Nuprl Lemma : unit-cube-cubical-type
∀[I:Cname List]
({unit-cube(I) ⊢ _} ~ {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)))))} )
Proof
Definitions occuring in Statement :
cubical-type: {X ⊢ _},
unit-cube: unit-cube(I),
name-comp: (f o g),
id-morph: 1,
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
uall: ∀[x:A]. B[x],
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],
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
unit-cube: unit-cube(I),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
pi2: snd(t),
functor-ob: functor-ob(F),
pi1: fst(t)
Lemmas referenced :
list_wf,
coordinate_name_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalAxiom,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin
Latex:
\mforall{}[I:Cname List]
(\{unit-cube(I) \mvdash{} \_\} \msim{} \{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))))\} )
Date html generated:
2016_06_16-PM-05_48_22
Last ObjectModification:
2015_12_28-PM-04_31_42
Theory : cubical!sets
Home
Index