Step
*
1
2
1
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. v ∈ X(e1))
27. iterated_classrel(es;B;A;f;init;X;e1;v1)
28. loc(pred(e2)) = loc(e2) ∈ Id
29. (pred(e2) < e2)
30. ∀e':E. (e' < e2) ⇒ ((e' = pred(e2) ∈ E) ∨ (e' < pred(e2))) supposing loc(e') = loc(e2) ∈ Id
31. (e1 < pred(e2))
⊢ R[v1;v2]
BY
{ (InstLemma `iterated_classrel_progress` [⌈Info⌉;⌈A⌉;⌈B⌉;⌈init⌉;⌈f⌉;⌈X⌉;⌈es⌉;⌈R⌉;⌈P⌉;⌈e1⌉;⌈pred(e2)⌉;⌈v1⌉;⌈v2⌉]⋅
THENA 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. v ∈ X(e1))
27. iterated_classrel(es;B;A;f;init;X;e1;v1)
28. loc(pred(e2)) = loc(e2) ∈ Id
29. (pred(e2) < e2)
30. ∀e':E. (e' < e2) ⇒ ((e' = pred(e2) ∈ E) ∨ (e' < pred(e2))) supposing loc(e') = loc(e2) ∈ Id
31. (e1 < pred(e2))
32. a : A@i
33. e : E@i
34. s : B@i
35. (e1 <loc e)@i
36. e ≤loc pred(e2) @i
37. a ∈ X(e)@i
38. iterated_classrel(es;B;A;f;init;X;pred(e);s)@i
39. P[a;s]@i
⊢ R[s;f a s]
2
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. v ∈ X(e1))
27. iterated_classrel(es;B;A;f;init;X;e1;v1)
28. loc(pred(e2)) = loc(e2) ∈ Id
29. (pred(e2) < e2)
30. ∀e':E. (e' < e2) ⇒ ((e' = pred(e2) ∈ E) ∨ (e' < pred(e2))) supposing loc(e') = loc(e2) ∈ Id
31. (e1 < pred(e2))
32. a : A@i
33. e : E@i
34. s : B@i
35. (e1 <loc e)@i
36. e ≤loc pred(e2) @i
37. a ∈ X(e)@i
38. iterated_classrel(es;B;A;f;init;X;pred(e);s)@i
39. P[a;s] ⇒ R[s;f a s]
40. ¬P[a;s]@i
⊢ s = (f a s) ∈ B
3
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. v ∈ X(e1))
27. iterated_classrel(es;B;A;f;init;X;e1;v1)
28. loc(pred(e2)) = loc(e2) ∈ Id
29. (pred(e2) < e2)
30. ∀e':E. (e' < e2) ⇒ ((e' = pred(e2) ∈ E) ∨ (e' < pred(e2))) supposing loc(e') = loc(e2) ∈ Id
31. (e1 < pred(e2))
32. ((∃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[v1;v2])
∧ ((∀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])))))
⇒ (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. \mneg{}(\mexists{}v:A. v \mmember{} X(e1))
27. iterated\_classrel(es;B;A;f;init;X;e1;v1)
28. loc(pred(e2)) = loc(e2)
29. (pred(e2) < e2)
30. \mforall{}e':E. (e' < e2) {}\mRightarrow{} ((e' = pred(e2)) \mvee{} (e' < pred(e2))) supposing loc(e') = loc(e2)
31. (e1 < pred(e2))
\mvdash{} R[v1;v2]
By
Latex:
(InstLemma `iterated\_classrel\_progress` [\mkleeneopen{}Info\mkleeneclose{};\mkleeneopen{}A\mkleeneclose{};\mkleeneopen{}B\mkleeneclose{};\mkleeneopen{}init\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}R\mkleeneclose{};\mkleeneopen{}P\mkleeneclose{};\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}pred(e2)\mkleeneclose{};
\mkleeneopen{}v1\mkleeneclose{};\mkleeneopen{}v2\mkleeneclose{}]\mcdot{}
THENA MaAuto
)
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