Step
*
1
1
1
of Lemma
loop-class-memory-program_wf
1. Info : Type
2. B : Type
3. valueall-type(B)
4. X : EClass(B ─→ B)
5. init : Id ─→ bag(B)
6. pr : Id ─→ hdataflow(Info;B ─→ B)
7. ∀es:EO+(Info). ∀e:E. (X(e) = (snd(pr loc(e)*(map(λx.info(x);before(e)))(info(e)))) ∈ bag(B ─→ B))
8. es : EO+(Info)@i'
9. e : E@i
10. ∀e1:E
((e1 < e)
⇒ (hdf-memory(pr loc(e1);init loc(e1))*(map(λx.info(x);before(e1)))
= hdf-memory(pr loc(e1)*(map(λx.info(x);before(e1)));loop-class-memory(X;init)(e1))
∈ hdataflow(Info;B)))
11. ¬↑first(e)
12. ↑pred(e) ∈b X
13. ¬↑first(e)
14. hdf-memory(pr loc(e);init loc(e))*(map(λx.info(x);before(pred(e))))
= hdf-memory(pr loc(e)*(map(λx.info(x);before(pred(e))));loop-class-memory(X;init)(pred(e)))
∈ hdataflow(Info;B)
15. v1 : bag(B)@i
16. loop-class-memory(X;init)(pred(e)) = v1 ∈ bag(B)@i
17. x : Info ─→ (hdataflow(Info;B ─→ B) × bag(B ─→ B))@i
18. z1 : hdataflow(Info;B ─→ B)@i
19. z2 : bag(B ─→ B)@i
20. (x info(pred(e))) = <z1, z2> ∈ (hdataflow(Info;B ─→ B) × bag(B ─→ B))@i
21. pr loc(pred(e))*(map(λx.info(x);before(pred(e)))) = (inl x) ∈ hdataflow(Info;B ─→ B)@i
22. ∪f∈z2.bag-map(f;v1) = {} ∈ bag(B)
⊢ v1 = ∪f∈z2.bag-map(f;v1) ∈ bag(B)
BY
{ (RepUR ``member-eclass`` (-11)
THEN Fold `class-ap` (-11)
THEN (RW assert_pushdownC (-11) THENA Auto)
THEN (RWO "7" (-11) THENA Auto)
THEN (HypSubst' (-2) (-11) THENA Auto)
THEN RepUR ``hdf-ap`` (-1)
THEN (HypSubst' (-4) (-1) THENA Auto)
THEN Reduce (-1)
THEN (RWO "assert-bag-null<" (-2) THENA Auto)
THEN (RWO "bag-combine-null" (-2) THENA Auto)
THEN (Assert ⌈0 < #(z2)⌉⋅ THENA Auto')
THEN (RWO "bag-member-iff-size<" (-1) THENA Auto)
THEN SquashExRepD
THEN (FHyp (-4) [-1] THENA Auto)
THEN RepUR ``bag-null`` (-1)
THEN (RWO "bag-map-null" (-1) THENA Auto)
THEN Fold `bag-null` (-1)
THEN (RWO "assert-bag-null" (-1) THENA Auto)
THEN (RWO "-1" 0 THENA Auto)
THEN Reduce 0
THEN RWO "bag-combine-empty-right" 0
THEN Auto) }
Latex:
Latex:
1. Info : Type
2. B : Type
3. valueall-type(B)
4. X : EClass(B {}\mrightarrow{} B)
5. init : Id {}\mrightarrow{} bag(B)
6. pr : Id {}\mrightarrow{} hdataflow(Info;B {}\mrightarrow{} B)
7. \mforall{}es:EO+(Info). \mforall{}e:E. (X(e) = (snd(pr loc(e)*(map(\mlambda{}x.info(x);before(e)))(info(e)))))
8. es : EO+(Info)@i'
9. e : E@i
10. \mforall{}e1:E
((e1 < e)
{}\mRightarrow{} (hdf-memory(pr loc(e1);init loc(e1))*(map(\mlambda{}x.info(x);before(e1)))
= hdf-memory(pr loc(e1)*(map(\mlambda{}x.info(x);before(e1)));loop-class-memory(X;init)(e1))))
11. \mneg{}\muparrow{}first(e)
12. \muparrow{}pred(e) \mmember{}\msubb{} X
13. \mneg{}\muparrow{}first(e)
14. hdf-memory(pr loc(e);init loc(e))*(map(\mlambda{}x.info(x);before(pred(e))))
= hdf-memory(pr loc(e)*(map(\mlambda{}x.info(x);before(pred(e))));loop-class-memory(X;init)(pred(e)))
15. v1 : bag(B)@i
16. loop-class-memory(X;init)(pred(e)) = v1@i
17. x : Info {}\mrightarrow{} (hdataflow(Info;B {}\mrightarrow{} B) \mtimes{} bag(B {}\mrightarrow{} B))@i
18. z1 : hdataflow(Info;B {}\mrightarrow{} B)@i
19. z2 : bag(B {}\mrightarrow{} B)@i
20. (x info(pred(e))) = <z1, z2>@i
21. pr loc(pred(e))*(map(\mlambda{}x.info(x);before(pred(e)))) = (inl x)@i
22. \mcup{}f\mmember{}z2.bag-map(f;v1) = \{\}
\mvdash{} v1 = \mcup{}f\mmember{}z2.bag-map(f;v1)
By
Latex:
(RepUR ``member-eclass`` (-11)
THEN Fold `class-ap` (-11)
THEN (RW assert\_pushdownC (-11) THENA Auto)
THEN (RWO "7" (-11) THENA Auto)
THEN (HypSubst' (-2) (-11) THENA Auto)
THEN RepUR ``hdf-ap`` (-1)
THEN (HypSubst' (-4) (-1) THENA Auto)
THEN Reduce (-1)
THEN (RWO "assert-bag-null<" (-2) THENA Auto)
THEN (RWO "bag-combine-null" (-2) THENA Auto)
THEN (Assert \mkleeneopen{}0 < \#(z2)\mkleeneclose{}\mcdot{} THENA Auto')
THEN (RWO "bag-member-iff-size<" (-1) THENA Auto)
THEN SquashExRepD
THEN (FHyp (-4) [-1] THENA Auto)
THEN RepUR ``bag-null`` (-1)
THEN (RWO "bag-map-null" (-1) THENA Auto)
THEN Fold `bag-null` (-1)
THEN (RWO "assert-bag-null" (-1) THENA Auto)
THEN (RWO "-1" 0 THENA Auto)
THEN Reduce 0
THEN RWO "bag-combine-empty-right" 0
THEN Auto)
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