Step
*
2
1
2
of Lemma
iterated_classrel_progress
1. Info : Type
2. A : Type
3. S : Type
4. init : Id ─→ bag(S)
5. f : A ─→ S ─→ S
6. X : EClass(A)
7. es : EO+(Info)@i'
8. R : S ─→ S ─→ ℙ@i'
9. P : A ─→ S ─→ ℙ@i'
10. e1 : E@i
11. e2 : E@i
12. ∀e':E
((e' < e2)
⇒ (∀v1,v2:S.
(single-valued-classrel(es;X;A)
⇒ (∀a:A. ∀s:S. Dec(P[a;s]))
⇒ (∀x,y:S. SqStable(R[x;y]))
⇒ Trans(S;x,y.R[x;y])
⇒ (∀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)))))
⇒ single-valued-bag(init loc(e1);S)
⇒ (e1 <loc e')
⇒ iterated_classrel(es;S;A;f;init;X;e1;v1)
⇒ 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))))))
13. v1 : S@i
14. v2 : S@i
15. single-valued-classrel(es;X;A)@i
16. ∀a:A. ∀s:S. Dec(P[a;s])@i
17. ∀x,y:S. SqStable(R[x;y])@i
18. Trans(S;x,y.R[x;y])@i
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. single-valued-bag(init loc(e1);S)@i
21. (e1 <loc e2)@i
22. iterated_classrel(es;S;A;f;init;X;e1;v1)@i
23. z : S@i
24. iterated_classrel(es;S;A;f;init;X;e1;z)
25. (∀a:A. (¬a ∈ X(e2))) ∧ (v2 = z ∈ S)@i
26. ¬↑first(e2)
27. e1 = pred(e2) ∈ E
28. v1 = z ∈ S
⊢ ((∃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])
∧ ((∀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])))))
⇒ (v1 = v2 ∈ S))
BY
{ (D 0
THEN (UnivCD THENA Auto)
THEN Auto
THEN Assert ⌈False⌉⋅
THEN Auto
THEN ExRepD
THEN (Assert ⌈e = e2 ∈ E⌉⋅ THENM (InstHyp [⌈a⌉] (-14)⋅ THEN Auto))
THEN D (-5)
THEN Auto
THEN (HypSubst' (-10) (-6) THENA Auto)
THEN (InstLemma `es-pred_property` [⌈es⌉;⌈e2⌉]⋅ THENA Auto)
THEN RepD
THEN (InstHyp [⌈e⌉] (-1)⋅ THENA Auto)
THEN D (-1)
THEN Auto) }
Latex:
1. Info : Type
2. A : Type
3. S : Type
4. init : Id {}\mrightarrow{} bag(S)
5. f : A {}\mrightarrow{} S {}\mrightarrow{} S
6. X : EClass(A)
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. e2 : E@i
12. \mforall{}e':E
((e' < e2)
{}\mRightarrow{} (\mforall{}v1,v2:S.
(single-valued-classrel(es;X;A)
{}\mRightarrow{} (\mforall{}a:A. \mforall{}s:S. Dec(P[a;s]))
{}\mRightarrow{} (\mforall{}x,y:S. SqStable(R[x;y]))
{}\mRightarrow{} Trans(S;x,y.R[x;y])
{}\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{} single-valued-bag(init loc(e1);S)
{}\mRightarrow{} (e1 <loc e')
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e1;v1)
{}\mRightarrow{} 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))))))
13. v1 : S@i
14. v2 : S@i
15. single-valued-classrel(es;X;A)@i
16. \mforall{}a:A. \mforall{}s:S. Dec(P[a;s])@i
17. \mforall{}x,y:S. SqStable(R[x;y])@i
18. Trans(S;x,y.R[x;y])@i
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. single-valued-bag(init loc(e1);S)@i
21. (e1 <loc e2)@i
22. iterated\_classrel(es;S;A;f;init;X;e1;v1)@i
23. z : S@i
24. iterated\_classrel(es;S;A;f;init;X;e1;z)
25. (\mforall{}a:A. (\mneg{}a \mmember{} X(e2))) \mwedge{} (v2 = z)@i
26. \mneg{}\muparrow{}first(e2)
27. e1 = pred(e2)
28. v1 = z
\mvdash{} ((\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])
\mwedge{} ((\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])))))
{}\mRightarrow{} (v1 = v2))
By
(D 0
THEN (UnivCD THENA Auto)
THEN Auto
THEN Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{}
THEN Auto
THEN ExRepD
THEN (Assert \mkleeneopen{}e = e2\mkleeneclose{}\mcdot{} THENM (InstHyp [\mkleeneopen{}a\mkleeneclose{}] (-14)\mcdot{} THEN Auto))
THEN D (-5)
THEN Auto
THEN (HypSubst' (-10) (-6) THENA Auto)
THEN (InstLemma `es-pred\_property` [\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{}]\mcdot{} THENA Auto)
THEN RepD
THEN (InstHyp [\mkleeneopen{}e\mkleeneclose{}] (-1)\mcdot{} THENA Auto)
THEN D (-1)
THEN Auto)
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