Nuprl Lemma : equiv-witness_wf
∀[G:j⊢]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:(T ⟶ A)}]. ∀[iseq:{G ⊢ _:IsEquiv(T;A;f)}].
(equiv-witness(f;iseq) ∈ {G ⊢ _:Equiv(T;A)})
Proof
Definitions occuring in Statement :
equiv-witness: equiv-witness(f;cntr),
cubical-equiv: Equiv(T;A),
is-cubical-equiv: IsEquiv(T;A;w),
cubical-fun: (A ⟶ B),
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,
equiv-witness: equiv-witness(f;cntr),
cubical-equiv: Equiv(T;A),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-term_wf,
is-cubical-equiv_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-fun_wf,
cubical-type_wf,
cubical_set_wf,
cubical-pair_wf,
cube-context-adjoin_wf,
csm-ap-type_wf,
cc-fst_wf,
cc-snd_wf-cubical-fun,
squash_wf,
true_wf,
equal_wf,
istype-universe,
csm-is-cubical-equiv,
csm-id-adjoin_wf,
subtype_rel_self,
iff_weakening_equal,
csm_id_adjoin_fst_type_lemma,
csm-ap-id-type,
cc_snd_csm_id_adjoin_lemma,
csm-ap-id-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
dependent_functionElimination,
lambdaEquality_alt,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :memTop
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,T:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[iseq:\{G \mvdash{} \_:IsEquiv(T;A;f)\}].
(equiv-witness(f;iseq) \mmember{} \{G \mvdash{} \_:Equiv(T;A)\})
Date html generated:
2020_05_20-PM-03_27_22
Last ObjectModification:
2020_04_06-PM-06_45_51
Theory : cubical!type!theory
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