Nuprl Lemma : cubical-universe-I-cube
∀[I:Cname List]
(c𝕌(I) ~ {p:AF:{AF:A:L:(Cname List) ⟶ name-morph(I;L) ⟶ Type × (L:(Cname List)
⟶ J:(Cname List)
⟶ f:name-morph(L;J)
⟶ a:name-morph(I;L)
⟶ (A L a)
⟶ (A J (a o f)))|
let A,F = AF
in (∀K:Cname List. ∀a:name-morph(I;K). ∀u:A K a. ((F K K 1 a u) = u ∈ (A K a)))
∧ (∀L,J,K:Cname List. ∀f:name-morph(L;J). ∀g:name-morph(J;K). ∀a:name-morph(I;L). ∀u:A L a.
((F L K (f o g) a u) = (F J K g (a o f) (F L J f a u)) ∈ (A K (a o (f o g)))))}
× (L:(Cname List)
⟶ f:name-morph(I;L)
⟶ J:(nameset(L) List)
⟶ x:nameset(L)
⟶ i:ℕ2
⟶ A-open-box(unit-cube(I);AF;L;f;J;x;i)
⟶ ((fst(AF)) L f))|
let AF,filler = p
in Kan-A-filler(unit-cube(I);AF;filler) ∧ uniform-Kan-A-filler(unit-cube(I);AF;filler)} )
Proof
Definitions occuring in Statement :
cubical-universe: c𝕌,
uniform-Kan-A-filler: uniform-Kan-A-filler(X;A;filler),
Kan-A-filler: Kan-A-filler(X;A;filler),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
unit-cube: unit-cube(I),
I-cube: X(I),
name-comp: (f o g),
id-morph: 1,
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
natural_number: $n,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-universe: c𝕌,
I-cube: X(I),
Kan-cubical-type: {X ⊢ _(Kan)},
functor-ob: functor-ob(F),
pi1: fst(t),
unit-cube: unit-cube(I),
cubical-type-at: A(a)
Lemmas referenced :
unit-cube-cubical-type,
list_wf,
coordinate_name_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalAxiom
Latex:
\mforall{}[I:Cname List]
(c\mBbbU{}(I) \msim{} \{p:AF:\{AF:A:L:(Cname List) {}\mrightarrow{} name-morph(I;L) {}\mrightarrow{} Type \mtimes{} (L:(Cname List)
{}\mrightarrow{} J:(Cname List)
{}\mrightarrow{} f:name-morph(L;J)
{}\mrightarrow{} a:name-morph(I;L)
{}\mrightarrow{} (A L a)
{}\mrightarrow{} (A J (a o f)))|
let A,F = AF
in (\mforall{}K:Cname List. \mforall{}a:name-morph(I;K). \mforall{}u:A K a. ((F K K 1 a u) = u))
\mwedge{} (\mforall{}L,J,K:Cname List. \mforall{}f:name-morph(L;J). \mforall{}g:name-morph(J;K).
\mforall{}a:name-morph(I;L). \mforall{}u:A L a.
((F L K (f o g) a u) = (F J K g (a o f) (F L J f a u))))\}
\mtimes{} (L:(Cname List)
{}\mrightarrow{} f:name-morph(I;L)
{}\mrightarrow{} J:(nameset(L) List)
{}\mrightarrow{} x:nameset(L)
{}\mrightarrow{} i:\mBbbN{}2
{}\mrightarrow{} A-open-box(unit-cube(I);AF;L;f;J;x;i)
{}\mrightarrow{} ((fst(AF)) L f))|
let AF,filler = p
in Kan-A-filler(unit-cube(I);AF;filler) \mwedge{} uniform-Kan-A-filler(unit-cube(I);AF;filler)\} \000C)
Date html generated:
2016_06_16-PM-08_06_53
Last ObjectModification:
2015_12_28-PM-04_11_57
Theory : cubical!sets
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