Nuprl Lemma : cubical-pi-family-comp
∀X,Delta:j⊢. ∀s:Delta j⟶ X. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:Delta(I). ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}.
∀w:cubical-pi-family(X;A;B;I;(s)a).
(λK,g. (w K f ⋅ g) ∈ cubical-pi-family(X;A;B;J;(s)f(a)))
Proof
Definitions occuring in Statement :
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
csm-ap: (s)x,
cube_set_map: A ⟶ B,
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
cube_set_map: A ⟶ B,
cube-cat: CubeCat,
I_cube: A(I),
I_set: A(I),
uall: ∀[x:A]. B[x],
cube-context-adjoin: X.A,
psc-adjoin: X.A,
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-pi-family: cubical-pi-family(X;A;B;I;a),
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
csm-ap: (s)x,
pscm-ap: (s)x,
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u)
Lemmas referenced :
presheaf-pi-family-comp,
cube-cat_wf,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cubical-type-sq-presheaf-type,
cat_comp_tuple_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
Error :memTop,
isectElimination
Latex:
\mforall{}X,Delta:j\mvdash{}. \mforall{}s:Delta j{}\mrightarrow{} X. \mforall{}I,J:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I. \mforall{}a:Delta(I). \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}.
\mforall{}w:cubical-pi-family(X;A;B;I;(s)a).
(\mlambda{}K,g. (w K f \mcdot{} g) \mmember{} cubical-pi-family(X;A;B;J;(s)f(a)))
Date html generated:
2020_05_20-PM-01_59_17
Last ObjectModification:
2020_04_03-PM-08_32_37
Theory : cubical!type!theory
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