Step
*
1
1
2
1
of Lemma
coW-trans_wf
.....wf.....
1. A : 𝕌'
2. B : A ⟶ Type
3. w1 : coW(A;a.B[a])
4. w2 : coW(A;a.B[a])
5. w3 : coW(A;a.B[a])
6. n : ℤ
7. 0 < n
8. ∀[X:win2strat(coW-game(a.B[a];w1;w2);n - 1)]. ∀[Y:win2strat(coW-game(a.B[a];w2;w3);n - 1)].
(coW-trans(X; Y) ∈ win2strat(coW-game(a.B[a];w1;w3);n - 1))
9. X : s:win2strat(coW-game(a.B[a];w1;w2);n - 1) ⋂ moves:{f:strat2play(coW-game(a.B[a];w1;w2);n - 1;s)|
||f|| = (2 * n) ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w1;w2))| Legal\000C2(moves[(2 * n) - 1];p)}
10. X ∈ win2strat(coW-game(a.B[a];w1;w2);n - 1)
11. X ∈ moves:{f:strat2play(coW-game(a.B[a];w1;w2);n - 1;X)| ||f|| = (2 * n) ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w1;w2))|
Legal2(moves[(2 * n) - 1];p)}
12. Y : s:win2strat(coW-game(a.B[a];w2;w3);n - 1) ⋂ moves:{f:strat2play(coW-game(a.B[a];w2;w3);n - 1;s)|
||f|| = (2 * n) ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w2;w3))| Lega\000Cl2(moves[(2 * n) - 1];p)}
13. Y ∈ win2strat(coW-game(a.B[a];w2;w3);n - 1)
14. Y ∈ moves:{f:strat2play(coW-game(a.B[a];w2;w3);n - 1;Y)| ||f|| = (2 * n) ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w2;w3))|
Legal2(moves[(2 * n) - 1];p)}
15. ∀k:ℕ. ((k ≤ n) ⇒ (X ∈ win2strat(coW-game(a.B[a];w1;w2);k)))
16. ∀k:ℕ. ((k ≤ n) ⇒ (Y ∈ win2strat(coW-game(a.B[a];w2;w3);k)))
17. moves : {f:strat2play(coW-game(a.B[a];w1;w3);n - 1;coW-trans(X; Y))| ||f|| = (2 * n) ∈ ℤ}
18. k : ℤ
19. 0 < k
20. k ≤ (n - 1)
21. m1 : strat2play(coW-game(a.B[a];w1;w3);k;coW-trans(X; Y))
22. ||m1|| = ((2 * k) + 2) ∈ ℤ
23. 1 ≤ k
24. (2 * 1) ≤ (2 * k)
25. ¬(||m1|| = 0 ∈ ℤ)
⊢ seq-truncate(m1;||m1|| - 2) ∈ {f:strat2play(coW-game(a.B[a];w1;w3);k - 1;coW-trans(X; Y))|
||f|| = ((2 * (k - 1)) + 2) ∈ ℤ}
BY
{ Fold `play-truncate` 0 }
1
1. A : 𝕌'
2. B : A ⟶ Type
3. w1 : coW(A;a.B[a])
4. w2 : coW(A;a.B[a])
5. w3 : coW(A;a.B[a])
6. n : ℤ
7. 0 < n
8. ∀[X:win2strat(coW-game(a.B[a];w1;w2);n - 1)]. ∀[Y:win2strat(coW-game(a.B[a];w2;w3);n - 1)].
