Nuprl Lemma : bijection-equiv_wf
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[g:B ⟶ A].
(∀[X:j⊢]. (bijection-equiv(X;A;B;f;g) ∈ {X ⊢ _:Equiv(discr(A);discr(B))})) supposing
((∀a:A. ((g (f a)) = a ∈ A)) and
(∀b:B. ((f (g b)) = b ∈ B)))
Proof
Definitions occuring in Statement :
bijection-equiv: bijection-equiv(X;A;B;f;g),
cubical-equiv: Equiv(T;A),
discrete-cubical-type: discr(T),
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cubical-term: {X ⊢ _:A},
discrete-cubical-type: discr(T),
cube-context-adjoin: X.A,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
pi2: snd(t),
subtype_rel: A ⊆r B,
guard: {T},
cubical-lam: cubical-lam(X;b),
cubical-app: app(w; u),
cubical-lambda: (λb),
cc-adjoin-cube: (v;u),
cc-fst: p,
csm-ap-term: (t)s,
cc-snd: q,
csm-ap-type: (AF)s,
cubical-type-at: A(a),
pi1: fst(t),
bijection-equiv: bijection-equiv(X;A;B;f;g),
is-cubical-equiv: IsEquiv(T;A;w),
squash: ↓T,
true: True,
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
fiber-point: fiber-point(t;c),
cubical-pair: cubical-pair(u;v),
cubical-fst: p.1,
csm-ap: (s)x,
fiber-member: fiber-member(p),
cubical-term-at: u(a)
Lemmas referenced :
I_cube_pair_redex_lemma,
cubical_type_at_pair_lemma,
pi2_wf,
I_cube_wf,
fset_wf,
nat_wf,
cube_set_restriction_pair_lemma,
cubical_type_ap_morph_pair_lemma,
names-hom_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
discrete-cubical-type_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
cubical-lam_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
cc-fst_wf,
csm-discrete-cubical-type,
cubical-type-at_wf,
subtype_rel_self,
cubical-term-equal,
cc-snd_wf,
equiv-witness_wf,
cubical_set_wf,
istype-universe,
cubical-lambda_wf,
contractible-type_wf,
cubical-fiber_wf,
csm-ap-term_wf,
cubical-fun_wf,
cubical-term_wf,
csm-cubical-fun,
contr-witness_wf,
fiber-point_wf,
cubical-refl_wf,
squash_wf,
true_wf,
cubical-type_wf,
path-type_wf,
cubical-type-cumulativity2,
csm-cubical-fiber,
equal_functionality_wrt_subtype_rel2,
subtype_rel_universe1,
equal-fiber-discrete,
member_wf,
iff_weakening_equal,
equal_wf,
csm-fiber-point,
fiber-path_wf,
discrete-path-endpoints,
cubical-app_wf_fun,
fiber-member_wf,
pi1_wf_top,
cubical-term-at_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
instantiate,
isectElimination,
cumulativity,
universeIsType,
productIsType,
lambdaFormation_alt,
because_Cache,
inhabitedIsType,
functionIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionExtensionality,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
independent_functionElimination,
productElimination,
rename,
applyLambdaEquality,
productEquality,
independent_pairEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[g:B {}\mrightarrow{} A].
(\mforall{}[X:j\mvdash{}]. (bijection-equiv(X;A;B;f;g) \mmember{} \{X \mvdash{} \_:Equiv(discr(A);discr(B))\})) supposing
((\mforall{}a:A. ((g (f a)) = a)) and
(\mforall{}b:B. ((f (g b)) = b)))
Date html generated:
2020_05_20-PM-03_42_53
Last ObjectModification:
2020_04_08-PM-10_06_29
Theory : cubical!type!theory
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