Step
*
3
1
2
1
1
2
1
1
of Lemma
consensus-ts5-archive-invariant
1. [V] : Type
2. ∀v1,v2:V. Dec(v1 = v2 ∈ V)@i
3. v2 : V@i
4. v' : V@i
5. ¬(v2 = v' ∈ V)@i
6. A : Id List@i
7. W : {a:Id| (a ∈ A)} List List@i
8. x1 : ConsensusState@i
9. x2 : Knowledge(ConsensusState)@i
10. [%3] : ts-init(consensus-ts5(V;A;W)) (ts-rel(consensus-ts5(V;A;W))^*) <x1, x2>@i
11. y1 : ConsensusState@i
12. y2 : Knowledge(ConsensusState)@i
13. [%5] : ts-init(consensus-ts5(V;A;W)) (ts-rel(consensus-ts5(V;A;W))^*) <y1, y2>@i
14. ∀v:V. ∀b:{a:Id| (a ∈ A)} . ∀i:ℤ.
(state x1 may consider v in inning i) supposing ((Estimate(x1;b)(i) = v ∈ V) and (↑i ∈ dom(Estimate(x1;b))))@i
15. a : {a:Id| (a ∈ A)} @i
16. ∀b:{a:Id| (a ∈ A)}
((¬(b = a ∈ Id))
⇒ ((Inning(y1;b) = Inning(x1;b) ∈ ℤ)
∧ (Estimate(y1;b) = Estimate(x1;b) ∈ i:ℤ fp-> V)
∧ (Knowledge(y2;b) = Knowledge(x2;b) ∈ b:Id fp-> ℤ × (ℤ × V + Top))))@i
17. Inning(y1;a) = Inning(x1;a) ∈ ℤ@i
18. Knowledge(y2;a) = Knowledge(x2;a) ∈ b:Id fp-> ℤ × (ℤ × V + Top)@i
19. ¬(Inning(x1;a) ∈ fpf-domain(Estimate(x1;a)))@i
20. v : V@i
21. b : {a:Id| (a ∈ A)} @i
22. i : ℤ@i
23. ↑i ∈ dom(Estimate(y1;b))
24. Estimate(y1;b)(i) = v ∈ V
25. may consider v in inning Inning(x1;a) based on knowledge (Knowledge(x2;a))
26. Estimate(y1;a) = Estimate(x1;a) ⊕ Inning(x1;a) : v ∈ i:ℤ fp-> V
27. ¬(i = Inning(x1;a) ∈ ℤ)
28. ↑i ∈ dom(Estimate(x1;b))
29. Estimate(x1;b)(i) = v ∈ V
30. ws : {a:Id| (a ∈ A)} List
31. (ws ∈ W)
32. ∀b:{a:Id| (a ∈ A)} . ((b ∈ ws) ⇒ (i ≤ Inning(x1;b)))
33. ∀ws':{a:Id| (a ∈ A)} List. ((ws' ∈ W) ⇒ (¬in state x1, ws' blocks ws from archiving v in inning i))
34. (ws ∈ W)
35. ∀b:{a:Id| (a ∈ A)} . ((b ∈ ws) ⇒ (i ≤ Inning(y1;b)))
36. ws' : {a:Id| (a ∈ A)} List@i
37. (ws' ∈ W)@i
38. ¬in state x1, ws' blocks ws from archiving v in inning i
⊢ ¬in state y1, ws' blocks ws from archiving v in inning i
BY
{ (RepeatFor 2 (ParallelLast)
THEN ExRepD
THEN (InstConcl [⌜j⌝]⋅ THEN Auto)
THEN (InstConcl [⌜v3⌝]⋅ THENA Auto)
THEN D 0
THEN Try (Trivial)
THEN ParallelOp -3
THEN ParallelLast) }
1
1. V : Type
2. ∀v1,v2:V. Dec(v1 = v2 ∈ V)@i
3. v2 : V@i
4. v' : V@i
5. ¬(v2 = v' ∈ V)@i
6. A : Id List@i
7. W : {a:Id| (a ∈ A)} List List@i
8. x1 : ConsensusState@i
9. x2 : Knowledge(ConsensusState)@i
10. ts-init(consensus-ts5(V;A;W)) (ts-rel(consensus-ts5(V;A;W))^*) <x1, x2>@i
11. y1 : ConsensusState@i
12. y2 : Knowledge(ConsensusState)@i
13. ts-init(consensus-ts5(V;A;W)) (ts-rel(consensus-ts5(V;A;W))^*) <y1, y2>@i
14. ∀v:V. ∀b:{a:Id| (a ∈ A)} . ∀i:ℤ.
