Nuprl Lemma : pscm-presheaf-lambda
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ∀[H:ps_context{j:l}(C)].
∀[s:psc_map{j:l}(C; H; X)].
(((λb))s = (λ(b)s+) ∈ {H ⊢ _:(ΠA B)s})
Proof
Definitions occuring in Statement :
presheaf-lambda: (λb),
presheaf-pi: ΠA B,
pscm+: tau+,
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],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
presheaf-term-at: u(a),
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
presheaf-pi: ΠA B,
all: ∀x:A. B[x],
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
squash: ↓T,
true: True,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
presheaf-lambda: (λb),
psc-adjoin-set: (v;u),
pscm+: tau+,
pscm-ap: (s)x,
psc-snd: q,
psc-fst: p,
pscm-comp: G o F,
pscm-adjoin: (s;u),
pi2: snd(t),
compose: f o g,
pi1: fst(t)
Lemmas referenced :
I_set_wf,
cat-ob_wf,
presheaf-term-equal,
pscm-ap-type_wf,
presheaf-pi_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
psc-adjoin_wf,
pscm-ap-term_wf,
presheaf-lambda_wf,
psc_map_wf,
presheaf-term_wf,
presheaf-type_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf,
presheaf-term-at_wf,
pscm-ap_wf,
presheaf_type_at_pair_lemma,
cat-arrow_wf,
presheaf-type-at_wf,
psc-restriction_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,
pscm-ap-type-at,
pscm-ap-term-at,
pscm-ap-restriction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
functionExtensionality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
because_Cache,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
Error :memTop,
dependent_functionElimination,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
functionIsType,
inhabitedIsType,
equalityIstype,
lambdaEquality_alt,
natural_numberEquality,
universeEquality,
productElimination,
independent_functionElimination
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{}[H:ps\_context\{j:l\}(C)]. \mforall{}[s:psc\_map\{j:l\}(C; H; X)].
(((\mlambda{}b))s = (\mlambda{}(b)s+))
Date html generated:
2020_05_20-PM-01_30_19
Last ObjectModification:
2020_04_02-PM-03_02_00
Theory : presheaf!models!of!type!theory
Home
Index