Nuprl Lemma : cubical-id-is-equiv_wf
∀X:j⊢. ∀T:{X ⊢ _}. (cubical-id-is-equiv(X;T) ∈ {X ⊢ _:IsEquiv(T;T;cubical-id-fun(X))})
Proof
Definitions occuring in Statement :
cubical-id-is-equiv: cubical-id-is-equiv(X;T),
is-cubical-equiv: IsEquiv(T;A;w),
cubical-id-fun: cubical-id-fun(X),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
all: ∀x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
cubical-id-is-equiv: cubical-id-is-equiv(X;T),
subtype_rel: A ⊆r B,
true: True,
is-cubical-equiv: IsEquiv(T;A;w),
uimplies: b supposing a,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
cubical-id-fun_wf,
cubical-fiber-id-fun,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-ap-type_wf,
cc-fst_wf,
cc-snd_wf,
csm-cubical-id-fun,
cubical-type_wf,
cubical_set_wf,
cubical-lambda_wf,
contractible-type_wf,
contr-witness_wf,
id-fiber-center_wf,
subset-cubical-term2,
sub_cubical_set_self,
id-fiber-contraction_wf,
subtype_rel-equal,
cubical-term_wf,
path-type_wf,
equal_wf,
cubical-fiber_wf,
csm-ap-term_wf,
cubical-fun_wf,
iff_weakening_equal,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
dependent_functionElimination,
applyEquality,
sqequalRule,
because_Cache,
universeIsType,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
inhabitedIsType,
universeEquality
Latex:
\mforall{}X:j\mvdash{}. \mforall{}T:\{X \mvdash{} \_\}. (cubical-id-is-equiv(X;T) \mmember{} \{X \mvdash{} \_:IsEquiv(T;T;cubical-id-fun(X))\})
Date html generated:
2020_05_20-PM-03_32_44
Last ObjectModification:
2020_04_07-PM-06_01_27
Theory : cubical!type!theory
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