Nuprl Lemma : csm-discrete-sigma
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X,Y:j⊢]. ∀[s:Y j⟶ X].
((Σ discr(A) discrete-family(A;a.B[a]))s = Σ discr(A) discrete-family(A;a.B[a]) ∈ {Y ⊢ _})
Proof
Definitions occuring in Statement :
discrete-family: discrete-family(A;a.B[a]),
discrete-cubical-type: discr(T),
cubical-sigma: Σ A B,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
discrete-cubical-type: discr(T),
csm-ap-type: (AF)s,
discrete-family: discrete-family(A;a.B[a]),
cc-snd: q,
cc-fst: p,
csm-comp: G o F,
csm-adjoin: (s;u),
compose: f o g,
csm-ap: (s)x,
pi2: snd(t)
Lemmas referenced :
csm-cubical-sigma,
discrete-cubical-type_wf,
discrete-family_wf,
cube_set_map_wf,
cubical_set_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
sqequalRule,
lambdaEquality_alt,
applyEquality,
universeIsType,
because_Cache,
instantiate,
functionIsType,
inhabitedIsType,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[X,Y:j\mvdash{}]. \mforall{}[s:Y j{}\mrightarrow{} X].
((\mSigma{} discr(A) discrete-family(A;a.B[a]))s = \mSigma{} discr(A) discrete-family(A;a.B[a]))
Date html generated:
2020_05_20-PM-03_38_35
Last ObjectModification:
2020_04_07-PM-04_29_21
Theory : cubical!type!theory
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