∈ [r0, r1]} . (f(r(2) * rmin(t;(r1/r(2)))) ∈ ℝ)
13. ∀t:{x:ℝ| x ∈ [r0, r1]} . (g((r(2) * rmax(t;(r1/r(2)))) - r1) ∈ ℝ)
14. ∀t:{x:ℝ| x ∈ [r0, r1]} . ∀n:ℕ+.
∃m:ℕ+
((|t - (r1/r(2))| ≤ (r1/r(m)))
⇒ (|f(r(2) * rmin(t;(r1/r(2)))) - g((r(2) * rmax(t;(r1/r(2)))) - r1)| ≤ (r1/r(n))))
15. ∀t:{x:ℝ| x ∈ [r0, r1]} . ∀n:ℕ+.
∃x:ℝ
(((t < ((r1/r(2)) + (r1/r(n)))) ∧ (x = f(r(2) * rmin(t;(r1/r(2))))))
∨ ((((r1/r(2)) - (r1/r(n))) < t) ∧ (x = g((r(2) * rmax(t;(r1/r(2)))) - r1))))
16. h : t:{x:ℝ| x ∈ [r0, r1]} ⟶ n:ℕ+ ⟶ ℝ
17. ∀t:{x:ℝ| x ∈ [r0, r1]} . ∀n:ℕ+.
(((t < ((r1/r(2)) + (r1/r(n)))) ∧ ((h t n) = f(r(2) * rmin(t;(r1/r(2))))))
∨ ((((r1/r(2)) - (r1/r(n))) < t) ∧ ((h t n) = g((r(2) * rmax(t;(r1/r(2)))) - r1))))
18. c : ∀t:{x:ℝ| x ∈ [r0, r1]} . cauchy(n.h t (n + 1))
19. ∀t:{x:ℝ| x ∈ [r0, r1]} . lim n→∞.h t (n + 1) = cauchy-limit(n.h t (n + 1);c t)
20. {x:ℝ| x ∈ [r0, (r1/r(2))]} ⊆r {x:ℝ| x ∈ [r0, r1]}
21. {x:ℝ| x ∈ [(r1/r(2)), r1]} ⊆r {x:ℝ| x ∈ [r0, r1]}
22. ∀t:{x:ℝ| x ∈ [r0, (r1/r(2))]} . (λt.cauchy-limit(n.h t (n + 1);c t)(t) = f(r(2) * t))
23. ∀t:{x:ℝ| x ∈ [(r1/r(2)), r1]} . (λt.cauchy-limit(n.h t (n + 1);c t)(t) = g((r(2) * t) - r1))
24. x : {x:ℝ| x ∈ [r0, r1]}
25. y : {x:ℝ| x ∈ [r0, r1]}
26. x = y
27. lim n→∞.h x (n + 1) = cauchy-limit(n.h x (n + 1);c x)
28. k : ℕ+
29. ∃N:ℕ [(∀n:ℕ. ((N ≤ n) ⇒ (|(h y (n + 1)) - cauchy-limit(n.h y (n + 1);c y)| ≤ (r1/r(2 * k)))))]
30. m : ℕ+
31. (|x - (r1/r(2))| ≤ (r1/r(m)))
⇒ (|f(r(2) * rmin(x;(r1/r(2)))) - g((r(2) * rmax(x;(r1/r(2)))) - r1)| ≤ (r1/r(2 * k)))
32. n : ℕ
33. m ≤ n
34. ((x < ((r1/r(2)) + (r1/r(n + 1)))) ∧ ((h x (n + 1)) = f(r(2) * rmin(x;(r1/r(2))))))
∨ ((((r1/r(2)) - (r1/r(n + 1))) < x) ∧ ((h x (n + 1)) = g((r(2) * rmax(x;(r1/r(2)))) - r1)))
35. ((y < ((r1/r(2)) + (r1/r(n + 1)))) ∧ ((h y (n + 1)) = f(r(2) * rmin(y;(r1/r(2))))))
∨ ((((r1/r(2)) - (r1/r(n + 1))) < y) ∧ ((h y (n + 1)) = g((r(2) * rmax(y;(r1/r(2)))) - r1)))
⊢ |(h x (n + 1)) - h y (n + 1)| ≤ (r1/r(2 * k))
Latex:
Latex:
1. f : [r0, r1] {}\mrightarrow{}\mBbbR{}
2. g : [r0, r1] {}\mrightarrow{}\mBbbR{}
3. \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . ((x = y) {}\mRightarrow{} (f(x) = f(y)))
4. \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . ((x = y) {}\mRightarrow{} (g(x) = g(y)))
5. f(r1) = g(r0)
6. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, (r1/r(2))]\} . (rmin(t;(r1/r(2))) = t)
7. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, (r1/r(2))]\} . (r(2) * t \mmember{} [r0, r1])
8. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, (r1/r(2))]\} . (f(r(2) * rmin(t;(r1/r(2)))) = f(r(2) * t))
9. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [(r1/r(2)), r1]\} . (rmax(t;(r1/r(2))) = t)
10. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [(r1/r(2)), r1]\} . ((r(2) * t) - r1 \mmember{} [r0, r1])
11. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [(r1/r(2)), r1]\} . (g((r(2) * rmax(t;(r1/r(2)))) - r1) = g((r(2) * t) - r1))
12. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . (f(r(2) * rmin(t;(r1/r(2)))) \mmember{} \mBbbR{})
13. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . (g((r(2) * rmax(t;(r1/r(2)))) - r1) \mmember{} \mBbbR{})
14. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . \mforall{}n:\mBbbN{}\msupplus{}.
