Nuprl Lemma : cubical-fun-family-discrete
∀[A,B:Type]. ∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[a:X(I)].
(cubical-fun-family(X; discr(A); discr(B); I; a)
= {w:J:fset(ℕ) ⟶ f:J ⟶ I ⟶ u:A ⟶ B| ∀J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀u:A. ((w J f u) = (w K f ⋅ g u) ∈ B)}
∈ Type)
Proof
Definitions occuring in Statement :
discrete-cubical-type: discr(T),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
I_cube: A(I),
cubical_set: CubicalSet,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: 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),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
cubical-type-at: A(a),
presheaf-type-at: A(a),
discrete-cubical-type: discr(T),
discrete-presheaf-type: discr(T),
cube-set-restriction: f(s),
psc-restriction: f(s),
cubical-type-ap-morph: (u a f),
presheaf-type-ap-morph: (u a f)
Lemmas referenced :
presheaf-fun-family-discrete,
cube-cat_wf,
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,
dependent_functionElimination,
Error :memTop
Latex:
\mforall{}[A,B:Type]. \mforall{}[X:j\mvdash{}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:X(I)].
(cubical-fun-family(X; discr(A); discr(B); I; a)
= \{w:J:fset(\mBbbN{}) {}\mrightarrow{} f:J {}\mrightarrow{} I {}\mrightarrow{} u:A {}\mrightarrow{} B|
\mforall{}J,K:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I. \mforall{}g:K {}\mrightarrow{} J. \mforall{}u:A. ((w J f u) = (w K f \mcdot{} g u))\} )
Date html generated:
2020_05_20-PM-02_34_59
Last ObjectModification:
2020_04_03-PM-08_45_24
Theory : cubical!type!theory
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