Step
*
1
1
1
2
of Lemma
Memory-class-trans
1. [Info] : Type
2. [B] : Type
3. [A] : Type
4. R : B ─→ B ─→ ℙ@i'
5. f : A ─→ B ─→ B@i
6. init : Id ─→ bag(B)@i
7. X : EClass(A)@i'
8. es : EO+(Info)@i'
9. e1 : E@i
10. e2 : E@i
11. v1 : B@i
12. v2 : B@i
13. Trans(B;x,y.R[x;y])@i
14. ∀s1,s2:B. SqStable(R[s1;s2])@i
15. ∀a:A. ∀e:E. (a ∈ X(e) ⇒ (∀s:B. R[s;f a s]))@i
16. single-valued-classrel(es;X;A)@i
17. single-valued-bag(init loc(e1);B)@i
18. ↑first(e1)
19. v1 ↓∈ init loc(e1)
20. ¬↑first(e2)
21. iterated_classrel(es;B;A;f;init;X;pred(e2);v2)
22. (e1 <loc e2)@i
23. ∃e:E. (e1 ≤loc e ∧ (e <loc e2) ∧ (∃a:A. a ∈ X(e)))@i
24. v : A
25. v ∈ X(e1)
26. iterated_classrel(es;B;A;f;init;X;e1;f v v1)
27. loc(pred(e2)) = loc(e2) ∈ Id
28. (pred(e2) < e2)
29. ∀e':E. (e' < e2) ⇒ ((e' = pred(e2) ∈ E) ∨ (e' < pred(e2))) supposing loc(e') = loc(e2) ∈ Id
30. (e1 < pred(e2))
⊢ R[v1;v2]
BY
{ (InstLemma `iterated_classrel_trans2`
[⌈Info⌉;⌈A⌉;⌈B⌉;⌈init⌉;⌈f⌉;⌈X⌉;⌈es⌉;⌈R⌉;⌈e1⌉;⌈pred(e2)⌉;⌈f v v1⌉;⌈v2⌉]⋅
THEN MaAuto
THEN (InstHyp [⌈v⌉;⌈e1⌉;⌈v1⌉] (-18)⋅ THENA Auto)
THEN (InstLemma `decidable__exists-classrel-between3` [⌈Info⌉;⌈A⌉;⌈X⌉;⌈es⌉;⌈e1⌉;⌈pred(e2)⌉]⋅ THENA Auto)
THEN D (-1)) }
1
1. [Info] : Type
2. [B] : Type
3. [A] : Type
4. R : B ─→ B ─→ ℙ@i'
5. f : A ─→ B ─→ B@i
6. init : Id ─→ bag(B)@i
7. X : EClass(A)@i'
8. es : EO+(Info)@i'
9. e1 : E@i
10. e2 : E@i
11. v1 : B@i
12. v2 : B@i
13. Trans(B;x,y.R[x;y])@i
14. ∀s1,s2:B. SqStable(R[s1;s2])@i
15. ∀a:A. ∀e:E. (a ∈ X(e) ⇒ (∀s:B. R[s;f a s]))@i
16. single-valued-classrel(es;X;A)@i
17. single-valued-bag(init loc(e1);B)@i
18. ↑first(e1)
19. v1 ↓∈ init loc(e1)
20. ¬↑first(e2)
21. iterated_classrel(es;B;A;f;init;X;pred(e2);v2)
22. (e1 <loc e2)@i
23. ∃e:E. (e1 ≤loc e ∧ (e <loc e2) ∧ (∃a:A. a ∈ X(e)))@i
24. v : A
25. v ∈ X(e1)
26. iterated_classrel(es;B;A;f;init;X;e1;f v v1)
27. loc(pred(e2)) = loc(e2) ∈ Id
28. (pred(e2) < e2)
29. ∀e':E. (e' < e2) ⇒ ((e' = pred(e2) ∈ E) ∨ (e' < pred(e2))) supposing loc(e') = loc(e2) ∈ Id
30. (e1 < pred(e2))
31. (∃e:E. ((e1 <loc e) ∧ e ≤loc pred(e2) ∧ (∃a:A. a ∈ X(e)))) ⇒ R[f v v1;v2]
32. (∀e:E. ((e1 <loc e) ⇒ e ≤loc pred(e2) ⇒ (∀a:A. (¬a ∈ X(e))))) ⇒ ((f v v1) = v2 ∈ B)
33. R[v1;f v v1]
34. ∃e:E. ((e1 <loc e) ∧ e ≤loc pred(e2) ∧ (↓∃v:A. v ∈ X(e)))
⊢ R[v1;v2]
2
1. [Info] : Type
2. [B] : Type
3. [A] : Type
4. R : B ─→ B ─→ ℙ@i'
5. f : A ─→ B ─→ B@i
6. init : Id ─→ bag(B)@i
7. X : EClass(A)@i'
8. es : EO+(Info)@i'
9. e1 : E@i
10. e2 : E@i
11. v1 : B@i
12. v2 : B@i
13. Trans(B;x,y.R[x;y])@i
14. ∀s1,s2:B. SqStable(R[s1;s2])@i
15. ∀a:A. ∀e:E. (a ∈ X(e) ⇒ (∀s:B. R[s;f a s]))@i
