Nuprl Lemma : groupoid-nerve-filler-fills
∀G:Groupoid. ∀I:Cname List. ∀J:nameset(I) List. ∀x:nameset(I). ∀i:ℕ2. ∀bx:open_box(cubical-nerve(cat(G));I;J;x;i).
fills-open_box(cubical-nerve(cat(G));I;bx;(λI,J,x,i,box. groupoid-nerve-filler(G;I;J;x;i;box)) I J x i bx)
Proof
Definitions occuring in Statement :
groupoid-nerve-filler: groupoid-nerve-filler(G;I;J;x;i;box),
cubical-nerve: cubical-nerve(X),
fills-open_box: fills-open_box(X;I;bx;cube),
open_box: open_box(X;I;J;x;i),
nameset: nameset(L),
coordinate_name: Cname,
groupoid-cat: cat(G),
groupoid: Groupoid,
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
apply: f a,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
groupoid-nerve-filler: groupoid-nerve-filler(G;I;J;x;i;box),
top: Top,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cons: [a / b],
prop: ℙ,
ge: i ≥ j ,
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_stable: SqStable(P),
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
true: True,
I-face: I-face(X;I),
face-name: face-name(f),
pi1: fst(t),
pi2: snd(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
fills-open_box: fills-open_box(X;I;bx;cube),
fills-faces: fills-faces(X;I;bx;L),
l_all: (∀x∈L.P[x]),
select: L[n],
groupoid-nerve-filler0: groupoid-nerve-filler0(I;x;box),
cubical-nerve: cubical-nerve(X),
is-face: is-face(X;I;bx;f),
face-cube: cube(f),
face-direction: direction(f),
face-dimension: dimension(f),
cube-set-restriction: f(s),
le: A ≤ B,
open_box: open_box(X;I;J;x;i),
l_exists: (∃x∈L. P[x]),
nil: [],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
I-cube: X(I),
cand: A c∧ B,
less_than: a < b,
name-morph: name-morph(I;J),
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
uext: uext(g),
isname: isname(z),
le_int: i ≤z j,
lt_int: i <z j,
less_than': less_than'(a;b),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
groupoid-nerve-filler2: groupoid-nerve-filler2(C;I;J;box),
poset-functor: poset-functor(J;K;f),
functor-comp: functor-comp(F;G),
mk-functor: mk-functor,
cat-ob: cat-ob(C),
poset-cat: poset-cat(J),
nat: ℕ,
subtract: n - m,
listp: A List+,
groupoid-nerve-filler1: groupoid-nerve-filler1(G;I;J;x;i;box)
Lemmas referenced :
open_box_wf,
cubical-nerve_wf,
groupoid-cat_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
groupoid_wf,
null_wf3,
top_wf,
istype-void,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
list-cases,
product_subtype_list,
open_box-nil,
subtype_rel_weakening,
nil_wf,
I-face_wf,
equal-wf-base,
equal_wf,
face-name_wf,
hd_wf,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
istype-int,
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_eq_lemma,
int_formula_prop_wf,
length_of_nil_lemma,
int_subtype_base,
length_of_cons_lemma,
reduce_hd_cons_lemma,
pi1_wf_top,
nameset_subtype_base,
set_subtype_base,
lelt_wf,
cubical-nerve-I-cube,
decidable__equal_int,
int_seg_subtype_special,
int_seg_cases,
intformless_wf,
int_formula_prop_less_lemma,
length_wf,
cat-functor_wf,
poset-cat_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
iota-wf,
poset-functor_wf,
face-map_wf2,
squash_wf,
true_wf,
istype-universe,
functor-comp-assoc,
functor-comp_wf,
small-category_wf,
poset-functor-comp,
subtype_rel_self,
iff_weakening_equal,
iota-identity,
poset-id-functor,
functor-comp-id,
cons_member,
l_member_wf,
member-list-diff,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
null_nil_lemma,
btrue_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
istype-less_than,
length-open_box-ge-3,
stuck-spread,
istype-base,
ob_pair_lemma,
poset-functor-extends-box-faces,
select_wf,
decidable__lt,
not_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
decidable__equal_int_seg,
name-comp_wf,
name-morph_wf,
nerve_box_label_wf,
member_singleton,
le_wf,
extd-nameset-nil,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
isname-name,
name-morph-flip_wf,
nerve_box_edge_wf,
subtype_rel-equal,
cat-arrow_wf,
or_wf,
name-morph-flip-face-map1,
eq_int_eq_true,
istype-false,
poset-extend-unique,
groupoid-nerve-filler2_wf,
poset_functor_extend-extends,
poset-functor-extends_wf,
cat-ob_wf,
cat-id_wf,
cat-comp_wf,
arrow_pair_lemma,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
poset_functor_extend-face-map,
assert_of_le_int,
extd-nameset_subtype_int,
length-open_box-le-3,
istype-assert,
poset-functor-extends-box-faces-1,
nerve_box_edge1_wf,
listp_properties,
length_wf_nat,
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,
list-subtype,
groupoid-nerve-filler1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
sqequalRule,
natural_numberEquality,
isect_memberEquality_alt,
voidElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation_alt,
equalityIsType3,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
baseClosed,
equalityIsType1,
hypothesis_subsumption,
setEquality,
productEquality,
closedConclusion,
imageMemberEquality,
imageElimination,
approximateComputation,
int_eqEquality,
independent_pairFormation,
independent_pairEquality,
intEquality,
applyLambdaEquality,
universeEquality,
hyp_replacement,
inlFormation_alt,
productIsType,
functionIsType,
dependent_set_memberEquality_alt,
equalityIsType4,
inrFormation_alt,
equalityIsType2,
setIsType,
dependent_pairEquality_alt,
functionEquality,
addEquality,
minusEquality
Latex:
\mforall{}G:Groupoid. \mforall{}I:Cname List. \mforall{}J:nameset(I) List. \mforall{}x:nameset(I). \mforall{}i:\mBbbN{}2.
\mforall{}bx:open\_box(cubical-nerve(cat(G));I;J;x;i).
fills-open\_box(cubical-nerve(cat(G));I;bx;(\mlambda{}I,J,x,i,box. groupoid-nerve-filler(G;I;J;x;i;box)) I J\000C x i bx)
Date html generated:
2019_11_05-PM-00_39_15
Last ObjectModification:
2018_11_08-PM-11_57_37
Theory : cubical!sets
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