Nuprl Lemma : presheaf-eta
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}].
((λapp((w)p; q)) = w ∈ {X ⊢ _:ΠA B})
Proof
Definitions occuring in Statement :
presheaf-app: app(w; u),
presheaf-lambda: (λb),
presheaf-pi: ΠA B,
psc-snd: q,
psc-fst: p,
psc-adjoin: X.A,
pscm-ap-term: (t)s,
presheaf-term: {X ⊢ _:A},
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,
psc-snd: q,
psc-fst: p,
pscm-ap-term: (t)s,
presheaf-app: app(w; u),
presheaf-lambda: (λb),
presheaf-pi: ΠA B,
pi1: fst(t),
pscm-ap: (s)x,
presheaf-term: {X ⊢ _:A},
all: ∀x:A. B[x],
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
squash: ↓T,
uimplies: b supposing a,
true: True,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
psc-adjoin-set: (v;u),
pi2: snd(t),
presheaf-type-at: A(a)
Lemmas referenced :
presheaf-term_wf,
presheaf-pi_wf,
presheaf-type_wf,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
ps_context_wf,
small-category_wf,
presheaf_type_at_pair_lemma,
I_set_wf,
cat-ob_wf,
presheaf-type-at_wf,
psc-restriction_wf,
cat-arrow_wf,
psc-adjoin-set_wf,
presheaf-type-ap-morph_wf,
cat-comp_wf,
subtype_rel-equal,
psc-restriction-comp,
psc-adjoin-set-restriction,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
presheaf_type_ap_morph_pair_lemma,
cat-id_wf,
psc-restriction-id,
cat-comp-ident1,
presheaf-term-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
functionExtensionality_alt,
setElimination,
rename,
dependent_functionElimination,
Error :memTop,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
functionIsType,
because_Cache,
equalityIstype,
independent_isectElimination,
lambdaEquality_alt,
natural_numberEquality,
equalityTransitivity,
universeEquality,
productElimination,
independent_functionElimination,
hyp_replacement,
functionExtensionality
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\}].
((\mlambda{}app((w)p; q)) = w)
Date html generated:
2020_05_20-PM-01_33_42
Last ObjectModification:
2020_04_02-PM-06_32_30
Theory : presheaf!models!of!type!theory
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