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of Lemma
coW-trans_wf
.....set predicate.....
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 ≤ (n - 1)
20. m1 : strat2play(coW-game(a.B[a];w1;w3);0;coW-trans(X; Y))
21. ||m1|| = 2 ∈ ℤ
22. z : copath(a.B[a];w3)
23. m1[1] = <(), z> ∈ Pos(coW-game(a.B[a];w1;w3))
24. m1[0] = InitialPos(coW-game(a.B[a];w1;w3)) ∈ Pos(coW-game(a.B[a];w1;w3))
25. copath-length(z) = 1 ∈ ℤ
26. copathAgree(a.B[a];w3;();z)
27. seq-add(seq-add(seq-nil();<(), ()>);<(), z>) ∈ {f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2 ∈ ℤ}
28. mm : strat2play(coW-game(a.B[a];w2;w3);0;Y)
29. ||mm|| = 2 ∈ ℤ
30. seq-add(seq-add(seq-nil();<(), ()>);<(), z>) = mm ∈ {f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2 ∈ ℤ}
31. Y ∈ win2strat(coW-game(a.B[a];w2;w3);0)
32. Y ∈ moves:{f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2 ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w2;w3))| Legal2(moves\000C[1];p)}
33. v1 : copath(a.B[a];w2)
34. v2 : copath(a.B[a];w3)
35. Legal2(mm[1];<v1, v2>)
36. (Y mm) = <v1, v2> ∈ {p:Pos(coW-game(a.B[a];w2;w3))| Legal2(mm[1];p)}
37. seq-add(seq-add(seq-nil();<(), ()>);<(), v1>) ∈ strat2play(coW-game(a.B[a];w1;w2);0;X)
⊢ let a,b = <seq-add(seq-add(seq-nil();<(), ()>);<(), v1>), mm>
in coWtransInvariant(a.B[a];w1;w2;w3;0;X;Y;a;b;m1)
BY
{ ((Assert mm[1] = <(), z> ∈ Pos(coW-game(a.B[a];w2;w3)) BY
(RWO "-8<" 0 THEN Auto))
THEN ((RWO "-1" (-4) THENA Auto) THEN RepUR ``sg-legal2 coW-game`` -4 THEN D -4)
THEN RepUR ``sg-pos coW-game`` -1
THEN D -5) }
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 ≤ (n - 1)
20. m1 : strat2play(coW-game(a.B[a];w1;w3);0;coW-trans(X; Y))
21. ||m1|| = 2 ∈ ℤ
22. z : copath(a.B[a];w3)
23. m1[1] = <(), z> ∈ Pos(coW-game(a.B[a];w1;w3))
24. m1[0] = InitialPos(coW-game(a.B[a];w1;w3)) ∈ Pos(coW-game(a.B[a];w1;w3))
25. copath-length(z) = 1 ∈ ℤ
26. copathAgree(a.B[a];w3;();z)
27. seq-add(seq-add(seq-nil();<(), ()>);<(), z>) ∈ {f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2 ∈ ℤ}
28. mm : strat2play(coW-game(a.B[a];w2;w3);0;Y)
29. ||mm|| = 2 ∈ ℤ
30. seq-add(seq-add(seq-nil();<(), ()>);<(), z>) = mm ∈ {f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2 ∈ ℤ}
31. Y ∈ win2strat(coW-game(a.B[a];w2;w3);0)
32. Y ∈ moves:{f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2 ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w2;w3))| Legal2(moves\000C[1];p)}
33. v1 : copath(a.B[a];w2)
34. v2 : copath(a.B[a];w3)
35. (copath-length(v1) = 1 ∈ ℤ) ∧ copathAgree(a.B[a];w2;();v1) ∧ (z = v2 ∈ copath(a.B[a];w3))
36. copath-length(v1) = copath-length(v2) ∈ ℤ
37. (Y mm) = <v1, v2> ∈ {p:Pos(coW-game(a.B[a];w2;w3))| Legal2(mm[1];p)}
38. seq-add(seq-add(seq-nil();<(), ()>);<(), v1>) ∈ strat2play(coW-game(a.B[a];w1;w2);0;X)
39. mm[1] = <(), z> ∈ (copath(a.B[a];w2) × copath(a.B[a];w3))
