Step
*
1
2
of Lemma
iterated-classrel-progress
1. Info : Type
2. A : Type
3. S : Type
4. init : Id ─→ bag(S)@i
5. f : A ─→ S ─→ S@i
6. X : EClass(A)@i'
7. es : EO+(Info)@i'
8. R : S ─→ S ─→ ℙ@i'
9. P : A ─→ S ─→ ℙ@i'
10. e1 : E@i
11. v1 : S@i
12. single-valued-classrel(es;X;A)@i
13. single-valued-bag(init loc(e1);S)@i
14. ∀a:A. ∀s:S. Dec(P[a;s])@i
15. Trans(S;x,y.R[x;y])@i
16. iterated-classrel(es;S;A;f;init;X;e1;v1)@i
17. e2 : E@i
18. ∀e':E
((e' < e2)
⇒ (∀a:A. ∀e:E. ∀s:S.
((e1 <loc e)
⇒ e ≤loc e'
⇒ a ∈ X(e)
⇒ iterated-classrel(es;S;A;f;init;X;pred(e);s)
⇒ ((P[a;s] ⇒ R[s;f a s]) ∧ ((¬P[a;s]) ⇒ (s = (f a s) ∈ S)))))
⇒ (e1 <loc e')
⇒ (∀v2:S
(iterated-classrel(es;S;A;f;init;X;e';v2)
⇒ (((∃e:E
∃s:S
((e1 <loc e)
∧ e ≤loc e'
∧ iterated-classrel(es;S;A;f;init;X;pred(e);s)
∧ (∃a:A. (a ∈ X(e) ∧ P[a;s]))))
⇒ R[v1;v2])
∧ ((∀e:E. ∀s:S.
((e1 <loc e)
⇒ e ≤loc e'
⇒ iterated-classrel(es;S;A;f;init;X;pred(e);s)
⇒ (∀a:A. ((¬a ∈ X(e)) ∨ (¬P[a;s])))))
⇒ (v1 = v2 ∈ S))))))
19. ∀a:A. ∀e:E. ∀s:S.
((e1 <loc e)
⇒ e ≤loc e2
⇒ a ∈ X(e)
⇒ iterated-classrel(es;S;A;f;init;X;pred(e);s)
⇒ ((P[a;s] ⇒ R[s;f a s]) ∧ ((¬P[a;s]) ⇒ (s = (f a s) ∈ S))))@i
20. (e1 <loc e2)@i
21. v2 : S@i
22. z : S@i
23. iterated-classrel(es;S;A;f;init;X;pred(e2);z)@i
24. (∃a:A. (a ∈ X(e2) ∧ (v2 = (f a z) ∈ S))) ∨ ((∀a:A. (¬a ∈ X(e2))) ∧ (v2 = z ∈ S))@i
25. ¬↑first(e2)
26. (e1 <loc pred(e2))
27. (∃e:E
∃s:S
((e1 <loc e) ∧ e ≤loc pred(e2) ∧ iterated-classrel(es;S;A;f;init;X;pred(e);s) ∧ (∃a:A. (a ∈ X(e) ∧ P[a;s]))))
⇒ R[v1;z]
28. (∀e:E. ∀s:S.
((e1 <loc e)
⇒ e ≤loc pred(e2)
⇒ iterated-classrel(es;S;A;f;init;X;pred(e);s)
⇒ (∀a:A. ((¬a ∈ X(e)) ∨ (¬P[a;s])))))
⇒ (v1 = z ∈ S)
29. (∃e:E
∃s:S. ((e1 <loc e) ∧ e ≤loc e2 ∧ iterated-classrel(es;S;A;f;init;X;pred(e);s) ∧ (∃a:A. (a ∈ X(e) ∧ P[a;s]))))
⇒ R[v1;v2]
30. ∀e:E. ∀s:S.
