Nuprl Lemma : groupoid-nerve-filler-uniform
∀G:Groupoid. uniform-Kan-filler(cubical-nerve(cat(G));λI,J,x,i,box. groupoid-nerve-filler(G;I;J;x;i;box))
Proof
Definitions occuring in Statement :
groupoid-nerve-filler: groupoid-nerve-filler(G;I;J;x;i;box),
cubical-nerve: cubical-nerve(X),
uniform-Kan-filler: uniform-Kan-filler(X;filler),
all: ∀x:A. B[x],
lambda: λx.A[x],
groupoid-cat: cat(G),
groupoid: Groupoid
Definitions unfolded in proof :
all: ∀x:A. B[x],
uniform-Kan-filler: uniform-Kan-filler(X;filler),
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
name-morph: name-morph(I;J),
nameset: nameset(L),
prop: ℙ,
cat-functor: Functor(C1;C2),
le: A ≤ B,
groupoid-nerve-filler0: groupoid-nerve-filler0(I;x;box),
groupoid-nerve-filler: groupoid-nerve-filler(G;I;J;x;i;box),
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
so_apply: x[s],
so_lambda: λ2x.t[x],
pi2: snd(t),
pi1: fst(t),
face-name: face-name(f),
I-face: I-face(X;I),
cons: [a / b],
true: True,
sq_type: SQType(T),
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
int_upper: {i...},
coordinate_name: Cname,
squash: ↓T,
sq_stable: SqStable(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
ge: i ≥ j ,
and: P ∧ Q,
guard: {T},
uiff: uiff(P;Q),
spreadn: spread3,
face-image: face-image(X;I;K;f;face),
cube-set-restriction: f(s),
open_box_image: open_box_image(X;I;K;f;bx),
cubical-nerve: cubical-nerve(X),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uext: uext(g),
compose: f o g,
name-comp: (f o g),
iota: iota(x),
bnot: ¬bb,
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
l_member: (x ∈ l),
cand: A c∧ B,
less_than: a < b,
nat: ℕ,
ext-eq: A ≡ B,
I-cube: X(I),
functor-ob: ob(F),
exposed-it: exposed-it,
respects-equality: respects-equality(S;T),
nerve_box_label: nerve_box_label(box;L),
face-direction: direction(f),
face-dimension: dimension(f),
face-cube: cube(f),
open_box: open_box(X;I;J;x;i),
l_all: (∀x∈L.P[x]),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
functor-comp: functor-comp(F;G),
poset-functor: poset-functor(J;K;f),
poset-cat: poset-cat(J),
groupoid-nerve-filler2: groupoid-nerve-filler2(C;I;J;box),
groupoid-nerve-filler1: groupoid-nerve-filler1(G;I;J;x;i;box),
cat-ob: cat-ob(C),
l_exists: (∃x∈L. P[x]),
name-morph-flip: flip(f;y),
istype: istype(T),
nerve_box_edge: nerve_box_edge(box;c;y),
functor-arrow: arrow(F),
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
listp: A List+,
nerve_box_edge1: nerve_box_edge1(G;box;x;i;j;c;y),
bor: p ∨bq,
nequal: a ≠ b ∈ T ,
nerve_box_edge': nerve_box_edge'(box; c; y),
label: ...$L... t,
small-category: SmallCategory,
cat-arrow: cat-arrow(C),
cat-id: cat-id(C)
Lemmas referenced :
decidable__assert,
null_wf3,
subtype_rel_list,
top_wf,
istype-void,
istype-assert,
isname_wf,
l_member_wf,
coordinate_name_wf,
nameset_wf,
name-morph_wf,
open_box_wf,
cubical-nerve_wf,
groupoid-cat_wf,
int_seg_wf,
list_wf,
groupoid_wf,
btrue_neq_bfalse,
bfalse_wf,
null_cons_lemma,
cons_wf,
assert_elim,
cat-comp_wf,
cat-id_wf,
cat-arrow_wf,
poset-cat_wf,
cat-ob_wf,
int_term_value_add_lemma,
itermAdd_wf,
non_neg_length,
lelt_wf,
set_subtype_base,
nameset_subtype_base,
null_nil_lemma,
pi1_wf_top,
pi2_wf,
reduce_hd_cons_lemma,
length_of_cons_lemma,
product_subtype_list,
int_subtype_base,
subtype_base_sq,
length_of_nil_lemma,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
sq_stable__le,
decidable__equal-coordinate_name,
sq_stable__l_member,
int_seg_properties,
hd_wf,
face-name_wf,
equal_wf,
equal-wf-base,
I-face_wf,
nil_wf,
subtype_rel_weakening,
open_box-nil,
map_nil_lemma,
list-cases,
cubical-nerve-I-cube,
iota-wf,
assert-isname,
map_cons_lemma,
name-morph_subtype_remove1,
poset-functor_wf,
cname_deq_wf,
list-diff_wf,
cat-functor_wf,
squash_wf,
true_wf,
istype-universe,
functor-comp-assoc,
