Step
*
1
2
1
1
3
1
of Lemma
get_face_image
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. (∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc))))))
∧ (map(f;J) ∈ nameset(K) List)
∧ (f x ∈ nameset(K))
∧ (f y ∈ nameset(map(f;J)))
14. ¬(map(λface.face-image(X;I;K;f;face);filter(λx.((dimension(face-image(X;I;K;f;x)) =z f y)
∧b (direction(face-image(X;I;K;f;x)) =z c));bx))
= []
∈ ({f@0:{f@0:I-face(X;K)| (f@0 ∈ map(λface.face-image(X;I;K;f;face);bx))} |
↑((dimension(f@0) =z f y) ∧b (direction(f@0) =z c))} List))
15. x2 : nameset(I)
16. x4 : ℕ2
17. x5 : X(I-[x2])
18. (<x2, x4, x5> ∈ bx)
⊢ (x2 =z y) ∧b (x4 =z c) = (f x2 =z f y) ∧b (x4 =z c)
BY
{ TACTIC:((BoolCase ⌜(x2 =z y)⌝⋅ THENA Auto) THEN (BoolCase ⌜(f x2 =z f y)⌝⋅ THENA Auto)) }
1
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. (∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc))))))
∧ (map(f;J) ∈ nameset(K) List)
∧ (f x ∈ nameset(K))
∧ (f y ∈ nameset(map(f;J)))
14. ¬(map(λface.face-image(X;I;K;f;face);filter(λx.((dimension(face-image(X;I;K;f;x)) =z f y)
∧b (direction(face-image(X;I;K;f;x)) =z c));bx))
= []
∈ ({f@0:{f@0:I-face(X;K)| (f@0 ∈ map(λface.face-image(X;I;K;f;face);bx))} |
↑((dimension(f@0) =z f y) ∧b (direction(f@0) =z c))} List))
15. x2 : nameset(I)
16. x4 : ℕ2
17. x5 : X(I-[x2])
18. (<x2, x4, x5> ∈ bx)
19. x2 = y ∈ ℤ
20. (f x2) = (f y) ∈ ℤ
⊢ (x4 =z c) = (x4 =z c)
2
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. (∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc))))))
∧ (map(f;J) ∈ nameset(K) List)
∧ (f x ∈ nameset(K))
∧ (f y ∈ nameset(map(f;J)))
14. ¬(map(λface.face-image(X;I;K;f;face);filter(λx.((dimension(face-image(X;I;K;f;x)) =z f y)
∧b (direction(face-image(X;I;K;f;x)) =z c));bx))
= []
∈ ({f@0:{f@0:I-face(X;K)| (f@0 ∈ map(λface.face-image(X;I;K;f;face);bx))} |
↑((dimension(f@0) =z f y) ∧b (direction(f@0) =z c))} List))
15. x2 : nameset(I)
16. f x2 ≠ f y
17. x4 : ℕ2
18. x5 : X(I-[x2])
19. (<x2, x4, x5> ∈ bx)
20. x2 = y ∈ ℤ
⊢ (x4 =z c) = ff
3
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. (∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc))))))
∧ (map(f;J) ∈ nameset(K) List)
∧ (f x ∈ nameset(K))
∧ (f y ∈ nameset(map(f;J)))
14. ¬(map(λface.face-image(X;I;K;f;face);filter(λx.((dimension(face-image(X;I;K;f;x)) =z f y)
∧b (direction(face-image(X;I;K;f;x)) =z c));bx))
= []
∈ ({f@0:{f@0:I-face(X;K)| (f@0 ∈ map(λface.face-image(X;I;K;f;face);bx))} |
↑((dimension(f@0) =z f y) ∧b (direction(f@0) =z c))} List))
15. x2 : nameset(I)
16. x2 ≠ y
17. x4 : ℕ2
18. x5 : X(I-[x2])
19. (<x2, x4, x5> ∈ bx)
20. (f x2) = (f y) ∈ ℤ
⊢ ff = (x4 =z c)
4
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. (∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc))))))
∧ (map(f;J) ∈ nameset(K) List)
∧ (f x ∈ nameset(K))
∧ (f y ∈ nameset(map(f;J)))
14. ¬(map(λface.face-image(X;I;K;f;face);filter(λx.((dimension(face-image(X;I;K;f;x)) =z f y)
∧b (direction(face-image(X;I;K;f;x)) =z c));bx))
= []
∈ ({f@0:{f@0:I-face(X;K)| (f@0 ∈ map(λface.face-image(X;I;K;f;face);bx))} |
↑((dimension(f@0) =z f y) ∧b (direction(f@0) =z c))} List))
15. x2 : nameset(I)
16. f x2 ≠ f y
17. x2 ≠ y
18. x4 : ℕ2
19. x5 : X(I-[x2])
20. (<x2, x4, x5> ∈ bx)
⊢ ff = ff
Latex:
Latex:
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : \mBbbN{}2
6. bx : open\_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : \mBbbN{}2
10. y : nameset(J)
11. nameset([x / J]) \msubseteq{}r name-morph-domain(f;I)
12. \mneg{}(filter(\mlambda{}f.((dimension(f) =\msubz{} y) \mwedge{}\msubb{} (direction(f) =\msubz{} c));bx) = [])
13. (\mforall{}fc:I-face(X;I). ((fc \mmember{} bx) {}\mRightarrow{} (\muparrow{}isname(f (fst(fc))))))
\mwedge{} (map(f;J) \mmember{} nameset(K) List)
\mwedge{} (f x \mmember{} nameset(K))
\mwedge{} (f y \mmember{} nameset(map(f;J)))
14. \mneg{}(map(\mlambda{}face.face-image(X;I;K;f;face);filter(\mlambda{}x.((dimension(face-image(X;I;K;f;x)) =\msubz{} f y)
\mwedge{}\msubb{} (direction(face-image(X;I;K;f;x)) =\msubz{} c));bx))
= [])
15. x2 : nameset(I)
16. x4 : \mBbbN{}2
17. x5 : X(I-[x2])
18. (<x2, x4, x5> \mmember{} bx)
\mvdash{} (x2 =\msubz{} y) \mwedge{}\msubb{} (x4 =\msubz{} c) = (f x2 =\msubz{} f y) \mwedge{}\msubb{} (x4 =\msubz{} c)
By
Latex:
TACTIC:((BoolCase \mkleeneopen{}(x2 =\msubz{} y)\mkleeneclose{}\mcdot{} THENA Auto) THEN (BoolCase \mkleeneopen{}(f x2 =\msubz{} f y)\mkleeneclose{}\mcdot{} THENA Auto))
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