Step
*
1
1
1
of Lemma
face-of-face-pairity
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. 1 < dim(c)
6. immediate-rc-face(k;a;b)
7. immediate-rc-face(k;b;c)
8. i1 : ℕk
9. dim(b i1) = 1 ∈ ℤ
10. ∀j:ℕk. ((¬(j = i1 ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
11. ((a i1) = [fst((b i1))] ∈ ℚInterval) ∨ ((a i1) = [snd((b i1))] ∈ ℚInterval)
12. i : ℕk
13. dim(c i) = 1 ∈ ℤ
14. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((b j) = (c j) ∈ ℚInterval))
15. ((b i) = [fst((c i))] ∈ ℚInterval) ∨ ((b i) = [snd((c i))] ∈ ℚInterval)
16. dim(a) = (dim(c) - 2) ∈ ℤ
17. ↑Inhabited(a)
18. dim(b) = (dim(c) - 1) ∈ ℤ
19. ↑Inhabited(b)
20. ↑Inhabited(c)
21. λj.if (j =z i1) then a i1 else c j fi ∈ ℚCube(k)
22. ↑Inhabited(λj.if (j =z i1) then a i1 else c j fi )
23. ¬(i1 = i ∈ ℤ)
24. dim(λj.if (j =z i1) then a i1 else c j fi ) = (dim(c) - 1) ∈ ℤ
⊢ ∃!b':ℚCube(k). (immediate-rc-face(k;a;b') ∧ immediate-rc-face(k;b';c) ∧ (¬(b' = b ∈ ℚCube(k))))
BY
{ (D 0 With ⌜λj.if (j =z i1) then a i1 else c j fi ⌝ THEN Auto) }
1
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. 1 < dim(c)
6. immediate-rc-face(k;a;b)
7. immediate-rc-face(k;b;c)
8. i1 : ℕk
9. dim(b i1) = 1 ∈ ℤ
10. ∀j:ℕk. ((¬(j = i1 ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
11. ((a i1) = [fst((b i1))] ∈ ℚInterval) ∨ ((a i1) = [snd((b i1))] ∈ ℚInterval)
12. i : ℕk
13. dim(c i) = 1 ∈ ℤ
14. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((b j) = (c j) ∈ ℚInterval))
15. ((b i) = [fst((c i))] ∈ ℚInterval) ∨ ((b i) = [snd((c i))] ∈ ℚInterval)
16. dim(a) = (dim(c) - 2) ∈ ℤ
17. ↑Inhabited(a)
18. dim(b) = (dim(c) - 1) ∈ ℤ
19. ↑Inhabited(b)
20. ↑Inhabited(c)
21. λj.if (j =z i1) then a i1 else c j fi ∈ ℚCube(k)
22. ↑Inhabited(λj.if (j =z i1) then a i1 else c j fi )
23. ¬(i1 = i ∈ ℤ)
24. dim(λj.if (j =z i1) then a i1 else c j fi ) = (dim(c) - 1) ∈ ℤ
⊢ immediate-rc-face(k;a;λj.if (j =z i1) then a i1 else c j fi )
2
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. 1 < dim(c)
6. immediate-rc-face(k;a;b)
7. immediate-rc-face(k;b;c)
8. i1 : ℕk
9. dim(b i1) = 1 ∈ ℤ
10. ∀j:ℕk. ((¬(j = i1 ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
11. ((a i1) = [fst((b i1))] ∈ ℚInterval) ∨ ((a i1) = [snd((b i1))] ∈ ℚInterval)
12. i : ℕk
13. dim(c i) = 1 ∈ ℤ
14. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((b j) = (c j) ∈ ℚInterval))
15. ((b i) = [fst((c i))] ∈ ℚInterval) ∨ ((b i) = [snd((c i))] ∈ ℚInterval)
16. dim(a) = (dim(c) - 2) ∈ ℤ
17. ↑Inhabited(a)
18. dim(b) = (dim(c) - 1) ∈ ℤ
19. ↑Inhabited(b)
20. ↑Inhabited(c)
21. λj.if (j =z i1) then a i1 else c j fi ∈ ℚCube(k)
22. ↑Inhabited(λj.if (j =z i1) then a i1 else c j fi )
23. ¬(i1 = i ∈ ℤ)
24. dim(λj.if (j =z i1) then a i1 else c j fi ) = (dim(c) - 1) ∈ ℤ
25. immediate-rc-face(k;a;λj.if (j =z i1) then a i1 else c j fi )
⊢ immediate-rc-face(k;λj.if (j =z i1) then a i1 else c j fi ;c)
3
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. 1 < dim(c)
6. immediate-rc-face(k;a;b)
7. immediate-rc-face(k;b;c)
8. i1 : ℕk
9. dim(b i1) = 1 ∈ ℤ
10. ∀j:ℕk. ((¬(j = i1 ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
11. ((a i1) = [fst((b i1))] ∈ ℚInterval) ∨ ((a i1) = [snd((b i1))] ∈ ℚInterval)
