Nuprl Lemma : presheaf-app_wf-pscm
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[H:ps_context{j:l}(C)].
∀[tau:psc_map{j:l}(C; H; X)]. ∀[u:{H ⊢ _:(A)tau}].
(app((w)tau; u) ∈ {H ⊢ _:((B)tau+)[u]})
Proof
Definitions occuring in Statement :
presheaf-app: app(w; u),
presheaf-pi: ΠA B,
pscm+: tau+,
pscm-id-adjoin: [u],
psc-adjoin: X.A,
pscm-ap-term: (t)s,
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,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
pscm+: tau+
Lemmas referenced :
presheaf-app_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
pscm-ap-type_wf,
presheaf-type-cumulativity2,
psc-adjoin_wf,
pscm+_wf,
subtype_rel_self,
psc_map_wf,
pscm-ap-term_wf,
presheaf-pi_wf,
subtype_rel-equal,
presheaf-term_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
pscm-presheaf-pi,
iff_weakening_equal,
presheaf-type_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,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:\mPi{}A B\}].
\mforall{}[H:ps\_context\{j:l\}(C)]. \mforall{}[tau:psc\_map\{j:l\}(C; H; X)]. \mforall{}[u:\{H \mvdash{} \_:(A)tau\}].
(app((w)tau; u) \mmember{} \{H \mvdash{} \_:((B)tau+)[u]\})
Date html generated:
2020_05_20-PM-01_31_09
Last ObjectModification:
2020_04_02-PM-03_02_42
Theory : presheaf!models!of!type!theory
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