Nuprl Lemma : universe-encode_wf
∀[G:j⊢]. ∀[T:{G ⊢ _}]. ∀[cT:G ⊢ CompOp(T)]. (encode(T;cT) ∈ {G ⊢ _:c𝕌})
Proof
Definitions occuring in Statement :
universe-encode: encode(T;cT),
cubical-universe: c𝕌,
composition-op: Gamma ⊢ CompOp(A),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-term: {X ⊢ _:A},
universe-encode: encode(T;cT),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
cubical-universe: c𝕌,
closed-cubical-universe: cc𝕌,
closed-type-to-type: closed-type-to-type(T),
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
squash: ↓T,
prop: ℙ,
names-hom: I ⟶ J,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
formal-cube: formal-cube(I),
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
csm-composition: (comp)sigma,
csm-comp: G o F,
csm-ap: (s)x,
compose: f o g,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s
Lemmas referenced :
cubical-universe_wf,
cubical-universe-at,
I_cube_wf,
fset_wf,
nat_wf,
names-hom_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
csm-ap-type_wf,
formal-cube_wf1,
context-map_wf,
csm-composition_wf,
cubical_type_ap_morph_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-ap-comp-type,
subtype_rel_self,
iff_weakening_equal,
cube_set_map_wf,
context-map-comp2,
subtype_rel-equal,
csm-comp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality_alt,
functionExtensionality,
sqequalRule,
Error :memTop,
lambdaFormation_alt,
universeIsType,
because_Cache,
functionIsType,
equalityIstype,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyEquality,
dependent_pairEquality_alt,
dependent_functionElimination,
lambdaEquality_alt,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
inhabitedIsType,
setElimination,
rename
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[T:\{G \mvdash{} \_\}]. \mforall{}[cT:G \mvdash{} CompOp(T)]. (encode(T;cT) \mmember{} \{G \mvdash{} \_:c\mBbbU{}\})
Date html generated:
2020_05_20-PM-07_10_00
Last ObjectModification:
2020_04_25-PM-08_16_46
Theory : cubical!type!theory
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