Nuprl Lemma : presheaf-term-at-comp
∀C:SmallCategory. ∀Gamma:ps_context{j:l}(C). ∀T:{Gamma ⊢ _}. ∀t:{Gamma ⊢ _:T}. ∀I:cat-ob(C). ∀rho:Gamma(I).
∀J:cat-ob(C). ∀f:cat-arrow(C) J I. ∀K:cat-ob(C). ∀g:cat-arrow(C) K J.
(t(cat-comp(C) K J I g f(rho)) = t(g(f(rho))) ∈ T(g(f(rho))))
Proof
Definitions occuring in Statement :
presheaf-term-at: u(a),
presheaf-term: {X ⊢ _:A},
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
presheaf-type-at_wf,
psc-restriction_wf,
presheaf-term-at_wf,
cat-comp_wf,
subtype_rel-equal,
psc-restriction-comp,
I_set_wf,
cat-ob_wf,
presheaf-term_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
subtype_rel_self,
iff_weakening_equal,
cat-arrow_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}C:SmallCategory. \mforall{}Gamma:ps\_context\{j:l\}(C). \mforall{}T:\{Gamma \mvdash{} \_\}. \mforall{}t:\{Gamma \mvdash{} \_:T\}. \mforall{}I:cat-ob(C).
\mforall{}rho:Gamma(I). \mforall{}J:cat-ob(C). \mforall{}f:cat-arrow(C) J I. \mforall{}K:cat-ob(C). \mforall{}g:cat-arrow(C) K J.
(t(cat-comp(C) K J I g f(rho)) = t(g(f(rho))))
Date html generated:
2020_05_20-PM-01_34_56
Last ObjectModification:
2020_04_02-PM-06_34_24
Theory : presheaf!models!of!type!theory
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