Nuprl Lemma : discrete-equiv-iff
∀A,B:Type. ({() ⊢ _:Equiv(discr(A);discr(B))} ⇐⇒ A ~ B)
Proof
Definitions occuring in Statement :
cubical-equiv: Equiv(T;A),
discrete-cubical-type: discr(T),
cubical-term: {X ⊢ _:A},
trivial-cube-set: (),
equipollent: A ~ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
prop: ℙ,
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T}
Lemmas referenced :
cubical-term_wf,
trivial-cube-set_wf,
cubical-equiv_wf,
discrete-cubical-type_wf,
equipollent_wf,
equiv-bijection_wf,
equiv-bijection-is_wf,
biject_wf,
bijection-equiv_wf,
bij_inv_wf,
set_wf,
all_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
cumulativity,
hypothesisEquality,
universeEquality,
rename,
dependent_pairEquality,
because_Cache,
functionExtensionality,
applyEquality,
productElimination,
sqequalRule,
independent_isectElimination,
functionEquality,
lambdaEquality,
productEquality,
setElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}A,B:Type. (\{() \mvdash{} \_:Equiv(discr(A);discr(B))\} \mLeftarrow{}{}\mRightarrow{} A \msim{} B)
Date html generated:
2017_10_05-AM-02_17_59
Last ObjectModification:
2017_03_02-PM-11_26_12
Theory : cubical!type!theory
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