Nuprl Lemma : equiv-contr_wf
∀[G:j⊢]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[a:{G ⊢ _:A}].
(equiv-contr(f;a) ∈ {G ⊢ _:Contractible(Fiber(equiv-fun(f);a))})
Proof
Definitions occuring in Statement :
equiv-contr: equiv-contr(f;a),
equiv-fun: equiv-fun(f),
cubical-equiv: Equiv(T;A),
cubical-fiber: Fiber(w;a),
contractible-type: Contractible(A),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-equiv: Equiv(T;A),
subtype_rel: A ⊆r B,
squash: ↓T,
all: ∀x:A. B[x],
true: True,
equiv-fun: equiv-fun(f),
is-cubical-equiv: IsEquiv(T;A;w),
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
csm-id: 1(X),
csm-ap-type: (AF)s,
cc-fst: p,
csm-id-adjoin: [u],
csm-ap: (s)x,
csm-adjoin: (s;u),
pi1: fst(t),
equiv-contr: equiv-contr(f;a),
cc-snd: q,
csm-comp: G o F,
compose: f o g,
pi2: snd(t),
rev_implies: P ⇐ Q
Lemmas referenced :
cubical-snd_wf,
cubical_set_cumulativity-i-j,
cubical-fun_wf,
is-cubical-equiv_wf,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
csm-ap-type_wf,
cc-fst_wf,
cc-snd_wf,
cubical-term_wf,
csm-cubical-fun,
cubical-equiv_wf,
cubical-type_wf,
cubical_set_wf,
csm-ap-term_wf,
contractible-type_wf,
cubical-fiber_wf,
csm-id-adjoin_wf,
equiv-fun_wf,
member_wf,
squash_wf,
true_wf,
istype-universe,
csm-cubical-pi,
iff_weakening_equal,
csm-contractible-type,
csm-adjoin_wf,
equal_wf,
csm-ap-type-fst-id-adjoin,
subtype_rel_self,
csm-cubical-fiber,
csm_ap_term_fst_adjoin_lemma,
cc_snd_csm_id_adjoin_lemma,
cc-snd-csm-adjoin-sq,
csm-id_wf,
subset-cubical-term2,
sub_cubical_set_self,
cubical-sigma_wf,
csm-ap-id-type,
csm-ap-id-term,
equal_functionality_wrt_subtype_rel2,
cubical-app_wf,
csm-comp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
lambdaEquality_alt,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename,
lambdaFormation_alt,
equalityIstype,
Error :memTop,
cumulativity
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,T:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(T;A)\}]. \mforall{}[a:\{G \mvdash{} \_:A\}].
(equiv-contr(f;a) \mmember{} \{G \mvdash{} \_:Contractible(Fiber(equiv-fun(f);a))\})
Date html generated:
2020_05_20-PM-03_27_10
Last ObjectModification:
2020_04_08-PM-04_45_44
Theory : cubical!type!theory
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