Nuprl Lemma : pscm-adjoin_wf
∀[C:SmallCategory]. ∀[Gamma,Delta:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢ _}]. ∀[sigma:psc_map{j:l}(C; Delta; Gamma)].
∀[u:{Delta ⊢ _:(A)sigma}].
((sigma;u) ∈ psc_map{[i | j]:l}(C; Delta; Gamma.A))
Proof
Definitions occuring in Statement :
pscm-adjoin: (s;u),
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
all: ∀x:A. B[x],
cat-comp: cat-comp(C),
compose: f o g
Lemmas referenced :
pscm-adjoin-wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
subtype_rel_self,
psc_map_wf,
presheaf-term_wf,
pscm-ap-type_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma,Delta:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[sigma:psc\_map\{j:l\}(C;
Delta;
Gamma
)].
\mforall{}[u:\{Delta \mvdash{} \_:(A)sigma\}].
((sigma;u) \mmember{} psc\_map\{[i | j]:l\}(C; Delta; Gamma.A))
Date html generated:
2020_05_20-PM-01_27_56
Last ObjectModification:
2020_04_02-PM-01_51_57
Theory : presheaf!models!of!type!theory
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