Proof of Nuprl Lemma : coW-trans

Step * 1 1 2 of Lemma coW-trans_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)}

20. ∀[moves:{f:strat2play(coW-game(a.B[a];w1;w3);k - 1;coW-trans(X; Y))| ||f|| = ((2 * (k - 1)) + 2) ∈ ℤ} ]

      (transMoves(X;Y;moves) ∈ {p:strat2play(coW-game(a.B[a];w1;w2);k - 1;X)
                                × strat2play(coW-game(a.B[a];w2;w3);k - 1;Y)|
                                let a,b = p

                                in coWtransInvariant(a.B[a];w1;w2;w3;k - 1;X;Y;a;b;moves)} )

    supposing (k - 1) ≤ (n - 1)
21. k ≤ (n - 1)
22. m1 : {f:strat2play(coW-game(a.B[a];w1;w3);k;coW-trans(X; Y))| ||f|| = ((2 * k) + 2) ∈ ℤ}
⊢ transMoves(X;Y;m1) ∈ {p:strat2play(coW-game(a.B[a];w1;w2);k;X) × strat2play(coW-game(a.B[a];w2;w3);k;Y)|
                        let a,b = p
                        in coWtransInvariant(a.B[a];w1;w2;w3;k;X;Y;a;b;m1)}
BY
{ (D -1
   THEN Unfold `play-len` -1
   THEN Unfold `transMoves` 0
   THEN (D -4 THENA Auto)
   THEN ((Assert 1 ≤ k BY Auto) THEN (Mul ⌜2⌝ (-1)⋅ THENA Auto))
   THEN (SplitOnConclITE THENA Auto)
   THEN Try ((Assert ⌜False⌝⋅ THEN Complete (Auto)))
   THEN D -4 With ⌜seq-truncate(m1;||m1|| - 2)⌝ ) }

1
.....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}

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)}

                                                   × strat2play(coW-game(a.B[a];w2;w3);k - 1;Y)|

let a,b = p

                                                   in coWtransInvariant(a.B[a];w1;w2;w3;k

                                                      - 1;X;Y;a;b;seq-truncate(m1;||m1|| - 2))}

⊢ let a,b = transMoves(X;Y;seq-truncate(m1;||m1|| - 2))
  in let x,z1 = if (||m1|| =_z 2) then <(), ()> else X a fi
     in let z2,y = if (||m1|| =_z 2) then <(), ()> else Y b fi
        in let u,v = m1[||m1|| - 1]
           in if copath-length(x) <z copath-length(u)
              then let M1 = seq-add(seq-add(a;<x, z1>);<u, z1>) in
                    let M2 = seq-add(seq-add(b;<z2, y>);<snd((X M1)), y>) in
                    <M1, M2>
              else let M2 = seq-add(seq-add(b;<z2, y>);<z2, v>) in
                    let M1 = seq-add(seq-add(a;<x, z1>);<x, fst((Y M2))>) in
                    <M1, M2>

              fi  ∈ {p:strat2play(coW-game(a.B[a];w1;w2);k;X) × strat2play(coW-game(a.B[a];w2;w3);k;Y)|

let a,b = p
in coWtransInvariant(a.B[a];w1;w2;w3;k;X;Y;a;b;m1)}

Latex:

Latex:
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. \mforall{}[moves:\{f:strat2play(coW-game(a.B[a];w1;w3);k - 1;coW-trans(X; Y))| ||f|| = ((2 * (k - 1)) + 2)\} ] (transMoves(X;Y;moves) \mmember{} \{p:strat2play(coW-game(a.B[a];w1;w2);k - 1;X) \mtimes{} strat2play(coW-game(a.B[a];w2;w3);k - 1;Y)| let a,b = p in coWtransInvariant(a.B[a];w1;w2;w3;k - 1;X;Y;a;b;moves)\} ) supposing (k - 1) \mleq{} (n - 1) 21. k \mleq{} (n - 1) 22. m1 : \{f:strat2play(coW-game(a.B[a];w1;w3);k;coW-trans(X; Y))| ||f|| = ((2 * k) + 2)\} \mvdash{} transMoves(X;Y;m1) \mmember{} \{p:strat2play(coW-game(a.B[a];w1;w2);k;X) \mtimes{} strat2play(coW-game(a.B[a];w2;w3);k;Y)| let a,b = p in coWtransInvariant(a.B[a];w1;w2;w3;k;X;Y;a;b;m1)\} By Latex: (D -1 THEN Unfold `play-len` -1 THEN Unfold `transMoves` 0 THEN (D -4 THENA Auto) THEN ((Assert 1 \mleq{} k BY Auto) THEN (Mul \mkleeneopen{}2\mkleeneclose{} (-1)\mcdot{} THENA Auto)) THEN (SplitOnConclITE THENA Auto) THEN Try ((Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{} THEN Complete (Auto))) THEN D -4 With \mkleeneopen{}seq-truncate(m1;||m1|| - 2)\mkleeneclose{} ) Home Index