Proof of Nuprl Lemma : three-cs-archive-invariant

Step * 1 2 1 2 1 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-type(three-consensus-ts(V;A;t;f))@i
11. ts-init(three-consensus-ts(V;A;t;f)) (ts-rel(three-consensus-ts(V;A;t;f))^*) s@i
12. y : ts-reachable(three-consensus-ts(V;A;t;f))@i
13. ∀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
14. a1 : {a:Id| (a ∈ A)} @i
15. e : consensus-rcv(V;A)@i
16. ∀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
17. ∀b:{a:Id| (a ∈ A)} . ((¬(b = a1 ∈ Id)) ⇒ ((y b) = (s b) ∈ (consensus-rcv(V;A) List)))@i
18. (y a1) = ((s a1) @ [e]) ∈ (consensus-rcv(V;A) List)@i
19. v : V@i
20. a : {a:Id| (a ∈ A)} @i
21. n : ℤ@i
22. L : consensus-rcv(V;A) List@i
23. L ≤ y a@i
24. archive-condition(V;A;t;f;n;v;L)@i
25. (∃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)))
26. a = a1 ∈ Id
27. L ≤ (s a1) @ [e]
28. L = ((s a1) @ [e]) ∈ (consensus-rcv(V;A) List)
29. archive-condition(V;A;t;f;n;v;(s a1) @ [e])
⊢ (∃a∈A. (||y a|| ≥ 1 ) ∧ (hd(y a) = Init[v] ∈ consensus-rcv(V;A)))
BY
{ (DVar `e' THEN All (Fold `cs-initial-rcv`)) }

1
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-type(three-consensus-ts(V;A;t;f))@i
11. ts-init(three-consensus-ts(V;A;t;f)) (ts-rel(three-consensus-ts(V;A;t;f))^*) s@i
12. y : ts-reachable(three-consensus-ts(V;A;t;f))@i
13. ∀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
14. a1 : {a:Id| (a ∈ A)} @i
15. x : V@i
16. ∀b:{a:Id| (a ∈ A)} . ∀i:ℕ. ∀v:V.
      ((Init[x] = 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))) ∧ (¬(Init[x] ∈ s a1))))@i

17. ∀b:{a:Id| (a ∈ A)} . ((¬(b = a1 ∈ Id)) ⇒ ((y b) = (s b) ∈ (consensus-rcv(V;A) List)))@i
18. (y a1) = ((s a1) @ [Init[x]]) ∈ (consensus-rcv(V;A) List)@i
19. v : V@i
20. a : {a:Id| (a ∈ A)} @i
21. n : ℤ@i
22. L : consensus-rcv(V;A) List@i
23. L ≤ y a@i
24. archive-condition(V;A;t;f;n;v;L)@i
25. (∃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)))
26. a = a1 ∈ Id
27. L ≤ (s a1) @ [Init[x]]
28. L = ((s a1) @ [Init[x]]) ∈ (consensus-rcv(V;A) List)
29. archive-condition(V;A;t;f;n;v;(s a1) @ [Init[x]])
⊢ (∃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-type(three-consensus-ts(V;A;t;f))@i
11. ts-init(three-consensus-ts(V;A;t;f)) (ts-rel(three-consensus-ts(V;A;t;f))^*) s@i
12. y : ts-reachable(three-consensus-ts(V;A;t;f))@i
13. ∀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
14. a1 : {a:Id| (a ∈ A)} @i
15. y1 : {b:Id| (b ∈ A)}  × ℕ × V@i
16. ∀b:{a:Id| (a ∈ A)} . ∀i:ℕ. ∀v:V.
      (((inr y1 ) = 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))) ∧ (¬(inr y1  ∈ s a1))))@i

17. ∀b:{a:Id| (a ∈ A)} . ((¬(b = a1 ∈ Id)) ⇒ ((y b) = (s b) ∈ (consensus-rcv(V;A) List)))@i
18. (y a1) = ((s a1) @ [inr y1 ]) ∈ (consensus-rcv(V;A) List)@i
19. v : V@i
20. a : {a:Id| (a ∈ A)} @i
21. n : ℤ@i
22. L : consensus-rcv(V;A) List@i
23. L ≤ y a@i
24. archive-condition(V;A;t;f;n;v;L)@i
25. (∃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)))
26. a = a1 ∈ Id
27. L ≤ (s a1) @ [inr y1 ]
28. L = ((s a1) @ [inr y1 ]) ∈ (consensus-rcv(V;A) List)
29. archive-condition(V;A;t;f;n;v;(s a1) @ [inr y1 ])
⊢ (∃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-type(three-consensus-ts(V;A;t;f))@i 11. ts-init(three-consensus-ts(V;A;t;f)) (ts-rel(three-consensus-ts(V;A;t;f))\^{}*) s@i 12. y : ts-reachable(three-consensus-ts(V;A;t;f))@i 13. \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 14. a1 : \{a:Id| (a \mmember{} A)\} @i 15. e : consensus-rcv(V;A)@i 16. \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 17. \mforall{}b:\{a:Id| (a \mmember{} A)\} . ((\mneg{}(b = a1)) {}\mRightarrow{} ((y b) = (s b)))@i 18. (y a1) = ((s a1) @ [e])@i 19. v : V@i 20. a : \{a:Id| (a \mmember{} A)\} @i 21. n : \mBbbZ{}@i 22. L : consensus-rcv(V;A) List@i 23. L \mleq{} y a@i 24. archive-condition(V;A;t;f;n;v;L)@i 25. (\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])) 26. a = a1 27. L \mleq{} (s a1) @ [e] 28. L = ((s a1) @ [e]) 29. archive-condition(V;A;t;f;n;v;(s a1) @ [e]) \mvdash{} (\mexists{}a\mmember{}A. (||y a|| \mgeq{} 1 ) \mwedge{} (hd(y a) = Init[v])) By (DVar `e' THEN All (Fold `cs-initial-rcv`)) Home Index