Nuprl Lemma : discrete-presheaf-term-is-constant
Not every term of a discrete presheaf type is constant, but when the
context is the Yoneda(I) -- Yoneda(I) -- then it is.⋅
∀[C:SmallCategory]. ∀[T:Type]. ∀[I:cat-ob(C)]. ∀[t:{Yoneda(I) ⊢ _:discr(T)}].
(t = discr(t(cat-id(C) I)) ∈ {Yoneda(I) ⊢ _:discr(T)})
Proof
Definitions occuring in Statement :
discrete-presheaf-term: discr(t),
discrete-presheaf-type: discr(T),
presheaf-term-at: u(a),
presheaf-term: {X ⊢ _:A},
Yoneda: Yoneda(I),
uall: ∀[x:A]. B[x],
apply: f a,
universe: Type,
equal: s = t ∈ T,
cat-id: cat-id(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
presheaf-term-at: u(a),
discrete-presheaf-term: discr(t),
implies: P ⇒ Q
Lemmas referenced :
discrete-presheaf-term-is-map,
Yoneda_wf,
ps-discrete-map-is-constant,
subtype_rel_weakening,
presheaf-term_wf2,
discrete-presheaf-type_wf,
subtype_rel_universe1,
psc_map_wf,
small-category-cumulativity-2,
discrete-set_wf,
ext-eq_inversion,
equal_functionality_wrt_subtype_rel2,
cat-ob_wf,
istype-universe,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
because_Cache,
applyEquality,
sqequalRule,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[T:Type]. \mforall{}[I:cat-ob(C)]. \mforall{}[t:\{Yoneda(I) \mvdash{} \_:discr(T)\}].
(t = discr(t(cat-id(C) I)))
Date html generated:
2020_05_20-PM-01_34_37
Last ObjectModification:
2020_04_03-PM-00_16_16
Theory : presheaf!models!of!type!theory
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