Step
*
1
1
1
1
1
2
2
1
of Lemma
rational-IVT
1. a : ℝ
2. b : ℝ
3. f : (ℤ × ℕ+) ⟶ (ℤ × ℕ+)
4. g : {x:ℝ| x ∈ [a, b]} ⟶ ℝ
5. a < b
6. (g[a] * g[b]) < r0
7. ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (g[x] = g[y]))
8. ∀r:ℤ × ℕ+. ((ratreal(r) ∈ [a, b]) ⇒ (g[ratreal(r)] = ratreal(f[r])))
9. ∀n:ℕ+. (ratreal(<(a (2 * n)) + 2, 4 * n>) = above a within 1/n)
10. ∀n:ℕ+. (ratreal(<(b (2 * n)) - 2, 4 * n>) = (below b within 1/n))
11. n : ℕ+
12. above a within 1/n ≤ (below b within 1/n)
13. (g[above a within 1/n] * g[(below b within 1/n)]) < r0
⊢ (ratreal(f[<(a (2 * n)) + 2, 4 * n>]) * ratreal(f[<(b (2 * n)) - 2, 4 * n>])) < r0
BY
{ ((InstLemma `rational-upper-approx-property` [⌜b⌝;⌜n⌝]⋅ THENA Auto)
THEN (InstLemma `rational-lower-approx-property` [⌜a⌝;⌜n⌝]⋅ THENA Auto)
THEN (Assert (ratreal(<(a (2 * n)) + 2, 4 * n>) ∈ [a, b]) ∧ (ratreal(<(b (2 * n)) - 2, 4 * n>) ∈ [a, b]) BY
((RWO "9 10" 0 THENA Auto) THEN All Reduce THEN Auto))) }
1
.....aux.....
1. a : ℝ
2. b : ℝ
3. f : (ℤ × ℕ+) ⟶ (ℤ × ℕ+)
4. g : {x:ℝ| (a ≤ x) ∧ (x ≤ b)} ⟶ ℝ
5. a < b
6. (g[a] * g[b]) < r0
7. ∀x,y:{x:ℝ| (a ≤ x) ∧ (x ≤ b)} . ((x = y) ⇒ (g[x] = g[y]))
8. ∀r:ℤ × ℕ+. (((a ≤ ratreal(r)) ∧ (ratreal(r) ≤ b)) ⇒ (g[ratreal(r)] = ratreal(f[r])))
9. ∀n:ℕ+. (ratreal(<(a (2 * n)) + 2, 4 * n>) = above a within 1/n)
10. ∀n:ℕ+. (ratreal(<(b (2 * n)) - 2, 4 * n>) = (below b within 1/n))
11. n : ℕ+
12. above a within 1/n ≤ (below b within 1/n)
13. (g[above a within 1/n] * g[(below b within 1/n)]) < r0
14. (above b within 1/n - (r1/r(n))) ≤ b
15. b ≤ above b within 1/n
16. (below a within 1/n) ≤ a
17. a ≤ ((below a within 1/n) + (r1/r(n)))
18. a ≤ above a within 1/n
⊢ above a within 1/n ≤ b
2
1. a : ℝ
2. b : ℝ
3. f : (ℤ × ℕ+) ⟶ (ℤ × ℕ+)
4. g : {x:ℝ| x ∈ [a, b]} ⟶ ℝ
5. a < b
6. (g[a] * g[b]) < r0
7. ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (g[x] = g[y]))
8. ∀r:ℤ × ℕ+. ((ratreal(r) ∈ [a, b]) ⇒ (g[ratreal(r)] = ratreal(f[r])))
9. ∀n:ℕ+. (ratreal(<(a (2 * n)) + 2, 4 * n>) = above a within 1/n)
10. ∀n:ℕ+. (ratreal(<(b (2 * n)) - 2, 4 * n>) = (below b within 1/n))
11. n : ℕ+
12. above a within 1/n ≤ (below b within 1/n)
13. (g[above a within 1/n] * g[(below b within 1/n)]) < r0
14. ((above b within 1/n - (r1/r(n))) ≤ b) ∧ (b ≤ above b within 1/n)
15. ((below a within 1/n) ≤ a) ∧ (a ≤ ((below a within 1/n) + (r1/r(n))))
16. (ratreal(<(a (2 * n)) + 2, 4 * n>) ∈ [a, b]) ∧ (ratreal(<(b (2 * n)) - 2, 4 * n>) ∈ [a, b])
⊢ (ratreal(f[<(a (2 * n)) + 2, 4 * n>]) * ratreal(f[<(b (2 * n)) - 2, 4 * n>])) < r0
Latex:
Latex:
1. a : \mBbbR{}
2. b : \mBbbR{}
3. f : (\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}) {}\mrightarrow{} (\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{})
4. g : \{x:\mBbbR{}| x \mmember{} [a, b]\} {}\mrightarrow{} \mBbbR{}
5. a < b
6. (g[a] * g[b]) < r0
7. \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) {}\mRightarrow{} (g[x] = g[y]))
8. \mforall{}r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. ((ratreal(r) \mmember{} [a, b]) {}\mRightarrow{} (g[ratreal(r)] = ratreal(f[r])))
9. \mforall{}n:\mBbbN{}\msupplus{}. (ratreal(<(a (2 * n)) + 2, 4 * n>) = above a within 1/n)
10. \mforall{}n:\mBbbN{}\msupplus{}. (ratreal(<(b (2 * n)) - 2, 4 * n>) = (below b within 1/n))
11. n : \mBbbN{}\msupplus{}
12. above a within 1/n \mleq{} (below b within 1/n)
13. (g[above a within 1/n] * g[(below b within 1/n)]) < r0
\mvdash{} (ratreal(f[<(a (2 * n)) + 2, 4 * n>]) * ratreal(f[<(b (2 * n)) - 2, 4 * n>])) < r0
By
Latex:
((InstLemma `rational-upper-approx-property` [\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}n\mkleeneclose{}]\mcdot{} THENA Auto)
THEN (InstLemma `rational-lower-approx-property` [\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}n\mkleeneclose{}]\mcdot{} THENA Auto)
THEN (Assert (ratreal(<(a (2 * n)) + 2, 4 * n>) \mmember{} [a, b]) \mwedge{} (ratreal(<(b (2 * n)) - 2, 4 * n>) \mmember{} [a\000C, b]) BY
((RWO "9 10" 0 THENA Auto) THEN All Reduce THEN Auto)))
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