Nuprl Lemma : Kan_id_filler_wf
∀X:CubicalSet. ∀A:{X ⊢ _(Kan)}. ∀a,b:{X ⊢ _:Kan-type(A)}.
  (Kan_id_filler(X;A;a;b) ∈ {filler: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)
                             ⟶ (Id_Kan-type(A) a b)(alpha)| 
                             Kan-A-filler(X;(Id_Kan-type(A) a b);filler)} )
Proof
Definitions occuring in Statement : 
Kan_id_filler: Kan_id_filler(X;A;a;b)
, 
cubical-identity: (Id_A 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-term: {X ⊢ _:AF}
, 
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
, 
uall: ∀[x:A]. B[x]
, 
nameset: nameset(L)
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
so_apply: x[s1;s2]
, 
so_lambda: λ2x y.t[x; y]
, 
cubical-path: cubical-path(X;A;a;b;I;alpha)
, 
pi1: fst(t)
, 
cubical-type-at: A(a)
, 
cubical-identity: (Id_A a b)
, 
Kan-A-filler: Kan-A-filler(X;A;filler)
, 
prop: ℙ
, 
Kan_id_filler: Kan_id_filler(X;A;a;b)
, 
I-path: I-path(X;A;a;b;I;alpha)
, 
pi2: snd(t)
, 
so_lambda: λ2x.t[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
false: False
, 
so_apply: x[s]
, 
top: Top
, 
named-path: named-path(X;A;a;b;I;alpha;z)
, 
squash: ↓T
, 
name-path-endpoints: name-path-endpoints(X;A;a;b;I;alpha;z;omega)
, 
and: P ∧ Q
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
or: 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)
, 
l_all: (∀x∈L.P[x])
, 
A-open-box: A-open-box(X;A;I;alpha;J;x;i)
, 
int_seg: {i..j-}
, 
guard: {T}
, 
lelt: i ≤ j < k
, 
sq_stable: SqStable(P)
, 
coordinate_name: Cname
, 
int_upper: {i...}
, 
decidable: Dec(P)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
less_than: a < b
, 
A-face: A-face(X;A;I;alpha)
, 
is-A-face: is-A-face(X;A;I;alpha;bx;f)
, 
spreadn: spread3, 
cubical-id-box: cubical-id-box(X;A;a;b;I;alpha;box)
, 
lift-id-faces: lift-id-faces(X;A;I;alpha;box)
, 
extend-A-open-box: extend-A-open-box(bx;f1;f2)
, 
uiff: uiff(P;Q)
, 
subtract: n - m
, 
sq_type: SQType(T)
, 
add-fresh-cname: I+
, 
iota': iota'(I)
, 
has-value: (a)↓
, 
lift-id-face: lift-id-face(X;A;I;alpha;face)
, 
assert: ↑b
, 
bnot: ¬bb
, 
bfalse: ff
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
it: ⋅
, 
unit: Unit
, 
bool: 𝔹
, 
deq: EqDecider(T)
, 
true: True
, 
cubical-type-ap-morph: (u a f)
, 
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p)
, 
cand: A c∧ B
, 
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w)
, 
path-eq: path-eq(X;A;I;alpha;p;q)
Lemmas referenced : 
cubical-term_wf, 
Kan-type_wf, 
Kan-cubical-type_wf, 
cubical-set_wf, 
Kan_id_filler_wf1, 
I-cube_wf, 
list_wf, 
nameset_wf, 
int_seg_wf, 
coordinate_name_wf, 
subtype_rel_list, 
cubical-identity_wf, 
A-open-box_wf, 
path-eq-equiv, 
path-eq_wf, 
I-path_wf, 
subtype_quotient, 
Kan-A-filler_wf, 
pi1_wf_top, 
not_wf, 
l_member_wf, 
subtype_rel_product, 
named-path_wf, 
istype-void, 
top_wf, 
Kanfiller_wf, 
cons_wf, 
fresh-cname_wf, 
cube-set-restriction_wf, 
iota_wf, 
cons_member, 
nameset_subtype, 
l_subset_right_cons_trivial, 
cubical-id-box_wf, 
select_wf, 
A-face_wf, 
int_seg_properties, 
length_wf, 
sq_stable__l_member, 
decidable__equal-coordinate_name, 
sq_stable__le, 
decidable__le, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
istype-int, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