(coW-trans(X; Y) ∈ win2strat(coW-game(a.B[a];w1;w3);n - 1))
9. X : s:win2strat(coW-game(a.B[a];w1;w2);n - 1) ⋂ moves:{f:strat2play(coW-game(a.B[a];w1;w2);n - 1;s)|
||f|| = (2 * n) ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w1;w2))| Legal\000C2(moves[(2 * n) - 1];p)}
10. X ∈ win2strat(coW-game(a.B[a];w1;w2);n - 1)
11. X ∈ moves:{f:strat2play(coW-game(a.B[a];w1;w2);n - 1;X)| ||f|| = (2 * n) ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w1;w2))|
Legal2(moves[(2 * n) - 1];p)}
12. Y : s:win2strat(coW-game(a.B[a];w2;w3);n - 1) ⋂ moves:{f:strat2play(coW-game(a.B[a];w2;w3);n - 1;s)|
||f|| = (2 * n) ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w2;w3))| Lega\000Cl2(moves[(2 * n) - 1];p)}
13. Y ∈ win2strat(coW-game(a.B[a];w2;w3);n - 1)
14. Y ∈ moves:{f:strat2play(coW-game(a.B[a];w2;w3);n - 1;Y)| ||f|| = (2 * n) ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w2;w3))|
Legal2(moves[(2 * n) - 1];p)}
15. ∀k:ℕ. ((k ≤ n) ⇒ (X ∈ win2strat(coW-game(a.B[a];w1;w2);k)))
16. ∀k:ℕ. ((k ≤ n) ⇒ (Y ∈ win2strat(coW-game(a.B[a];w2;w3);k)))
17. moves : {f:strat2play(coW-game(a.B[a];w1;w3);n - 1;coW-trans(X; Y))| ||f|| = (2 * n) ∈ ℤ}
18. k : ℤ
19. 0 < k
20. k ≤ (n - 1)
21. m1 : strat2play(coW-game(a.B[a];w1;w3);k;coW-trans(X; Y))
22. ||m1|| = ((2 * k) + 2) ∈ ℤ
23. 1 ≤ k
24. (2 * 1) ≤ (2 * k)
25. ¬(||m1|| = 0 ∈ ℤ)
⊢ play-truncate(m1;||m1|| - 2) ∈ {f:strat2play(coW-game(a.B[a];w1;w3);k - 1;coW-trans(X; Y))|
||f|| = ((2 * (k - 1)) + 2) ∈ ℤ}
Latex:
Latex:
.....wf.....
1. A : \mBbbU{}'
2. B : A {}\mrightarrow{} Type
3. w1 : coW(A;a.B[a])
4. w2 : coW(A;a.B[a])
5. w3 : coW(A;a.B[a])
6. n : \mBbbZ{}
7. 0 < n
8. \mforall{}[X:win2strat(coW-game(a.B[a];w1;w2);n - 1)]. \mforall{}[Y:win2strat(coW-game(a.B[a];w2;w3);n - 1)].
(coW-trans(X; Y) \mmember{} win2strat(coW-game(a.B[a];w1;w3);n - 1))
9. X : s:win2strat(coW-game(a.B[a];w1;w2);n - 1)
\mcap{} moves:\{f:strat2play(coW-game(a.B[a];w1;w2);n - 1;s)| ||f|| = (2 * n)\}
{}\mrightarrow{} \{p:Pos(coW-game(a.B[a];w1;w2))| Legal2(moves[(2 * n) - 1];p)\}
10. X \mmember{} win2strat(coW-game(a.B[a];w1;w2);n - 1)
11. X \mmember{} moves:\{f:strat2play(coW-game(a.B[a];w1;w2);n - 1;X)| ||f|| = (2 * n)\}
{}\mrightarrow{} \{p:Pos(coW-game(a.B[a];w1;w2))| Legal2(moves[(2 * n) - 1];p)\}
12. Y : s:win2strat(coW-game(a.B[a];w2;w3);n - 1)
\mcap{} moves:\{f:strat2play(coW-game(a.B[a];w2;w3);n - 1;s)| ||f|| = (2 * n)\}
{}\mrightarrow{} \{p:Pos(coW-game(a.B[a];w2;w3))| Legal2(moves[(2 * n) - 1];p)\}
13. Y \mmember{} win2strat(coW-game(a.B[a];w2;w3);n - 1)
14. Y \mmember{} moves:\{f:strat2play(coW-game(a.B[a];w2;w3);n - 1;Y)| ||f|| = (2 * n)\}
{}\mrightarrow{} \{p:Pos(coW-game(a.B[a];w2;w3))| Legal2(moves[(2 * n) - 1];p)\}
15. \mforall{}k:\mBbbN{}. ((k \mleq{} n) {}\mRightarrow{} (X \mmember{} win2strat(coW-game(a.B[a];w1;w2);k)))
16. \mforall{}k:\mBbbN{}. ((k \mleq{} n) {}\mRightarrow{} (Y \mmember{} win2strat(coW-game(a.B[a];w2;w3);k)))
17. moves : \{f:strat2play(coW-game(a.B[a];w1;w3);n - 1;coW-trans(X; Y))| ||f|| = (2 * n)\}
18. k : \mBbbZ{}
19. 0 < k
20. k \mleq{} (n - 1)
21. m1 : strat2play(coW-game(a.B[a];w1;w3);k;coW-trans(X; Y))
22. ||m1|| = ((2 * k) + 2)
23. 1 \mleq{} k
24. (2 * 1) \mleq{} (2 * k)
25. \mneg{}(||m1|| = 0)
\mvdash{} seq-truncate(m1;||m1|| - 2) \mmember{} \{f:strat2play(coW-game(a.B[a];w1;w3);k - 1;coW-trans(X; Y))|
||f|| = ((2 * (k - 1)) + 2)\}
By
Latex:
Fold `play-truncate` 0
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