Nuprl Lemma : fiber-discrete-equal
∀[B:Type]. ∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[f:{X ⊢ _:(A ⟶ discr(B))}]. ∀[z:{X ⊢ _:discr(B)}]. ∀[fbr:{X ⊢ _:Fiber(f;z)}].
(app(f; fiber-member(fbr)) = z ∈ {X ⊢ _:discr(B)})
Proof
Definitions occuring in Statement :
fiber-member: fiber-member(p),
cubical-fiber: Fiber(w;a),
discrete-cubical-type: discr(T),
cubical-app: app(w; u),
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
fiber-path_wf,
discrete-path-endpoints,
cubical-app_wf_fun,
discrete-cubical-type_wf,
fiber-member_wf,
istype-cubical-term,
cubical-fiber_wf,
cubical-fun_wf,
cubical-type_wf,
cubical_set_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality
Latex:
\mforall{}[B:Type]. \mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[f:\{X \mvdash{} \_:(A {}\mrightarrow{} discr(B))\}]. \mforall{}[z:\{X \mvdash{} \_:discr(B)\}].
\mforall{}[fbr:\{X \mvdash{} \_:Fiber(f;z)\}].
(app(f; fiber-member(fbr)) = z)
Date html generated:
2020_05_20-PM-03_37_44
Last ObjectModification:
2020_04_20-PM-07_37_40
Theory : cubical!type!theory
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