Nuprl Lemma : cubical-type-equal2
∀[X:j⊢]. ∀[A,B:{X ⊢ _}].
A = B ∈ {X ⊢ _}
supposing A
= B
∈ (A:I:fset(ℕ) ⟶ X(I) ⟶ Type × (I:fset(ℕ) ⟶ J:fset(ℕ) ⟶ f:J ⟶ I ⟶ a:X(I) ⟶ (A I a) ⟶ (A J f(a))))
Proof
Definitions occuring in Statement :
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
cube-cat: CubeCat,
all: ∀x:A. B[x],
I_cube: A(I),
I_set: A(I),
cube-set-restriction: f(s),
psc-restriction: f(s)
Lemmas referenced :
presheaf-type-equal2,
cube-cat_wf,
cubical-type-sq-presheaf-type,
cat_ob_pair_lemma,
cat_arrow_triple_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
sqequalRule,
Error :memTop,
dependent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A,B:\{X \mvdash{} \_\}]. A = B supposing A = B
Date html generated:
2020_05_20-PM-01_46_27
Last ObjectModification:
2020_04_03-PM-07_57_47
Theory : cubical!type!theory
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