Nuprl Lemma : discrete-sigma-bijection
∀[A:Type]. ∀[B:A ⟶ Type]. ∀X:j⊢. {X ⊢ _:Σ discr(A) discrete-family(A;a.B[a])} ~ {X ⊢ _:discr(a:A × B[a])}
Proof
Definitions occuring in Statement :
discrete-family: discrete-family(A;a.B[a]),
discrete-cubical-type: discr(T),
cubical-sigma: Σ A B,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
implies: P ⇒ Q,
surject: Surj(A;B;f),
prop: ℙ,
uimplies: b supposing a
Lemmas referenced :
discrete-pair_wf,
cubical_set_cumulativity-i-j,
cubical-term_wf,
cubical-sigma_wf,
discrete-cubical-type_wf,
discrete-family_wf,
biject_wf,
cubical_set_wf,
istype-universe,
discrete-pair-injection,
discrete-pair-inv_wf,
discrete-pair-inv-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
dependent_pairFormation_alt,
lambdaEquality_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
applyEquality,
universeIsType,
hypothesis,
independent_pairFormation,
equalityIstype,
productEquality,
because_Cache,
inhabitedIsType,
functionIsType,
universeEquality,
dependent_functionElimination,
independent_isectElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}X:j\mvdash{}. \{X \mvdash{} \_:\mSigma{} discr(A) discrete-family(A;a.B[a])\} \msim{} \{X \mvdash{} \_:discr(a:A \mtimes{} B[a])\}
Date html generated:
2020_05_20-PM-03_41_23
Last ObjectModification:
2020_04_06-PM-07_13_26
Theory : cubical!type!theory
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