Step
*
2
4
of Lemma
archive-consensus-accum-num
1. V : Type
2. A : Id List
3. t : ℕ+
4. f : (V List) ─→ V
5. v0 : V
6. IdDeq ∈ EqDecider({b:Id| (b ∈ A)} )
7. ys : consensus-rcv(V;A) List@i
8. y2 : {b:Id| (b ∈ A)} @i
9. y4 : ℕ@i
10. y5 : V@i
11. Vote[y2;y4;y5] ∈ consensus-rcv(V;A)
12. v1 : 𝔹@i
13. v3 : ℤ@i
14. v5 : {a:Id| (a ∈ A)} List@i
15. v7 : V List@i
16. v8 : V@i
17. consensus-accum-num-state(t;f;v0;ys) = <v1, v3, v5, v7, v8> ∈ (𝔹 × ℤ × {a:Id| (a ∈ A)} List × V List × V)@i
18. (((2 * t) + 1) ≤ ||remove-repeats(IdDeq;map(λp.(fst(p));votes-from-inning(v3 - 2;ys)))||) supposing
((votes-from-inning(v3 - 1;ys) = [] ∈ (({b:Id| (b ∈ A)} × V) List)) and
1 < v3)@i
19. (v3 - 1) ≤ y4
20. y4 ≤ (v3 - 1)
21. ¬(||remove-repeats(IdDeq;v5 @ [y2])|| = ((2 * t) + 1) ∈ ℤ)
22. ∀[v@0:V]. ∀[i:ℤ]. ↑v1 supposing archive-condition(V;A;t;f;i;v@0;ys)@i
23. filter(λr.v3 - 1 <z inning(r);ys) = [] ∈ (consensus-rcv(V;A) List)@i
24. ||v5|| = ||v7|| ∈ ℤ@i
25. zip(v5;v7) = votes-from-inning(v3 - 1;ys) ∈ (({b:Id| (b ∈ A)} × V) List)@i
26. 0 ≤ v3@i
27. 1 ≤ v3 supposing ¬↑null(ys)@i
28. ||values-for-distinct(IdDeq;votes-from-inning(v3 - 1;ys))|| ≤ (2 * t)@i
29. filter(λr.v3 - 1 <z inning(r);ys @ [Vote[y2;y4;y5]]) = [] ∈ (consensus-rcv(V;A) List)@i
30. ||v5 @ [y2]|| = ||v7 @ [y5]|| ∈ ℤ@i
31. zip(v5 @ [y2];v7 @ [y5]) = votes-from-inning(v3 - 1;ys @ [Vote[y2;y4;y5]]) ∈ (({b:Id| (b ∈ A)} × V) List)@i
32. 0 ≤ v3@i
33. 1 ≤ v3 supposing ¬↑null(ys @ [Vote[y2;y4;y5]])@i
34. ||values-for-distinct(IdDeq;votes-from-inning(v3 - 1;ys @ [Vote[y2;y4;y5]]))|| ≤ (2 * t)@i
35. v@0 : V
36. i : ℤ
37. 0 < i
38. ||values-for-distinct(IdDeq;votes-from-inning(i - 1;ys))|| ≤ (2 * t)
39. ↑null(filter(λr.i - 1 <z inning(r);ys))
40. ((2 * t) + 1) ≤ ||values-for-distinct(IdDeq;votes-from-inning(i - 1;ys @ [Vote[y2;y4;y5]]))||
41. (f values-for-distinct(IdDeq;votes-from-inning(i - 1;ys @ [Vote[y2;y4;y5]]))) = v@0 ∈ V
⊢ False
BY
{ Assert ⌈y4 = (i - 1) ∈ ℤ⌉⋅ }
1
.....assertion.....
