Step
*
1
2
2
1
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. (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'')
BY
{ (InstLemma `es-pred_property` [⌈es⌉;⌈e⌉]⋅
THEN Auto
THEN (InstHyp [⌈e''⌉] (-1)⋅ THENA Auto)
THEN D (-1)
THEN Auto
THEN Try (Complete ((BackThruSomeHyp THEN Auto THEN D 0 THEN Auto)))
THEN (D 0 THENA Auto)
THEN SquashExRepD
THEN MaUseClassRel (-1)
THEN OnMaybeHyp 12 (\h. (D h
THEN BLemma `assert-member-eclass`
THEN Auto
THEN D 0
THEN InstConcl [⌈f⌉]⋅
THEN Auto))) }
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. (e' <loc pred(e))
15. \mexists{}w:B. w \mmember{} eclass3(X;loop-class-memory(X;init))(e')
16. \mforall{}e'':E
((e'' <loc pred(e))
{}\mRightarrow{} (e' <loc e'')
{}\mRightarrow{} (\mneg{}\mdownarrow{}\mexists{}w:B. w \mmember{} eclass3(X;loop-class-memory(X;init))(e'')))
17. v \mmember{} eclass3(X;loop-class-memory(X;init))(e')
18. (e' <loc e)
19. \mexists{}w:B. w \mmember{} eclass3(X;loop-class-memory(X;init))(e')
20. e'' : E@i
21. (e'' <loc e)@i
22. (e' <loc e'')@i
\mvdash{} \mneg{}\mdownarrow{}\mexists{}w:B. w \mmember{} eclass3(X;loop-class-memory(X;init))(e'')
By
Latex:
(InstLemma `es-pred\_property` [\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e\mkleeneclose{}]\mcdot{}
THEN Auto
THEN (InstHyp [\mkleeneopen{}e''\mkleeneclose{}] (-1)\mcdot{} THENA Auto)
THEN D (-1)
THEN Auto
THEN Try (Complete ((BackThruSomeHyp THEN Auto THEN D 0 THEN Auto)))
THEN (D 0 THENA Auto)
THEN SquashExRepD
THEN MaUseClassRel (-1)
THEN OnMaybeHyp 12 (\mbackslash{}h. (D h
THEN BLemma `assert-member-eclass`
THEN Auto
THEN D 0
THEN InstConcl [\mkleeneopen{}f\mkleeneclose{}]\mcdot{}
THEN Auto)))
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