(state x1 may consider v in inning i) supposing ((Estimate(x1;b)(i) = v ∈ V) and (↑i ∈ dom(Estimate(x1;b))))@i
15. a : {a:Id| (a ∈ A)} @i
16. ∀b:{a:Id| (a ∈ A)}
((¬(b = a ∈ Id))
⇒ ((Inning(y1;b) = Inning(x1;b) ∈ ℤ)
∧ (Estimate(y1;b) = Estimate(x1;b) ∈ i:ℤ fp-> V)
∧ (Knowledge(y2;b) = Knowledge(x2;b) ∈ b:Id fp-> ℤ × (ℤ × V + Top))))@i
17. Inning(y1;a) = Inning(x1;a) ∈ ℤ@i
18. Knowledge(y2;a) = Knowledge(x2;a) ∈ b:Id fp-> ℤ × (ℤ × V + Top)@i
19. ¬(Inning(x1;a) ∈ fpf-domain(Estimate(x1;a)))@i
20. v : V@i
21. b : {a:Id| (a ∈ A)} @i
22. i : ℤ@i
23. ↑i ∈ dom(Estimate(y1;b))
24. Estimate(y1;b)(i) = v ∈ V
25. may consider v in inning Inning(x1;a) based on knowledge (Knowledge(x2;a))
26. Estimate(y1;a) = Estimate(x1;a) ⊕ Inning(x1;a) : v ∈ i:ℤ fp-> V
27. ¬(i = Inning(x1;a) ∈ ℤ)
28. ↑i ∈ dom(Estimate(x1;b))
29. Estimate(x1;b)(i) = v ∈ V
30. ws : {a:Id| (a ∈ A)} List
31. (ws ∈ W)
32. ∀b:{a:Id| (a ∈ A)} . ((b ∈ ws) ⇒ (i ≤ Inning(x1;b)))
33. ∀ws':{a:Id| (a ∈ A)} List. ((ws' ∈ W) ⇒ (¬in state x1, ws' blocks ws from archiving v in inning i))
34. (ws ∈ W)
35. ∀b:{a:Id| (a ∈ A)} . ((b ∈ ws) ⇒ (i ≤ Inning(y1;b)))
36. ws' : {a:Id| (a ∈ A)} List@i
37. (ws' ∈ W)@i
38. j : ℤ@i
39. 0 ≤ j@i
40. j < i@i
41. v3 : V@i
42. ¬(v3 = v ∈ V)@i
43. ∀b:{a:Id| (a ∈ A)}
(((b ∈ ws) ∧ (b ∈ ws'))
⇒ ((↑j ∈ dom(Estimate(y1;b)))
∧ (Estimate(y1;b)(j) = v3 ∈ V)
∧ (∀k:ℤ. (((j < k ∧ k < i) ∧ (↑k ∈ dom(Estimate(y1;b)))) ⇒ (Estimate(y1;b)(k) = v3 ∈ V)))))@i
44. 0 ≤ j
45. j < i
46. b1 : {a:Id| (a ∈ A)} @i
47. (b1 ∈ ws) ∧ (b1 ∈ ws')@i
48. (↑j ∈ dom(Estimate(y1;b1)))
∧ (Estimate(y1;b1)(j) = v3 ∈ V)
∧ (∀k:ℤ. (((j < k ∧ k < i) ∧ (↑k ∈ dom(Estimate(y1;b1)))) ⇒ (Estimate(y1;b1)(k) = v3 ∈ V)))
⊢ (↑j ∈ dom(Estimate(x1;b1)))
∧ (Estimate(x1;b1)(j) = v3 ∈ V)
∧ (∀k:ℤ. (((j < k ∧ k < i) ∧ (↑k ∈ dom(Estimate(x1;b1)))) ⇒ (Estimate(x1;b1)(k) = v3 ∈ V)))
Latex:
Latex:
1. [V] : Type
2. \mforall{}v1,v2:V. Dec(v1 = v2)@i
3. v2 : V@i
4. v' : V@i
5. \mneg{}(v2 = v')@i
6. A : Id List@i
7. W : \{a:Id| (a \mmember{} A)\} List List@i
8. x1 : ConsensusState@i
9. x2 : Knowledge(ConsensusState)@i
10. [\%3] : ts-init(consensus-ts5(V;A;W))
(ts-rel(consensus-ts5(V;A;W))\^{}*)
<x1, x2>@i
11. y1 : ConsensusState@i
12. y2 : Knowledge(ConsensusState)@i
13. [\%5] : ts-init(consensus-ts5(V;A;W))
(ts-rel(consensus-ts5(V;A;W))\^{}*)
<y1, y2>@i
14. \mforall{}v:V. \mforall{}b:\{a:Id| (a \mmember{} A)\} . \mforall{}i:\mBbbZ{}.
