Nuprl Lemma : poset-functor-extends-box-faces
∀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).
(((¬↑null(J)) ∧ (∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ Cname))))
⇒ (∀fc∈bx.poset-functor-extends(cat(G);I-[dimension(fc)];λx.nerve_box_label(bx;((dimension(fc):=direction(fc)) o x));
λz,f. nerve_box_edge(bx;((dimension(fc):=direction(fc)) o f);z);cube(fc))))
Proof
Definitions occuring in Statement :
nerve_box_edge: nerve_box_edge(box;c;y),
nerve_box_label: nerve_box_label(box;L),
cubical-nerve: cubical-nerve(X),
poset-functor-extends: poset-functor-extends(C;I;L;E;F),
open_box: open_box(X;I;J;x;i),
face-cube: cube(f),
face-direction: direction(f),
face-dimension: dimension(f),
name-comp: (f o g),
face-map: (x:=i),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
groupoid-cat: cat(G),
groupoid: Groupoid,
list-diff: as-bs,
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
null: null(as),
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
l_all: (∀x∈L.P[x]),
member: t ∈ T,
uall: ∀[x:A]. B[x],
open_box: open_box(X;I;J;x;i),
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
nameset: nameset(L),
lelt: i ≤ j < k,
sq_stable: SqStable(P),
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
less_than: a < b,
subtype_rel: A ⊆r B,
poset-functor-extends: poset-functor-extends(C;I;L;E;F),
name-morph: name-morph(I;J),
respects-equality: respects-equality(S;T),
so_lambda: λ2x.t[x],
so_apply: x[s],
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
uext: uext(g),
ifthenelse: if b then t else f fi ,
btrue: tt,
isname: isname(z),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
I-face: I-face(X;I),
face-direction: direction(f),
face-dimension: dimension(f),
face-cube: cube(f),
pi1: fst(t),
pi2: snd(t),
poset-cat: poset-cat(J),
cat-ob: cat-ob(C),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
assert: ↑b,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
name-morph-flip: flip(f;y),
l_member: (x ∈ l),
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j
Lemmas referenced :
name-comp_wf,
list-diff_wf,
coordinate_name_wf,
cname_deq_wf,
cons_wf,
face-dimension_wf,
cubical-nerve_wf,
groupoid-cat_wf,
select_wf,
I-face_wf,
int_seg_properties,
length_wf,
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,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
nil_wf,
face-map_wf2,
face-direction_wf,
name-morph_wf,
extd-nameset-respects-equality,
int_seg_wf,
nameset_wf,
istype-assert,
null_wf3,
subtype_rel_list,
top_wf,
l_exists_wf,
l_member_wf,
not_wf,
equal_wf,
open_box_wf,
list_wf,
groupoid_wf,
nerve_box_label_same,
set_subtype_base,
lelt_wf,
int_subtype_base,
select_member,
subtype_base_sq,
bool_wf,
bool_subtype_base,
squash_wf,
true_wf,
istype-universe,
eq_int_eq_true,
btrue_wf,
subtype_rel_self,
iff_weakening_equal,
decidable__equal_int,
bfalse_wf,
int_seg_subtype_special,
int_seg_cases,
cat-ob_wf,
cubical-nerve-I-cube,
functor-ob_wf,
poset-cat_wf,
name-morph-ext,
name-morph_subtype,
nameset_subtype,
list-diff-subset,
member-list-diff,
member_singleton,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
istype-le,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
uext-ap-name,
le_wf,
respects-equality-set-trivial2,
extd-nameset-nil,
nerve_box_edge_same,
eq-cname_wf,
assert-eq-cname,
iff_weakening_uiff,
assert_wf,
assert_elim,
bnot_wf,
btrue_neq_bfalse,
not_assert_elim,
cat-arrow_wf,
nerve_box_label_wf,
name-morph-flip_wf,
nsub2_subtype_extd-nameset,
istype-less_than,
isname-name,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
or_wf,
equal-wf-T-base,
small-category_wf,
functor-arrow_wf,
extd-nameset_wf,
extd-nameset_subtype_base,
member-poset-cat-arrow,
subtype_rel_set,
poset-cat-arrow-flip,
subtype_rel-equal,
int_seg_subtype_nat,
istype-false,
cat-functor_wf,
nat_properties,
respects-equality-product,
I-cube_wf,
respects-equality-trivial,
respects-equality_weakening,
istype-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
because_Cache,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
dependent_functionElimination,
independent_functionElimination,
inhabitedIsType,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
applyEquality,
setIsType,
equalityIstype,
sqequalBase,
equalitySymmetry,
productIsType,
functionIsType,
functionExtensionality,
equalityTransitivity,
inrFormation_alt,
intEquality,
instantiate,
cumulativity,
universeEquality,
hypothesis_subsumption,
hyp_replacement,
promote_hyp,
equalityElimination,
dependent_set_memberEquality_alt,
applyLambdaEquality,
inlFormation_alt,
baseApply,
closedConclusion,
dependent_pairEquality_alt,
independent_pairEquality,
productEquality
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).
(((\mneg{}\muparrow{}null(J)) \mwedge{} (\mexists{}j1\mmember{}J. (\mexists{}j2\mmember{}J. \mneg{}(j1 = j2))))
{}\mRightarrow{} (\mforall{}fc\mmember{}bx.poset-functor-extends(cat(G);I-[dimension(fc)];
\mlambda{}x.nerve\_box\_label(bx;((dimension(fc):=direction(fc)) o x));
\mlambda{}z,f. nerve\_box\_edge(bx;((dimension(fc):=direction(fc)) o f);z);
cube(fc))))
Date html generated:
2019_11_05-PM-00_36_11
Last ObjectModification:
2018_12_15-PM-08_45_14
Theory : cubical!sets
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