Step
*
1
2
1
2
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. 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(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(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)))
∈ I-face(X;K)
BY
{ ((InstLemma `hd_map` [⌜{f:I-face(X;I)| (f ∈ bx)} ⌝;⌜I-face(X;K)⌝;
⌜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)⌝;
⌜λface.face-image(X;I;K;f;face)⌝]⋅
THENA ((MemTypeCD THENL [Complete (Auto); TACTIC:skip{[tactic]}; TACTIC:Complete (Auto)]) ORELSE Auto)
)
THENM (Reduce (-1) THEN Try (((NthHypEq (-1) THEN EqCD) THEN Try (Trivial))))
) }
1
.....set predicate.....
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. 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)
⊢ 0 < ||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)||
2
.....subterm..... T:t
1:n
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. 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)
16. 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)
⊢ I-face(X;K) = I-face(X;K) ∈ Type
3
.....subterm..... T:t
3:n
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. 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)
16. 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)
⊢ face-image(X;I;K;f;hd(filter(λf.((dimension(f) =z y) ∧b (direction(f) =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)
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. 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)
\mvdash{} 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{}f.((dimension(f) =\msubz{} y) \mwedge{}\msubb{} (direction(f) =\msubz{} c));bx)))
By
Latex:
((InstLemma `hd\_map` [\mkleeneopen{}\{f:I-face(X;I)| (f \mmember{} bx)\} \mkleeneclose{};\mkleeneopen{}I-face(X;K)\mkleeneclose{};
\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)\mkleeneclose{};\mkleeneopen{}\mlambda{}face.face-image(X;I;K;f;face)\mkleeneclose{}]\mcdot{}
THENA ((MemTypeCD THENL [Complete (Auto); TACTIC:skip\{[tactic]\}; TACTIC:Complete (Auto)])
ORELSE Auto
)
)
THENM (Reduce (-1) THEN Try (((NthHypEq (-1) THEN EqCD) THEN Try (Trivial))))
)
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