Nuprl Lemma : equiv-discrete-type
∀A,B:Type. ∀f:A ⟶ B. (Bij(A;B;f) ⇒ (∀X:j⊢. {X ⊢ _:Equiv(discr(A);discr(B))}))
Proof
Definitions occuring in Statement :
cubical-equiv: Equiv(T;A),
discrete-cubical-type: discr(T),
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
biject: Bij(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
biject-inverse,
bijection-equiv_wf,
cubical_set_cumulativity-i-j,
cubical_set_wf,
biject_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
instantiate,
cumulativity,
independent_isectElimination,
applyEquality,
sqequalRule,
universeIsType,
functionIsType,
inhabitedIsType,
universeEquality
Latex:
\mforall{}A,B:Type. \mforall{}f:A {}\mrightarrow{} B. (Bij(A;B;f) {}\mRightarrow{} (\mforall{}X:j\mvdash{}. \{X \mvdash{} \_:Equiv(discr(A);discr(B))\}))
Date html generated:
2020_05_20-PM-03_43_05
Last ObjectModification:
2020_04_06-PM-07_14_26
Theory : cubical!type!theory
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