Nuprl Lemma : pscm-dependent_wf
∀C:SmallCategory. ∀X,Delta:ps_context{j:l}(C). ∀A:{X ⊢ _}. ∀s:psc_map{j:l}(C; Delta; X).
((s)dep ∈ psc_map{j:l}(C; Delta.(A)s; X.A))
Proof
Definitions occuring in Statement :
pscm-dependent: (s)dep,
psc-adjoin: X.A,
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
ps_context: __⊢,
all: ∀x:A. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
pscm-dependent: (s)dep,
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,
cat-comp: cat-comp(C),
compose: f o g,
presheaf-type: {X ⊢ _},
pscm-ap-type: (AF)s,
typed-psc-snd: tq,
typed-psc-fst: tp{i:l},
pscm-comp: G o F,
psc-snd: q,
psc-fst: p,
pscm-ap: (s)x
Lemmas referenced :
typed-psc-fst_wf,
pscm-ap-type_wf,
typed-psc-snd_wf,
pscm-adjoin_wf,
ps_context_cumulativity2,
small-category-cumulativity-2,
psc-adjoin_wf,
presheaf-type-cumulativity2,
pscm-comp_wf,
subtype_rel_self,
psc_map_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
instantiate,
applyEquality,
sqequalRule,
setElimination,
rename,
productElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType
Latex:
\mforall{}C:SmallCategory. \mforall{}X,Delta:ps\_context\{j:l\}(C). \mforall{}A:\{X \mvdash{} \_\}. \mforall{}s:psc\_map\{j:l\}(C; Delta; X).
((s)dep \mmember{} psc\_map\{j:l\}(C; Delta.(A)s; X.A))
Date html generated:
2020_05_20-PM-01_29_26
Last ObjectModification:
2020_04_02-PM-06_28_49
Theory : presheaf!models!of!type!theory
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