Step
*
3
1
2
1
of Lemma
loop-class-memory-fun-eq
1. Info : Type
2. B : Type
3. X : EClass(B ─→ B)
4. init : Id ─→ bag(B)
5. es : EO+(Info)
6. e : E
7. ¬↑pred(e) ∈b X
8. ¬↑first(e)
9. ∀l:Id. (1 ≤ #(init l))
10. ∀l:Id. single-valued-bag(init l;B)
11. single-valued-classrel(es;X;B ─→ B)
12. x : E@i
13. (x <loc e)@i
14. ↑0 <z #(eclass3(X;loop-class-memory(X;init)) es x)@i
15. ∀e'':E. ((x <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))@i
16. (last(λe'.0 <z #(eclass3(X;loop-class-memory(X;init)) es e')) e)
= (inl x)
∈ ((∃e':{E| ((e' <loc e)
∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'))
∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc e) ∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e')))})))@i
17. (x <loc pred(e))
18. x1 : E@i
19. (x1 <loc pred(e))@i
20. ↑0 <z #(eclass3(X;loop-class-memory(X;init)) es x1)@i
21. ∀e'':E. ((x1 <loc e'') ⇒ (e'' <loc pred(e)) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))@i
22. (last(λe'.0 <z #(eclass3(X;loop-class-memory(X;init)) es e')) pred(e))
= (inl x1)
∈ ((∃e':{E| ((e' <loc pred(e))
∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'))
∧ (∀e'':E
((e' <loc e'') ⇒ (e'' <loc pred(e)) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc pred(e)) ∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e')))})))@i
⊢ eclass3(X;loop-class-memory(X;init))(x) = eclass3(X;loop-class-memory(X;init))(x1) ∈ B
BY
{ Assert ⌈x = x1 ∈ E⌉⋅ }
1
.....assertion.....
1. Info : Type
2. B : Type
3. X : EClass(B ─→ B)
4. init : Id ─→ bag(B)
5. es : EO+(Info)
6. e : E
7. ¬↑pred(e) ∈b X
8. ¬↑first(e)
9. ∀l:Id. (1 ≤ #(init l))
10. ∀l:Id. single-valued-bag(init l;B)
11. single-valued-classrel(es;X;B ─→ B)
12. x : E@i
13. (x <loc e)@i
14. ↑0 <z #(eclass3(X;loop-class-memory(X;init)) es x)@i
15. ∀e'':E. ((x <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))@i
16. (last(λe'.0 <z #(eclass3(X;loop-class-memory(X;init)) es e')) e)
= (inl x)
∈ ((∃e':{E| ((e' <loc e)
∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'))
∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc e) ∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e')))})))@i
17. (x <loc pred(e))
18. x1 : E@i
19. (x1 <loc pred(e))@i
20. ↑0 <z #(eclass3(X;loop-class-memory(X;init)) es x1)@i
21. ∀e'':E. ((x1 <loc e'') ⇒ (e'' <loc pred(e)) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))@i
22. (last(λe'.0 <z #(eclass3(X;loop-class-memory(X;init)) es e')) pred(e))
= (inl x1)
∈ ((∃e':{E| ((e' <loc pred(e))
∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'))
∧ (∀e'':E
((e' <loc e'') ⇒ (e'' <loc pred(e)) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc pred(e)) ∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e')))})))@i
⊢ x = x1 ∈ E
2
1. Info : Type
2. B : Type
3. X : EClass(B ─→ B)
4. init : Id ─→ bag(B)
5. es : EO+(Info)
6. e : E
7. ¬↑pred(e) ∈b X
8. ¬↑first(e)
9. ∀l:Id. (1 ≤ #(init l))
10. ∀l:Id. single-valued-bag(init l;B)
11. single-valued-classrel(es;X;B ─→ B)
12. x : E@i
13. (x <loc e)@i
14. ↑0 <z #(eclass3(X;loop-class-memory(X;init)) es x)@i
15. ∀e'':E. ((x <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))@i
16. (last(λe'.0 <z #(eclass3(X;loop-class-memory(X;init)) es e')) e)
= (inl x)
∈ ((∃e':{E| ((e' <loc e)
∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'))
∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc e) ∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e')))})))@i
17. (x <loc pred(e))
18. x1 : E@i
19. (x1 <loc pred(e))@i
20. ↑0 <z #(eclass3(X;loop-class-memory(X;init)) es x1)@i
21. ∀e'':E. ((x1 <loc e'') ⇒ (e'' <loc pred(e)) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))@i
22. (last(λe'.0 <z #(eclass3(X;loop-class-memory(X;init)) es e')) pred(e))
= (inl x1)
∈ ((∃e':{E| ((e' <loc pred(e))
∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'))
∧ (∀e'':E
((e' <loc e'') ⇒ (e'' <loc pred(e)) ⇒ (¬↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc pred(e)) ∧ (↑0 <z #(eclass3(X;loop-class-memory(X;init)) es e')))})))@i
23. x = x1 ∈ E
⊢ eclass3(X;loop-class-memory(X;init))(x) = eclass3(X;loop-class-memory(X;init))(x1) ∈ B
Latex:
Latex:
1. Info : Type
2. B : Type
3. X : EClass(B {}\mrightarrow{} B)
4. init : Id {}\mrightarrow{} bag(B)
5. es : EO+(Info)
6. e : E
7. \mneg{}\muparrow{}pred(e) \mmember{}\msubb{} X
8. \mneg{}\muparrow{}first(e)
9. \mforall{}l:Id. (1 \mleq{} \#(init l))
10. \mforall{}l:Id. single-valued-bag(init l;B)
11. single-valued-classrel(es;X;B {}\mrightarrow{} B)
12. x : E@i
13. (x <loc e)@i
14. \muparrow{}0 <z \#(eclass3(X;loop-class-memory(X;init)) es x)@i
15. \mforall{}e'':E
((x <loc e'') {}\mRightarrow{} (e'' <loc e) {}\mRightarrow{} (\mneg{}\muparrow{}0 <z \#(eclass3(X;loop-class-memory(X;init)) es e'')))@i
16. (last(\mlambda{}e'.0 <z \#(eclass3(X;loop-class-memory(X;init)) es e')) e) = (inl x)@i
17. (x <loc pred(e))
18. x1 : E@i
19. (x1 <loc pred(e))@i
20. \muparrow{}0 <z \#(eclass3(X;loop-class-memory(X;init)) es x1)@i
21. \mforall{}e'':E
((x1 <loc e'')
{}\mRightarrow{} (e'' <loc pred(e))
{}\mRightarrow{} (\mneg{}\muparrow{}0 <z \#(eclass3(X;loop-class-memory(X;init)) es e'')))@i
22. (last(\mlambda{}e'.0 <z \#(eclass3(X;loop-class-memory(X;init)) es e')) pred(e)) = (inl x1)@i
\mvdash{} eclass3(X;loop-class-memory(X;init))(x) = eclass3(X;loop-class-memory(X;init))(x1)
By
Latex:
Assert \mkleeneopen{}x = x1\mkleeneclose{}\mcdot{}
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