Nuprl Lemma : Kan_id_filler_uniform
∀X:CubicalSet. ∀A:{X ⊢ _(Kan)}. ∀a,b:{X ⊢ _:Kan-type(A)}.
uniform-Kan-A-filler(X;(Id_Kan-type(A) a b);Kan_id_filler(X;A;a;b))
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)},
uniform-Kan-A-filler: uniform-Kan-A-filler(X;A;filler),
cubical-term: {X ⊢ _:AF},
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,
name-morph: name-morph(I;J),
nameset: nameset(L),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
sq_stable: SqStable(P),
squash: ↓T,
nat: ℕ,
guard: {T},
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_upper: {i...},
coordinate_name: Cname,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
name-morph-domain: name-morph-domain(f;I),
rev_implies: P ⇐ Q,
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
l_subset: l_subset(T;as;bs),
bnot: ¬bb,
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
extend-name-morph: f[z1:=z2],
rev_uimplies: rev_uimplies(P;Q),
isname: isname(z),
less_than': less_than'(a;b),
less_than: a < b,
nil: [],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
cons: [a / b],
pi2: snd(t),
cubical-type-ap-morph: (u a f),
cubical-identity: (Id_A a b),
Kan_id_filler: Kan_id_filler(X;A;a;b),
path-eq: path-eq(X;A;I;alpha;p;q),
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p),
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
cubical-id-box: cubical-id-box(X;A;a;b;I;alpha;box),
A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx),
extend-A-open-box: extend-A-open-box(bx;f1;f2),
term-A-face: term-A-face(a;I;alpha;i),
A-face-image: A-face-image(X;A;I;K;f;alpha;face),
A-face: A-face(X;A;I;alpha),
spreadn: spread3,
le: A ≤ B,
lift-id-faces: lift-id-faces(X;A;I;alpha;box),
compose: f o g,
cubical-type-at: A(a),
pi1: fst(t),
l_all: (∀x∈L.P[x]),
A-face-name: A-face-name(f),
cubical-path: cubical-path(X;A;a;b;I;alpha),
quotient: x,y:A//B[x; y],
I-path: I-path(X;A;a;b;I;alpha),
lift-id-face: lift-id-face(X;A;I;alpha;face),
set-path-name: set-path-name(X;A;I;alpha;x;p),
named-path: named-path(X;A;a;b;I;alpha;z),
respects-equality: respects-equality(S;T)
Lemmas referenced :
istype-assert,
isname_wf,
l_member_wf,
coordinate_name_wf,
subtype_rel_list,
nameset_wf,
name-morph_wf,
A-open-box_wf,
cubical-identity_wf,
Kan-type_wf,
int_seg_wf,
list_wf,
I-cube_wf,
cubical-term_wf,
Kan-cubical-type_wf,
cubical-set_wf,
equal_wf,
map_wf,
list-subtype,
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,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
decidable__equal-coordinate_name,
sq_stable__l_member,
int_seg_properties,
nat_properties,
sq_stable__le,
select_wf,
int_subtype_base,
le_wf,
set_subtype_base,
subtype_base_sq,
assert-isname,
cons_wf,
cons_member,
filter_wf5,
member_filter_2,
bool_subtype_base,
bool_wf,
assert_elim,
extd-nameset_wf,
and_wf,
not_wf,
fresh-cname_wf,
l_subset_cons_same,
l_subset_right_cons_trivial,
nameset_subtype,
assert_wf,
not_functionality_wrt_uiff,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
assert-eq-cname,
eqtt_to_assert,
eq-cname_wf,
set_wf,
assert_of_le_int,
map_cons_lemma,
istype-void,
iff_weakening_uiff,
member_wf,
decidable__equal_int,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
spread_cons_lemma,
product_subtype_list,
map_nil_lemma,
list-cases,
colength_wf_list,
nat_wf,
equal-wf-T-base,
less_than_wf,
ge_wf,
int_formula_prop_less_lemma,
intformless_wf,
fresh-cname-not-equal2,
equal-cubical-identity-at,
Kan_id_filler_wf1,
cube-set-restriction_wf,
A-open-box-image_wf,
I-path-morph_wf,
I-path_wf,
Kanfiller-uniform,
iota_wf,
cubical-id-box_wf,
extend-name-morph_wf,
fresh-cname-not-member2,
isname-name,
ifthenelse_wf,
isname-nameset,
squash_wf,
true_wf,
istype-universe,
extend-name-morph-iota,
name-comp_wf,
iff_weakening_equal,
subtype_rel_self,
cube-set-restriction-comp,
cubical-type_wf,
cubical-type-at_wf,
Kanfiller_wf,
cubical-type-ap-morph_wf,
subtype_rel-equal,
rename-one-name_wf,
cubical-type-ap-morph-id,
rename-one-same,
istype-int,
A-open-box-equal,
filter_cons_lemma,
length_wf,
lelt_wf,
select_member,
A-face_wf,
equal-wf-base,
false_wf,
cubical-term-at-morph,
face-map_wf2,
nil_wf,
cname_deq_wf,
list-diff_wf,
add-remove-fresh-sq,
name-comp-id-right,
l_subset_refl,
name-morph_subtype,
id-morph_wf,
iota-identity,
cubical-term-at_wf,
face-map_wf,
extend-name-morph-irrelevant,
cube-set-restriction-id,
map-map,
istype-le,
istype-less_than,
A-face-name_wf,
decidable__lt,
path-eq_wf,
equal-wf-base-T,
ite_rw_false,
bfalse_wf,
assert_of_ff,
extd-nameset_subtype_base,
name-morph_subtype_remove1,
member-list-diff,
extd-nameset-respects-equality,
nameset_subtype_extd-nameset,
rename-one-iota,
subtype_rel_weakening,
ext-eq_weakening,
list-diff-cons-single,
rename-one-extend-id,
iota-face-map,
list_subtype_base,
face-map-comp,
name-comp-assoc,
extend-name-morph-rename-one,
cubical-type-ap-morph-comp,
rename-one-extend-name-morph,
l_subset_wf,
fills-A-open-box_wf,
subtype_rel_set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
setElimination,
rename,
sqequalRule,
functionIsType,
inhabitedIsType,
universeIsType,
independent_isectElimination,
lambdaEquality_alt,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
independent_pairFormation,
independent_functionElimination,
lambdaFormation,
setEquality,
equalitySymmetry,
equalityTransitivity,
lambdaEquality,
voidEquality,
voidElimination,
functionExtensionality,
hyp_replacement,
equalityElimination,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
intEquality,
cumulativity,
instantiate,
productElimination,
dependent_set_memberEquality,
inlFormation,
promote_hyp,
isect_memberEquality_alt,
dependent_pairFormation_alt,
equalityIstype,
addEquality,
hypothesis_subsumption,
axiomSqEquality,
intWeakElimination,
inlFormation_alt,
dependent_set_memberEquality_alt,
universeEquality,
equalityIsType1,
setIsType,
sqequalBase,
inrFormation_alt,
unionIsType,
equalityIsType2,
closedConclusion,
inrFormation,
equalityIsType3,
dependent_pairEquality_alt,
productIsType,
productEquality,
independent_pairEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
baseApply
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_(Kan)\}. \mforall{}a,b:\{X \mvdash{} \_:Kan-type(A)\}.
uniform-Kan-A-filler(X;(Id\_Kan-type(A) a b);Kan\_id\_filler(X;A;a;b))
Date html generated:
2019_11_06-PM-00_51_50
Last ObjectModification:
2018_12_20-PM-00_18_49
Theory : cubical!sets
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