Nuprl Lemma : lift-id-faces-wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[alpha:X(I)]. ∀[box:A-open-box(X;(Id_A a b);I;alpha;J;x;i)].
(lift-id-faces(X;A;I;alpha;box) ∈ A-open-box(X;A;I+;iota'(I)(alpha);J;x;i))
Proof
Definitions occuring in Statement :
lift-id-faces: lift-id-faces(X;A;I;alpha;box),
cubical-identity: (Id_A a b),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
cubical-term: {X ⊢ _:AF},
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
iota': iota'(I),
add-fresh-cname: I+,
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
and: P ∧ Q,
lift-id-faces: lift-id-faces(X;A;I;alpha;box),
cand: A c∧ B,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
A-face: A-face(X;A;I;alpha),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
A-adjacent-compatible: A-adjacent-compatible(X;A;I;alpha;L),
pairwise: (∀x,y∈L. P[x; y]),
sq_stable: SqStable(P),
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
spreadn: spread3,
lift-id-face: lift-id-face(X;A;I;alpha;face),
cubical-path: cubical-path(X;A;a;b;I;alpha),
pi1: fst(t),
cubical-type-at: A(a),
cubical-identity: (Id_A a b),
iff: P ⇐⇒ Q,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
cubical-type-ap-morph: (u a f),
pi2: snd(t),
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p),
I-path: I-path(X;A;a;b;I;alpha),
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w),
sq_type: SQType(T),
iota': iota'(I),
has-value: (a)↓,
add-fresh-cname: I+,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
named-path: named-path(X;A;a;b;I;alpha;z),
top: Top,
deq: EqDecider(T),
A-face-name: A-face-name(f),
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x])
Lemmas referenced :
map_wf,
A-face_wf,
cubical-identity_wf,
add-fresh-cname_wf,
cube-set-restriction_wf,
iota'_wf,
lift-id-face_wf,
nameset_wf,
subtype_rel_list,
coordinate_name_wf,
A-adjacent-compatible_wf,
l_member_wf,
istype-void,
l_subset_wf,
l_exists_wf,
equal_wf,
int_seg_wf,
A-face-name_wf,
nameset_subtype,
subtype-add-fresh-cname,
l_all_wf2,
not_wf,
subtract_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
istype-le,
istype-less_than,
pi1_wf_top,
subtype_rel_product,
cubical-type-at_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
face-map_wf2,
top_wf,
pairwise_wf2,
A-open-box_wf,
I-cube_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
length-map,
length_wf,
select_wf,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
A-face-compatible_wf,
select-map,
path-eq-equiv,
path-eq_wf,
I-path_wf,
subtype_quotient,
set-path-name_wf,
fresh-cname_wf,
subtype_rel_sets_simple,
member-list-diff,
true_wf,
squash_wf,
name-morph_wf,
name-comp_wf,
subtype_rel_wf,
list-diff2-sym,
iff_weakening_equal,
subtype_rel_self,
cubical-type-ap-morph_wf,
list-diff2,
cube-set-restriction-comp,
face-maps-commute,
cubical-path-same-name,
I-path-morph_wf,
named-path-equal-implies,
fresh-cname-not-member2,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
coordinate_name-value-type,
set-value-type,
value-type-has-value,
set_wf,
cons_member,
or_wf,
member_singleton,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert-deq-member,
eqtt_to_assert,
bool_wf,
deq-member_wf,
list-diff-cons,
and_wf,
iota_wf,
subtype_rel-equal,
iota-face-map,
cubical-type-ap-rename-one-equal,
fresh-cname-not-member-list-diff,
istype-universe,
list-diff-cons-single,
list_subtype_base,
rename-one-name_wf,
iota-two-face-maps,
name-comp-assoc,
rename-one-iota,
cubical-type-ap-morph-comp,
int_formula_prop_eq_lemma,
intformeq_wf,
deq_wf,
trivial-equal,
extend-name-morph_wf,
extended-face-map,
bfalse_wf,
bor_wf,
deq_member_nil_lemma,
deq_member_cons_lemma,
l_subset-l_contains,
cons-l_contains,
l_contains_weakening,
l_contains_transitivity,
map-length,
equal_functionality_wrt_subtype_rel2,
lelt_wf,
product_subtype_base,
nameset_subtype_base,
decidable__equal_int,
l_subset_right_cons_trivial,
length-map-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
productElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
universeIsType,
independent_pairFormation,
lambdaFormation_alt,
because_Cache,
applyEquality,
independent_isectElimination,
sqequalRule,
productIsType,
functionIsType,
productEquality,
imageElimination,
independent_pairEquality,
inhabitedIsType,
setIsType,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
voidElimination,
instantiate,
cumulativity,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
equalityIstype,
lambdaEquality,
lambdaFormation,
universeEquality,
hyp_replacement,
setEquality,
applyLambdaEquality,
promote_hyp,
intEquality,
callbyvalueReduce,
inrFormation,
inlFormation,
addLevel,
dependent_pairFormation,
equalityElimination,
dependent_set_memberEquality,
equalityIsType1,
closedConclusion,
voidEquality,
isect_memberEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List].
\mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2]. \mforall{}[alpha:X(I)]. \mforall{}[box:A-open-box(X;(Id\_A a b);I;alpha;J;x;i)].
(lift-id-faces(X;A;I;alpha;box) \mmember{} A-open-box(X;A;I+;iota'(I)(alpha);J;x;i))
Date html generated:
2020_05_21-AM-11_12_29
Last ObjectModification:
2020_01_03-PM-06_02_22
Theory : cubical!sets
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