Step
*
1
of Lemma
three-cs-archive-condition
1. V : Type@i'
2. A : Id List@i
3. t : ℕ+@i
4. f : (V List) ─→ V@i
5. s : {a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List)@i
6. ts-init(three-consensus-ts(V;A;t;f)) (ts-rel(three-consensus-ts(V;A;t;f))^*) s@i
7. y : {a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List)@i
8. ∀v:V. ∀a:{a:Id| (a ∈ A)} . ∀n:ℕ+. ∀L:consensus-rcv(V;A) List.
(L ≤ s a
⇒ archive-condition(V;A;t;f;n;v;L)
⇒ (∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||
∃L:consensus-rcv(V;A) List. (L ≤ s as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L))))))))@i
9. ∃a:{a:Id| (a ∈ A)}
∃e:consensus-rcv(V;A)
((∀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 a)))))
∧ (∀b:{a:Id| (a ∈ A)} . ((¬(b = a ∈ Id)) ⇒ ((y b) = (s b) ∈ (consensus-rcv(V;A) List))))
∧ ((y a) = ((s a) @ [e]) ∈ (consensus-rcv(V;A) List)))@i
10. v : V@i
11. a : {a:Id| (a ∈ A)} @i
12. n : ℕ+@i
13. L : consensus-rcv(V;A) List@i
14. L ≤ y a@i
15. archive-condition(V;A;t;f;n;v;L)@i
⊢ ∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||. ∃L:consensus-rcv(V;A) List. (L ≤ y as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L))))))
BY
{ Assert ⌈(∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||. ∃L:consensus-rcv(V;A) List. (L ≤ s as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L)))))))
⇒ (∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||
∃L:consensus-rcv(V;A) List. (L ≤ y as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L)))))))⌉⋅ }
1
.....assertion.....
1. V : Type@i'
2. A : Id List@i
3. t : ℕ+@i
4. f : (V List) ─→ V@i
5. s : {a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List)@i
6. ts-init(three-consensus-ts(V;A;t;f)) (ts-rel(three-consensus-ts(V;A;t;f))^*) s@i
7. y : {a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List)@i
8. ∀v:V. ∀a:{a:Id| (a ∈ A)} . ∀n:ℕ+. ∀L:consensus-rcv(V;A) List.
(L ≤ s a
⇒ archive-condition(V;A;t;f;n;v;L)
⇒ (∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||
∃L:consensus-rcv(V;A) List. (L ≤ s as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L))))))))@i
9. ∃a:{a:Id| (a ∈ A)}
∃e:consensus-rcv(V;A)
((∀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 a)))))
∧ (∀b:{a:Id| (a ∈ A)} . ((¬(b = a ∈ Id)) ⇒ ((y b) = (s b) ∈ (consensus-rcv(V;A) List))))
∧ ((y a) = ((s a) @ [e]) ∈ (consensus-rcv(V;A) List)))@i
10. v : V@i
11. a : {a:Id| (a ∈ A)} @i
12. n : ℕ+@i
13. L : consensus-rcv(V;A) List@i
14. L ≤ y a@i
15. archive-condition(V;A;t;f;n;v;L)@i
⊢ (∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||. ∃L:consensus-rcv(V;A) List. (L ≤ s as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L)))))))
⇒ (∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||. ∃L:consensus-rcv(V;A) List. (L ≤ y as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L)))))))
2
1. V : Type@i'
2. A : Id List@i
3. t : ℕ+@i
4. f : (V List) ─→ V@i
5. s : {a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List)@i
6. ts-init(three-consensus-ts(V;A;t;f)) (ts-rel(three-consensus-ts(V;A;t;f))^*) s@i
7. y : {a:Id| (a ∈ A)} ─→ (consensus-rcv(V;A) List)@i
8. ∀v:V. ∀a:{a:Id| (a ∈ A)} . ∀n:ℕ+. ∀L:consensus-rcv(V;A) List.
