Nuprl Lemma : csm-Kan-cubical-sigma
∀[X,Delta:CubicalSet]. ∀[s:Delta ⟶ X]. ∀[A:{X ⊢ _(Kan)}]. ∀[B:{X.Kan-type(A) ⊢ _(Kan)}].
((KanΣ A B)s = KanΣ (A)s (B)(s o p;q) ∈ {Delta ⊢ _(Kan)})
Proof
Definitions occuring in Statement :
Kan-cubical-sigma: KanΣ A B,
csm-Kan-cubical-type: (AK)s,
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
csm-comp: G o F,
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
csm-Kan-cubical-type: (AK)s,
Kan-cubical-sigma: KanΣ A B,
Kan-cubical-type: {X ⊢ _(Kan)},
Kan-type: Kan-type(Ak),
pi1: fst(t),
all: ∀x:A. B[x],
uimplies: b supposing a,
nameset: nameset(L),
Kan_sigma_filler: Kan_sigma_filler(A;B),
Kanfiller: filler(x;i;bx),
let: let,
pi2: snd(t),
csm-ap: (s)x,
cc-adjoin-cube: (v;u),
cc-snd: q,
cc-fst: p,
csm-comp: G o F,
csm-adjoin: (s;u),
type-cat: TypeCat,
trans-comp: t1 o t2,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
compose: f o g,
cubical-type-at: A(a),
cubical-sigma: Σ A B,
csm-ap-type: (AF)s
Lemmas referenced :
Kan-cubical-type_wf,
cube-context-adjoin_wf,
Kan-type_wf,
cube-set-map_wf,
cubical-set_wf,
csm-ap-type_wf,
cc-snd_wf,
cc-fst_wf,
csm-ap-comp-type,
cubical-term_wf,
csm-cubical-sigma,
subtype_rel_self,
cubical-type_wf,
list_wf,
coordinate_name_wf,
I-cube_wf,
nameset_wf,
int_seg_wf,
A-open-box_wf,
subtype_rel_list,
cubical-type-at_wf,
ap_mk_nat_trans_lemma,
istype-void,
cat_comp_tuple_lemma,
cubical-sigma_wf,
Kan_sigma_filler_wf,
csm-ap_wf,
csm-A-open-box,
Kan-cubical-sigma_wf,
csm-Kan-cubical-type_wf,
Kan-cubical-type-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
sqequalRule,
hyp_replacement,
lambdaEquality,
equalitySymmetry,
equalityTransitivity,
applyEquality,
cumulativity,
applyLambdaEquality,
dependent_pairEquality_alt,
setElimination,
rename,
productElimination,
dependent_functionElimination,
instantiate,
functionIsType,
natural_numberEquality,
independent_isectElimination,
lambdaEquality_alt,
functionExtensionality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}[X,Delta:CubicalSet]. \mforall{}[s:Delta {}\mrightarrow{} X]. \mforall{}[A:\{X \mvdash{} \_(Kan)\}]. \mforall{}[B:\{X.Kan-type(A) \mvdash{} \_(Kan)\}].
((Kan\mSigma{} A B)s = Kan\mSigma{} (A)s (B)(s o p;q))
Date html generated:
2019_11_05-PM-00_31_20
Last ObjectModification:
2018_11_10-PM-03_21_50
Theory : cubical!sets
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