(state x1 may consider v in inning i) supposing
((Estimate(x1;b)(i) = v) and
(\muparrow{}i \mmember{} dom(Estimate(x1;b))))@i
15. a : \{a:Id| (a \mmember{} A)\} @i
16. \mforall{}b:\{a:Id| (a \mmember{} A)\}
((\mneg{}(b = a))
{}\mRightarrow{} ((Inning(y1;b) = Inning(x1;b))
\mwedge{} (Estimate(y1;b) = Estimate(x1;b))
\mwedge{} (Knowledge(y2;b) = Knowledge(x2;b))))@i
17. Inning(y1;a) = Inning(x1;a)@i
18. Knowledge(y2;a) = Knowledge(x2;a)@i
19. \mneg{}(Inning(x1;a) \mmember{} fpf-domain(Estimate(x1;a)))@i
20. v : V@i
21. b : \{a:Id| (a \mmember{} A)\} @i
22. i : \mBbbZ{}@i
23. \muparrow{}i \mmember{} dom(Estimate(y1;b))
24. Estimate(y1;b)(i) = v
25. may consider v in inning Inning(x1;a) based on knowledge (Knowledge(x2;a))
26. Estimate(y1;a) = Estimate(x1;a) \moplus{} Inning(x1;a) : v
27. \mneg{}(i = Inning(x1;a))
28. \muparrow{}i \mmember{} dom(Estimate(x1;b))
29. Estimate(x1;b)(i) = v
30. ws : \{a:Id| (a \mmember{} A)\} List
31. (ws \mmember{} W)
32. \mforall{}b:\{a:Id| (a \mmember{} A)\} . ((b \mmember{} ws) {}\mRightarrow{} (i \mleq{} Inning(x1;b)))
33. \mforall{}ws':\{a:Id| (a \mmember{} A)\} List
((ws' \mmember{} W) {}\mRightarrow{} (\mneg{}in state x1, ws' blocks ws from archiving v in inning i))
34. (ws \mmember{} W)
35. \mforall{}b:\{a:Id| (a \mmember{} A)\} . ((b \mmember{} ws) {}\mRightarrow{} (i \mleq{} Inning(y1;b)))
36. ws' : \{a:Id| (a \mmember{} A)\} List@i
37. (ws' \mmember{} W)@i
38. \mneg{}in state x1, ws' blocks ws from archiving v in inning i
\mvdash{} \mneg{}in state y1, ws' blocks ws from archiving v in inning i
By
Latex:
(RepeatFor 2 (ParallelLast)
THEN ExRepD
THEN (InstConcl [\mkleeneopen{}j\mkleeneclose{}]\mcdot{} THEN Auto)
THEN (InstConcl [\mkleeneopen{}v3\mkleeneclose{}]\mcdot{} THENA Auto)
THEN D 0
THEN Try (Trivial)
THEN ParallelOp -3
THEN ParallelLast)
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