Nuprl Lemma : presheaf-type-iso-inverse2
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)].
((presheaf-type-iso(X) o presheaf-type-rev-iso(X)) = (λx.x) ∈ ({X ⊢ _} ⟶ {X ⊢ _}))
Proof
Definitions occuring in Statement :
presheaf-type-rev-iso: presheaf-type-rev-iso(X),
presheaf-type-iso: presheaf-type-iso(X),
presheaf-type: {X ⊢ _},
ps_context: __⊢,
compose: f o g,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
compose: f o g,
presheaf-type: {X ⊢ _},
presheaf-type-rev-iso: presheaf-type-rev-iso(X),
presheaf-type-iso: presheaf-type-iso(X),
pi1: fst(t),
pi2: snd(t),
mk-presheaf: mk-presheaf,
all: ∀x:A. B[x],
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
I_set_wf,
cat-ob_wf,
cat-arrow_wf,
psc-restriction_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
cat-id_wf,
subtype_rel-equal,
equal_wf,
squash_wf,
true_wf,
istype-universe,
psc-restriction-id,
subtype_rel_self,
iff_weakening_equal,
cat-comp_wf,
psc-restriction-comp,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
sqequalRule,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
productElimination,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
dependent_pairEquality_alt,
applyEquality,
hypothesisEquality,
isectElimination,
because_Cache,
functionIsType,
universeIsType,
instantiate,
productIsType,
equalityIstype,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)].
((presheaf-type-iso(X) o presheaf-type-rev-iso(X)) = (\mlambda{}x.x))
Date html generated:
2020_05_20-PM-01_25_44
Last ObjectModification:
2020_04_01-AM-11_00_50
Theory : presheaf!models!of!type!theory
Home
Index