Nuprl Lemma : presheaf-beta
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ∀[u:{X ⊢ _:A}].
(app((λb); u) = (b)[u] ∈ {X ⊢ _:(B)[u]})
Proof
Definitions occuring in Statement :
presheaf-app: app(w; u),
presheaf-lambda: (λb),
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 ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
presheaf-term: {X ⊢ _:A},
uimplies: b supposing a,
presheaf-lambda: (λb),
presheaf-app: app(w; u),
pscm-ap-term: (t)s,
pscm-ap: (s)x,
pscm-id-adjoin: [u],
pscm-id: 1(X),
pscm-adjoin: (s;u),
psc-adjoin-set: (v;u),
all: ∀x:A. B[x],
implies: P ⇒ Q,
presheaf-type: {X ⊢ _},
presheaf-type-at: A(a),
pi1: fst(t),
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
presheaf-term-equal,
pscm-ap-type_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-id-adjoin_wf,
pscm-ap-term_wf,
presheaf-app_wf,
presheaf-lambda_wf,
pscm-ap-type-at,
I_set_wf,
presheaf-term_wf,
presheaf-term_wf2,
small-category-cumulativity-2,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
psc-adjoin-set_wf,
subtype_rel_self,
psc-restriction-id,
subtype_rel-equal,
presheaf-type-at_wf,
psc-restriction_wf,
cat-id_wf,
presheaf_type_at_pair_lemma,
cat-ob_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
independent_isectElimination,
Error :memTop,
functionExtensionality_alt,
lambdaFormation_alt,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
productElimination,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
natural_numberEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[b:\{X.A \mvdash{} \_:B\}].
\mforall{}[u:\{X \mvdash{} \_:A\}].
(app((\mlambda{}b); u) = (b)[u])
Date html generated:
2020_05_20-PM-01_33_51
Last ObjectModification:
2020_04_02-PM-06_33_28
Theory : presheaf!models!of!type!theory
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