Nuprl Lemma : cubical-term-at-morph
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
((u(a) a f) = u(f(a)) ∈ A(f(a)))
Proof
Definitions occuring in Statement :
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
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,
uall: ∀[x:A]. B[x],
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),
cubical-term-at: u(a),
presheaf-term-at: u(a)
Lemmas referenced :
presheaf-term-at-morph,
cube-cat_wf,
cubical-type-sq-presheaf-type,
cubical-term-sq-presheaf-term,
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{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:X(I)]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I].
((u(a) a f) = u(f(a)))
Date html generated:
2020_05_20-PM-01_52_14
Last ObjectModification:
2020_04_03-PM-08_27_48
Theory : cubical!type!theory
Home
Index