Step
*
3
1
2
1
of Lemma
partition-refines-cons
.....subterm..... T:t
1:n
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. q : partition([left-endpoint(I), a])
26. r : partition([a, right-endpoint(I)])
27. x : q refines []
28. x1 : r refines bs
29. x2 : ∃x:ℝ. ((x = a) ∧ (p = (q @ [x / r]) ∈ (ℝ List)))
30. x3 : ||r|| + ||q|| < ||p||
31. x4 : partition-choice(full-partition(I;p))
32. x5 : is-partition-choice(full-partition([left-endpoint(I), a];q);x4)
33. i1 : ℕ||full-partition([a, right-endpoint(I)];r)|| - 1
⊢ x4 (i1 + ||q|| + 1) ∈ ℝ
BY
{ DoSubsume }
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)])
25. q : partition([left-endpoint(I), a])
26. r : partition([a, right-endpoint(I)])
27. x : q refines []
28. x1 : r refines bs
29. x2 : ∃x:ℝ. ((x = a) ∧ (p = (q @ [x / r]) ∈ (ℝ List)))
30. x3 : ||r|| + ||q|| < ||p||
31. x4 : partition-choice(full-partition(I;p))
32. x5 : is-partition-choice(full-partition([left-endpoint(I), a];q);x4)
33. i1 : ℕ||full-partition([a, right-endpoint(I)];r)|| - 1
⊢ x4 (i1 + ||q|| + 1) ∈ {x:ℝ| x ∈ [full-partition(I;p)[i1 + ||q|| + 1], full-partition(I;p)[(i1 + ||q|| + 1) + 1]]}
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. q : partition([left-endpoint(I), a])
26. r : partition([a, right-endpoint(I)])
27. x : q refines []
28. x1 : r refines bs
29. x2 : ∃x:ℝ. ((x = a) ∧ (p = (q @ [x / r]) ∈ (ℝ List)))
30. x3 : ||r|| + ||q|| < ||p||
31. x4 : partition-choice(full-partition(I;p))
32. x5 : is-partition-choice(full-partition([left-endpoint(I), a];q);x4)
33. i1 : ℕ||full-partition([a, right-endpoint(I)];r)|| - 1
34. (x4 (i1 + ||q|| + 1))
= (x4 (i1 + ||q|| + 1))
∈ {x:ℝ| x ∈ [full-partition(I;p)[i1 + ||q|| + 1], full-partition(I;p)[(i1 + ||q|| + 1) + 1]]}
⊢ {x:ℝ| x ∈ [full-partition(I;p)[i1 + ||q|| + 1], full-partition(I;p)[(i1 + ||q|| + 1) + 1]]} ⊆r ℝ
Latex:
Latex:
.....subterm..... T:t
1:n
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)])
25. q : partition([left-endpoint(I), a])
26. r : partition([a, right-endpoint(I)])
27. x : q refines []
28. x1 : r refines bs
29. x2 : \mexists{}x:\mBbbR{}. ((x = a) \mwedge{} (p = (q @ [x / r])))
30. x3 : ||r|| + ||q|| < ||p||
31. x4 : partition-choice(full-partition(I;p))
32. x5 : is-partition-choice(full-partition([left-endpoint(I), a];q);x4)
33. i1 : \mBbbN{}||full-partition([a, right-endpoint(I)];r)|| - 1
\mvdash{} x4 (i1 + ||q|| + 1) \mmember{} \mBbbR{}
By
Latex:
DoSubsume
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