Nuprl Lemma : nerve-box-face_wf
∀[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:cat-ob(poset-cat(I))].
nerve-box-face(box;L) ∈ {f:I-face(cubical-nerve(C);I)| (f ∈ box) ∧ (direction(f) = (L dimension(f)) ∈ ℕ2)}
supposing ((L x) = i ∈ ℕ2) ∨ (¬↑null(J))
Proof
Definitions occuring in Statement :
nerve-box-face: nerve-box-face(box;L),
cubical-nerve: cubical-nerve(X),
poset-cat: poset-cat(J),
open_box: open_box(X;I;J;x;i),
face-direction: direction(f),
face-dimension: dimension(f),
I-face: I-face(X;I),
nameset: nameset(L),
coordinate_name: Cname,
cat-ob: cat-ob(C),
small-category: SmallCategory,
l_member: (x ∈ l),
null: null(as),
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
poset-cat: poset-cat(J),
cat-ob: cat-ob(C),
pi1: fst(t),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
open_box: open_box(X;I;J;x;i),
all: ∀x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
uiff: uiff(P;Q),
so_apply: x[s],
int_seg: {i..j-},
prop: ℙ,
nerve-box-face: nerve-box-face(box;L),
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
cand: A c∧ B,
guard: {T},
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
l_exists: (∃x∈L. P[x]),
less_than: a < b,
I-face: I-face(X;I),
face-dimension: dimension(f),
face-direction: direction(f),
face-name: face-name(f),
pi2: snd(t),
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
sq_type: SQType(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
cons: [a / b],
bfalse: ff,
rev_implies: P ⇐ Q
Lemmas referenced :
non_null_filter_iff,
I-face_wf,
cubical-nerve_wf,
eq_int_wf,
face-direction_wf,
face-dimension_wf,
hd-wf-not-null,
filter_wf5,
int_seg_wf,
extd-nameset_subtype_int,
nil_wf,
coordinate_name_wf,
l_member_wf,
hd_member,
member_filter,
assert_of_eq_int,
int_seg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
equal_wf,
extd-nameset-nil,
or_wf,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
nameset_wf,
top_wf,
name-morph_wf,
open_box_wf,
list_wf,
small-category_wf,
select_wf,
length_wf,
and_wf,
pi1_wf_top,
subtype_rel_product,
subtype_base_sq,
nameset_subtype_base,
pi2_wf,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
cons_member,
cons_wf,
set_subtype_base,
int_subtype_base,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
productElimination,
lambdaEquality,
applyEquality,
because_Cache,
independent_isectElimination,
natural_numberEquality,
setEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
lambdaFormation,
imageMemberEquality,
baseClosed,
imageElimination,
applyLambdaEquality,
productEquality,
axiomEquality,
instantiate,
cumulativity,
independent_pairEquality,
promote_hyp,
hypothesis_subsumption,
inlFormation
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:cat-ob(poset-cat(I))].
nerve-box-face(box;L) \mmember{} \{f:I-face(cubical-nerve(C);I)|
(f \mmember{} box) \mwedge{} (direction(f) = (L dimension(f)))\}
supposing ((L x) = i) \mvee{} (\mneg{}\muparrow{}null(J))
Date html generated:
2017_10_05-PM-03_36_29
Last ObjectModification:
2017_07_28-AM-11_25_14
Theory : cubical!sets
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