subtype_rel_self,
iff_weakening_equal,
small-category_wf,
functor-comp_wf,
poset-functor-comp,
name-morph-ext,
name-comp_wf,
isname-name,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
list-diff-subset,
nameset_subtype,
le_wf,
select_wf,
nat_properties,
list-subtype,
map_wf,
name-morph-domain_wf,
subtype_rel_transitivity,
assert_wf,
cons_member,
name-morph-domain_subtype,
poset-functors-equal,
cube-set-restriction_wf,
groupoid-nerve-filler_wf,
open_box_image_wf,
null-map,
assert_of_null,
length_wf,
btrue_wf,
not_assert_elim,
iff_weakening_uiff,
equal-wf-T-base,
extd-nameset-nil,
nerve_box_label_same,
nerve-box-face_wf,
istype-less_than,
istype-le,
I-cube_wf,
length-open_box_image,
select-map,
face-image_wf,
less_than_wf,
ob_pair_lemma,
ob_mk_functor_lemma,
functor-ob_wf,
name-morph_subtype,
cat_ob_pair_lemma,
length_wf_nat,
lt_int_wf,
assert_of_lt_int,
length-open_box-ge-3,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
arrow_pair_lemma,
arrow_mk_functor_lemma,
extd-nameset-respects-equality,
length-open_box-le-3,
poset_functor_extend-flip,
nerve_box_label_wf,
nerve_box_edge_wf,
subtype_rel_set,
name-morph-flip_wf,
subtype_rel-equal,
decidable__l_exists,
decidable__cand,
select_member,
not-assert-isname,
assert-eq-cname,
eq-cname_wf,
nsub2_subtype_extd-nameset,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
decidable__equal_int,
name-morph-one-one,
isname-nameset,
not_wf,
or_wf,
extd-nameset_subtype_int,
poset_functor_extend_wf,
nerve_box_edge_same,
nerve-box-common-face_wf,
member_wf,
groupoid-nerve-functor-flip,
name-comp-flip,
respects-equality-set-trivial2,
l_exists_iff,
poset_functor_extend_same,
member-map,
functor-arrow-id,
functor-arrow_wf,
le_reflexive,
assert_of_le_int,
le_int_wf,
assert_witness,
cat_id_tuple_lemma,
cat_arrow_triple_lemma,
equal_functionality_wrt_subtype_rel2,
poset-cat-arrow-flip,
member-poset-cat-arrow,
poset-cat-arrow_subtype,
nerve_box_edge1_wf,
listp_properties,
istype-false,
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,
l_exists_wf,
nerve-box-common-face_wf2,
nerve_box_edge_same1,
iff_imp_equal_bool,
iff_functionality_wrt_iff,
istype-base,
respects-equality_weakening,
respects-equality-trivial,
respects-equality-product,
bnot_wf,
hd-map,
bool_cases,
iff_transitivity,
assert_of_bnot,
hd_member,
bor_wf,
eq_int_wf,
assert_of_bor,
assert_of_eq_int,
neg_assert_of_eq_int,
not-equal-2,
nequal_wf,
int_seg_subtype_special,
int_seg_cases,
groupoid-square1_wf,
nerve_box_edge'_wf,
name-morph-flip-flip,
name-morph-flips-commute,
cat-square-commutes_wf,
nerve_box_edge_wf2,
member_map,
l_member_subtype,
groupoid-square2_wf,
bor-btrue
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
independent_isectElimination,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
unionElimination,
setElimination,
rename,
functionIsType,
inhabitedIsType,
universeIsType,
natural_numberEquality,
equalityIstype,
productIsType,
dependent_set_memberEquality_alt,
applyLambdaEquality,
hypothesis_subsumption,
promote_hyp,
intEquality,
cumulativity,
instantiate,
independent_pairEquality,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
productElimination,
closedConclusion,
productEquality,
setEquality,
universeEquality,
equalityElimination,
functionExtensionality,
Error :memTop,
setIsType,
sqequalBase,
equalityIsType3,
inrFormation_alt,
equalityIsType1,
hyp_replacement,
dependent_pairEquality_alt,
inlFormation_alt,
functionEquality,
addEquality,
minusEquality,
unionIsType,
baseApply
Latex:
\mforall{}G:Groupoid
uniform-Kan-filler(cubical-nerve(cat(G));\mlambda{}I,J,x,i,box. groupoid-nerve-filler(G;I;J;x;i;box))
Date html generated:
2020_05_21-AM-11_05_56
Last ObjectModification:
2019_12_10-PM-03_20_02
Theory : cubical!sets
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