(L ≤ s a
⇒ archive-condition(V;A;t;f;n;v;L)
⇒ (∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||
∃L:consensus-rcv(V;A) List. (L ≤ s as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L))))))))@i
9. ∃a:{a:Id| (a ∈ A)}
∃e:consensus-rcv(V;A)
((∀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 a)))))
∧ (∀b:{a:Id| (a ∈ A)} . ((¬(b = a ∈ Id)) ⇒ ((y b) = (s b) ∈ (consensus-rcv(V;A) List))))
∧ ((y a) = ((s a) @ [e]) ∈ (consensus-rcv(V;A) List)))@i
10. v : V@i
11. a : {a:Id| (a ∈ A)} @i
12. n : ℕ+@i
13. L : consensus-rcv(V;A) List@i
14. L ≤ y a@i
15. archive-condition(V;A;t;f;n;v;L)@i
16. (∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||. ∃L:consensus-rcv(V;A) List. (L ≤ s as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L)))))))
⇒ (∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||. ∃L:consensus-rcv(V;A) List. (L ≤ y as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L)))))))
⊢ ∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||. ∃L:consensus-rcv(V;A) List. (L ≤ y as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L))))))
Latex:
1. V : Type@i'
2. A : Id List@i
3. t : \mBbbN{}\msupplus{}@i
4. f : (V List) {}\mrightarrow{} V@i
5. s : \{a:Id| (a \mmember{} A)\} {}\mrightarrow{} (consensus-rcv(V;A) List)@i
6. ts-init(three-consensus-ts(V;A;t;f))
(ts-rel(three-consensus-ts(V;A;t;f))\^{}*)
s@i
7. y : \{a:Id| (a \mmember{} A)\} {}\mrightarrow{} (consensus-rcv(V;A) List)@i
8. \mforall{}v:V. \mforall{}a:\{a:Id| (a \mmember{} A)\} . \mforall{}n:\mBbbN{}\msupplus{}. \mforall{}L:consensus-rcv(V;A) List.
(L \mleq{} s a
{}\mRightarrow{} archive-condition(V;A;t;f;n;v;L)
{}\mRightarrow{} (\mexists{}as:\{a:Id| (a \mmember{} A)\} List
(no\_repeats(\{a:Id| (a \mmember{} A)\} ;as)
\mwedge{} (||as|| = ((2 * t) + 1))
\mwedge{} (\mexists{}vs:V List
((||vs|| = ||as||)
\mwedge{} (v = (f vs))
\mwedge{} (\mforall{}i:\mBbbN{}||as||
\mexists{}L:consensus-rcv(V;A) List
(L \mleq{} s as[i] \mwedge{} archive-condition(V;A;t;f;n - 1;vs[i];L))))))))@i
9. \mexists{}a:\{a:Id| (a \mmember{} A)\}
\mexists{}e:consensus-rcv(V;A)
((\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 a)))))
\mwedge{} (\mforall{}b:\{a:Id| (a \mmember{} A)\} . ((\mneg{}(b = a)) {}\mRightarrow{} ((y b) = (s b))))
\mwedge{} ((y a) = ((s a) @ [e])))@i
10. v : V@i
11. a : \{a:Id| (a \mmember{} A)\} @i
12. n : \mBbbN{}\msupplus{}@i
13. L : consensus-rcv(V;A) List@i
14. L \mleq{} y a@i
15. archive-condition(V;A;t;f;n;v;L)@i
\mvdash{} \mexists{}as:\{a:Id| (a \mmember{} A)\} List
(no\_repeats(\{a:Id| (a \mmember{} A)\} ;as)
\mwedge{} (||as|| = ((2 * t) + 1))
\mwedge{} (\mexists{}vs:V List
((||vs|| = ||as||)
\mwedge{} (v = (f vs))
\mwedge{} (\mforall{}i:\mBbbN{}||as||
\mexists{}L:consensus-rcv(V;A) List. (L \mleq{} y as[i] \mwedge{} archive-condition(V;A;t;f;n - 1;vs[i];L))))))
By
Assert \mkleeneopen{}(\mexists{}as:\{a:Id| (a \mmember{} A)\} List
(no\_repeats(\{a:Id| (a \mmember{} A)\} ;as)
\mwedge{} (||as|| = ((2 * t) + 1))
\mwedge{} (\mexists{}vs:V List
((||vs|| = ||as||)
\mwedge{} (v = (f vs))
\mwedge{} (\mforall{}i:\mBbbN{}||as||
\mexists{}L:consensus-rcv(V;A) List
(L \mleq{} s as[i] \mwedge{} archive-condition(V;A;t;f;n - 1;vs[i];L)))))))
{}\mRightarrow{} (\mexists{}as:\{a:Id| (a \mmember{} A)\} List
(no\_repeats(\{a:Id| (a \mmember{} A)\} ;as)
\mwedge{} (||as|| = ((2 * t) + 1))
\mwedge{} (\mexists{}vs:V List
((||vs|| = ||as||)
\mwedge{} (v = (f vs))
\mwedge{} (\mforall{}i:\mBbbN{}||as||
\mexists{}L:consensus-rcv(V;A) List
(L \mleq{} y as[i] \mwedge{} archive-condition(V;A;t;f;n - 1;vs[i];L)))))))\mkleeneclose{}\mcdot{}
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