Nuprl Lemma : sub_cubical_set-cumulativity1
∀[Y,X:j⊢]. sub_cubical_set{[j | i]:l}(Y; X) supposing sub_cubical_set{j:l}(Y; X)
Proof
Definitions occuring in Statement :
sub_cubical_set: Y ⊆ X,
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sub_cubical_set: Y ⊆ X,
subtype_rel: A ⊆r B,
guard: {T},
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
all: ∀x:A. B[x],
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g
Lemmas referenced :
subtype_rel_self,
cube_set_map_wf,
cubical_set_cumulativity-i-j,
sub_cubical_set_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
equalityTransitivity,
hypothesis,
equalitySymmetry,
applyEquality,
sqequalRule,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
axiomEquality,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[Y,X:j\mvdash{}]. sub\_cubical\_set\{[j | i]:l\}(Y; X) supposing sub\_cubical\_set\{j:l\}(Y; X)
Date html generated:
2020_05_20-PM-01_43_05
Last ObjectModification:
2020_04_06-PM-00_14_43
Theory : cubical!type!theory
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