Nuprl Lemma : Kan-cubical-sigma_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _(Kan)}]. ∀[B:{X.Kan-type(A) ⊢ _(Kan)}]. (KanΣ A B ∈ {X ⊢ _(Kan)})
Proof
Definitions occuring in Statement :
Kan-cubical-sigma: KanΣ A B,
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
cube-context-adjoin: X.A,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Kan-cubical-sigma: KanΣ A B,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
Kan-cubical-type: {X ⊢ _(Kan)},
and: P ∧ Q,
cand: A c∧ B
Lemmas referenced :
Kan_sigma_filler_wf,
set_wf,
list_wf,
coordinate_name_wf,
I-cube_wf,
nameset_wf,
int_seg_wf,
A-open-box_wf,
cubical-sigma_wf,
Kan-type_wf,
cube-context-adjoin_wf,
subtype_rel_list,
cubical-type-at_wf,
Kan-A-filler_wf,
Kan_sigma_filler_uniform,
uniform-Kan-A-filler_wf,
equal_wf,
Kan-cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
functionEquality,
because_Cache,
natural_numberEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
sqequalRule,
functionExtensionality,
lambdaFormation,
dependent_set_memberEquality,
dependent_pairEquality,
independent_pairFormation,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
productElimination,
productEquality,
equalityTransitivity,
independent_functionElimination,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_(Kan)\}]. \mforall{}[B:\{X.Kan-type(A) \mvdash{} \_(Kan)\}]. (Kan\mSigma{} A B \mmember{} \{X \mvdash{} \_(Kan)\})
Date html generated:
2016_10_28-AM-11_39_50
Last ObjectModification:
2016_07_12-AM-11_51_57
Theory : cubical!sets
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