Nuprl Lemma : equal-fiber-discrete
∀[B:Type]. ∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[f:{X ⊢ _:(A ⟶ discr(B))}]. ∀[z:{X ⊢ _:discr(B)}]. ∀[a,b:{X ⊢ _:Fiber(f;z)}].
(a = b ∈ {X ⊢ _:Fiber(f;z)} ⇐⇒ a.1 = b.1 ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
cubical-fiber: Fiber(w;a),
discrete-cubical-type: discr(T),
cubical-fst: p.1,
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
cubical-path-app: pth @ r
Lemmas referenced :
equal-fiber-when-discrete,
discrete-cubical-type_wf,
cubical-term_wf,
pathtype_wf,
cubical-fiber_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-fun_wf,
cubical-type_wf,
cubical_set_wf,
istype-universe,
discrete-pathtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation_alt,
universeIsType,
instantiate,
cumulativity,
inhabitedIsType,
applyEquality,
sqequalRule,
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{}[a,b:\{X \mvdash{} \_:Fiber(f;z)\}].
(a = b \mLeftarrow{}{}\mRightarrow{} a.1 = b.1)
Date html generated:
2020_05_20-PM-03_37_31
Last ObjectModification:
2020_04_07-PM-04_28_31
Theory : cubical!type!theory
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