Nuprl Lemma : groupoid-nerve-filler0_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-filler0(I;x;box) ∈ cubical-nerve(cat(G))(I) supposing ↑null(J)
Proof
Definitions occuring in Statement :
groupoid-nerve-filler0: groupoid-nerve-filler0(I;x;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,
null: null(as),
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
top: Top,
bfalse: ff,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
nameset: nameset(L),
l_subset: l_subset(T;as;bs),
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
groupoid-nerve-filler0: groupoid-nerve-filler0(I;x;box),
groupoid: Groupoid,
groupoid-cat: cat(G),
pi1: fst(t),
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
I-face: I-face(X;I),
pi2: snd(t),
l_all: (∀x∈L.P[x]),
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
less_than: a < b,
true: True,
decidable: Dec(P),
uiff: uiff(P;Q),
select: L[n]
Lemmas referenced :
nameset_wf,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
open_box_wf,
cubical-nerve_wf,
groupoid-cat_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
groupoid_wf,
cons_member,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
member-list-diff,
l_member_wf,
not_wf,
or_wf,
equal_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
iota_wf,
name-morph_subtype,
nameset_subtype,
cubical-nerve-I-cube,
sq_stable__l_member,
decidable__equal-coordinate_name,
functor-comp_wf,
poset-cat_wf,
poset-functor_wf,
I-face_wf,
length_of_nil_lemma,
int_seg_properties,
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,
reduce_hd_cons_lemma,
le_wf,
length_of_cons_lemma,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
intformnot_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
lelt_wf,
length_wf,
member_singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_isectElimination,
lambdaEquality,
because_Cache,
setElimination,
rename,
natural_numberEquality,
lambdaFormation,
independent_functionElimination,
addLevel,
orFunctionality,
productEquality,
instantiate,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
cumulativity,
dependent_set_memberEquality,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
addEquality,
equalityElimination
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-filler0(I;x;box) \mmember{} cubical-nerve(cat(G))(I) supposing \muparrow{}null(J)
Date html generated:
2017_10_05-PM-03_43_38
Last ObjectModification:
2017_07_28-AM-11_27_03
Theory : cubical!sets
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