Nuprl Lemma : equiv-bijection-equiv
∀[A,B:Type]. ∀[e:A ~ B].
((((λe.<equiv-bijection(e), equiv-bijection-is(e)>) o (λe.bijection-equiv(();A;B;fst(e);bij_inv(snd(e))))) e)
= e
∈ A ~ B)
Proof
Definitions occuring in Statement :
equiv-bijection-is: equiv-bijection-is(e),
equiv-bijection: equiv-bijection(e),
bijection-equiv: bijection-equiv(X;A;B;f;g),
trivial-cube-set: (),
equipollent: A ~ B,
bij_inv: bij_inv(bi),
compose: f o g,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
apply: f a,
lambda: λx.A[x],
pair: <a, b>,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bijection-equiv: bijection-equiv(X;A;B;f;g),
equiv-bijection: equiv-bijection(e),
compose: f o g,
equiv-witness: equiv-witness(f;cntr),
equiv-fun: equiv-fun(f),
discrete-fun: discrete-fun(f),
cubical-lam: cubical-lam(X;b),
cubical-lambda: (λb),
cubical-pair: cubical-pair(u;v),
cubical-fst: p.1,
cubical-term-at: u(a),
pi1: fst(t),
trivial-cube-set: (),
cc-adjoin-cube: (v;u),
equiv-bijection-is: equiv-bijection-is(e),
pi2: snd(t),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
equiv-contr: equiv-contr(f;a),
fiber-point: fiber-point(t;c),
contr-witness: contr-witness(X;c;p),
cubical-snd: p.2,
cubical-app: app(w; u),
discrete-cubical-term: discr(t),
uall: ∀[x:A]. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
inject: Inj(A;B;f),
bij_inv: bij_inv(bi),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
cube_set_restriction_pair_lemma,
equipollent_wf,
equal_wf,
squash_wf,
true_wf,
eta_conv,
iff_weakening_equal,
biject_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isect_memberFormation,
isectElimination,
cumulativity,
hypothesisEquality,
axiomEquality,
because_Cache,
universeEquality,
productElimination,
dependent_pairEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
independent_pairEquality,
functionExtensionality,
productEquality,
functionEquality,
lambdaFormation,
equalityElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[e:A \msim{} B].
((((\mlambda{}e.<equiv-bijection(e), equiv-bijection-is(e)>)
o (\mlambda{}e.bijection-equiv(();A;B;fst(e);bij\_inv(snd(e)))))
e)
= e)
Date html generated:
2017_10_05-AM-02_18_12
Last ObjectModification:
2017_03_02-PM-11_26_24
Theory : cubical!type!theory
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