Step
*
1
of Lemma
partition-refines-cons
1. I : Interval
2. icompact(I)
3. a : ℝ
4. bs : ℝ List
5. partitions(I;[a / bs])
6. partitions([left-endpoint(I), a];[])
7. partitions([a, right-endpoint(I)];bs)
8. left-endpoint(I) ≤ a
9. a ≤ right-endpoint(I)
10. partition-mesh([left-endpoint(I), a];[]) ≤ partition-mesh(I;[a / bs])
11. partition-mesh([a, right-endpoint(I)];bs) ≤ partition-mesh(I;[a / bs])
12. 0 < ||bs|| ⇒ (a < hd(bs))
13. p : partition(I)
14. p refines [a / bs]
15. i : ℕ||p||
16. a = p[i]
17. p ~ firstn(i;p) @ [p[i]] @ nth_tl(1 + i;p)
18. ||firstn(i;p)|| ≤ ||full-partition(I;p)||
19. icompact([a, right-endpoint(I)])
20. bs ∈ partition([a, right-endpoint(I)])
21. icompact([left-endpoint(I), a])
22. [] ∈ partition([left-endpoint(I), a])
23. firstn(i;p) ∈ partition([left-endpoint(I), a])
24. nth_tl(i + 1;p) ∈ partition([a, right-endpoint(I)])
⊢ firstn(i;p) refines []
∧ nth_tl(i + 1;p) refines bs
∧ (∃x:ℝ. ((x = a) ∧ (p = (firstn(i;p) @ [x / nth_tl(i + 1;p)]) ∈ (ℝ List))))
∧ ||nth_tl(i + 1;p)|| + ||firstn(i;p)|| < ||p||
∧ (∀x:partition-choice(full-partition(I;p))
(is-partition-choice(full-partition([left-endpoint(I), a];firstn(i;p));x)
∧ is-partition-choice(full-partition([a, right-endpoint(I)];nth_tl(i + 1;
p));λi@0.(x (i@0 + ||firstn(i;p)|| + 1)))))
BY
{ Auto }
1
1. I : Interval
2. icompact(I)
3. a : ℝ
4. bs : ℝ List
5. partitions(I;[a / bs])
6. partitions([left-endpoint(I), a];[])
7. partitions([a, right-endpoint(I)];bs)
8. left-endpoint(I) ≤ a
9. a ≤ right-endpoint(I)
10. partition-mesh([left-endpoint(I), a];[]) ≤ partition-mesh(I;[a / bs])
11. partition-mesh([a, right-endpoint(I)];bs) ≤ partition-mesh(I;[a / bs])
12. 0 < ||bs|| ⇒ (a < hd(bs))
13. p : partition(I)
14. p refines [a / bs]
15. i : ℕ||p||
16. a = p[i]
17. p ~ firstn(i;p) @ [p[i]] @ nth_tl(1 + i;p)
18. ||firstn(i;p)|| ≤ ||full-partition(I;p)||
19. icompact([a, right-endpoint(I)])
20. bs ∈ partition([a, right-endpoint(I)])
21. icompact([left-endpoint(I), a])
22. [] ∈ partition([left-endpoint(I), a])
23. firstn(i;p) ∈ partition([left-endpoint(I), a])
24. nth_tl(i + 1;p) ∈ partition([a, right-endpoint(I)])
⊢ firstn(i;p) refines []
2
1. I : Interval
2. icompact(I)
3. a : ℝ
4. bs : ℝ List
5. partitions(I;[a / bs])
6. partitions([left-endpoint(I), a];[])
7. partitions([a, right-endpoint(I)];bs)
8. left-endpoint(I) ≤ a
9. a ≤ right-endpoint(I)
10. partition-mesh([left-endpoint(I), a];[]) ≤ partition-mesh(I;[a / bs])
11. partition-mesh([a, right-endpoint(I)];bs) ≤ partition-mesh(I;[a / bs])
12. 0 < ||bs|| ⇒ (a < hd(bs))
13. p : partition(I)
14. p refines [a / bs]
15. i : ℕ||p||
16. a = p[i]
