Step
*
1
of Lemma
three-cs-archive-invariant
1. [V] : Type
2. eq : EqDecider(V)@i
3. A : Id List@i
4. t : ℕ+@i
5. f : (V List) ─→ V@i
6. ∀vs:V List. (f vs ∈ vs) supposing ||vs|| ≥ 1 @i
7. ∀x,y:V. Dec(x = y ∈ V)
8. ∀s:{a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List). ∀v:V.
Dec((∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A))))
9. ts-reachable(three-consensus-ts(V;A;t;f)) ⊆r ({a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List))
10. s : ts-reachable(three-consensus-ts(V;A;t;f))@i
11. y : ts-reachable(three-consensus-ts(V;A;t;f))@i
12. ∀v:V. ∀a:{a:Id| (a ∈ A)} . ∀n:ℤ. ∀L:consensus-rcv(V;A) List.
(L ≤ s a
⇒ archive-condition(V;A;t;f;n;v;L)
⇒ (∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A))))@i
13. a1 : {a:Id| (a ∈ A)} @i
14. e : consensus-rcv(V;A)@i
15. ∀b:{a:Id| (a ∈ A)} . ∀i:ℕ. ∀v:V.
((e = Vote[b;i;v] ∈ consensus-rcv(V;A))
⇒ ((∃L:consensus-rcv(V;A) List. (L ≤ s b ∧ archive-condition(V;A;t;f;i;v;L))) ∧ (¬(e ∈ s a1))))@i
16. ∀b:{a:Id| (a ∈ A)} . ((¬(b = a1 ∈ Id)) ⇒ ((y b) = (s b) ∈ (consensus-rcv(V;A) List)))@i
17. (y a1) = ((s a1) @ [e]) ∈ (consensus-rcv(V;A) List)@i
18. v : V@i
19. a : {a:Id| (a ∈ A)} @i
20. n : ℤ@i
21. L : consensus-rcv(V;A) List@i
22. L ≤ y a@i
23. archive-condition(V;A;t;f;n;v;L)@i
⊢ (∃a∈A. (||y a|| ≥ 1 ) ∧ (hd(y a) = Init[v] ∈ consensus-rcv(V;A)))
BY
{ Assert ⌈(∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A)))
⇒ (∃a∈A. (||y a|| ≥ 1 ) ∧ (hd(y a) = Init[v] ∈ consensus-rcv(V;A)))⌉⋅ }
1
.....assertion.....
1. [V] : Type
2. eq : EqDecider(V)@i
3. A : Id List@i
4. t : ℕ+@i
5. f : (V List) ─→ V@i
6. ∀vs:V List. (f vs ∈ vs) supposing ||vs|| ≥ 1 @i
7. ∀x,y:V. Dec(x = y ∈ V)
8. ∀s:{a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List). ∀v:V.
Dec((∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A))))
9. ts-reachable(three-consensus-ts(V;A;t;f)) ⊆r ({a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List))
10. s : ts-reachable(three-consensus-ts(V;A;t;f))@i
11. y : ts-reachable(three-consensus-ts(V;A;t;f))@i
12. ∀v:V. ∀a:{a:Id| (a ∈ A)} . ∀n:ℤ. ∀L:consensus-rcv(V;A) List.
(L ≤ s a
⇒ archive-condition(V;A;t;f;n;v;L)
⇒ (∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A))))@i
13. a1 : {a:Id| (a ∈ A)} @i
14. e : consensus-rcv(V;A)@i
15. ∀b:{a:Id| (a ∈ A)} . ∀i:ℕ. ∀v:V.
((e = Vote[b;i;v] ∈ consensus-rcv(V;A))
⇒ ((∃L:consensus-rcv(V;A) List. (L ≤ s b ∧ archive-condition(V;A;t;f;i;v;L))) ∧ (¬(e ∈ s a1))))@i
16. ∀b:{a:Id| (a ∈ A)} . ((¬(b = a1 ∈ Id)) ⇒ ((y b) = (s b) ∈ (consensus-rcv(V;A) List)))@i
17. (y a1) = ((s a1) @ [e]) ∈ (consensus-rcv(V;A) List)@i
18. v : V@i
19. a : {a:Id| (a ∈ A)} @i
20. n : ℤ@i
21. L : consensus-rcv(V;A) List@i
22. L ≤ y a@i
23. archive-condition(V;A;t;f;n;v;L)@i
⊢ (∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A)))
⇒ (∃a∈A. (||y a|| ≥ 1 ) ∧ (hd(y a) = Init[v] ∈ consensus-rcv(V;A)))
2
1. [V] : Type
2. eq : EqDecider(V)@i
3. A : Id List@i
4. t : ℕ+@i
5. f : (V List) ─→ V@i
6. ∀vs:V List. (f vs ∈ vs) supposing ||vs|| ≥ 1 @i
7. ∀x,y:V. Dec(x = y ∈ V)
8. ∀s:{a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List). ∀v:V.