16. single-valued-classrel(es;X;A)@i
17. single-valued-bag(init loc(e1);B)@i
18. ↑first(e1)
19. v1 ↓∈ init loc(e1)
20. ¬↑first(e2)
21. iterated_classrel(es;B;A;f;init;X;pred(e2);v2)
22. (e1 <loc e2)@i
23. ∃e:E. (e1 ≤loc e ∧ (e <loc e2) ∧ (∃a:A. a ∈ X(e)))@i
24. v : A
25. v ∈ X(e1)
26. iterated_classrel(es;B;A;f;init;X;e1;f v v1)
27. loc(pred(e2)) = loc(e2) ∈ Id
28. (pred(e2) < e2)
29. ∀e':E. (e' < e2) ⇒ ((e' = pred(e2) ∈ E) ∨ (e' < pred(e2))) supposing loc(e') = loc(e2) ∈ Id
30. (e1 < pred(e2))
31. (∃e:E. ((e1 <loc e) ∧ e ≤loc pred(e2) ∧ (∃a:A. a ∈ X(e)))) ⇒ R[f v v1;v2]
32. (∀e:E. ((e1 <loc e) ⇒ e ≤loc pred(e2) ⇒ (∀a:A. (¬a ∈ X(e))))) ⇒ ((f v v1) = v2 ∈ B)
33. R[v1;f v v1]
34. ¬(∃e:E. ((e1 <loc e) ∧ e ≤loc pred(e2) ∧ (↓∃v:A. v ∈ X(e))))
⊢ R[v1;v2]
Latex:
Latex:
1. [Info] : Type
2. [B] : Type
3. [A] : Type
4. R : B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}@i'
5. f : A {}\mrightarrow{} B {}\mrightarrow{} B@i
6. init : Id {}\mrightarrow{} bag(B)@i
7. X : EClass(A)@i'
8. es : EO+(Info)@i'
9. e1 : E@i
10. e2 : E@i
11. v1 : B@i
12. v2 : B@i
13. Trans(B;x,y.R[x;y])@i
14. \mforall{}s1,s2:B. SqStable(R[s1;s2])@i
15. \mforall{}a:A. \mforall{}e:E. (a \mmember{} X(e) {}\mRightarrow{} (\mforall{}s:B. R[s;f a s]))@i
16. single-valued-classrel(es;X;A)@i
17. single-valued-bag(init loc(e1);B)@i
18. \muparrow{}first(e1)
19. v1 \mdownarrow{}\mmember{} init loc(e1)
20. \mneg{}\muparrow{}first(e2)
21. iterated\_classrel(es;B;A;f;init;X;pred(e2);v2)
22. (e1 <loc e2)@i
23. \mexists{}e:E. (e1 \mleq{}loc e \mwedge{} (e <loc e2) \mwedge{} (\mexists{}a:A. a \mmember{} X(e)))@i
24. v : A
25. v \mmember{} X(e1)
26. iterated\_classrel(es;B;A;f;init;X;e1;f v v1)
27. loc(pred(e2)) = loc(e2)
28. (pred(e2) < e2)
29. \mforall{}e':E. (e' < e2) {}\mRightarrow{} ((e' = pred(e2)) \mvee{} (e' < pred(e2))) supposing loc(e') = loc(e2)
30. (e1 < pred(e2))
\mvdash{} R[v1;v2]
By
Latex:
(InstLemma `iterated\_classrel\_trans2`
[\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{}e1\mkleeneclose{};\mkleeneopen{}pred(e2)\mkleeneclose{};\mkleeneopen{}f v v1\mkleeneclose{};\mkleeneopen{}v2\mkleeneclose{}]\mcdot{}
THEN MaAuto
THEN (InstHyp [\mkleeneopen{}v\mkleeneclose{};\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}v1\mkleeneclose{}] (-18)\mcdot{} THENA Auto)
THEN (InstLemma `decidable\_\_exists-classrel-between3` [\mkleeneopen{}Info\mkleeneclose{};\mkleeneopen{}A\mkleeneclose{};\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}pred(e2)\mkleeneclose{}]\mcdot{}
THENA Auto
)
THEN D (-1))
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