1. V : Type
2. A : Id List
3. t : ℕ+
4. f : (V List) ─→ V
5. v0 : V
6. IdDeq ∈ EqDecider({b:Id| (b ∈ A)} )
7. ys : consensus-rcv(V;A) List@i
8. y2 : {b:Id| (b ∈ A)} @i
9. y4 : ℕ@i
10. y5 : V@i
11. Vote[y2;y4;y5] ∈ consensus-rcv(V;A)
12. v1 : 𝔹@i
13. v3 : ℤ@i
14. v5 : {a:Id| (a ∈ A)} List@i
15. v7 : V List@i
16. v8 : V@i
17. consensus-accum-num-state(t;f;v0;ys) = <v1, v3, v5, v7, v8> ∈ (𝔹 × ℤ × {a:Id| (a ∈ A)} List × V List × V)@i
18. (((2 * t) + 1) ≤ ||remove-repeats(IdDeq;map(λp.(fst(p));votes-from-inning(v3 - 2;ys)))||) supposing
((votes-from-inning(v3 - 1;ys) = [] ∈ (({b:Id| (b ∈ A)} × V) List)) and
1 < v3)@i
19. (v3 - 1) ≤ y4
20. y4 ≤ (v3 - 1)
21. ¬(||remove-repeats(IdDeq;v5 @ [y2])|| = ((2 * t) + 1) ∈ ℤ)
22. ∀[v@0:V]. ∀[i:ℤ]. ↑v1 supposing archive-condition(V;A;t;f;i;v@0;ys)@i
23. filter(λr.v3 - 1 <z inning(r);ys) = [] ∈ (consensus-rcv(V;A) List)@i
24. ||v5|| = ||v7|| ∈ ℤ@i
25. zip(v5;v7) = votes-from-inning(v3 - 1;ys) ∈ (({b:Id| (b ∈ A)} × V) List)@i
26. 0 ≤ v3@i
27. 1 ≤ v3 supposing ¬↑null(ys)@i
28. ||values-for-distinct(IdDeq;votes-from-inning(v3 - 1;ys))|| ≤ (2 * t)@i
29. filter(λr.v3 - 1 <z inning(r);ys @ [Vote[y2;y4;y5]]) = [] ∈ (consensus-rcv(V;A) List)@i
30. ||v5 @ [y2]|| = ||v7 @ [y5]|| ∈ ℤ@i
31. zip(v5 @ [y2];v7 @ [y5]) = votes-from-inning(v3 - 1;ys @ [Vote[y2;y4;y5]]) ∈ (({b:Id| (b ∈ A)} × V) List)@i
32. 0 ≤ v3@i
33. 1 ≤ v3 supposing ¬↑null(ys @ [Vote[y2;y4;y5]])@i
34. ||values-for-distinct(IdDeq;votes-from-inning(v3 - 1;ys @ [Vote[y2;y4;y5]]))|| ≤ (2 * t)@i
35. v@0 : V
36. i : ℤ
37. 0 < i
38. ||values-for-distinct(IdDeq;votes-from-inning(i - 1;ys))|| ≤ (2 * t)
39. ↑null(filter(λr.i - 1 <z inning(r);ys))
40. ((2 * t) + 1) ≤ ||values-for-distinct(IdDeq;votes-from-inning(i - 1;ys @ [Vote[y2;y4;y5]]))||
41. (f values-for-distinct(IdDeq;votes-from-inning(i - 1;ys @ [Vote[y2;y4;y5]]))) = v@0 ∈ V
⊢ y4 = (i - 1) ∈ ℤ
2
1. V : Type
2. A : Id List
3. t : ℕ+
4. f : (V List) ─→ V
5. v0 : V
6. IdDeq ∈ EqDecider({b:Id| (b ∈ A)} )
7. ys : consensus-rcv(V;A) List@i
8. y2 : {b:Id| (b ∈ A)} @i
9. y4 : ℕ@i
10. y5 : V@i
11. Vote[y2;y4;y5] ∈ consensus-rcv(V;A)
12. v1 : 𝔹@i
13. v3 : ℤ@i
14. v5 : {a:Id| (a ∈ A)} List@i
15. v7 : V List@i
16. v8 : V@i
17. consensus-accum-num-state(t;f;v0;ys) = <v1, v3, v5, v7, v8> ∈ (𝔹 × ℤ × {a:Id| (a ∈ A)} List × V List × V)@i
18. (((2 * t) + 1) ≤ ||remove-repeats(IdDeq;map(λp.(fst(p));votes-from-inning(v3 - 2;ys)))||) supposing
((votes-from-inning(v3 - 1;ys) = [] ∈ (({b:Id| (b ∈ A)} × V) List)) and
1 < v3)@i
19. (v3 - 1) ≤ y4
20. y4 ≤ (v3 - 1)
21. ¬(||remove-repeats(IdDeq;v5 @ [y2])|| = ((2 * t) + 1) ∈ ℤ)
22. ∀[v@0:V]. ∀[i:ℤ]. ↑v1 supposing archive-condition(V;A;t;f;i;v@0;ys)@i
23. filter(λr.v3 - 1 <z inning(r);ys) = [] ∈ (consensus-rcv(V;A) List)@i
24. ||v5|| = ||v7|| ∈ ℤ@i
25. zip(v5;v7) = votes-from-inning(v3 - 1;ys) ∈ (({b:Id| (b ∈ A)} × V) List)@i
26. 0 ≤ v3@i
27. 1 ≤ v3 supposing ¬↑null(ys)@i
