Nuprl Lemma : universe-type-at
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[K:fset(ℕ)]. ∀[f:K ⟶ I].
(universe-type(t;I;a)(f) = decode(t)(f(a)) ∈ Type)
Proof
Definitions occuring in Statement :
universe-decode: decode(t),
universe-type: universe-type(t;I;a),
cubical-universe: c𝕌,
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
names-hom: I ⟶ J,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
formal-cube: formal-cube(I),
true: True,
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
names-hom_wf,
I_cube_wf,
fset_wf,
nat_wf,
istype-cubical-universe-term,
cubical_set_wf,
formal-cube_wf1,
subtype_rel_self,
cubical-type-at_wf,
universe-decode_wf,
cube-set-restriction_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type_wf,
universe-decode-type,
iff_weakening_equal,
csm-universe-decode,
csm-ap-type-at,
csm-ap-context-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
dependent_functionElimination,
instantiate,
universeEquality,
applyEquality,
sqequalRule,
natural_numberEquality,
Error :memTop,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[t:\{X \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:X(I)]. \mforall{}[K:fset(\mBbbN{})]. \mforall{}[f:K {}\mrightarrow{} I].
(universe-type(t;I;a)(f) = decode(t)(f(a)))
Date html generated:
2020_05_20-PM-07_12_15
Last ObjectModification:
2020_04_25-PM-09_28_18
Theory : cubical!type!theory
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