Nuprl Lemma : csm-comp-context-map
∀[Gamma,Delta:j⊢]. ∀[sigma:Delta j⟶ Gamma]. ∀[I:fset(ℕ)]. ∀[rho:Delta(I)].
(sigma o <rho> = <(sigma)rho> ∈ formal-cube(I) j⟶ Gamma)
Proof
Definitions occuring in Statement :
csm-comp: G o F,
csm-ap: (s)x,
context-map: <rho>,
cube_set_map: A ⟶ B,
formal-cube: formal-cube(I),
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
subtype_rel: A ⊆r B,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-ob: cat-ob(C),
pi1: fst(t),
cube-cat: CubeCat,
I_cube: A(I),
functor-ob: ob(F),
I_set: A(I),
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
op-cat: op-cat(C),
spreadn: spread4,
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
formal-cube: formal-cube(I),
names-hom: I ⟶ J,
Yoneda: Yoneda(I),
all: ∀x:A. B[x],
cat-comp: cat-comp(C),
compose: f o g,
functor-arrow: arrow(F),
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
csm-comp: G o F,
pscm-comp: G o F,
context-map: <rho>,
ps-context-map: <rho>,
csm-ap: (s)x,
pscm-ap: (s)x
Lemmas referenced :
pscm-comp-context-map,
cube-cat_wf,
subtype_rel_self,
cat-ob_wf,
I_set_wf,
I_cube_wf,
fset_wf,
nat_wf,
cube_set_map_wf,
cubical_set_wf
Rules used in proof :
cut,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
sqequalRule,
isect_memberFormation_alt,
hypothesisEquality,
applyEquality,
universeIsType,
because_Cache
Latex:
\mforall{}[Gamma,Delta:j\mvdash{}]. \mforall{}[sigma:Delta j{}\mrightarrow{} Gamma]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[rho:Delta(I)].
(sigma o <rho> = <(sigma)rho>)
Date html generated:
2020_05_20-PM-01_54_09
Last ObjectModification:
2020_04_20-AM-10_45_15
Theory : cubical!type!theory
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