Nuprl Lemma : pscm-presheaf-lam
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A,B:{X ⊢ _}]. ∀[b:{X.A ⊢ _:(B)p}]. ∀[H:ps_context{j:l}(C)].
∀[s:psc_map{j:l}(C; H; X)].
((presheaf-lam(X;b))s = presheaf-lam(H;(b)s+) ∈ {H ⊢ _:((A ⟶ B))s})
Proof
Definitions occuring in Statement :
presheaf-lam: presheaf-lam(X;b),
presheaf-fun: (A ⟶ B),
pscm+: tau+,
psc-fst: p,
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-lam: presheaf-lam(X;b),
member: t ∈ T,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
all: ∀x:A. B[x],
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
presheaf-type: {X ⊢ _},
pscm-ap-type: (AF)s,
psc-fst: p,
psc-snd: q,
pscm-comp: G o F,
pscm-adjoin: (s;u),
pscm-ap: (s)x,
compose: f o g,
pi1: fst(t)
Lemmas referenced :
pscm-presheaf-lambda,
pscm-ap-type_wf,
ps_context_cumulativity2,
psc-adjoin_wf,
presheaf-type-cumulativity2,
psc-fst_wf,
presheaf-term_wf,
squash_wf,
true_wf,
small-category-cumulativity-2,
psc_map_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
equal_wf,
istype-universe,
pscm-presheaf-pi,
pscm-presheaf-fun,
subtype_rel_self,
iff_weakening_equal,
presheaf-pi_wf,
presheaf-fun-as-presheaf-pi
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeEquality,
dependent_functionElimination,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A,B:\{X \mvdash{} \_\}]. \mforall{}[b:\{X.A \mvdash{} \_:(B)p\}].
\mforall{}[H:ps\_context\{j:l\}(C)]. \mforall{}[s:psc\_map\{j:l\}(C; H; X)].
((presheaf-lam(X;b))s = presheaf-lam(H;(b)s+))
Date html generated:
2020_05_20-PM-01_30_26
Last ObjectModification:
2020_04_02-PM-05_56_45
Theory : presheaf!models!of!type!theory
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