Nuprl Lemma : singleton-contr_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. (singleton-contr(X;A;a) ∈ {X ⊢ _:Contractible(Singleton(a))})
Proof
Definitions occuring in Statement :
singleton-contr: singleton-contr(X;A;a),
singleton-type: Singleton(a),
contractible-type: Contractible(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,
singleton-contr: singleton-contr(X;A;a),
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
prop: ℙ
Lemmas referenced :
contr-witness_wf,
singleton-type_wf,
singleton-center_wf,
singleton-contraction_wf2,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
cc-snd_wf,
cubical-term_wf,
csm-singleton-type,
cubical-type_wf,
cubical_set_wf,
path-type_wf,
squash_wf,
true_wf,
csm-singleton-center
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
hypothesis,
instantiate,
applyEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a:\{X \mvdash{} \_:A\}].
(singleton-contr(X;A;a) \mmember{} \{X \mvdash{} \_:Contractible(Singleton(a))\})
Date html generated:
2020_05_20-PM-03_30_12
Last ObjectModification:
2020_04_08-AM-11_40_29
Theory : cubical!type!theory
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