12. i : ℕk
13. dim(c i) = 1 ∈ ℤ
14. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((b j) = (c j) ∈ ℚInterval))
15. ((b i) = [fst((c i))] ∈ ℚInterval) ∨ ((b i) = [snd((c i))] ∈ ℚInterval)
16. dim(a) = (dim(c) - 2) ∈ ℤ
17. ↑Inhabited(a)
18. dim(b) = (dim(c) - 1) ∈ ℤ
19. ↑Inhabited(b)
20. ↑Inhabited(c)
21. λj.if (j =z i1) then a i1 else c j fi ∈ ℚCube(k)
22. ↑Inhabited(λj.if (j =z i1) then a i1 else c j fi )
23. ¬(i1 = i ∈ ℤ)
24. dim(λj.if (j =z i1) then a i1 else c j fi ) = (dim(c) - 1) ∈ ℤ
25. immediate-rc-face(k;a;λj.if (j =z i1) then a i1 else c j fi )
26. immediate-rc-face(k;λj.if (j =z i1) then a i1 else c j fi ;c)
⊢ ¬((λj.if (j =z i1) then a i1 else c j fi ) = b ∈ ℚCube(k))
4
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. 1 < dim(c)
6. immediate-rc-face(k;a;b)
7. immediate-rc-face(k;b;c)
8. i1 : ℕk
9. dim(b i1) = 1 ∈ ℤ
10. ∀j:ℕk. ((¬(j = i1 ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
11. ((a i1) = [fst((b i1))] ∈ ℚInterval) ∨ ((a i1) = [snd((b i1))] ∈ ℚInterval)
12. i : ℕk
13. dim(c i) = 1 ∈ ℤ
14. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((b j) = (c j) ∈ ℚInterval))
15. ((b i) = [fst((c i))] ∈ ℚInterval) ∨ ((b i) = [snd((c i))] ∈ ℚInterval)
16. dim(a) = (dim(c) - 2) ∈ ℤ
17. ↑Inhabited(a)
18. dim(b) = (dim(c) - 1) ∈ ℤ
19. ↑Inhabited(b)
20. ↑Inhabited(c)
21. λj.if (j =z i1) then a i1 else c j fi ∈ ℚCube(k)
22. ↑Inhabited(λj.if (j =z i1) then a i1 else c j fi )
23. ¬(i1 = i ∈ ℤ)
24. dim(λj.if (j =z i1) then a i1 else c j fi ) = (dim(c) - 1) ∈ ℤ
25. immediate-rc-face(k;a;λj.if (j =z i1) then a i1 else c j fi )
26. immediate-rc-face(k;λj.if (j =z i1) then a i1 else c j fi ;c)
27. ¬((λj.if (j =z i1) then a i1 else c j fi ) = b ∈ ℚCube(k))
28. y : ℚCube(k)
29. immediate-rc-face(k;a;y)
30. immediate-rc-face(k;y;c)
31. ¬(y = b ∈ ℚCube(k))
⊢ y = (λj.if (j =z i1) then a i1 else c j fi ) ∈ ℚCube(k)
Latex:
Latex:
1. k : \mBbbN{}
2. a : \mBbbQ{}Cube(k)
3. b : \mBbbQ{}Cube(k)
4. c : \mBbbQ{}Cube(k)
5. 1 < dim(c)
6. immediate-rc-face(k;a;b)
7. immediate-rc-face(k;b;c)
8. i1 : \mBbbN{}k
9. dim(b i1) = 1
10. \mforall{}j:\mBbbN{}k. ((\mneg{}(j = i1)) {}\mRightarrow{} ((a j) = (b j)))
11. ((a i1) = [fst((b i1))]) \mvee{} ((a i1) = [snd((b i1))])
12. i : \mBbbN{}k
13. dim(c i) = 1
14. \mforall{}j:\mBbbN{}k. ((\mneg{}(j = i)) {}\mRightarrow{} ((b j) = (c j)))
15. ((b i) = [fst((c i))]) \mvee{} ((b i) = [snd((c i))])
16. dim(a) = (dim(c) - 2)
17. \muparrow{}Inhabited(a)
18. dim(b) = (dim(c) - 1)
19. \muparrow{}Inhabited(b)
20. \muparrow{}Inhabited(c)
21. \mlambda{}j.if (j =\msubz{} i1) then a i1 else c j fi \mmember{} \mBbbQ{}Cube(k)
22. \muparrow{}Inhabited(\mlambda{}j.if (j =\msubz{} i1) then a i1 else c j fi )
23. \mneg{}(i1 = i)
24. dim(\mlambda{}j.if (j =\msubz{} i1) then a i1 else c j fi ) = (dim(c) - 1)
\mvdash{} \mexists{}!b':\mBbbQ{}Cube(k). (immediate-rc-face(k;a;b') \mwedge{} immediate-rc-face(k;b';c) \mwedge{} (\mneg{}(b' = b)))
By
Latex:
(D 0 With \mkleeneopen{}\mlambda{}j.if (j =\msubz{} i1) then a i1 else c j fi \mkleeneclose{} THEN Auto)
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