Step
*
1
1
1
2
3
2
of Lemma
Memory-class-progress
1. [Info] : Type
2. [B] : Type
3. [A] : Type
4. R : B ─→ B ─→ ℙ@i'
5. P : A ─→ B ─→ ℙ@i'
6. f : A ─→ B ─→ B@i
7. init : Id ─→ bag(B)@i
8. X : EClass(A)@i'
9. es : EO+(Info)@i'
10. e1 : E@i
11. e2 : E@i
12. v1 : B@i
13. v2 : B@i
14. ∀a:A. ∀s:B. Dec(P[a;s])@i
15. Trans(B;x,y.R[x;y])@i
16. ∀s1,s2:B. SqStable(R[s1;s2])@i
17. ∀a:A. ∀e:E. ∀s:B.
(e1 ≤loc e
⇒ (e <loc e2)
⇒ a ∈ X(e)
⇒ s ∈ Memory-class(f;init;X)(e)
⇒ ((P[a;s] ⇒ R[s;f a s]) ∧ ((¬P[a;s]) ⇒ (s = (f a s) ∈ B))))@i
18. single-valued-classrel(es;X;A)@i
19. single-valued-bag(init loc(e1);B)@i
20. ↑first(e1)
21. v1 ↓∈ init loc(e1)
22. ¬↑first(e2)
23. iterated_classrel(es;B;A;f;init;X;pred(e2);v2)
24. (e1 <loc e2)@i
25. ∃e:E. ∃a:A. ∃s:B. (e1 ≤loc e ∧ (e <loc e2) ∧ s ∈ Memory-class(f;init;X)(e) ∧ a ∈ X(e) ∧ P[a;s])@i
26. v : A
27. v ∈ X(e1)
28. iterated_classrel(es;B;A;f;init;X;e1;f v v1)
29. loc(pred(e2)) = loc(e2) ∈ Id
30. (pred(e2) < e2)
31. ∀e':E. (e' < e2) ⇒ ((e' = pred(e2) ∈ E) ∨ (e' < pred(e2))) supposing loc(e') = loc(e2) ∈ Id
32. (e1 < pred(e2))
33. (∃e:E
∃s:B
((e1 <loc e) ∧ e ≤loc pred(e2) ∧ iterated_classrel(es;B;A;f;init;X;pred(e);s) ∧ (∃a:A. (a ∈ X(e) ∧ P[a;s]))))
⇒ R[f v v1;v2]
34. (∀e:E. ∀s:B.
((e1 <loc e)
⇒ e ≤loc pred(e2)
⇒ iterated_classrel(es;B;A;f;init;X;pred(e);s)
⇒ (∀a:A. ((¬a ∈ X(e)) ∨ (¬P[a;s])))))
⇒ ((f v v1) = v2 ∈ B)
35. P[v;v1] ⇒ R[v1;f v v1]
36. (¬P[v;v1]) ⇒ (v1 = (f v v1) ∈ B)
37. ¬(∃e:E
∃s:B
((e1 <loc e) ∧ e ≤loc pred(e2) ∧ iterated_classrel(es;B;A;f;init;X;pred(e);s) ∧ (∃v:A. (v ∈ X(e) ∧ P[v;s]))))
⊢ R[v1;v2]
BY
{ (D (-4)
THENA (Auto
THEN (InstHyp [⌈a⌉;⌈s⌉] (-29)⋅ THENA Auto)
THEN D (-1)
THEN Try (Complete ((OrRight THEN Auto)))
THEN (OrLeft THENA Auto)
THEN (D 0 THENA Auto)
THEN D (-9)
THEN InstConcl [⌈e⌉;⌈s⌉]⋅
THEN MaAuto)
) }
1
1. [Info] : Type
2. [B] : Type
3. [A] : Type
4. R : B ─→ B ─→ ℙ@i'
5. P : A ─→ B ─→ ℙ@i'
6. f : A ─→ B ─→ B@i
7. init : Id ─→ bag(B)@i
8. X : EClass(A)@i'
9. es : EO+(Info)@i'
10. e1 : E@i
11. e2 : E@i
12. v1 : B@i
13. v2 : B@i
14. ∀a:A. ∀s:B. Dec(P[a;s])@i
15. Trans(B;x,y.R[x;y])@i
16. ∀s1,s2:B. SqStable(R[s1;s2])@i
17. ∀a:A. ∀e:E. ∀s:B.
