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Step * 1 2 of Lemma loop-class-memory-no-input


1. Info : Type
2. B : Type
3. X : EClass(B ─→ B)
4. init : Id ─→ bag(B)
5. es : EO+(Info)
6. e : E@i
7. ∀e1:E
     ((e1 < e)
     ⇒ ((¬↑first(e1)) ⇒ (¬↑pred(e1) ∈b X))
     ⇒ (loop-class-memory(X;init)(e1) = Prior(loop-class-memory(X;init))?init(e1) ∈ bag(B)))
8. (¬↑first(e)) ⇒ (¬↑pred(e) ∈b X)@i
9. ¬0 < #(init loc(e))
⊢ loop-class-memory(X;init)(e) = Prior(loop-class-memory(X;init))?init(e) ∈ bag(B)
BY
{ ((Assert ⌈¬↓∃v:B. v ∈ loop-class-memory(X;init)(e)⌉⋅
    THENA (ParallelLast THEN FLemma `loop-class-memory-exists` [-1] THEN Auto)
    )
   THEN (Assert ⌈¬↓∃v:B. v ∈ Prior(loop-class-memory(X;init))?init(e)⌉⋅
         THENA (ParallelLast THEN FLemma `loop-class-memory-exists-prior` [-1] THEN Auto)
         )
   THEN RepUR ``classrel`` (-2)
   THEN RepUR ``classrel`` (-1)
   THEN Fold `class-ap` (-2)
   THEN Fold `class-ap` (-1)
   THEN (RWO "bag-size-bag-member<" (-2) THENA Auto)
   THEN (RWO "bag-size-bag-member<" (-1) THENA Auto)
   THEN (Assert ⌈↑(#(loop-class-memory(X;init)(e)) =z 0)⌉⋅ THENA (RW assert_pushdownC 0 THEN Auto'))
   THEN (Assert ⌈↑(#(Prior(loop-class-memory(X;init))?init(e)) =z 0)⌉⋅ THENA (RW assert_pushdownC 0 THEN Auto'))
   THEN (RWO "null-bag-size<" (-2) THENA Auto)
   THEN (RWO "null-bag-size<" (-1) THENA Auto)
   THEN RepUR ``bag-null`` (-2)
   THEN RepUR ``bag-null`` (-1)
   THEN (RWO "null-bag" (-2) THENA Auto)
   THEN RWO "null-bag" (-1)
   THEN Auto) }

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Latex:

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Latex:

1.  Info  :  Type
2.  B  :  Type
3.  X  :  EClass(B  {}\mrightarrow{}  B)
4.  init  :  Id  {}\mrightarrow{}  bag(B)
5.  es  :  EO+(Info)
6.  e  :  E@i
7.  \mforall{}e1:E
          ((e1  <  e)
          {}\mRightarrow{}  ((\mneg{}\muparrow{}first(e1))  {}\mRightarrow{}  (\mneg{}\muparrow{}pred(e1)  \mmember{}\msubb{}  X))
          {}\mRightarrow{}  (loop-class-memory(X;init)(e1)  =  Prior(loop-class-memory(X;init))?init(e1)))
8.  (\mneg{}\muparrow{}first(e))  {}\mRightarrow{}  (\mneg{}\muparrow{}pred(e)  \mmember{}\msubb{}  X)@i
9.  \mneg{}0  <  \#(init  loc(e))
\mvdash{}  loop-class-memory(X;init)(e)  =  Prior(loop-class-memory(X;init))?init(e)


By

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Latex:
((Assert  \mkleeneopen{}\mneg{}\mdownarrow{}\mexists{}v:B.  v  \mmember{}  loop-class-memory(X;init)(e)\mkleeneclose{}\mcdot{}
    THENA  (ParallelLast  THEN  FLemma  `loop-class-memory-exists`  [-1]  THEN  Auto)
    )
  THEN  (Assert  \mkleeneopen{}\mneg{}\mdownarrow{}\mexists{}v:B.  v  \mmember{}  Prior(loop-class-memory(X;init))?init(e)\mkleeneclose{}\mcdot{}
              THENA  (ParallelLast  THEN  FLemma  `loop-class-memory-exists-prior`  [-1]  THEN  Auto)
              )
  THEN  RepUR  ``classrel``  (-2)
  THEN  RepUR  ``classrel``  (-1)
  THEN  Fold  `class-ap`  (-2)
  THEN  Fold  `class-ap`  (-1)
  THEN  (RWO  "bag-size-bag-member<"  (-2)  THENA  Auto)
  THEN  (RWO  "bag-size-bag-member<"  (-1)  THENA  Auto)
  THEN  (Assert  \mkleeneopen{}\muparrow{}(\#(loop-class-memory(X;init)(e))  =\msubz{}  0)\mkleeneclose{}\mcdot{}  THENA  (RW  assert\_pushdownC  0  THEN  Auto'))
  THEN  (Assert  \mkleeneopen{}\muparrow{}(\#(Prior(loop-class-memory(X;init))?init(e))  =\msubz{}  0)\mkleeneclose{}\mcdot{}
              THENA  (RW  assert\_pushdownC  0  THEN  Auto')
              )
  THEN  (RWO  "null-bag-size<"  (-2)  THENA  Auto)
  THEN  (RWO  "null-bag-size<"  (-1)  THENA  Auto)
  THEN  RepUR  ``bag-null``  (-2)
  THEN  RepUR  ``bag-null``  (-1)
  THEN  (RWO  "null-bag"  (-2)  THENA  Auto)
  THEN  RWO  "null-bag"  (-1)
  THEN  Auto)

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