Nuprl Lemma : universe-decode-encode
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[cT:X ⊢ CompOp(T)]. (decode(encode(T;cT)) = T ∈ {X ⊢ _})
Proof
Definitions occuring in Statement :
universe-decode: decode(t),
universe-encode: encode(T;cT),
composition-op: Gamma ⊢ CompOp(A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
cubical-type: {X ⊢ _},
universe-decode: decode(t),
universe-encode: encode(T;cT),
cubical-term-at: u(a),
cubical-type-at: A(a),
pi1: fst(t),
csm-ap-type: (AF)s,
context-map: <rho>,
csm-ap: (s)x,
functor-arrow: arrow(F),
cube-set-restriction: f(s),
universe-type: universe-type(t;I;a),
I_cube: A(I),
functor-ob: ob(F),
formal-cube: formal-cube(I),
names-hom: I ⟶ J,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-type-equal2,
universe-decode_wf,
universe-encode_wf,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
I_cube_wf,
fset_wf,
nat_wf,
cube-set-restriction-id,
cubical-type-at_wf,
formal-cube_wf1,
universe-type_wf,
nh-id_wf,
names-hom_wf,
cubical_type_ap_morph_pair_lemma,
subtype_rel-equal,
cube-set-restriction_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-comp,
subtype_rel_self,
iff_weakening_equal,
nh-id-right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
universeIsType,
instantiate,
applyEquality,
sqequalRule,
setElimination,
rename,
productElimination,
dependent_pairEquality_alt,
functionExtensionality,
because_Cache,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
inhabitedIsType,
functionIsType
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:\{X \mvdash{} \_\}]. \mforall{}[cT:X \mvdash{} CompOp(T)]. (decode(encode(T;cT)) = T)
Date html generated:
2020_05_20-PM-07_17_17
Last ObjectModification:
2020_04_25-PM-09_42_03
Theory : cubical!type!theory
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