Step
*
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1
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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. (∀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)))
13. ¬(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))
14. 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)
= filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
∈ ({f:I-face(X;I)| (f ∈ bx)} List)
15. hd(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)))
= face-image(X;I;K;f;hd(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)))
∈ I-face(X;K)
⊢ (¬(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)))
⇒ (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)
= filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
∈ ({f:I-face(X;I)| (f ∈ bx)} List))
⇒ (hd(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx))
= hd(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))
∈ I-face(X;I))
BY
{ ((GenConcl ⌜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)
= L
∈ ({f:I-face(X;I)| (f ∈ bx)} List)⌝⋅
THENA Auto
)
THEN (GenConcl ⌜filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= LL
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List)⌝⋅
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. (∀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)))
13. ¬(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))
14. 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)
= filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
∈ ({f:I-face(X;I)| (f ∈ bx)} List)
15. hd(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)))
= face-image(X;I;K;f;hd(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)))
∈ I-face(X;K)
16. L : {f:I-face(X;I)| (f ∈ bx)} List
17. 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)
= L
∈ ({f:I-face(X;I)| (f ∈ bx)} List)
18. LL : {f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List
19. filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= LL
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List)
⊢ (¬(LL = [] ∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List)))
⇒ (L = LL ∈ ({f:I-face(X;I)| (f ∈ bx)} List))
⇒ (hd(LL) = hd(L) ∈ I-face(X;I))
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. (\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)))
13. \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))
= [])
14. 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)
= filter(\mlambda{}f.((dimension(f) =\msubz{} y) \mwedge{}\msubb{} (direction(f) =\msubz{} c));bx)
15. hd(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)))
= face-image(X;I;K;f;hd(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)))
\mvdash{} (\mneg{}(filter(\mlambda{}f.((dimension(f) =\msubz{} y) \mwedge{}\msubb{} (direction(f) =\msubz{} c));bx) = []))
{}\mRightarrow{} (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)
= filter(\mlambda{}f.((dimension(f) =\msubz{} y) \mwedge{}\msubb{} (direction(f) =\msubz{} c));bx))
{}\mRightarrow{} (hd(filter(\mlambda{}f.((dimension(f) =\msubz{} y) \mwedge{}\msubb{} (direction(f) =\msubz{} c));bx))
= hd(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)))
By
Latex:
((GenConcl \mkleeneopen{}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)
= L\mkleeneclose{}\mcdot{}
THENA Auto
)
THEN (GenConcl \mkleeneopen{}filter(\mlambda{}f.((dimension(f) =\msubz{} y) \mwedge{}\msubb{} (direction(f) =\msubz{} c));bx) = LL\mkleeneclose{}\mcdot{} THENA Auto)
)
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