Nuprl Lemma : discrete-presheaf-term-at-morph
∀[C:SmallCategory]. ∀[T:Type]. ∀[X:ps_context{j:l}(C)]. ∀[t:{X ⊢ _:discr(T)}].
∀I,J:cat-ob(C). ∀f:cat-arrow(C) J I. ∀a:X(I). (t(a) = t(f(a)) ∈ T)
Proof
Definitions occuring in Statement :
discrete-presheaf-type: discr(T),
presheaf-term-at: u(a),
presheaf-term: {X ⊢ _:A},
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
universe: Type,
equal: s = t ∈ T,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
presheaf-term: {X ⊢ _:A},
discrete-presheaf-type: discr(T),
presheaf-term-at: u(a),
guard: {T}
Lemmas referenced :
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
I_set_wf,
cat-arrow_wf,
cat-ob_wf,
presheaf-term_wf,
discrete-presheaf-type_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
universeIsType,
isectElimination,
hypothesisEquality,
applyEquality,
inhabitedIsType,
lambdaEquality_alt,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
because_Cache,
instantiate,
universeEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[T:Type]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[t:\{X \mvdash{} \_:discr(T)\}].
\mforall{}I,J:cat-ob(C). \mforall{}f:cat-arrow(C) J I. \mforall{}a:X(I). (t(a) = t(f(a)))
Date html generated:
2020_05_20-PM-01_34_15
Last ObjectModification:
2020_04_02-PM-06_33_19
Theory : presheaf!models!of!type!theory
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