Nuprl Lemma : nerve_box_edge_same
∀[C:SmallCategory]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[box:open_box(cubical-nerve(C);I;J;x;i)]. ∀[y:nameset(I)]. ∀[c:{c:name-morph(I;[])| (c y) = 0 ∈ ℕ2} ].
∀f:I-face(cubical-nerve(C);I)
(arrow(cube(f)) c flip(c;y) (λx.Ax))
= nerve_box_edge(box;c;y)
∈ (cat-arrow(C) nerve_box_label(box;c) nerve_box_label(box;flip(c;y)))
supposing (f ∈ box) ∧ (direction(f) = (c dimension(f)) ∈ ℕ2) ∧ (direction(f) = (flip(c;y) dimension(f)) ∈ ℕ2)
supposing (∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ Cname))) ∨ (((c x) = i ∈ ℕ2) ∧ (¬↑null(J)))
Proof
Definitions occuring in Statement :
nerve_box_edge: nerve_box_edge(box;c;y),
nerve_box_label: nerve_box_label(box;L),
cubical-nerve: cubical-nerve(X),
open_box: open_box(X;I;J;x;i),
face-cube: cube(f),
face-direction: direction(f),
face-dimension: dimension(f),
I-face: I-face(X;I),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
functor-arrow: arrow(F),
cat-arrow: cat-arrow(C),
small-category: SmallCategory,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
null: null(as),
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
or: P ∨ Q,
prop: ℙ,
and: P ∧ Q,
name-morph: name-morph(I;J),
subtype_rel: A ⊆r B,
top: Top,
guard: {T},
so_lambda: λ2x.t[x],
nameset: nameset(L),
so_apply: x[s],
all: ∀x:A. B[x],
open_box: open_box(X;I;J;x;i),
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
less_than: a < b
Lemmas referenced :
nerve_box_edge_same1,
equal_wf,
int_seg_wf,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
nameset_wf,
top_wf,
l_exists_wf,
coordinate_name_wf,
l_member_wf,
I-face_wf,
cubical-nerve_wf,
face-direction_wf,
face-dimension_wf,
name-morph-flip_wf,
name-morph_wf,
nil_wf,
or_wf,
set_wf,
equal-wf-T-base,
extd-nameset-nil,
open_box_wf,
list_wf,
small-category_wf,
decidable__equal-coordinate_name,
select_wf,
int_seg_properties,
length_wf,
sq_stable__l_member,
sq_stable__le,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
unionElimination,
inlFormation,
productEquality,
natural_numberEquality,
applyEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
inrFormation,
setEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
baseClosed,
productElimination,
independent_functionElimination,
imageMemberEquality,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
applyLambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(C);I;J;x;i)]. \mforall{}[y:nameset(I)]. \mforall{}[c:\{c:name-morph(I;[])| (c y) = 0\} ].
\mforall{}f:I-face(cubical-nerve(C);I)
(arrow(cube(f)) c flip(c;y) (\mlambda{}x.Ax)) = nerve\_box\_edge(box;c;y)
supposing (f \mmember{} box)
\mwedge{} (direction(f) = (c dimension(f)))
\mwedge{} (direction(f) = (flip(c;y) dimension(f)))
supposing (\mexists{}j1\mmember{}J. (\mexists{}j2\mmember{}J. \mneg{}(j1 = j2))) \mvee{} (((c x) = i) \mwedge{} (\mneg{}\muparrow{}null(J)))
Date html generated:
2017_10_05-PM-03_38_29
Last ObjectModification:
2017_07_28-AM-11_25_52
Theory : cubical!sets
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