((e1 <loc e) ⇒ e ≤loc e2 ⇒ iterated-classrel(es;S;A;f;init;X;pred(e);s) ⇒ (∀a:A. ((¬a ∈ X(e)) ∨ (¬P[a;s]))))@i
⊢ v1 = v2 ∈ S
BY
{ ((D (-3) THENA (RepeatFor 4 (ParallelLast) THEN Auto))
THEN D (-7)
THEN Auto
THEN ExRepD
THEN (InstHyp [⌈e2⌉;⌈z⌉;⌈a⌉] (-2)⋅ THENA Auto)
THEN D (-1)
THEN Auto
THEN (InstHyp [⌈a⌉;⌈e2⌉;⌈z⌉] (-15)⋅ THENA Auto)
THEN RepD
THEN D (-1)
THEN Auto) }
Latex:
1. Info : Type
2. A : Type
3. S : Type
4. init : Id {}\mrightarrow{} bag(S)@i
5. f : A {}\mrightarrow{} S {}\mrightarrow{} S@i
6. X : EClass(A)@i'
7. es : EO+(Info)@i'
8. R : S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}@i'
9. P : A {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}@i'
10. e1 : E@i
11. v1 : S@i
12. single-valued-classrel(es;X;A)@i
13. single-valued-bag(init loc(e1);S)@i
14. \mforall{}a:A. \mforall{}s:S. Dec(P[a;s])@i
15. Trans(S;x,y.R[x;y])@i
16. iterated-classrel(es;S;A;f;init;X;e1;v1)@i
17. e2 : E@i
18. \mforall{}e':E
((e' < e2)
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E. \mforall{}s:S.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e'
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e);s)
{}\mRightarrow{} ((P[a;s] {}\mRightarrow{} R[s;f a s]) \mwedge{} ((\mneg{}P[a;s]) {}\mRightarrow{} (s = (f a s))))))
{}\mRightarrow{} (e1 <loc e')
{}\mRightarrow{} (\mforall{}v2:S
(iterated-classrel(es;S;A;f;init;X;e';v2)
{}\mRightarrow{} (((\mexists{}e:E
\mexists{}s:S
((e1 <loc e)
\mwedge{} e \mleq{}loc e'
\mwedge{} iterated-classrel(es;S;A;f;init;X;pred(e);s)
\mwedge{} (\mexists{}a:A. (a \mmember{} X(e) \mwedge{} P[a;s]))))
{}\mRightarrow{} R[v1;v2])
\mwedge{} ((\mforall{}e:E. \mforall{}s:S.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e'
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e);s)
{}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s])))))
{}\mRightarrow{} (v1 = v2))))))
19. \mforall{}a:A. \mforall{}e:E. \mforall{}s:S.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e);s)
{}\mRightarrow{} ((P[a;s] {}\mRightarrow{} R[s;f a s]) \mwedge{} ((\mneg{}P[a;s]) {}\mRightarrow{} (s = (f a s)))))@i
20. (e1 <loc e2)@i
21. v2 : S@i
22. z : S@i
23. iterated-classrel(es;S;A;f;init;X;pred(e2);z)@i
24. (\mexists{}a:A. (a \mmember{} X(e2) \mwedge{} (v2 = (f a z)))) \mvee{} ((\mforall{}a:A. (\mneg{}a \mmember{} X(e2))) \mwedge{} (v2 = z))@i
25. \mneg{}\muparrow{}first(e2)
26. (e1 <loc pred(e2))
27. (\mexists{}e:E
\mexists{}s:S
((e1 <loc e)
\mwedge{} e \mleq{}loc pred(e2)
\mwedge{} iterated-classrel(es;S;A;f;init;X;pred(e);s)
\mwedge{} (\mexists{}a:A. (a \mmember{} X(e) \mwedge{} P[a;s]))))
{}\mRightarrow{} R[v1;z]
28. (\mforall{}e:E. \mforall{}s:S.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc pred(e2)
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e);s)
{}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s])))))
{}\mRightarrow{} (v1 = z)
29. (\mexists{}e:E
\mexists{}s:S
((e1 <loc e)
\mwedge{} e \mleq{}loc e2
\mwedge{} iterated-classrel(es;S;A;f;init;X;pred(e);s)
\mwedge{} (\mexists{}a:A. (a \mmember{} X(e) \mwedge{} P[a;s]))))
{}\mRightarrow{} R[v1;v2]
30. \mforall{}e:E. \mforall{}s:S.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e);s)
{}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s]))))@i
\mvdash{} v1 = v2
By
((D (-3) THENA (RepeatFor 4 (ParallelLast) THEN Auto))
THEN D (-7)
THEN Auto
THEN ExRepD
THEN (InstHyp [\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{};\mkleeneopen{}a\mkleeneclose{}] (-2)\mcdot{} THENA Auto)
THEN D (-1)
THEN Auto
THEN (InstHyp [\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{}] (-15)\mcdot{} THENA Auto)
THEN RepD
THEN D (-1)
THEN Auto)
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