Nuprl Lemma : cubical-pi-family_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. (cubical-pi-family(X;A;B;I;a) ∈ Type)
Proof
Definitions occuring in Statement :
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
cube-context-adjoin: X.A,
psc-adjoin: X.A,
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),
cube-cat: CubeCat,
all: ∀x:A. B[x],
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u)
Lemmas referenced :
presheaf-pi-family_wf,
cube-cat_wf,
cubical-type-sq-presheaf-type,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cat_comp_tuple_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:\{X.A \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:X(I)].
(cubical-pi-family(X;A;B;I;a) \mmember{} Type)
Date html generated:
2020_05_20-PM-01_59_07
Last ObjectModification:
2020_04_03-PM-08_32_27
Theory : cubical!type!theory
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