17. p ~ firstn(i;p) @ [p[i]] @ nth_tl(1 + i;p)
18. ||firstn(i;p)|| ≤ ||full-partition(I;p)||
19. icompact([a, right-endpoint(I)])
20. bs ∈ partition([a, right-endpoint(I)])
21. icompact([left-endpoint(I), a])
22. [] ∈ partition([left-endpoint(I), a])
23. firstn(i;p) ∈ partition([left-endpoint(I), a])
24. nth_tl(i + 1;p) ∈ partition([a, right-endpoint(I)])
25. firstn(i;p) refines []
⊢ nth_tl(i + 1;p) refines bs
3
1. I : Interval
2. icompact(I)
3. a : ℝ
4. bs : ℝ List
5. partitions(I;[a / bs])
6. partitions([left-endpoint(I), a];[])
7. partitions([a, right-endpoint(I)];bs)
8. left-endpoint(I) ≤ a
9. a ≤ right-endpoint(I)
10. partition-mesh([left-endpoint(I), a];[]) ≤ partition-mesh(I;[a / bs])
11. partition-mesh([a, right-endpoint(I)];bs) ≤ partition-mesh(I;[a / bs])
12. 0 < ||bs|| ⇒ (a < hd(bs))
13. p : partition(I)
14. p refines [a / bs]
15. i : ℕ||p||
16. a = p[i]
17. p ~ firstn(i;p) @ [p[i]] @ nth_tl(1 + i;p)
18. ||firstn(i;p)|| ≤ ||full-partition(I;p)||
19. icompact([a, right-endpoint(I)])
20. bs ∈ partition([a, right-endpoint(I)])
21. icompact([left-endpoint(I), a])
22. [] ∈ partition([left-endpoint(I), a])
23. firstn(i;p) ∈ partition([left-endpoint(I), a])
24. nth_tl(i + 1;p) ∈ partition([a, right-endpoint(I)])
25. firstn(i;p) refines []
26. nth_tl(i + 1;p) refines bs
⊢ ∃x:ℝ. ((x = a) ∧ (p = (firstn(i;p) @ [x / nth_tl(i + 1;p)]) ∈ (ℝ List)))
4
1. I : Interval
2. icompact(I)
3. a : ℝ
4. bs : ℝ List
5. partitions(I;[a / bs])
6. partitions([left-endpoint(I), a];[])
7. partitions([a, right-endpoint(I)];bs)
8. left-endpoint(I) ≤ a
9. a ≤ right-endpoint(I)
10. partition-mesh([left-endpoint(I), a];[]) ≤ partition-mesh(I;[a / bs])
11. partition-mesh([a, right-endpoint(I)];bs) ≤ partition-mesh(I;[a / bs])
12. 0 < ||bs|| ⇒ (a < hd(bs))
13. p : partition(I)
14. p refines [a / bs]
15. i : ℕ||p||
16. a = p[i]
17. p ~ firstn(i;p) @ [p[i]] @ nth_tl(1 + i;p)
18. ||firstn(i;p)|| ≤ ||full-partition(I;p)||
19. icompact([a, right-endpoint(I)])
20. bs ∈ partition([a, right-endpoint(I)])
21. icompact([left-endpoint(I), a])
22. [] ∈ partition([left-endpoint(I), a])
23. firstn(i;p) ∈ partition([left-endpoint(I), a])
24. nth_tl(i + 1;p) ∈ partition([a, right-endpoint(I)])
25. firstn(i;p) refines []
26. nth_tl(i + 1;p) refines bs
27. ∃x:ℝ. ((x = a) ∧ (p = (firstn(i;p) @ [x / nth_tl(i + 1;p)]) ∈ (ℝ List)))
28. ||nth_tl(i + 1;p)|| + ||firstn(i;p)|| < ||p||
29. x : partition-choice(full-partition(I;p))
⊢ is-partition-choice(full-partition([left-endpoint(I), a];firstn(i;p));x)
5
1. I : Interval
2. icompact(I)
3. a : ℝ
4. bs : ℝ List
5. partitions(I;[a / bs])
6. partitions([left-endpoint(I), a];[])
