Step
*
2
2
of Lemma
iterated_classrel_mem
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 : A ─→ S ─→ S ─→ ℙ@i'
9. e1 : E@i
10. e2 : E@i
11. ∀e':E
((e' < e2)
⇒ (∀v1,v2:S. ∀a:A.
(single-valued-classrel(es;X;A)
⇒ single-valued-bag(init loc(e1);S)
⇒ (∀s1,s2:S. ∀a:A. SqStable(R[a;s1;s2]))
⇒ (∀a1,a2:A. ∀s:S. ∀e,e2:E.
(e1 ≤loc e
⇒ (e <loc e2)
⇒ e2 ≤loc e'
⇒ a1 ∈ X(e)
⇒ iterated_classrel(es;S;A;f;init;X;e;s)
⇒ a2 ∈ X(e2)
⇒ R[a1;s;f a2 s]))
⇒ (∀a1,a2:A. ∀s1,s2:S. ∀e1,e2:E.
((e1 <loc e2)
⇒ a1 ∈ X(e1)
⇒ iterated_classrel(es;S;A;f;init;X;e1;s1)
⇒ a2 ∈ X(e2)
⇒ iterated_classrel(es;S;A;f;init;X;pred(e2);s2)
⇒ R[a1;s1;s2]
⇒ R[a1;s1;f a2 s2]))
⇒ (e1 <loc e')
⇒ a ∈ X(e1)
⇒ iterated_classrel(es;S;A;f;init;X;e1;v1)
⇒ iterated_classrel(es;S;A;f;init;X;e';v2)
⇒ (((∃e:E. ((e1 <loc e) ∧ e ≤loc e' ∧ (∃a:A. a ∈ X(e)))) ⇒ R[a;v1;v2])
∧ ((∀e:E. ((e1 <loc e) ⇒ e ≤loc e' ⇒ (∀a:A. (¬a ∈ X(e))))) ⇒ (v1 = v2 ∈ S))))))
12. v1 : S@i
13. v2 : S@i
14. a : A@i
15. single-valued-classrel(es;X;A)@i
16. single-valued-bag(init loc(e1);S)@i
17. ∀s1,s2:S. ∀a:A. SqStable(R[a;s1;s2])@i
18. ∀a1,a2:A. ∀s:S. ∀e,e':E.
(e1 ≤loc e
⇒ (e <loc e')
⇒ e' ≤loc e2
⇒ a1 ∈ X(e)
⇒ iterated_classrel(es;S;A;f;init;X;e;s)
⇒ a2 ∈ X(e')
⇒ R[a1;s;f a2 s])@i
19. ∀a1,a2:A. ∀s1,s2:S. ∀e1,e2:E.
((e1 <loc e2)
⇒ a1 ∈ X(e1)
⇒ iterated_classrel(es;S;A;f;init;X;e1;s1)
⇒ a2 ∈ X(e2)
⇒ iterated_classrel(es;S;A;f;init;X;pred(e2);s2)
⇒ R[a1;s1;s2]
⇒ R[a1;s1;f a2 s2])@i
20. (e1 <loc e2)@i
21. a ∈ X(e1)@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;pred(e2);z)@i
25. (∃a:A. (a ∈ X(e2) ∧ (v2 = (f a z) ∈ S))) ∨ ((∀a:A. (¬a ∈ X(e2))) ∧ (v2 = z ∈ S))@i
26. ¬↑first(e2)
27. (pred(e2) <loc e1)
⊢ ((∃e:E. ((e1 <loc e) ∧ e ≤loc e2 ∧ (∃a:A. a ∈ X(e)))) ⇒ R[a;v1;v2])
∧ ((∀e:E. ((e1 <loc e) ⇒ e ≤loc e2 ⇒ (∀a:A. (¬a ∈ X(e))))) ⇒ (v1 = v2 ∈ S))
BY
{ ((Assert ⌈False⌉⋅ THEN Auto)
THEN (InstLemma `es-pred_property` [⌈es⌉;⌈e2⌉]⋅ THEN Auto)
THEN InstHyp [⌈e1⌉] (-1)⋅
THEN 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 : A {}\mrightarrow{} S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}@i'
9. e1 : E@i
10. e2 : E@i
11. \mforall{}e':E
((e' < e2)
{}\mRightarrow{} (\mforall{}v1,v2:S. \mforall{}a:A.