\mexists{}m:\mBbbN{}\msupplus{}
((|t - (r1/r(2))| \mleq{} (r1/r(m)))
{}\mRightarrow{} (|f(r(2) * rmin(t;(r1/r(2)))) - g((r(2) * rmax(t;(r1/r(2)))) - r1)| \mleq{} (r1/r(n))))
15. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . \mforall{}n:\mBbbN{}\msupplus{}.
\mexists{}x:\mBbbR{}
(((t < ((r1/r(2)) + (r1/r(n)))) \mwedge{} (x = f(r(2) * rmin(t;(r1/r(2))))))
\mvee{} ((((r1/r(2)) - (r1/r(n))) < t) \mwedge{} (x = g((r(2) * rmax(t;(r1/r(2)))) - r1))))
16. h : t:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} {}\mrightarrow{} n:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbR{}
17. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . \mforall{}n:\mBbbN{}\msupplus{}.
(((t < ((r1/r(2)) + (r1/r(n)))) \mwedge{} ((h t n) = f(r(2) * rmin(t;(r1/r(2))))))
\mvee{} ((((r1/r(2)) - (r1/r(n))) < t) \mwedge{} ((h t n) = g((r(2) * rmax(t;(r1/r(2)))) - r1))))
18. c : \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . cauchy(n.h t (n + 1))
19. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . lim n\mrightarrow{}\minfty{}.h t (n + 1) = cauchy-limit(n.h t (n + 1);c t)
20. \{x:\mBbbR{}| x \mmember{} [r0, (r1/r(2))]\} \msubseteq{}r \{x:\mBbbR{}| x \mmember{} [r0, r1]\}
21. \{x:\mBbbR{}| x \mmember{} [(r1/r(2)), r1]\} \msubseteq{}r \{x:\mBbbR{}| x \mmember{} [r0, r1]\}
22. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [r0, (r1/r(2))]\} . (\mlambda{}t.cauchy-limit(n.h t (n + 1);c t)(t) = f(r(2) * t))
23. \mforall{}t:\{x:\mBbbR{}| x \mmember{} [(r1/r(2)), r1]\} . (\mlambda{}t.cauchy-limit(n.h t (n + 1);c t)(t) = g((r(2) * t) - r1))
24. x : \{x:\mBbbR{}| x \mmember{} [r0, r1]\}
25. y : \{x:\mBbbR{}| x \mmember{} [r0, r1]\}
26. x = y
27. lim n\mrightarrow{}\minfty{}.h x (n + 1) = cauchy-limit(n.h x (n + 1);c x)
28. k : \mBbbN{}\msupplus{}
29. \mexists{}N:\mBbbN{} [(\mforall{}n:\mBbbN{}. ((N \mleq{} n) {}\mRightarrow{} (|(h y (n + 1)) - cauchy-limit(n.h y (n + 1);c y)| \mleq{} (r1/r(2 * k)))))]
30. m : \mBbbN{}\msupplus{}
31. (|x - (r1/r(2))| \mleq{} (r1/r(m)))
{}\mRightarrow{} (|f(r(2) * rmin(x;(r1/r(2)))) - g((r(2) * rmax(x;(r1/r(2)))) - r1)| \mleq{} (r1/r(2 * k)))
32. n : \mBbbN{}
33. m \mleq{} n
\mvdash{} |(h x (n + 1)) - h y (n + 1)| \mleq{} (r1/r(2 * k))
By
Latex:
((InstHyp [\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}n + 1\mkleeneclose{}] 17 \mcdot{} THENA Auto) THEN (InstHyp [\mkleeneopen{}y\mkleeneclose{};\mkleeneopen{}n + 1\mkleeneclose{}] 17 \mcdot{} THENA Auto))
Home
Index