Step
*
1
2
2
1
of Lemma
loop-class-memory-exists
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)
⇒ uiff(0 < #(init loc(e1));↓∃v:B. v ∈ loop-class-memory(X;init)(e1)))
8. 0 < #(init loc(e))
9. ¬↑first(e)
10. v : B
11. v ∈ loop-class-memory(X;init)(pred(e))
12. ¬↑pred(e) ∈b X
13. e' : E
14. es-p-local-pred(es;λe'.(↓∃w:B. w ∈ eclass3(X;loop-class-memory(X;init))(e'))) pred(e) e'
15. v ∈ eclass3(X;loop-class-memory(X;init))(e')
⊢ ↓∃v:B. v ∈ loop-class-memory(X;init)(e)
BY
{ (D 0
THEN InstConcl [⌈v⌉]⋅
THEN Auto
THEN RecUnfold `loop-class-memory` 0
THEN MaUseClassRel 0
THEN D 0
THEN OrLeft
THEN Auto
THEN InstConcl [⌈e'⌉]⋅
THEN Auto
THEN All (RepUR ``es-p-local-pred``)
THEN Auto
THEN Auto) }
1
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)
⇒ uiff(0 < #(init loc(e1));↓∃v:B. v ∈ loop-class-memory(X;init)(e1)))
8. 0 < #(init loc(e))
9. ¬↑first(e)
10. v : B
11. v ∈ loop-class-memory(X;init)(pred(e))
12. ¬↑pred(e) ∈b X
13. e' : E
14. (e' <loc pred(e))
15. ∃w:B. w ∈ eclass3(X;loop-class-memory(X;init))(e')
16. ∀e'':E. ((e'' <loc pred(e))
⇒ (e' <loc e'')
⇒ (¬↓∃w:B. w ∈ eclass3(X;loop-class-memory(X;init))(e'')))
17. v ∈ eclass3(X;loop-class-memory(X;init))(e')
18. (e' <loc e)
19. ∃w:B. w ∈ eclass3(X;loop-class-memory(X;init))(e')
20. e'' : E@i
21. (e'' <loc e)@i
22. (e' <loc e'')@i
⊢ ¬↓∃w:B. w ∈ eclass3(X;loop-class-memory(X;init))(e'')
Latex:
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{} uiff(0 < \#(init loc(e1));\mdownarrow{}\mexists{}v:B. v \mmember{} loop-class-memory(X;init)(e1)))
8. 0 < \#(init loc(e))
9. \mneg{}\muparrow{}first(e)
10. v : B
11. v \mmember{} loop-class-memory(X;init)(pred(e))
12. \mneg{}\muparrow{}pred(e) \mmember{}\msubb{} X
13. e' : E
14. es-p-local-pred(es;\mlambda{}e'.(\mdownarrow{}\mexists{}w:B. w \mmember{} eclass3(X;loop-class-memory(X;init))(e'))) pred(e) e'
15. v \mmember{} eclass3(X;loop-class-memory(X;init))(e')
\mvdash{} \mdownarrow{}\mexists{}v:B. v \mmember{} loop-class-memory(X;init)(e)
By
Latex:
(D 0
THEN InstConcl [\mkleeneopen{}v\mkleeneclose{}]\mcdot{}
THEN Auto
THEN RecUnfold `loop-class-memory` 0
THEN MaUseClassRel 0
THEN D 0
THEN OrLeft
THEN Auto
THEN InstConcl [\mkleeneopen{}e'\mkleeneclose{}]\mcdot{}
THEN Auto
THEN All (RepUR ``es-p-local-pred``)
THEN Auto
THEN Auto)
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