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
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