Nuprl Lemma : cubical-type-equal3
∀[X:j⊢]. ∀[A,B:{X ⊢ _}].
(A = B ∈ {X ⊢ _}) supposing
((∀I:fset(ℕ). ∀[rho:X(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[u:A(rho)]. ((u rho f) = (u rho f) ∈ A(f(rho)))) and
(∀I:fset(ℕ). ∀[rho:X(I)]. (A(rho) = B(rho) ∈ Type)))
Proof
Definitions occuring in Statement :
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
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],
all: ∀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),
cubical-type-at: A(a),
presheaf-type-at: A(a),
cube-set-restriction: f(s),
psc-restriction: f(s),
cubical-type-ap-morph: (u a f),
presheaf-type-ap-morph: (u a f)
Lemmas referenced :
presheaf-type-equal3,
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
((\mforall{}I:fset(\mBbbN{})
\mforall{}[rho:X(I)]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[u:A(rho)]. ((u rho f) = (u rho f))) and
(\mforall{}I:fset(\mBbbN{}). \mforall{}[rho:X(I)]. (A(rho) = B(rho))))
Date html generated:
2020_05_20-PM-01_48_51
Last ObjectModification:
2020_04_03-PM-08_26_27
Theory : cubical!type!theory
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