Nuprl Lemma : cubical-universe-cumulativity2
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}].  (t ∈ {X ⊢ _:c𝕌'})
Proof
Definitions occuring in Statement : 
cubical-universe: c𝕌
, 
cubical-term: {X ⊢ _:A}
, 
cubical_set: CubicalSet
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
cubical-universe: c𝕌
, 
cubical-term: {X ⊢ _:A}
, 
closed-cubical-universe: cc𝕌
, 
closed-type-to-type: closed-type-to-type(T)
, 
all: ∀x:A. B[x]
, 
subtype_rel: A ⊆r B
, 
squash: ↓T
, 
prop: ℙ
, 
true: True
, 
uimplies: b supposing a
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
names-hom: I ⟶ J
, 
I_cube: A(I)
, 
functor-ob: ob(F)
, 
pi1: fst(t)
, 
formal-cube: formal-cube(I)
Lemmas referenced : 
cubical_type_at_pair_lemma, 
cubical_type_ap_morph_pair_lemma, 
fibrant-type-cumulativity, 
formal-cube_wf1, 
I_cube_wf, 
fset_wf, 
nat_wf, 
equal_wf, 
squash_wf, 
true_wf, 
istype-universe, 
fibrant-type_wf_formal-cube, 
cube-set-restriction_wf, 
subtype_rel_self, 
iff_weakening_equal, 
names-hom_wf, 
csm-fibrant-type_wf, 
context-map_wf, 
istype-cubical-universe-term, 
cubical_set_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
cut, 
sqequalHypSubstitution, 
setElimination, 
thin, 
rename, 
dependent_set_memberEquality_alt, 
sqequalRule, 
introduction, 
extract_by_obid, 
dependent_functionElimination, 
Error :memTop, 
hypothesis, 
functionExtensionality, 
applyEquality, 
hypothesisEquality, 
isectElimination, 
lambdaFormation_alt, 
instantiate, 
lambdaEquality_alt, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
universeIsType, 
universeEquality, 
because_Cache, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
independent_isectElimination, 
productElimination, 
independent_functionElimination, 
inhabitedIsType, 
functionIsType, 
equalityIstype
Latex:
\mforall{}[X:j\mvdash{}].  \mforall{}[t:\{X  \mvdash{}  \_:c\mBbbU{}\}].    (t  \mmember{}  \{X  \mvdash{}  \_:c\mBbbU{}'\})
Date html generated:
2020_05_20-PM-07_09_29
Last ObjectModification:
2020_04_25-PM-03_21_16
Theory : cubical!type!theory
Home
Index