Nuprl Lemma : Kan_sigma_filler_uniform
∀X:CubicalSet. ∀A:{X ⊢ _(Kan)}. ∀B:{X.Kan-type(A) ⊢ _(Kan)}.
uniform-Kan-A-filler(X;Σ Kan-type(A) Kan-type(B);Kan_sigma_filler(A;B))
Proof
Definitions occuring in Statement :
Kan_sigma_filler: Kan_sigma_filler(A;B),
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
uniform-Kan-A-filler: uniform-Kan-A-filler(X;A;filler),
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-set: CubicalSet,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uniform-Kan-A-filler: uniform-Kan-A-filler(X;A;filler),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
and: P ∧ Q,
name-morph: name-morph(I;J),
prop: ℙ,
uiff: uiff(P;Q),
l_subset: l_subset(T;as;bs),
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
nat: ℕ,
squash: ↓T,
sq_stable: SqStable(P),
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
iff: P ⇐⇒ Q,
name-morph-domain: name-morph-domain(f;I),
rev_implies: P ⇐ Q,
Kan_sigma_filler: Kan_sigma_filler(A;B),
let: let,
cubical-type-ap-morph: (u a f),
cubical-sigma: Σ A B,
pi2: snd(t),
pi1: fst(t),
sigma-box-fst: sigma-box-fst(bx),
A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx),
compose: f o g,
A-face: A-face(X;A;I;alpha),
A-face-image: A-face-image(X;A;I;K;f;alpha;face),
spreadn: spread3,
true: True,
l_all: (∀x∈L.P[x]),
cube-set-restriction: f(s),
cube-context-adjoin: X.A,
cc-adjoin-cube: (v;u),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sigma-box-snd: sigma-box-snd(bx),
isname: isname(z),
rev_uimplies: rev_uimplies(P;Q),
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),
cubical-type-at: A(a),
is-A-face: is-A-face(X;A;I;alpha;bx;f)
Lemmas referenced :
list-subtype,
coordinate_name_wf,
subtype_rel_list,
nameset_wf,
assert-isname,
l_member_wf,
map_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
select_wf,
sq_stable__le,
nat_properties,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-assert,
isname_wf,
name-morph_wf,
A-open-box_wf,
cubical-sigma_wf,
Kan-type_wf,
cube-context-adjoin_wf,
int_seg_wf,
list_wf,
I-cube_wf,
Kan-cubical-type_wf,
cubical-set_wf,
cons_member,
member_filter_2,
filter_wf5,
cons_wf,
cubical-sigma-at,
Kanfiller-uniform,
sigma-box-fst_wf,
cubical-type-at_wf,
cc-adjoin-cube_wf,
cube-set-restriction_wf,
A-open-box-equal,
A-open-box-image_wf,
equal_wf,
Kanfiller_wf,
A-face_wf,
squash_wf,
true_wf,
istype-universe,
set_wf,
map-map,
istype-le,
istype-less_than,
length_wf,
list-diff_wf,
cname_deq_wf,
nil_wf,
face-map_wf2,
name-morph_subtype_remove1,
isname-nameset,
cubical-type-ap-morph_wf,
subtype_rel-equal,
face-map-comp,
cubical-type_wf,
iff_weakening_equal,
cube-set-restriction-comp,
sigma-box-snd_wf,
cc-adjoin-cube-restriction,
subtype_rel_self,
subtype_rel_set,
fills-A-open-box_wf,
subtype_rel_universe1,
A-adjacent-compatible_wf,
not_wf,
l_subset_wf,
all_wf,
l_exists_wf,
A-face-name_wf,
nameset_subtype,
l_all_wf2,
subtract_wf,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
decidable__lt,
pairwise_wf2,
assert_of_le_int,
length-map-sq,
top_wf,
select-map,
is-A-face_wf,
name-comp_wf,
cubical-type-ap-morph-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
productElimination,
dependent_set_memberEquality_alt,
universeIsType,
setEquality,
because_Cache,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
instantiate,
cumulativity,
intEquality,
natural_numberEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
functionIsType,
setIsType,
dependent_pairEquality_alt,
hyp_replacement,
universeEquality,
productIsType,
productEquality,
closedConclusion,
independent_pairEquality,
spreadEquality
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_(Kan)\}. \mforall{}B:\{X.Kan-type(A) \mvdash{} \_(Kan)\}.
uniform-Kan-A-filler(X;\mSigma{} Kan-type(A) Kan-type(B);Kan\_sigma\_filler(A;B))
Date html generated:
2019_11_05-PM-00_31_12
Last ObjectModification:
2018_12_10-PM-00_56_44
Theory : cubical!sets
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