⊢ let a,b = <seq-add(seq-add(seq-nil();<(), ()>);<(), v1>), mm>
in coWtransInvariant(a.B[a];w1;w2;w3;0;X;Y;a;b;m1)
2
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 ≤ (n - 1)
20. m1 : strat2play(coW-game(a.B[a];w1;w3);0;coW-trans(X; Y))
21. ||m1|| = 2 ∈ ℤ
22. z : copath(a.B[a];w3)
23. m1[1] = <(), z> ∈ Pos(coW-game(a.B[a];w1;w3))
24. m1[0] = InitialPos(coW-game(a.B[a];w1;w3)) ∈ Pos(coW-game(a.B[a];w1;w3))
25. copath-length(z) = 1 ∈ ℤ
26. copathAgree(a.B[a];w3;();z)
27. seq-add(seq-add(seq-nil();<(), ()>);<(), z>) ∈ {f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2 ∈ ℤ}
28. mm : strat2play(coW-game(a.B[a];w2;w3);0;Y)
29. ||mm|| = 2 ∈ ℤ
30. seq-add(seq-add(seq-nil();<(), ()>);<(), z>) = mm ∈ {f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2 ∈ ℤ}
31. Y ∈ win2strat(coW-game(a.B[a];w2;w3);0)
32. Y ∈ moves:{f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2 ∈ ℤ} ⟶ {p:Pos(coW-game(a.B[a];w2;w3))| Legal2(moves\000C[1];p)}
33. v1 : copath(a.B[a];w2)
34. v2 : copath(a.B[a];w3)
35. (() = v1 ∈ copath(a.B[a];w2)) ∧ (copath-length(v2) = (copath-length(z) + 1) ∈ ℤ) ∧ copathAgree(a.B[a];w3;z;v2)
36. copath-length(v1) = copath-length(v2) ∈ ℤ
37. (Y mm) = <v1, v2> ∈ {p:Pos(coW-game(a.B[a];w2;w3))| Legal2(mm[1];p)}
38. seq-add(seq-add(seq-nil();<(), ()>);<(), v1>) ∈ strat2play(coW-game(a.B[a];w1;w2);0;X)
39. mm[1] = <(), z> ∈ (copath(a.B[a];w2) × copath(a.B[a];w3))
⊢ let a,b = <seq-add(seq-add(seq-nil();<(), ()>);<(), v1>), mm>
in coWtransInvariant(a.B[a];w1;w2;w3;0;X;Y;a;b;m1)
Latex:
Latex:
.....set predicate.....
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 \mleq{} (n - 1)
20. m1 : strat2play(coW-game(a.B[a];w1;w3);0;coW-trans(X; Y))
21. ||m1|| = 2
22. z : copath(a.B[a];w3)
23. m1[1] = <(), z>
24. m1[0] = InitialPos(coW-game(a.B[a];w1;w3))
25. copath-length(z) = 1
26. copathAgree(a.B[a];w3;();z)
27. seq-add(seq-add(seq-nil();<(), ()>);<(), z>) \mmember{} \{f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| \000C= 2\}
28. mm : strat2play(coW-game(a.B[a];w2;w3);0;Y)
29. ||mm|| = 2
30. seq-add(seq-add(seq-nil();<(), ()>);<(), z>) = mm
31. Y \mmember{} win2strat(coW-game(a.B[a];w2;w3);0)
32. Y \mmember{} moves:\{f:strat2play(coW-game(a.B[a];w2;w3);0;Y)| ||f|| = 2\} {}\mrightarrow{} \{p:Pos(coW-game(a.B[a];w2;w3\000C))|
Legal2(moves[1];p)\}
33. v1 : copath(a.B[a];w2)
34. v2 : copath(a.B[a];w3)
35. Legal2(mm[1];<v1, v2>)
36. (Y mm) = <v1, v2>
37. seq-add(seq-add(seq-nil();<(), ()>);<(), v1>) \mmember{} strat2play(coW-game(a.B[a];w1;w2);0;X)
\mvdash{} let a,b = <seq-add(seq-add(seq-nil();<(), ()>);<(), v1>), mm>
in coWtransInvariant(a.B[a];w1;w2;w3;0;X;Y;a;b;m1)
By
Latex:
((Assert mm[1] = <(), z> BY
(RWO "-8<" 0 THEN Auto))
THEN ((RWO "-1" (-4) THENA Auto) THEN RepUR ``sg-legal2 coW-game`` -4 THEN D -4)
THEN RepUR ``sg-pos coW-game`` -1
THEN D -5)
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