Dec((∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A))))
9. ts-reachable(three-consensus-ts(V;A;t;f)) ⊆r ({a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List))
10. s : ts-reachable(three-consensus-ts(V;A;t;f))@i
11. y : ts-reachable(three-consensus-ts(V;A;t;f))@i
12. ∀v:V. ∀a:{a:Id| (a ∈ A)} . ∀n:ℤ. ∀L:consensus-rcv(V;A) List.
(L ≤ s a
⇒ archive-condition(V;A;t;f;n;v;L)
⇒ (∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A))))@i
13. a1 : {a:Id| (a ∈ A)} @i
14. e : consensus-rcv(V;A)@i
15. ∀b:{a:Id| (a ∈ A)} . ∀i:ℕ. ∀v:V.
((e = Vote[b;i;v] ∈ consensus-rcv(V;A))
⇒ ((∃L:consensus-rcv(V;A) List. (L ≤ s b ∧ archive-condition(V;A;t;f;i;v;L))) ∧ (¬(e ∈ s a1))))@i
16. ∀b:{a:Id| (a ∈ A)} . ((¬(b = a1 ∈ Id)) ⇒ ((y b) = (s b) ∈ (consensus-rcv(V;A) List)))@i
17. (y a1) = ((s a1) @ [e]) ∈ (consensus-rcv(V;A) List)@i
18. v : V@i
19. a : {a:Id| (a ∈ A)} @i
20. n : ℤ@i
21. L : consensus-rcv(V;A) List@i
22. L ≤ y a@i
23. archive-condition(V;A;t;f;n;v;L)@i
24. (∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A)))
⇒ (∃a∈A. (||y a|| ≥ 1 ) ∧ (hd(y a) = Init[v] ∈ consensus-rcv(V;A)))
⊢ (∃a∈A. (||y a|| ≥ 1 ) ∧ (hd(y a) = Init[v] ∈ consensus-rcv(V;A)))
Latex:
1. [V] : Type
2. eq : EqDecider(V)@i
3. A : Id List@i
4. t : \mBbbN{}\msupplus{}@i
5. f : (V List) {}\mrightarrow{} V@i
6. \mforall{}vs:V List. (f vs \mmember{} vs) supposing ||vs|| \mgeq{} 1 @i
7. \mforall{}x,y:V. Dec(x = y)
8. \mforall{}s:\{a:Id| (a \mmember{} A)\} {}\mrightarrow{} (consensus-rcv(V;A) List). \mforall{}v:V.
Dec((\mexists{}a\mmember{}A. (||s a|| \mgeq{} 1 ) \mwedge{} (hd(s a) = Init[v])))
9. ts-reachable(three-consensus-ts(V;A;t;f)) \msubseteq{}r (\{a:Id| (a \mmember{} A)\} {}\mrightarrow{} (consensus-rcv(V;A) List))
10. s : ts-reachable(three-consensus-ts(V;A;t;f))@i
11. y : ts-reachable(three-consensus-ts(V;A;t;f))@i
12. \mforall{}v:V. \mforall{}a:\{a:Id| (a \mmember{} A)\} . \mforall{}n:\mBbbZ{}. \mforall{}L:consensus-rcv(V;A) List.
(L \mleq{} s a
{}\mRightarrow{} archive-condition(V;A;t;f;n;v;L)
{}\mRightarrow{} (\mexists{}a\mmember{}A. (||s a|| \mgeq{} 1 ) \mwedge{} (hd(s a) = Init[v])))@i
13. a1 : \{a:Id| (a \mmember{} A)\} @i
14. e : consensus-rcv(V;A)@i
15. \mforall{}b:\{a:Id| (a \mmember{} A)\} . \mforall{}i:\mBbbN{}. \mforall{}v:V.
((e = Vote[b;i;v])
{}\mRightarrow{} ((\mexists{}L:consensus-rcv(V;A) List. (L \mleq{} s b \mwedge{} archive-condition(V;A;t;f;i;v;L)))
\mwedge{} (\mneg{}(e \mmember{} s a1))))@i
16. \mforall{}b:\{a:Id| (a \mmember{} A)\} . ((\mneg{}(b = a1)) {}\mRightarrow{} ((y b) = (s b)))@i
17. (y a1) = ((s a1) @ [e])@i
18. v : V@i
19. a : \{a:Id| (a \mmember{} A)\} @i
20. n : \mBbbZ{}@i
21. L : consensus-rcv(V;A) List@i
22. L \mleq{} y a@i
23. archive-condition(V;A;t;f;n;v;L)@i
\mvdash{} (\mexists{}a\mmember{}A. (||y a|| \mgeq{} 1 ) \mwedge{} (hd(y a) = Init[v]))
By
Assert \mkleeneopen{}(\mexists{}a\mmember{}A. (||s a|| \mgeq{} 1 ) \mwedge{} (hd(s a) = Init[v]))
{}\mRightarrow{} (\mexists{}a\mmember{}A. (||y a|| \mgeq{} 1 ) \mwedge{} (hd(y a) = Init[v]))\mkleeneclose{}\mcdot{}
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