Nuprl Lemma : nerve_box_label_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)]. ∀[L:name-morph(I;[])].
∀f:I-face(cubical-nerve(C);I)
((ob(cube(f)) L) = nerve_box_label(box;L) ∈ cat-ob(C)) supposing
((f ∈ box) and
(direction(f) = (L dimension(f)) ∈ ℕ2))
supposing ((L x) = i ∈ ℕ2) ∨ (¬↑null(J))
Proof
Definitions occuring in Statement :
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: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
functor-ob: ob(F),
cat-ob: cat-ob(C),
small-category: SmallCategory,
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,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nerve_box_label: nerve_box_label(box;L),
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
and: P ∧ Q,
open_box: open_box(X;I;J;x;i),
top: Top,
nameset: nameset(L),
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
not: ¬A,
I-face: I-face(X;I),
face-dimension: dimension(f),
face-direction: direction(f),
pi2: snd(t),
face-name: face-name(f),
coordinate_name: Cname,
int_upper: {i...},
sq_type: SQType(T),
guard: {T},
int_seg: {i..j-},
false: False,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
cubical-nerve: cubical-nerve(X),
cube-set-restriction: f(s),
label: ...$L... t,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
poset-functor: poset-functor(J;K;f),
functor-comp: functor-comp(F;G),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
isname: isname(z),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
bfalse: ff,
le: A ≤ B,
less_than': less_than'(a;b),
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
nerve-box-face_wf,
subtype_rel_self,
nameset_wf,
extd-nameset_wf,
nil_wf,
coordinate_name_wf,
all_wf,
assert_wf,
isname_wf,
equal_wf,
set_wf,
I-face_wf,
cubical-nerve_wf,
l_member_wf,
int_seg_wf,
face-direction_wf,
face-dimension_wf,
extd-nameset-nil,
or_wf,
not_wf,
null_wf3,
subtype_rel_list,
top_wf,
name-morph_wf,
open_box_wf,
list_wf,
small-category_wf,
decidable__equal-coordinate_name,
pairwise-implies,
face-name_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
lelt_wf,
face-cube_wf,
squash_wf,
true_wf,
cubical-set_wf,
cubical-nerve-I-cube,
name-morph_subtype,
list-diff_wf,
cname_deq_wf,
cons_wf,
nameset_subtype,
list-diff-subset,
cat-functor_wf,
poset-cat_wf,
functor-ob_wf,
cat-ob_wf,
adjacent-compatible-iff,
I-cube_wf,
functor-comp_wf,
poset-functor_wf,
member_wf,
iff_weakening_equal,
face-map_wf2,
subtype_rel_wf,
list-diff2,
list-diff2-sym,
ob_mk_functor_lemma,
ob_pair_lemma,
cat_ob_pair_lemma,
face-map-comp-id,
decidable__equal_int,
int_seg_properties,
false_wf,
int_seg_subtype,
int_seg_cases,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
member-nameset-diff,
intformeq_wf,
intformnot_wf,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
decidable__le,
member_singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
applyEquality,
sqequalRule,
setEquality,
functionEquality,
hypothesis,
lambdaEquality,
functionExtensionality,
independent_isectElimination,
productEquality,
setElimination,
rename,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
voidElimination,
voidEquality,
productElimination,
unionElimination,
cumulativity,
promote_hyp,
instantiate,
intEquality,
independent_pairEquality,
imageElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
universeEquality,
independent_pairFormation,
hypothesis_subsumption,
addEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
dependent_set_memberEquality,
addLevel,
impliesFunctionality
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{}[L:name-morph(I;[])].
\mforall{}f:I-face(cubical-nerve(C);I)
((ob(cube(f)) L) = nerve\_box\_label(box;L)) supposing
((f \mmember{} box) and
(direction(f) = (L dimension(f))))
supposing ((L x) = i) \mvee{} (\mneg{}\muparrow{}null(J))
Date html generated:
2017_10_05-PM-03_37_07
Last ObjectModification:
2017_07_28-AM-11_25_29
Theory : cubical!sets
Home
Index