Nuprl Lemma : csm-cubical-pi-family
∀X,Delta:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta j⟶ X. ∀I:fset(ℕ). ∀a:Delta(I).
(cubical-pi-family(X;A;B;I;(s)a) = cubical-pi-family(Delta;(A)s;(B)(s o p;q);I;a) ∈ Type)
Proof
Definitions occuring in Statement :
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-comp: G o F,
csm-ap: (s)x,
cube_set_map: A ⟶ B,
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
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_set_map: A ⟶ B,
cube-cat: CubeCat,
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),
csm-ap-type: (AF)s,
pscm-ap-type: (AF)s,
csm-adjoin: (s;u),
pscm-adjoin: (s;u),
csm-comp: G o F,
pscm-comp: G o F,
cc-fst: p,
psc-fst: p,
cc-snd: q,
psc-snd: q
Lemmas referenced :
pscm-presheaf-pi-family,
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,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
isectElimination,
Error :memTop
Latex:
\mforall{}X,Delta:j\mvdash{}. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}s:Delta j{}\mrightarrow{} X. \mforall{}I:fset(\mBbbN{}). \mforall{}a:Delta(I).
(cubical-pi-family(X;A;B;I;(s)a) = cubical-pi-family(Delta;(A)s;(B)(s o p;q);I;a))
Date html generated:
2020_05_20-PM-01_59_26
Last ObjectModification:
2020_04_03-PM-08_32_47
Theory : cubical!type!theory
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