Nuprl Lemma : is-equiv-witness_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[c:{X.A ⊢ _:Fiber((w)p;q)}].
∀[p:{X.A.Fiber((w)p;q) ⊢ _:(Path_(Fiber((w)p;q))p (c)p q)}].
(is-equiv-witness(X;A;c;p) ∈ {X ⊢ _:IsEquiv(T;A;w)})
Proof
Definitions occuring in Statement :
is-equiv-witness: is-equiv-witness(G;A;c;p),
is-cubical-equiv: IsEquiv(T;A;w),
cubical-fiber: Fiber(w;a),
path-type: (Path_A a b),
cubical-fun: (A ⟶ B),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
is-equiv-witness: is-equiv-witness(G;A;c;p),
is-cubical-equiv: IsEquiv(T;A;w),
member: t ∈ T,
subtype_rel: A ⊆r B,
squash: ↓T,
all: ∀x:A. B[x],
true: True
Lemmas referenced :
contr-witness_wf,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical-fiber_wf,
csm-ap-term_wf,
cubical-term_wf,
csm-cubical-fun,
cc-fst_wf,
cubical-lambda_wf,
contractible-type_wf,
csm-ap-type_wf,
path-type_wf,
cc-snd_wf,
cubical_set_cumulativity-i-j,
cubical-fun_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
universeIsType
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T,A:\{X \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[c:\{X.A \mvdash{} \_:Fiber((w)p;q)\}].
\mforall{}[p:\{X.A.Fiber((w)p;q) \mvdash{} \_:(Path\_(Fiber((w)p;q))p (c)p q)\}].
(is-equiv-witness(X;A;c;p) \mmember{} \{X \mvdash{} \_:IsEquiv(T;A;w)\})
Date html generated:
2020_05_20-PM-03_25_55
Last ObjectModification:
2020_04_07-PM-04_49_44
Theory : cubical!type!theory
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