Step
*
1
2
1
1
2
1
3
1
2
of Lemma
Riemann-sum-alt-req
1. a : ℝ
2. b : ℝ
3. a ≤ b
4. f : [a, b] ⟶ℝ
5. ∀x,y:ℝ. ((x ∈ [a, b]) ⇒ (x = y) ⇒ ((f y) = (f x)))@i
6. k : ℕ+
7. icompact([a, b])
8. ∀i:ℕk. (((r(k - i) * a) + (r(i) * b)/r(k)) ∈ [a, b])
9. (∀x∈full-partition([a, b];uniform-partition([a, b];k)).x ∈ [a, b])
10. i : ℤ@i
11. 0 ≤ i@i
12. i ≤ (k - 1)@i
13. ∀i:ℕk + 1
([a / (mklist(k - 1;λi.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i]
= ((r(k - i) * a) + (r(i) * b)/r(k)))
⊢ ((f ((r(k - i) * a) + (r(i) * b)/r(k))) * (b - a/r(k)))
= ((f [a / (mklist(k - 1;λi.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i])
* ([a / (mklist(k - 1;λi.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i + 1] - [a /
(mklist(k
- 1;λi.(((r(k) - r(i + 1))
* a)
+ (r(i + 1)
* b)/r(k)))
@ [b])][i]))
BY
{ (BLemma `rmul_functionality`
THEN Auto
THEN Try (Complete ((Reduce 0
THEN (RWW "length-append mklist_length mklist_select" 0 THENA Auto)
THEN Reduce 0
THEN Auto)))) }
1
1. a : ℝ
2. b : ℝ
3. a ≤ b
4. f : [a, b] ⟶ℝ
5. ∀x,y:ℝ. ((x ∈ [a, b]) ⇒ (x = y) ⇒ ((f y) = (f x)))@i
6. k : ℕ+
7. icompact([a, b])
8. ∀i:ℕk. (((r(k - i) * a) + (r(i) * b)/r(k)) ∈ [a, b])
9. (∀x∈full-partition([a, b];uniform-partition([a, b];k)).x ∈ [a, b])
10. i : ℤ@i
11. 0 ≤ i@i
12. i ≤ (k - 1)@i
13. ∀i:ℕk + 1
([a / (mklist(k - 1;λi.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i]
= ((r(k - i) * a) + (r(i) * b)/r(k)))
14. i1 : ℕk - 1@i
⊢ (((r(k) - r(i1 + 1)) * a) + (r(i1 + 1) * b)/r(k)) ∈ {x:ℝ| (a ≤ x) ∧ (x ≤ b)}
2
1. a : ℝ
2. b : ℝ
3. a ≤ b
4. f : [a, b] ⟶ℝ
5. ∀x,y:ℝ. ((x ∈ [a, b]) ⇒ (x = y) ⇒ ((f y) = (f x)))@i
6. k : ℕ+
7. icompact([a, b])
8. ∀i:ℕk. (((r(k - i) * a) + (r(i) * b)/r(k)) ∈ [a, b])
9. (∀x∈full-partition([a, b];uniform-partition([a, b];k)).x ∈ [a, b])
10. i : ℤ@i
11. 0 ≤ i@i
12. i ≤ (k - 1)@i
13. ∀i:ℕk + 1
([a / (mklist(k - 1;λi.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i]
= ((r(k - i) * a) + (r(i) * b)/r(k)))
⊢ (f ((r(k - i) * a) + (r(i) * b)/r(k)))
= (f [a / (mklist(k - 1;λi.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i])
3
1. a : ℝ
2. b : ℝ
3. a ≤ b
4. f : [a, b] ⟶ℝ
5. ∀x,y:ℝ. ((x ∈ [a, b]) ⇒ (x = y) ⇒ ((f y) = (f x)))@i
6. k : ℕ+
7. icompact([a, b])
8. ∀i:ℕk. (((r(k - i) * a) + (r(i) * b)/r(k)) ∈ [a, b])
9. (∀x∈full-partition([a, b];uniform-partition([a, b];k)).x ∈ [a, b])
10. i : ℤ@i
11. 0 ≤ i@i
12. i ≤ (k - 1)@i
13. ∀i:ℕk + 1
([a / (mklist(k - 1;λi.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i]
= ((r(k - i) * a) + (r(i) * b)/r(k)))
⊢ (b - a/r(k))
= ([a / (mklist(k - 1;λi.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i + 1] - [a /
(mklist(k
- 1;λi.(((r(k) - r(i + 1))
* a)
+ (r(i + 1)
* b)/r(k)))
@ [b])][i])
Latex:
Latex:
1. a : \mBbbR{}
2. b : \mBbbR{}
3. a \mleq{} b
4. f : [a, b] {}\mrightarrow{}\mBbbR{}
5. \mforall{}x,y:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} (x = y) {}\mRightarrow{} ((f y) = (f x)))@i
6. k : \mBbbN{}\msupplus{}
7. icompact([a, b])
8. \mforall{}i:\mBbbN{}k. (((r(k - i) * a) + (r(i) * b)/r(k)) \mmember{} [a, b])
9. (\mforall{}x\mmember{}full-partition([a, b];uniform-partition([a, b];k)).x \mmember{} [a, b])
10. i : \mBbbZ{}@i
11. 0 \mleq{} i@i
12. i \mleq{} (k - 1)@i
13. \mforall{}i:\mBbbN{}k + 1
([a / (mklist(k - 1;\mlambda{}i.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i]
= ((r(k - i) * a) + (r(i) * b)/r(k)))
\mvdash{} ((f ((r(k - i) * a) + (r(i) * b)/r(k))) * (b - a/r(k)))
= ((f [a / (mklist(k - 1;\mlambda{}i.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i])
* ([a / (mklist(k - 1;\mlambda{}i.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i + 1]
- [a / (mklist(k - 1;\mlambda{}i.(((r(k) - r(i + 1)) * a) + (r(i + 1) * b)/r(k))) @ [b])][i]))
By
Latex:
(BLemma `rmul\_functionality`
THEN Auto
THEN Try (Complete ((Reduce 0
THEN (RWW "length-append mklist\_length mklist\_select" 0 THENA Auto)
THEN Reduce 0
THEN Auto))))
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