Nuprl Lemma : cubical-type-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[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)))].
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-equal,
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:\{X \mvdash{} \_\}]. \mforall{}[B:A:I:fset(\mBbbN{}) {}\mrightarrow{} X(I) {}\mrightarrow{} Type \mtimes{} (I:fset(\mBbbN{})
{}\mrightarrow{} J:fset(\mBbbN{})
{}\mrightarrow{} f:J {}\mrightarrow{} I
{}\mrightarrow{} a:X(I)
{}\mrightarrow{} (A I a)
{}\mrightarrow{} (A J f(a)))].
A = B supposing A = B
Date html generated:
2020_05_20-PM-01_46_16
Last ObjectModification:
2020_04_03-PM-07_58_08
Theory : cubical!type!theory
Home
Index