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Step * 1 1 1 of Lemma adjacent-partition-points


1. I : Interval
2. icompact(I)
3. p : partition(I)
4. ∀[i:ℕ||full-partition(I;p)|| - 1]
     (frs-non-dec(full-partition(I;p)) ⇒ r0≤full-partition(I;p)[i + 1] - full-partition(I;p)[i]≤partition-mesh(I;p))
5. frs-non-dec(full-partition(I;p))
6. ∀i:ℕ||p|| + 1. r0≤full-partition(I;p)[i + 1] - full-partition(I;p)[i]≤partition-mesh(I;p)
7. r0≤p @ [right-endpoint(I)][0] - left-endpoint(I)≤partition-mesh(I;p)
8. r0≤[left-endpoint(I) / (p @ [right-endpoint(I)])][||p|| + 1] - [left-endpoint(I) / 
                                                                   (p @ [right-endpoint(I)])][||p||]≤partition-mesh(I;p)
⊢ (¬0 < ||p||) ⇒ r0≤right-endpoint(I) - left-endpoint(I)≤partition-mesh(I;p)
BY
{ ((D 0 THENA Auto) THEN RepeatFor 2 (DVar `p') THEN All Reduce  THEN Try (Complete ((D (-1) THEN Auto'))) THEN Auto) }

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Latex:

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Latex:

1.  I  :  Interval
2.  icompact(I)
3.  p  :  partition(I)
4.  \mforall{}[i:\mBbbN{}||full-partition(I;p)||  -  1]
          (frs-non-dec(full-partition(I;p))
          {}\mRightarrow{}  r0\mleq{}full-partition(I;p)[i  +  1]  -  full-partition(I;p)[i]\mleq{}partition-mesh(I;p))
5.  frs-non-dec(full-partition(I;p))
6.  \mforall{}i:\mBbbN{}||p||  +  1.  r0\mleq{}full-partition(I;p)[i  +  1]  -  full-partition(I;p)[i]\mleq{}partition-mesh(I;p)
7.  r0\mleq{}p  @  [right-endpoint(I)][0]  -  left-endpoint(I)\mleq{}partition-mesh(I;p)
8.  r0\mleq{}[left-endpoint(I)  /  (p  @  [right-endpoint(I)])][||p||  +  1] 
-  [left-endpoint(I)  /  (p  @  [right-endpoint(I)])][||p||]\mleq{}partition-mesh(I;p)
\mvdash{}  (\mneg{}0  <  ||p||)  {}\mRightarrow{}  r0\mleq{}right-endpoint(I)  -  left-endpoint(I)\mleq{}partition-mesh(I;p)


By

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Latex:
((D  0  THENA  Auto)
  THEN  RepeatFor  2  (DVar  `p')
  THEN  All  Reduce 
  THEN  Try  (Complete  ((D  (-1)  THEN  Auto')))
  THEN  Auto)

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