Nuprl Lemma : csm-universe-type
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[K:fset(ℕ)]. ∀[f:K ⟶ I].
((universe-type(t;I;a))<f> = universe-type(t;K;f(a)) ∈ {formal-cube(K) ⊢ _})
Proof
Definitions occuring in Statement :
universe-type: universe-type(t;I;a),
cubical-universe: c𝕌,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
context-map: <rho>,
formal-cube: formal-cube(I),
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],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
universe-type: universe-type(t;I;a),
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
cubical-universe: c𝕌,
pi1: fst(t),
subtype_rel: A ⊆r B,
names-hom: I ⟶ J,
I_cube: A(I),
functor-ob: ob(F),
formal-cube: formal-cube(I),
cubical-type-ap-morph: (u a f),
pi2: snd(t),
closed-type-to-type: closed-type-to-type(T),
closed-cubical-universe: cc𝕌,
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
csm-ap-type: (AF)s
Lemmas referenced :
cubical-term-at-morph,
cubical-universe_wf,
cubical-universe-at,
pi1_wf_top,
cubical-type_wf,
formal-cube_wf1,
names-hom_wf,
I_cube_wf,
fset_wf,
nat_wf,
istype-cubical-universe-term,
cubical_set_wf,
cubical-term-at_wf,
csm-ap-type_wf,
context-map_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesisEquality,
hypothesis,
sqequalRule,
Error :memTop,
applyLambdaEquality,
productElimination,
independent_pairEquality,
universeIsType,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
independent_functionElimination,
applyEquality
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> = universe-type(t;K;f(a)))
Date html generated:
2020_05_20-PM-07_11_59
Last ObjectModification:
2020_04_25-PM-09_24_55
Theory : cubical!type!theory
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