7. partitions([a, right-endpoint(I)];bs)
8. left-endpoint(I) ≤ a
9. a ≤ right-endpoint(I)
10. partition-mesh([left-endpoint(I), a];[]) ≤ partition-mesh(I;[a / bs])
11. partition-mesh([a, right-endpoint(I)];bs) ≤ partition-mesh(I;[a / bs])
12. 0 < ||bs|| ⇒ (a < hd(bs))
13. p : partition(I)
14. p refines [a / bs]
15. i : ℕ||p||
16. a = p[i]
17. p ~ firstn(i;p) @ [p[i]] @ nth_tl(1 + i;p)
18. ||firstn(i;p)|| ≤ ||full-partition(I;p)||
19. icompact([a, right-endpoint(I)])
20. bs ∈ partition([a, right-endpoint(I)])
21. icompact([left-endpoint(I), a])
22. [] ∈ partition([left-endpoint(I), a])
23. firstn(i;p) ∈ partition([left-endpoint(I), a])
24. nth_tl(i + 1;p) ∈ partition([a, right-endpoint(I)])
25. firstn(i;p) refines []
26. nth_tl(i + 1;p) refines bs
27. ∃x:ℝ. ((x = a) ∧ (p = (firstn(i;p) @ [x / nth_tl(i + 1;p)]) ∈ (ℝ List)))
28. ||nth_tl(i + 1;p)|| + ||firstn(i;p)|| < ||p||
29. x : partition-choice(full-partition(I;p))
30. is-partition-choice(full-partition([left-endpoint(I), a];firstn(i;p));x)
⊢ is-partition-choice(full-partition([a, right-endpoint(I)];nth_tl(i + 1;p));λi@0.(x (i@0 + ||firstn(i;p)|| + 1)))
Latex:
Latex:
1. I : Interval
2. icompact(I)
3. a : \mBbbR{}
4. bs : \mBbbR{} List
5. partitions(I;[a / bs])
6. partitions([left-endpoint(I), a];[])
7. partitions([a, right-endpoint(I)];bs)
8. left-endpoint(I) \mleq{} a
9. a \mleq{} right-endpoint(I)
10. partition-mesh([left-endpoint(I), a];[]) \mleq{} partition-mesh(I;[a / bs])
11. partition-mesh([a, right-endpoint(I)];bs) \mleq{} partition-mesh(I;[a / bs])
12. 0 < ||bs|| {}\mRightarrow{} (a < hd(bs))
13. p : partition(I)
14. p refines [a / bs]
15. i : \mBbbN{}||p||
16. a = p[i]
17. p \msim{} firstn(i;p) @ [p[i]] @ nth\_tl(1 + i;p)
18. ||firstn(i;p)|| \mleq{} ||full-partition(I;p)||
19. icompact([a, right-endpoint(I)])
20. bs \mmember{} partition([a, right-endpoint(I)])
21. icompact([left-endpoint(I), a])
22. [] \mmember{} partition([left-endpoint(I), a])
23. firstn(i;p) \mmember{} partition([left-endpoint(I), a])
24. nth\_tl(i + 1;p) \mmember{} partition([a, right-endpoint(I)])
\mvdash{} firstn(i;p) refines []
\mwedge{} nth\_tl(i + 1;p) refines bs
\mwedge{} (\mexists{}x:\mBbbR{}. ((x = a) \mwedge{} (p = (firstn(i;p) @ [x / nth\_tl(i + 1;p)]))))
\mwedge{} ||nth\_tl(i + 1;p)|| + ||firstn(i;p)|| < ||p||
\mwedge{} (\mforall{}x:partition-choice(full-partition(I;p))
(is-partition-choice(full-partition([left-endpoint(I), a];firstn(i;p));x)
\mwedge{} is-partition-choice(full-partition([a, right-endpoint(I)];nth\_tl(i + 1;
p));\mlambda{}i@0.(x
(i@0
+ ||firstn(i;p)||
+ 1)))))
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