Nuprl Lemma : dcff-inj-injection
∀[A,B:Type]. ∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[a:X(I)].
Inj(cubical-fun-family(X; discr(A); discr(B); I; a);A ⟶ B;λw.dcff-inj(I;w))
Proof
Definitions occuring in Statement :
dcff-inj: dcff-inj(I;w),
discrete-cubical-type: discr(T),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
inject: Inj(A;B;f),
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
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),
dcff-inj: dcff-inj(I;w),
psdcff-inj: psdcff-inj(I;w)
Lemmas referenced :
psdcff-inj-injection,
cube-cat_wf,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cat_id_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)].
Inj(cubical-fun-family(X; discr(A); discr(B); I; a);A {}\mrightarrow{} B;\mlambda{}w.dcff-inj(I;w))
Date html generated:
2020_05_20-PM-02_35_18
Last ObjectModification:
2020_04_03-PM-08_45_39
Theory : cubical!type!theory
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