(single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e1);S)
{}\mRightarrow{} (\mforall{}s1,s2:S. \mforall{}a:A. SqStable(R[a;s1;s2]))
{}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s:S. \mforall{}e,e2:E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e2)
{}\mRightarrow{} e2 \mleq{}loc e'
{}\mRightarrow{} a1 \mmember{} X(e)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e;s)
{}\mRightarrow{} a2 \mmember{} X(e2)
{}\mRightarrow{} R[a1;s;f a2 s]))
{}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e1,e2:E.
((e1 <loc e2)
{}\mRightarrow{} a1 \mmember{} X(e1)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e1;s1)
{}\mRightarrow{} a2 \mmember{} X(e2)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;pred(e2);s2)
{}\mRightarrow{} R[a1;s1;s2]
{}\mRightarrow{} R[a1;s1;f a2 s2]))
{}\mRightarrow{} (e1 <loc e')
{}\mRightarrow{} a \mmember{} X(e1)
{}\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. ((e1 <loc e) \mwedge{} e \mleq{}loc e' \mwedge{} (\mexists{}a:A. a \mmember{} X(e)))) {}\mRightarrow{} R[a;v1;v2])
\mwedge{} ((\mforall{}e:E. ((e1 <loc e) {}\mRightarrow{} e \mleq{}loc e' {}\mRightarrow{} (\mforall{}a:A. (\mneg{}a \mmember{} X(e))))) {}\mRightarrow{} (v1 = v2))))))
12. v1 : S@i
13. v2 : S@i
14. a : A@i
15. single-valued-classrel(es;X;A)@i
16. single-valued-bag(init loc(e1);S)@i
17. \mforall{}s1,s2:S. \mforall{}a:A. SqStable(R[a;s1;s2])@i
18. \mforall{}a1,a2:A. \mforall{}s:S. \mforall{}e,e':E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e')
{}\mRightarrow{} e' \mleq{}loc e2
{}\mRightarrow{} a1 \mmember{} X(e)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e;s)
{}\mRightarrow{} a2 \mmember{} X(e')
{}\mRightarrow{} R[a1;s;f a2 s])@i
19. \mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e1,e2:E.
((e1 <loc e2)
{}\mRightarrow{} a1 \mmember{} X(e1)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e1;s1)
{}\mRightarrow{} a2 \mmember{} X(e2)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;pred(e2);s2)
{}\mRightarrow{} R[a1;s1;s2]
{}\mRightarrow{} R[a1;s1;f a2 s2])@i
20. (e1 <loc e2)@i
21. a \mmember{} X(e1)@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;pred(e2);z)@i
25. (\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
26. \mneg{}\muparrow{}first(e2)
27. (pred(e2) <loc e1)
\mvdash{} ((\mexists{}e:E. ((e1 <loc e) \mwedge{} e \mleq{}loc e2 \mwedge{} (\mexists{}a:A. a \mmember{} X(e)))) {}\mRightarrow{} R[a;v1;v2])
\mwedge{} ((\mforall{}e:E. ((e1 <loc e) {}\mRightarrow{} e \mleq{}loc e2 {}\mRightarrow{} (\mforall{}a:A. (\mneg{}a \mmember{} X(e))))) {}\mRightarrow{} (v1 = v2))
By
((Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{} THEN Auto)
THEN (InstLemma `es-pred\_property` [\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{}]\mcdot{} THEN Auto)
THEN InstHyp [\mkleeneopen{}e1\mkleeneclose{}] (-1)\mcdot{}
THEN Auto
THEN D (-1)
THEN Auto)
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