Nuprl Lemma : groupoid-nerve-filler1_wf
∀[G:Groupoid]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[box:open_box(cubical-nerve(cat(G));I;J;x;i)].
groupoid-nerve-filler1(G;I;J;x;i;box) ∈ cubical-nerve(cat(G))(I) supposing (¬↑null(J)) ∧ (||box|| ≤ 3)
Proof
Definitions occuring in Statement :
groupoid-nerve-filler1: groupoid-nerve-filler1(G;I;J;x;i;box),
cubical-nerve: cubical-nerve(X),
open_box: open_box(X;I;J;x;i),
I-cube: X(I),
nameset: nameset(L),
coordinate_name: Cname,
groupoid-cat: cat(G),
groupoid: Groupoid,
length: ||as||,
null: null(as),
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
open_box: open_box(X;I;J;x;i),
nameset: nameset(L),
groupoid-nerve-filler1: groupoid-nerve-filler1(G;I;J;x;i;box),
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
guard: {T},
name-morph: name-morph(I;J),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
cons: [a / b],
bfalse: ff,
nat: ℕ,
int_seg: {i..j-},
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
subtract: n - m,
less_than': less_than'(a;b),
listp: A List+,
groupoid-cat: cat(G)
Lemmas referenced :
length-open_box-le-3,
cubical-nerve_wf,
groupoid-cat_wf,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
nameset_wf,
top_wf,
le_wf,
length_wf,
I-face_wf,
open_box_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
groupoid_wf,
cubical-nerve-I-cube,
poset_functor_extend-is-functor,
nerve_box_label_wf,
decidable__assert,
equal_wf,
extd-nameset-nil,
name-morph_wf,
nil_wf,
equal-wf-T-base,
nerve_box_edge1_wf,
hd_wf,
listp_properties,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
cat-arrow_wf,
name-morph-flip_wf,
groupoid-edges-commute1,
hd_member,
list-subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
setElimination,
rename,
natural_numberEquality,
unionElimination,
inlFormation,
inrFormation,
lambdaFormation,
baseClosed,
promote_hyp,
hypothesis_subsumption,
addEquality,
independent_pairFormation,
intEquality,
minusEquality,
dependent_set_memberEquality,
setEquality
Latex:
\mforall{}[G:Groupoid]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(cat(G));I;J;x;i)].
groupoid-nerve-filler1(G;I;J;x;i;box) \mmember{} cubical-nerve(cat(G))(I)
supposing (\mneg{}\muparrow{}null(J)) \mwedge{} (||box|| \mleq{} 3)
Date html generated:
2017_10_05-PM-03_43_44
Last ObjectModification:
2017_07_28-AM-11_27_08
Theory : cubical!sets
Home
Index