Proof of Nuprl Lemma : word-rel-diamond

Step * 1 1 4 4 of Lemma word-rel-diamond

1. [X] : Type
2. n : ℕ
3. x : (X + X) List
4. ∀x1:(X + X) List
     (||x1|| < ||x||
     ⇒ (∀y,z:(X + X) List.
           (word-rel(X;x1;y)
           ⇒ word-rel(X;x1;z)
           ⇒ ((y = z ∈ ((X + X) List)) ∨ (∃w:(X + X) List. (word-rel(X;y;w) ∧ word-rel(X;z;w)))))))
5. y : (X + X) List
6. z : (X + X) List
7. x@0 : X + X
8. y@0 : X + X
9. u : X + X
10. v : (X + X) List
11. b1 : (X + X) List
12. x@0 = -y@0
13. x = [u / (v @ [x@0; [y@0 / b1]])] ∈ ((X + X) List)
14. y = [u / (v @ b1)] ∈ ((X + X) List)
15. x1 : X + X
16. y1 : X + X
17. u1 : X + X
18. v1 : (X + X) List
19. b : (X + X) List
20. x1 = -y1
21. x = [u1 / (v1 @ [x1; [y1 / b]])] ∈ ((X + X) List)
22. z = [u1 / (v1 @ b)] ∈ ((X + X) List)
23. u = u1 ∈ (X + X)
24. (v @ [x@0; [y@0 / b1]]) = (v1 @ [x1; [y1 / b]]) ∈ ((X + X) List)

25. ((v @ b1) = (v1 @ b) ∈ ((X + X) List)) ∨ (∃w:(X + X) List. (word-rel(X;v @ b1;w) ∧ word-rel(X;v1 @ b;w)))

⊢ ([u / (v @ b1)] = [u1 / (v1 @ b)] ∈ ((X + X) List))
∨ (∃w:(X + X) List. (word-rel(X;[u / (v @ b1)];w) ∧ word-rel(X;[u1 / (v1 @ b)];w)))
BY
{ ParallelLast }

1
1. X : Type
2. n : ℕ
3. x : (X + X) List
4. ∀x1:(X + X) List
     (||x1|| < ||x||
     ⇒ (∀y,z:(X + X) List.
           (word-rel(X;x1;y)
           ⇒ word-rel(X;x1;z)
           ⇒ ((y = z ∈ ((X + X) List)) ∨ (∃w:(X + X) List. (word-rel(X;y;w) ∧ word-rel(X;z;w)))))))
5. y : (X + X) List
6. z : (X + X) List
7. x@0 : X + X
8. y@0 : X + X
9. u : X + X
10. v : (X + X) List
11. b1 : (X + X) List
12. x@0 = -y@0
13. x = [u / (v @ [x@0; [y@0 / b1]])] ∈ ((X + X) List)
14. y = [u / (v @ b1)] ∈ ((X + X) List)
15. x1 : X + X
16. y1 : X + X
17. u1 : X + X
18. v1 : (X + X) List
19. b : (X + X) List
20. x1 = -y1
21. x = [u1 / (v1 @ [x1; [y1 / b]])] ∈ ((X + X) List)
22. z = [u1 / (v1 @ b)] ∈ ((X + X) List)
23. u = u1 ∈ (X + X)
24. (v @ [x@0; [y@0 / b1]]) = (v1 @ [x1; [y1 / b]]) ∈ ((X + X) List)
25. (v @ b1) = (v1 @ b) ∈ ((X + X) List)
⊢ [u / (v @ b1)] = [u1 / (v1 @ b)] ∈ ((X + X) List)

2
1. [X] : Type
2. n : ℕ
3. x : (X + X) List
4. ∀x1:(X + X) List
     (||x1|| < ||x||
     ⇒ (∀y,z:(X + X) List.
           (word-rel(X;x1;y)
           ⇒ word-rel(X;x1;z)
           ⇒ ((y = z ∈ ((X + X) List)) ∨ (∃w:(X + X) List. (word-rel(X;y;w) ∧ word-rel(X;z;w)))))))
5. y : (X + X) List
6. z : (X + X) List
7. x@0 : X + X
8. y@0 : X + X
9. u : X + X
10. v : (X + X) List
11. b1 : (X + X) List
12. x@0 = -y@0
13. x = [u / (v @ [x@0; [y@0 / b1]])] ∈ ((X + X) List)
14. y = [u / (v @ b1)] ∈ ((X + X) List)
15. x1 : X + X
16. y1 : X + X
17. u1 : X + X
18. v1 : (X + X) List
19. b : (X + X) List
20. x1 = -y1
21. x = [u1 / (v1 @ [x1; [y1 / b]])] ∈ ((X + X) List)
22. z = [u1 / (v1 @ b)] ∈ ((X + X) List)
23. u = u1 ∈ (X + X)
24. (v @ [x@0; [y@0 / b1]]) = (v1 @ [x1; [y1 / b]]) ∈ ((X + X) List)
25. ∃w:(X + X) List. (word-rel(X;v @ b1;w) ∧ word-rel(X;v1 @ b;w))
⊢ ∃w:(X + X) List. (word-rel(X;[u / (v @ b1)];w) ∧ word-rel(X;[u1 / (v1 @ b)];w))

Latex:

Latex:
1. [X] : Type 2. n : \mBbbN{} 3. x : (X + X) List 4. \mforall{}x1:(X + X) List (||x1|| < ||x|| {}\mRightarrow{} (\mforall{}y,z:(X + X) List. (word-rel(X;x1;y) {}\mRightarrow{} word-rel(X;x1;z) {}\mRightarrow{} ((y = z) \mvee{} (\mexists{}w:(X + X) List. (word-rel(X;y;w) \mwedge{} word-rel(X;z;w))))))) 5. y : (X + X) List 6. z : (X + X) List 7. x@0 : X + X 8. y@0 : X + X 9. u : X + X 10. v : (X + X) List 11. b1 : (X + X) List 12. x@0 = -y@0 13. x = [u / (v @ [x@0; [y@0 / b1]])] 14. y = [u / (v @ b1)] 15. x1 : X + X 16. y1 : X + X 17. u1 : X + X 18. v1 : (X + X) List 19. b : (X + X) List 20. x1 = -y1 21. x = [u1 / (v1 @ [x1; [y1 / b]])] 22. z = [u1 / (v1 @ b)] 23. u = u1 24. (v @ [x@0; [y@0 / b1]]) = (v1 @ [x1; [y1 / b]]) 25. ((v @ b1) = (v1 @ b)) \mvee{} (\mexists{}w:(X + X) List. (word-rel(X;v @ b1;w) \mwedge{} word-rel(X;v1 @ b;w))) \mvdash{} ([u / (v @ b1)] = [u1 / (v1 @ b)]) \mvee{} (\mexists{}w:(X + X) List. (word-rel(X;[u / (v @ b1)];w) \mwedge{} word-rel(X;[u1 / (v1 @ b)];w))) By Latex: ParallelLast Home Index