Nuprl Lemma : equiv-bijection-is_wf
∀[A,B:Type]. ∀[e:{() ⊢ _:Equiv(discr(A);discr(B))}]. (equiv-bijection-is(e) ∈ Bij(A;B;equiv-bijection(e)))
Proof
Definitions occuring in Statement :
equiv-bijection-is: equiv-bijection-is(e),
equiv-bijection: equiv-bijection(e),
cubical-equiv: Equiv(T;A),
discrete-cubical-type: discr(T),
cubical-term: {X ⊢ _:A},
trivial-cube-set: (),
biject: Bij(A;B;f),
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
equiv-bijection: equiv-bijection(e),
member: t ∈ T,
equiv-bijection-is: equiv-bijection-is(e),
biject: Bij(A;B;f),
surject: Surj(A;B;f),
inject: Inj(A;B;f),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
unit: Unit,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
trivial-cube-set: (),
cubical-type-at: A(a),
discrete-cubical-type: discr(T),
uimplies: b supposing a,
contractible-type: Contractible(A),
cubical-fiber: Fiber(w;a),
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
discrete-cubical-term: discr(t),
equiv-fun: equiv-fun(f),
cubical-fst: p.1,
csm-id: 1(X),
csm-ap-term: (t)s,
csm-ap: (s)x,
cubical-term-at: u(a),
cc-snd: q,
cc-fst: p,
csm-ap-type: (AF)s,
csm-id-adjoin: [u],
csm-comp: G o F,
cubical-sigma: Σ A B,
cc-adjoin-cube: (v;u),
csm-adjoin: (s;u),
compose: f o g,
pi2: snd(t),
exists: ∃x:A. B[x],
cubical-app: app(w; u)
Lemmas referenced :
equiv-contr_wf,
discrete-cubical-type_wf,
trivial-cube-set_wf,
istype-cubical-term,
cubical-equiv_wf,
istype-universe,
cubical-term-at_wf,
discrete-fun_wf,
equiv-fun_wf,
empty-fset_wf,
nat_wf,
it_wf,
subtype_rel_self,
I_cube_wf,
cubical-term_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
path-type_wf,
cc-fst_wf,
csm-ap-term_wf,
discrete-cubical-term_wf,
cubical-app_wf_fun,
cubical-fun_wf,
csm-cubical-fun,
cubical-term-eqcd,
cc-snd_wf,
csm-id-adjoin_wf,
discrete-fun-at-app,
cubical-fst_wf,
cubical-fiber_wf,
cubical-pi_wf,
cubical-type-cumulativity2,
cubical-pair_wf,
cubical-refl_wf,
csm-path-type,
equal_wf,
squash_wf,
true_wf,
cubical-type_wf,
iff_weakening_equal,
csm-discrete-cubical-type,
csm-discrete-cubical-term,
csm-cubical-app,
csm_id_adjoin_fst_term_lemma,
cc_snd_csm_id_adjoin_lemma,
cubical_type_at_pair_lemma,
discrete-fun-app-invariant,
fset_wf,
cubical-term-equal,
cubical-snd_wf,
csm-cubical-pi,
csm_id_adjoin_fst_type_lemma,
csm-adjoin_wf,
csm-comp_wf,
csm-id_wf,
cubical_set_cumulativity-i-j,
cubical_set_wf,
csm-ap-id-type,
subtype_rel-equal,
equal_functionality_wrt_subtype_rel2,
cubical-app_wf,
cubical-sigma_wf,
csm-ap-id-term,
sigma-path-fst,
cubical-fst-pair,
discrete-path-endpoints
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
independent_pairEquality,
lambdaEquality_alt,
universeIsType,
inhabitedIsType,
instantiate,
universeEquality,
axiomEquality,
equalityIstype,
applyEquality,
functionEquality,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
cumulativity,
dependent_functionElimination,
independent_isectElimination,
hyp_replacement,
independent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
functionExtensionality,
rename,
independent_pairFormation,
applyLambdaEquality,
dependent_pairEquality_alt
Latex:
\mforall{}[A,B:Type]. \mforall{}[e:\{() \mvdash{} \_:Equiv(discr(A);discr(B))\}].
(equiv-bijection-is(e) \mmember{} Bij(A;B;equiv-bijection(e)))
Date html generated:
2020_05_20-PM-03_43_39
Last ObjectModification:
2020_04_21-AM-01_00_10
Theory : cubical!type!theory
Home
Index