Nuprl Lemma : universe-decode-restriction
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. ∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[rho:X(J)].
(decode(t)(f(rho)) = universe-type(t;J;rho)(f(1)) ∈ 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),
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nh-id: 1,
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],
universe-type: universe-type(t;I;a),
universe-decode: decode(t),
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
pi1: fst(t),
cubical-universe: c𝕌,
closed-cubical-universe: cc𝕌,
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
closed-type-to-type: closed-type-to-type(T),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
pi2: snd(t),
I_cube: A(I),
functor-ob: ob(F),
formal-cube: formal-cube(I),
names-hom: I ⟶ J,
squash: ↓T,
context-map: <rho>,
csm-ap: (s)x,
functor-arrow: arrow(F),
cube-set-restriction: f(s),
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
cubical_type_at_pair_lemma,
cubical-term-at-morph,
cubical-universe_wf,
cubical-universe-at,
cubical-term-at_wf,
I_cube_wf,
names-hom_wf,
fset_wf,
nat_wf,
istype-cubical-universe-term,
cubical_set_wf,
cubical_type_ap_morph_pair_lemma,
pi2_wf,
cubical-type_wf,
formal-cube_wf1,
composition-op_wf,
cubical-type-cumulativity2,
pi1_wf_top,
csm-ap-type_wf,
context-map_wf,
csm-composition_wf,
cubical-type-at_wf,
nh-id_wf,
csm-ap-type-at,
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
nh-id-left,
nh-comp_wf,
subtype_rel_self,
iff_weakening_equal,
nh-id-right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
instantiate,
isectElimination,
because_Cache,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
productElimination,
equalityIstype,
independent_functionElimination,
universeIsType,
applyLambdaEquality,
lambdaEquality_alt,
applyEquality,
independent_pairEquality,
dependent_pairEquality_alt,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
natural_numberEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[t:\{X \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[f:I {}\mrightarrow{} J]. \mforall{}[rho:X(J)].
(decode(t)(f(rho)) = universe-type(t;J;rho)(f(1)))
Date html generated:
2020_05_20-PM-07_11_43
Last ObjectModification:
2020_04_25-PM-09_14_20
Theory : cubical!type!theory
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