Step
*
1
2
1
1
of Lemma
partition-refines-cons
.....assertion.....
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. ∀i:ℕ||[a / bs]||. (∃y∈p. [a / bs][i] = y)
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. j : ℕ||bs||
27. k : ℕ||p||
28. bs[(j + 1) - 1] = p[k]
⊢ i < k
BY
{ ((Subst' (j + 1) - 1 ~ j -1 THENA Auto) THEN SupposeNot THEN Assert ⌜(p[k] ≤ p[i]) ∧ (a < bs[j])⌝⋅) }
1
.....assertion.....
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. ∀i:ℕ||[a / bs]||. (∃y∈p. [a / bs][i] = y)
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. j : ℕ||bs||
27. k : ℕ||p||
28. bs[j] = p[k]
29. ¬i < k
⊢ (p[k] ≤ p[i]) ∧ (a < bs[j])
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. ∀i:ℕ||[a / bs]||. (∃y∈p. [a / bs][i] = y)
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. j : ℕ||bs||
27. k : ℕ||p||
28. bs[j] = p[k]
29. ¬i < k
30. (p[k] ≤ p[i]) ∧ (a < bs[j])
⊢ i < k
Latex:
Latex:
.....assertion.....
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. \mforall{}i:\mBbbN{}||[a / bs]||. (\mexists{}y\mmember{}p. [a / bs][i] = y)
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)])
25. firstn(i;p) refines []
26. j : \mBbbN{}||bs||
27. k : \mBbbN{}||p||
28. bs[(j + 1) - 1] = p[k]
\mvdash{} i < k
By
Latex:
((Subst' (j + 1) - 1 \msim{} j -1 THENA Auto) THEN SupposeNot THEN Assert \mkleeneopen{}(p[k] \mleq{} p[i]) \mwedge{} (a < bs[j])\mkleeneclose{}\mcdot{})
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