Nuprl Lemma : Kan_id_filler_wf1
∀X:CubicalSet. ∀A:{X ⊢ _(Kan)}. ∀a,b:{X ⊢ _:Kan-type(A)}.
(Kan_id_filler(X;A;a;b) ∈ I:(Cname List)
⟶ alpha:X(I)
⟶ J:(nameset(I) List)
⟶ x:nameset(I)
⟶ i:ℕ2
⟶ A-open-box(X;(Id_Kan-type(A) a b);I;alpha;J;x;i)
⟶ I-path(X;Kan-type(A);a;b;I;alpha))
Proof
Definitions occuring in Statement :
Kan_id_filler: Kan_id_filler(X;A;a;b),
cubical-identity: (Id_A a b),
I-path: I-path(X;A;a;b;I;alpha),
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
cubical-term: {X ⊢ _:AF},
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,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nameset: nameset(L),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
Kan_id_filler: Kan_id_filler(X;A;a;b),
implies: P ⇒ Q,
and: P ∧ Q,
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
sq_stable: SqStable(P),
squash: ↓T,
false: False,
not: ¬A,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
named-path: named-path(X;A;a;b;I;alpha;z),
I-path: I-path(X;A;a;b;I;alpha),
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
fills-A-faces: fills-A-faces(X;A;I;alpha;bx;L),
extend-A-open-box: extend-A-open-box(bx;f1;f2),
lift-id-faces: lift-id-faces(X;A;I;alpha;box),
fills-A-open-box: fills-A-open-box(X;A;I;alpha;bx;cube),
cubical-id-box: cubical-id-box(X;A;a;b;I;alpha;box),
cons: [a / b],
select: L[n],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
int_upper: {i...},
coordinate_name: Cname,
guard: {T},
ge: i ≥ j ,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
subtract: n - m,
true: True,
iota': iota'(I),
name-path-endpoints: name-path-endpoints(X;A;a;b;I;alpha;z;omega),
is-A-face: is-A-face(X;A;I;alpha;bx;f),
term-A-face: term-A-face(a;I;alpha;i),
spreadn: spread3,
less_than: a < b,
cubical-term-at: u(a)
Lemmas referenced :
cubical-set_wf,
Kan-cubical-type_wf,
Kan-type_wf,
cubical-term_wf,
I-cube_wf,
list_wf,
int_seg_wf,
coordinate_name_wf,
nameset_wf,
subtype_rel_list,
cubical-identity_wf,
A-open-box_wf,
l_subset_cons_same,
fresh-cname_wf,
cons_wf,
sq_stable__l_subset,
decidable__equal-coordinate_name,
equal_wf,
l_member_wf,
not_wf,
set_wf,
named-path_wf,
name-path-endpoints_wf,
fills-A-open-box_wf,
cubical-type-at_wf,
cubical-id-box_wf,
l_subset_right_cons_trivial,
nameset_subtype,
cons_member,
iota_wf,
cube-set-restriction_wf,
Kanfiller_wf,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
length_wf,
decidable__lt,
int_seg_properties,
A-face_wf,
non_neg_length,
map-length,
false_wf,
length_of_cons_lemma,
lift-id-face_wf,
map_wf,
iota'_wf,
add-fresh-cname_wf,
cube-set-restriction-id,
iota'-identity,
iff_weakening_equal,
face-map_wf,
cube-set-restriction-comp,
true_wf,
squash_wf,
add-remove-fresh-sq,
cubical-type-ap-morph_wf,
subtype_rel_weakening,
ext-eq_weakening,
cubical-term-at_wf
Rules used in proof :
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
hypothesis,
sqequalHypSubstitution,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
natural_numberEquality,
sqequalRule,
rename,
setElimination,
independent_isectElimination,
applyEquality,
because_Cache,
dependent_functionElimination,
lambdaEquality,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalitySymmetry,
equalityTransitivity,
voidElimination,
dependent_pairEquality,
setEquality,
dependent_set_memberEquality,
inlFormation,
independent_pairFormation,
computeAll,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
addEquality,
voidEquality,
isect_memberEquality,
applyLambdaEquality,
hyp_replacement,
universeEquality,
promote_hyp
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_(Kan)\}. \mforall{}a,b:\{X \mvdash{} \_:Kan-type(A)\}.
(Kan\_id\_filler(X;A;a;b) \mmember{} 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;(Id\_Kan-type(A) a b);I;alpha;J;x;i)
{}\mrightarrow{} I-path(X;Kan-type(A);a;b;I;alpha))
Date html generated:
2018_05_23-PM-07_07_27
Last ObjectModification:
2018_05_17-PM-07_00_23
Theory : cubical!sets
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