Nuprl Lemma : pscm-adjoin-wf
∀[C:SmallCategory]. ∀[Gamma,Delta:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢' _}]. ∀[sigma:psc_map{j:l}(C; Delta; Gamma)].
∀[u:{Delta ⊢ _:(A)sigma}].
((sigma;u) ∈ psc_map{[i | j]:l}(C; Delta; Gamma.A))
Proof
Definitions occuring in Statement :
pscm-adjoin: (s;u),
psc-adjoin: X.A,
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,
pscm-adjoin: (s;u),
I_set: A(I),
functor-ob: ob(F),
pi1: fst(t),
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
uimplies: b supposing a,
all: ∀x:A. B[x],
psc-adjoin-set: (v;u),
presheaf-term-at: u(a),
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
psc-map-is,
ps_context_cumulativity2,
small-category-cumulativity-2,
psc-adjoin-wf,
presheaf-type-cumulativity2,
small-category-subtype,
pscm-ap_wf,
subtype_rel-equal,
presheaf-type-at_wf,
pscm-ap-type_wf,
pscm-ap-type-at,
I_set_wf,
cat-ob_wf,
cat-arrow_wf,
psc-adjoin_wf,
psc-restriction_wf,
presheaf-term_wf,
psc_map_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
presheaf-type-ap-morph_wf,
presheaf-term-at_wf,
presheaf-term-at-morph,
pscm-ap-restriction,
subtype_rel_self,
iff_weakening_equal,
pscm-presheaf-type-ap-morph,
psc-adjoin-set-restriction,
psc-adjoin-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
dependent_pairEquality_alt,
setElimination,
rename,
independent_isectElimination,
Error :memTop,
universeIsType,
lambdaFormation_alt,
inhabitedIsType,
functionIsType,
equalityIstype,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma,Delta:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{}' \_\}].
\mforall{}[sigma:psc\_map\{j:l\}(C; Delta; Gamma)]. \mforall{}[u:\{Delta \mvdash{} \_:(A)sigma\}].
((sigma;u) \mmember{} psc\_map\{[i | j]:l\}(C; Delta; Gamma.A))
Date html generated:
2020_05_20-PM-01_27_52
Last ObjectModification:
2020_04_02-PM-01_47_57
Theory : presheaf!models!of!type!theory
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