28. ||values-for-distinct(IdDeq;votes-from-inning(v3 - 1;ys))|| ≤ (2 * t)@i
29. filter(λr.v3 - 1 <z inning(r);ys @ [Vote[y2;y4;y5]]) = [] ∈ (consensus-rcv(V;A) List)@i
30. ||v5 @ [y2]|| = ||v7 @ [y5]|| ∈ ℤ@i
31. zip(v5 @ [y2];v7 @ [y5]) = votes-from-inning(v3 - 1;ys @ [Vote[y2;y4;y5]]) ∈ (({b:Id| (b ∈ A)} × V) List)@i
32. 0 ≤ v3@i
33. 1 ≤ v3 supposing ¬↑null(ys @ [Vote[y2;y4;y5]])@i
34. ||values-for-distinct(IdDeq;votes-from-inning(v3 - 1;ys @ [Vote[y2;y4;y5]]))|| ≤ (2 * t)@i
35. v@0 : V
36. i : ℤ
37. 0 < i
38. ||values-for-distinct(IdDeq;votes-from-inning(i - 1;ys))|| ≤ (2 * t)
39. ↑null(filter(λr.i - 1 <z inning(r);ys))
40. ((2 * t) + 1) ≤ ||values-for-distinct(IdDeq;votes-from-inning(i - 1;ys @ [Vote[y2;y4;y5]]))||
41. (f values-for-distinct(IdDeq;votes-from-inning(i - 1;ys @ [Vote[y2;y4;y5]]))) = v@0 ∈ V
42. y4 = (i - 1) ∈ ℤ
⊢ False
Latex:
1. V : Type
2. A : Id List
3. t : \mBbbN{}\msupplus{}
4. f : (V List) {}\mrightarrow{} V
5. v0 : V
6. IdDeq \mmember{} EqDecider(\{b:Id| (b \mmember{} A)\} )
7. ys : consensus-rcv(V;A) List@i
8. y2 : \{b:Id| (b \mmember{} A)\} @i
9. y4 : \mBbbN{}@i
10. y5 : V@i
11. Vote[y2;y4;y5] \mmember{} consensus-rcv(V;A)
12. v1 : \mBbbB{}@i
13. v3 : \mBbbZ{}@i
14. v5 : \{a:Id| (a \mmember{} A)\} List@i
15. v7 : V List@i
16. v8 : V@i
17. consensus-accum-num-state(t;f;v0;ys) = <v1, v3, v5, v7, v8>@i
18. (((2 * t) + 1) \mleq{} ||remove-repeats(IdDeq;map(\mlambda{}p.(fst(p));votes-from-inning(v3
- 2;ys)))||) supposing
((votes-from-inning(v3 - 1;ys) = []) and
1 < v3)@i
19. (v3 - 1) \mleq{} y4
20. y4 \mleq{} (v3 - 1)
21. \mneg{}(||remove-repeats(IdDeq;v5 @ [y2])|| = ((2 * t) + 1))
22. \mforall{}[v@0:V]. \mforall{}[i:\mBbbZ{}]. \muparrow{}v1 supposing archive-condition(V;A;t;f;i;v@0;ys)@i
23. filter(\mlambda{}r.v3 - 1 <z inning(r);ys) = []@i
24. ||v5|| = ||v7||@i
25. zip(v5;v7) = votes-from-inning(v3 - 1;ys)@i
26. 0 \mleq{} v3@i
27. 1 \mleq{} v3 supposing \mneg{}\muparrow{}null(ys)@i
28. ||values-for-distinct(IdDeq;votes-from-inning(v3 - 1;ys))|| \mleq{} (2 * t)@i
29. filter(\mlambda{}r.v3 - 1 <z inning(r);ys @ [Vote[y2;y4;y5]]) = []@i
30. ||v5 @ [y2]|| = ||v7 @ [y5]||@i
31. zip(v5 @ [y2];v7 @ [y5]) = votes-from-inning(v3 - 1;ys @ [Vote[y2;y4;y5]])@i
32. 0 \mleq{} v3@i
33. 1 \mleq{} v3 supposing \mneg{}\muparrow{}null(ys @ [Vote[y2;y4;y5]])@i
34. ||values-for-distinct(IdDeq;votes-from-inning(v3 - 1;ys @ [Vote[y2;y4;y5]]))|| \mleq{} (2 * t)@i
35. v@0 : V
36. i : \mBbbZ{}
37. 0 < i
38. ||values-for-distinct(IdDeq;votes-from-inning(i - 1;ys))|| \mleq{} (2 * t)
39. \muparrow{}null(filter(\mlambda{}r.i - 1 <z inning(r);ys))
40. ((2 * t) + 1) \mleq{} ||values-for-distinct(IdDeq;votes-from-inning(i - 1;ys @ [Vote[y2;y4;y5]]))||
41. (f values-for-distinct(IdDeq;votes-from-inning(i - 1;ys @ [Vote[y2;y4;y5]]))) = v@0
\mvdash{} False
By
Assert \mkleeneopen{}y4 = (i - 1)\mkleeneclose{}\mcdot{}
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