Nuprl Lemma : universe-decode-type
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. ∀[I:fset(ℕ)]. ∀[rho:X(I)]. (decode((t)<rho>) = universe-type(t;I;rho) ∈ {formal-cube(I) ⊢ _})
Proof
Definitions occuring in Statement :
universe-decode: decode(t),
universe-type: universe-type(t;I;a),
cubical-universe: c𝕌,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
context-map: <rho>,
formal-cube: formal-cube(I),
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
universe-decode: decode(t),
csm-ap-type: (AF)s,
cubical-term-at: u(a),
cubical-type-at: A(a),
context-map: <rho>,
csm-ap: (s)x,
functor-arrow: arrow(F),
cube-set-restriction: f(s),
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
pi1: fst(t),
pi2: snd(t),
formal-cube: formal-cube(I),
universe-type: universe-type(t;I;a),
prop: ℙ,
squash: ↓T,
true: True,
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),
subtype_rel: A ⊆r B,
names-hom: I ⟶ J,
I_cube: A(I),
functor-ob: ob(F),
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q
Lemmas referenced :
csm-universe-decode,
cubical-type-equal2,
formal-cube_wf1,
universe-type_wf,
csm-ap-type_wf,
universe-decode_wf,
context-map_wf,
I_cube_wf,
fset_wf,
nat_wf,
istype-cubical-universe-term,
cubical_set_wf,
names-hom_wf,
cube-set-restriction_wf,
I_cube_pair_redex_lemma,
cubical-term-at-morph,
cubical-universe_wf,
cubical-universe-at,
pi1_wf_top,
cubical-type_wf,
cubical-type-at_wf,
nh-id_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-term-at_wf,
cubical_type_ap_morph_pair_lemma,
cubical_type_at_pair_lemma,
nh-id-left,
subtype_rel_self,
composition-op_wf,
cubical-type-cumulativity2,
csm-composition_wf,
istype-cubical-type-at,
pi2_wf,
csm-ap-type-at,
cube_set_restriction_pair_lemma,
nh-id-right,
cubical-type-ap-morph_wf,
arrow_pair_lemma,
csm-cubical-type-ap-morph
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
independent_isectElimination,
universeIsType,
instantiate,
inhabitedIsType,
lambdaFormation_alt,
setElimination,
rename,
productElimination,
equalityIstype,
equalityTransitivity,
independent_functionElimination,
dependent_pairEquality_alt,
functionIsType,
because_Cache,
applyEquality,
functionExtensionality,
applyLambdaEquality,
independent_pairEquality,
hyp_replacement,
lambdaEquality_alt,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
closedConclusion
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[t:\{X \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[rho:X(I)]. (decode((t)<rho>) = universe-type(t;I;rho))
Date html generated:
2020_05_20-PM-07_11_24
Last ObjectModification:
2020_04_25-PM-09_22_28
Theory : cubical!type!theory
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