name-path-endpoints_wf, 
fresh-cname-not-member2, 
length_of_cons_lemma, 
length-map, 
add-member-int_seg2, 
subtract_wf, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
itermAdd_wf, 
int_term_value_add_lemma, 
istype-le, 
istype-less_than, 
select-cons-tl, 
select-map, 
subtype_base_sq, 
int_subtype_base, 
decidable__equal_int, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
is-A-face_wf, 
coordinate_name-value-type, 
set-value-type, 
value-type-has-value, 
lift-id-face_wf, 
set-path-name_wf, 
list-diff_wf, 
cname_deq_wf, 
nil_wf, 
face-map_wf2, 
subtype_rel_sets_simple, 
member-list-diff, 
set_subtype_base, 
le_wf, 
member_singleton, 
assert-bnot, 
bool_wf, 
bool_cases_sqequal, 
bool_subtype_base, 
deq-member_wf, 
eqff_to_assert, 
assert-deq-member, 
eqtt_to_assert, 
bfalse_wf, 
bor_wf, 
deq_member_nil_lemma, 
deq_member_cons_lemma, 
iff_weakening_equal, 
subtype_rel_self, 
list-diff-cons, 
istype-universe, 
true_wf, 
squash_wf, 
equal_wf, 
cube-set-restriction-comp, 
subtype_rel-equal, 
fresh-cname-not-equal2, 
iota-face-map, 
cubical-type-at_wf, 
cubical-type-ap-morph_wf, 
subtype_rel_set, 
fresh-cname-not-member-list-diff, 
quotient-member-eq, 
named-path-morph_wf, 
cubical-type_wf, 
rename-one-name_wf, 
extend-name-morph_wf, 
cubical-type-ap-morph-comp, 
name-comp_wf, 
name-morph_wf, 
rename-one-extend-name-morph, 
fresh-cname-not-equal, 
extend-face-map-same, 
l_subset_refl, 
l_subset_wf, 
name-morph_subtype, 
rename-one-iota, 
extended-face-map1, 
list_subtype_base, 
name-comp-assoc
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation_alt, 
cut, 
sqequalHypSubstitution, 
hypothesis, 
inhabitedIsType, 
hypothesisEquality, 
universeIsType, 
introduction, 
extract_by_obid, 
isectElimination, 
thin, 
dependent_functionElimination, 
natural_numberEquality, 
rename, 
setElimination, 
lambdaEquality_alt, 
independent_isectElimination, 
because_Cache, 
sqequalRule, 
applyEquality, 
functionExtensionality, 
equalitySymmetry, 
equalityTransitivity, 
dependent_set_memberEquality_alt, 
applyLambdaEquality, 
productElimination, 
setEquality, 
setIsType, 
functionIsType, 
isect_memberEquality_alt, 
voidElimination, 
independent_functionElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
independent_pairFormation, 
inlFormation_alt, 
equalityIstype, 
unionElimination, 
approximateComputation, 
dependent_pairFormation_alt, 
int_eqEquality, 
closedConclusion, 
addEquality, 
productIsType, 
instantiate, 
cumulativity, 
intEquality, 
hyp_replacement, 
callbyvalueReduce, 
promote_hyp, 
equalityIsType1, 
baseApply, 
equalityIsType4, 
equalityElimination, 
universeEquality, 
axiomEquality, 
dependent_pairEquality_alt
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{}  \{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;(Id\_Kan-type(A)  a  b);I;alpha;J;x;i)
                                                          {}\mrightarrow{}  (Id\_Kan-type(A)  a  b)(alpha)| 
                                                          Kan-A-filler(X;(Id\_Kan-type(A)  a  b);filler)\}  )
Date html generated:
2019_11_06-PM-00_45_50
Last ObjectModification:
2018_12_10-PM-05_00_24
Theory : cubical!sets
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