Nuprl Lemma : discrete-presheaf-term-is-map
∀[C:SmallCategory]. ∀[T:Type]. ∀[X:ps_context{j:l}(C)]. {X ⊢ _:discr(T)} ≡ psc_map{[i | j]:l}(C; X; discrete-set(T))
Proof
Definitions occuring in Statement :
discrete-presheaf-type: discr(T),
presheaf-term: {X ⊢ _:A},
psc_map: A ⟶ B,
discrete-set: discrete-set(A),
ps_context: __⊢,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
universe: Type,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
presheaf-term: {X ⊢ _:A},
discrete-presheaf-type: discr(T),
all: ∀x:A. B[x],
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
discrete-set: discrete-set(A),
type-cat: TypeCat,
compose: f o g,
I_set: A(I),
squash: ↓T,
prop: ℙ,
ps_context: __⊢,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
psc-restriction: f(s),
functor-arrow: arrow(F),
cat-ob: cat-ob(C),
pi1: fst(t),
presheaf-type-at: A(a)
Lemmas referenced :
presheaf-term_wf,
discrete-presheaf-type_wf,
psc_map_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
discrete-set_wf,
ps_context_wf,
istype-universe,
small-category_wf,
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
cat_ob_op_lemma,
cat_arrow_triple_lemma,
ob_pair_lemma,
op-cat-arrow,
cat_comp_tuple_lemma,
arrow_pair_lemma,
equal_wf,
squash_wf,
true_wf,
functor-arrow_wf,
op-cat_wf,
type-cat_wf,
subtype_rel-equal,
cat-ob_wf,
cat-arrow_wf,
subtype_rel_self,
iff_weakening_equal,
I_set_wf,
functor-ob_wf,
psc-restriction_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaEquality_alt,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
cumulativity,
productElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
universeEquality,
setElimination,
rename,
dependent_functionElimination,
Error :memTop,
dependent_set_memberEquality_alt,
because_Cache,
lambdaFormation_alt,
functionExtensionality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
functionIsType,
equalityIstype,
applyLambdaEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[T:Type]. \mforall{}[X:ps\_context\{j:l\}(C)].
\{X \mvdash{} \_:discr(T)\} \mequiv{} psc\_map\{[i | j]:l\}(C; X; discrete-set(T))
Date html generated:
2020_05_20-PM-01_34_26
Last ObjectModification:
2020_04_03-AM-01_44_44
Theory : presheaf!models!of!type!theory
Home
Index