Nuprl Lemma : Kan_sigma_filler_wf
∀X:CubicalSet. ∀A:{X ⊢ _(Kan)}. ∀B:{X.Kan-type(A) ⊢ _(Kan)}.
(Kan_sigma_filler(A;B) ∈ {filler:I:(Cname List)
⟶ alpha:X(I)
⟶ J:(nameset(I) List)
⟶ x:nameset(I)
⟶ i:ℕ2
⟶ A-open-box(X;Σ Kan-type(A) Kan-type(B);I;alpha;J;x;i)
⟶ Σ Kan-type(A) Kan-type(B)(alpha)|
Kan-A-filler(X;Σ Kan-type(A) Kan-type(B);filler)} )
Proof
Definitions occuring in Statement :
Kan_sigma_filler: Kan_sigma_filler(A;B),
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
Kan-A-filler: Kan-A-filler(X;A;filler),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-type-at: A(a),
I-cube: X(I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
Kan_sigma_filler: Kan_sigma_filler(A;B),
uall: ∀[x:A]. B[x],
nameset: nameset(L),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
and: P ∧ Q,
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
top: Top,
let: let,
Kan-A-filler: Kan-A-filler(X;A;filler),
fills-A-open-box: fills-A-open-box(X;A;I;alpha;bx;cube),
fills-A-faces: fills-A-faces(X;A;I;alpha;bx;L),
l_all: (∀x∈L.P[x]),
spreadn: spread3,
is-A-face: is-A-face(X;A;I;alpha;bx;f),
A-face: A-face(X;A;I;alpha),
less_than: a < b,
prop: ℙ,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
int_upper: {i...},
coordinate_name: Cname,
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
sigma-box-snd: sigma-box-snd(bx),
sigma-box-fst: sigma-box-fst(bx),
pi1: fst(t),
pi2: snd(t),
respects-equality: respects-equality(S;T),
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
cube-set-restriction: f(s),
cubical-type-ap-morph: (u a f),
cubical-sigma: Σ A B,
true: True,
cubical-type-at: A(a)
Lemmas referenced :
Kan-cubical-type_wf,
cube-context-adjoin_wf,
Kan-type_wf,
cubical-set_wf,
I-cube_wf,
list_wf,
int_seg_wf,
coordinate_name_wf,
nameset_wf,
subtype_rel_list,
cubical-sigma_wf,
A-open-box_wf,
cubical-type-at_wf,
sigma-box-snd_wf,
cc-adjoin-cube_wf,
sigma-box-fst_wf,
Kanfiller_wf,
decidable__equal-coordinate_name,
sq_stable__l_subset,
istype-void,
cubical-sigma-at,
length_wf,
A-face_wf,
cubical-type-ap-morph_wf,
face-map_wf2,
cube-set-restriction_wf,
nil_wf,
cons_wf,
cname_deq_wf,
list-diff_wf,
sq_stable__equal,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
sq_stable__le,
sq_stable__l_member,
int_seg_properties,
select_wf,
fills-A-open-box_wf,
length-map,
select-map,
top_wf,
subtype-respects-equality,
equal_functionality_wrt_subtype_rel2,
subtype_rel_self,
subtype_rel-equal,
cc-adjoin-cube-restriction,
is-A-face_wf,
Kan-A-filler_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
dependent_set_memberEquality_alt,
sqequalHypSubstitution,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
sqequalRule,
inhabitedIsType,
rename,
setElimination,
independent_isectElimination,
applyEquality,
dependent_functionElimination,
lambdaEquality_alt,
equalitySymmetry,
equalityTransitivity,
equalityIsType1,
dependent_pairEquality_alt,
imageElimination,
baseClosed,
imageMemberEquality,
because_Cache,
independent_functionElimination,
productElimination,
voidElimination,
isect_memberEquality_alt,
applyLambdaEquality,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
equalityIstype,
hyp_replacement,
spreadEquality,
productEquality,
productIsType
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_(Kan)\}. \mforall{}B:\{X.Kan-type(A) \mvdash{} \_(Kan)\}.
(Kan\_sigma\_filler(A;B) \mmember{} \{filler:I:(Cname List)
{}\mrightarrow{} alpha:X(I)
{}\mrightarrow{} J:(nameset(I) List)
{}\mrightarrow{} x:nameset(I)
{}\mrightarrow{} i:\mBbbN{}2
{}\mrightarrow{} A-open-box(X;\mSigma{} Kan-type(A) Kan-type(B);I;alpha;J;x;i)
{}\mrightarrow{} \mSigma{} Kan-type(A) Kan-type(B)(alpha)|
Kan-A-filler(X;\mSigma{} Kan-type(A) Kan-type(B);filler)\} )
Date html generated:
2019_11_05-PM-00_30_26
Last ObjectModification:
2018_12_10-AM-09_29_41
Theory : cubical!sets
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