Nuprl Lemma : csm-Kan-cubical-identity
∀[X,Delta:CubicalSet]. ∀[s:Delta ⟶ X]. ∀[A:{X ⊢ _(Kan)}]. ∀[a,b:{X ⊢ _:Kan-type(A)}].
((Kan(Id_A a b))s = Kan(Id_(A)s (a)s (b)s) ∈ {Delta ⊢ _(Kan)})
Proof
Definitions occuring in Statement :
Kan-cubical-identity: Kan(Id_A a b),
csm-Kan-cubical-type: (AK)s,
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:AF},
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
nameset: nameset(L),
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
all: ∀x:A. B[x],
prop: ℙ,
cubical-type: {X ⊢ _},
Kan-cubical-identity: Kan(Id_A a b),
csm-Kan-cubical-type: (AK)s,
Kan-cubical-type: {X ⊢ _(Kan)},
true: True,
top: Top,
squash: ↓T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
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),
csm-ap-type: (AF)s,
cubical-identity: (Id_A a b),
Kan-type: Kan-type(Ak),
ge: i ≥ j ,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
int_upper: {i...},
coordinate_name: Cname,
less_than: a < b,
nat_plus: ℕ+,
cand: A c∧ B,
cons: [a / b],
select: L[n],
lelt: i ≤ j < k,
int_seg: {i..j-},
false: False,
not: ¬A,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
pi2: snd(t),
Kanfiller: filler(x;i;bx),
Kan_id_filler: Kan_id_filler(X;A;a;b),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
A-face: A-face(X;A;I;alpha),
cubical-id-box: cubical-id-box(X;A;a;b;I;alpha;box),
extend-A-open-box: extend-A-open-box(bx;f1;f2),
term-A-face: term-A-face(a;I;alpha;i),
csm-ap: (s)x,
cubical-term-at: u(a),
csm-ap-term: (t)s,
lift-id-faces: lift-id-faces(X;A;I;alpha;box),
quotient: x,y:A//B[x; y],
I-path: I-path(X;A;a;b;I;alpha),
named-path: named-path(X;A;a;b;I;alpha;z),
path-eq: path-eq(X;A;I;alpha;p;q),
lift-id-face: lift-id-face(X;A;I;alpha;face),
spreadn: spread3,
set-path-name: set-path-name(X;A;I;alpha;x;p),
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T)
Lemmas referenced :
cubical-set_wf,
cube-set-map_wf,
Kan-cubical-type_wf,
cubical-type-at_wf,
subtype_rel_list,
A-open-box_wf,
int_seg_wf,
nameset_wf,
iff_weakening_equal,
subtype_rel_self,
csm-cubical-identity,
cube-set-restriction_wf,
name-morph_wf,
I-cube_wf,
coordinate_name_wf,
list_wf,
istype-universe,
true_wf,
squash_wf,
equal_wf,
cubical-identity_wf,
cubical-type-equal,
istype-void,
type-csm-Kan-cubical-type,
csm-ap-type_wf,
cubical-term_wf,
subtype_rel-equal,
Kan-type_wf,
csm-ap-term_wf,
Kan-cubical-identity_wf,
csm-Kan-cubical-type_wf,
Kan-cubical-type-equal,
path-eq-equiv,
path-eq_wf,
csm-ap_wf,
I-path_wf,
subtype_quotient,
csm-A-open-box,
Kan_id_filler_wf1,
csm-I-path,
equal-I-paths,
cubical-id-box_wf,
A-open-box-equal,
l_subset_right_cons_trivial,
nameset_subtype,
l_member_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
sq_stable__le,
decidable__equal-coordinate_name,
sq_stable__l_member,
nat_properties,
select_wf,
length_wf,
false_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
intformand_wf,
add-is-int-iff,
nat_plus_properties,
istype-less_than,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
int_seg_properties,
length_wf_nat,
add_nat_plus,
length_of_cons_lemma,
istype-le,
iota_wf,
csm-ap-restriction,
cons_wf,
fresh-cname_wf,
not_wf,
A-face_wf,
csm-type-at,
cubical-type_wf,
face-map_wf2,
nil_wf,
cname_deq_wf,
list-diff_wf,
add-remove-fresh-sq,
cube-set-restriction-comp,
cubical-term-at_wf,
iota-identity,
fresh-cname-not-member2,
cube-set-restriction-id,
name-morph_subtype,
l_subset_refl,
map_wf,
member-list-diff,
csm-cubical-type-ap-morph,
cubical-type-ap-morph-comp,
rename-one-name_wf,
extend-name-morph_wf,
id-morph_wf,
list-diff-cons-single,
fresh-cname-not-equal,
name-comp_wf,
extend-name-morph-rename-one,
l_subset_wf,
rename-one-extend-id,
subtype_base_sq,
list_subtype_base,
set_subtype_base,
le_wf,
int_subtype_base,
rename-one-iota,
iota-face-map,
fresh-cname-not-equal2,
cubical-type-ap-morph_wf,
extend-name-morph-iota,
name-comp-id-left
Rules used in proof :
isectIsTypeImplies,
axiomEquality,
functionIsType,
dependent_pairEquality_alt,
independent_functionElimination,
productElimination,
dependent_functionElimination,
cumulativity,
functionEquality,
productEquality,
universeEquality,
universeIsType,
instantiate,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
rename,
setElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
voidElimination,
isect_memberEquality_alt,
sqequalRule,
because_Cache,
imageElimination,
lambdaEquality_alt,
independent_isectElimination,
applyEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
functionExtensionality,
lambdaEquality,
voidEquality,
isect_memberEquality,
productIsType,
equalityIstype,
int_eqEquality,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
applyLambdaEquality,
Error :memTop,
approximateComputation,
unionElimination,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
dependent_pairFormation_alt,
independent_pairFormation,
hyp_replacement,
setEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
sqequalBase,
intEquality
Latex:
\mforall{}[X,Delta:CubicalSet]. \mforall{}[s:Delta {}\mrightarrow{} X]. \mforall{}[A:\{X \mvdash{} \_(Kan)\}]. \mforall{}[a,b:\{X \mvdash{} \_:Kan-type(A)\}].
((Kan(Id\_A a b))s = Kan(Id\_(A)s (a)s (b)s))
Date html generated:
2020_05_21-AM-11_14_14
Last ObjectModification:
2020_01_15-PM-01_40_28
Theory : cubical!sets
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