Nuprl Lemma : cubical-universe-at-cumulativity
∀[I:fset(ℕ)]. ∀[a:Top]. (c𝕌(a) ⊆r c𝕌'(a))
Proof
Definitions occuring in Statement :
cubical-universe: c𝕌,
cubical-type-at: A(a),
fset: fset(T),
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
cubical-type: {X ⊢ _},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q
Lemmas referenced :
istype-top,
fset_wf,
nat_wf,
composition-op_wf,
formal-cube_wf1,
cubical-type-cumulativity2,
cubical-type_wf,
cubical-universe-at,
subtype_rel_dep_function,
I_cube_wf,
subtype_rel_universe1,
names-hom_wf,
cube-set-restriction_wf,
nh-id_wf,
subtype_rel-equal,
cube-set-restriction-id,
nh-comp_wf,
cube-set-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
axiomEquality,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
Error :memTop,
lambdaEquality_alt,
productElimination,
dependent_pairEquality_alt,
instantiate,
applyEquality,
productIsType,
setElimination,
rename,
dependent_set_memberEquality_alt,
functionExtensionality,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
functionIsType,
equalityIstype,
dependent_functionElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:Top]. (c\mBbbU{}(a) \msubseteq{}r c\mBbbU{}'(a))
Date html generated:
2020_05_20-PM-07_09_14
Last ObjectModification:
2020_04_25-PM-01_33_56
Theory : cubical!type!theory
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