(e1 ≤loc e
⇒ (e <loc e2)
⇒ a ∈ X(e)
⇒ s ∈ Memory-class(f;init;X)(e)
⇒ ((P[a;s] ⇒ R[s;f a s]) ∧ ((¬P[a;s]) ⇒ (s = (f a s) ∈ B))))@i
18. single-valued-classrel(es;X;A)@i
19. single-valued-bag(init loc(e1);B)@i
20. ↑first(e1)
21. v1 ↓∈ init loc(e1)
22. ¬↑first(e2)
23. iterated_classrel(es;B;A;f;init;X;pred(e2);v2)
24. (e1 <loc e2)@i
25. ∃e:E. ∃a:A. ∃s:B. (e1 ≤loc e ∧ (e <loc e2) ∧ s ∈ Memory-class(f;init;X)(e) ∧ a ∈ X(e) ∧ P[a;s])@i
26. v : A
27. v ∈ X(e1)
28. iterated_classrel(es;B;A;f;init;X;e1;f v v1)
29. loc(pred(e2)) = loc(e2) ∈ Id
30. (pred(e2) < e2)
31. ∀e':E. (e' < e2) ⇒ ((e' = pred(e2) ∈ E) ∨ (e' < pred(e2))) supposing loc(e') = loc(e2) ∈ Id
32. (e1 < pred(e2))
33. (∃e:E
∃s:B
((e1 <loc e) ∧ e ≤loc pred(e2) ∧ iterated_classrel(es;B;A;f;init;X;pred(e);s) ∧ (∃a:A. (a ∈ X(e) ∧ P[a;s]))))
⇒ R[f v v1;v2]
34. P[v;v1] ⇒ R[v1;f v v1]
35. (¬P[v;v1]) ⇒ (v1 = (f v v1) ∈ B)
36. ¬(∃e:E
∃s:B
((e1 <loc e) ∧ e ≤loc pred(e2) ∧ iterated_classrel(es;B;A;f;init;X;pred(e);s) ∧ (∃v:A. (v ∈ X(e) ∧ P[v;s]))))
37. (f v v1) = v2 ∈ B
⊢ R[v1;v2]
Latex:
Latex:
1. [Info] : Type
2. [B] : Type
3. [A] : Type
4. R : B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}@i'
5. P : A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}@i'
6. f : A {}\mrightarrow{} B {}\mrightarrow{} B@i
7. init : Id {}\mrightarrow{} bag(B)@i
8. X : EClass(A)@i'
9. es : EO+(Info)@i'
10. e1 : E@i
11. e2 : E@i
12. v1 : B@i
13. v2 : B@i
14. \mforall{}a:A. \mforall{}s:B. Dec(P[a;s])@i
15. Trans(B;x,y.R[x;y])@i
16. \mforall{}s1,s2:B. SqStable(R[s1;s2])@i
17. \mforall{}a:A. \mforall{}e:E. \mforall{}s:B.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e2)
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} s \mmember{} Memory-class(f;init;X)(e)
{}\mRightarrow{} ((P[a;s] {}\mRightarrow{} R[s;f a s]) \mwedge{} ((\mneg{}P[a;s]) {}\mRightarrow{} (s = (f a s)))))@i
18. single-valued-classrel(es;X;A)@i
19. single-valued-bag(init loc(e1);B)@i
20. \muparrow{}first(e1)
21. v1 \mdownarrow{}\mmember{} init loc(e1)
22. \mneg{}\muparrow{}first(e2)
23. iterated\_classrel(es;B;A;f;init;X;pred(e2);v2)
24. (e1 <loc e2)@i
25. \mexists{}e:E
\mexists{}a:A. \mexists{}s:B. (e1 \mleq{}loc e \mwedge{} (e <loc e2) \mwedge{} s \mmember{} Memory-class(f;init;X)(e) \mwedge{} a \mmember{} X(e) \mwedge{} P[a;s])@i
26. v : A
27. v \mmember{} X(e1)
28. iterated\_classrel(es;B;A;f;init;X;e1;f v v1)
29. loc(pred(e2)) = loc(e2)
30. (pred(e2) < e2)
31. \mforall{}e':E. (e' < e2) {}\mRightarrow{} ((e' = pred(e2)) \mvee{} (e' < pred(e2))) supposing loc(e') = loc(e2)
32. (e1 < pred(e2))
33. (\mexists{}e:E
\mexists{}s:B
((e1 <loc e)
\mwedge{} e \mleq{}loc pred(e2)
\mwedge{} iterated\_classrel(es;B;A;f;init;X;pred(e);s)
\mwedge{} (\mexists{}a:A. (a \mmember{} X(e) \mwedge{} P[a;s]))))
{}\mRightarrow{} R[f v v1;v2]
34. (\mforall{}e:E. \mforall{}s:B.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc pred(e2)
{}\mRightarrow{} iterated\_classrel(es;B;A;f;init;X;pred(e);s)
{}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s])))))
{}\mRightarrow{} ((f v v1) = v2)
35. P[v;v1] {}\mRightarrow{} R[v1;f v v1]
36. (\mneg{}P[v;v1]) {}\mRightarrow{} (v1 = (f v v1))
37. \mneg{}(\mexists{}e:E
\mexists{}s:B
((e1 <loc e)
\mwedge{} e \mleq{}loc pred(e2)
\mwedge{} iterated\_classrel(es;B;A;f;init;X;pred(e);s)
\mwedge{} (\mexists{}v:A. (v \mmember{} X(e) \mwedge{} P[v;s]))))
\mvdash{} R[v1;v2]
By
Latex:
(D (-4)
THENA (Auto
THEN (InstHyp [\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}s\mkleeneclose{}] (-29)\mcdot{} THENA Auto)
THEN D (-1)
THEN Try (Complete ((OrRight THEN Auto)))
THEN (OrLeft THENA Auto)
THEN (D 0 THENA Auto)
THEN D (-9)
THEN InstConcl [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}s\mkleeneclose{}]\mcdot{}
THEN MaAuto)
)
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