Nuprl Lemma : cubical-id-box_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)].
(cubical-id-box(X;A;a;b;I;alpha;box) ∈ A-open-box(X;A;[fresh-cname(I) /
I];iota(fresh-cname(I))(alpha);[fresh-cname(I) / J];x;i))
Proof
Definitions occuring in Statement :
cubical-id-box: cubical-id-box(X;A;a;b;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(x),
fresh-cname: fresh-cname(I),
nameset: nameset(L),
coordinate_name: Cname,
cons: [a / b],
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,
cubical-id-box: cubical-id-box(X;A;a;b;I;alpha;box),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
false: False,
l_member: (x ∈ l),
nat: ℕ,
less_than': less_than'(a;b),
top: Top,
select: L[n],
cons: [a / b],
cand: A c∧ B,
nat_plus: ℕ+,
guard: {T},
uiff: uiff(P;Q),
sq_stable: SqStable(P),
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
A-face-name: A-face-name(f),
term-A-face: term-A-face(a;I;alpha;i),
pi1: fst(t),
pi2: snd(t),
l_all: (∀x∈L.P[x]),
lift-id-faces: lift-id-faces(X;A;I;alpha;box),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
A-face: A-face(X;A;I;alpha),
lift-id-face: lift-id-face(X;A;I;alpha;face),
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
spreadn: spread3,
iff: P ⇐⇒ Q,
I-path: I-path(X;A;a;b;I;alpha),
named-path: named-path(X;A;a;b;I;alpha;z),
name-path-endpoints: name-path-endpoints(X;A;a;b;I;alpha;z;omega),
true: True,
rev_implies: P ⇐ Q,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
iota': iota'(I),
cubical-term-at: u(a)
Lemmas referenced :
extend-A-open-box_wf,
cons_wf,
coordinate_name_wf,
fresh-cname_wf,
cube-set-restriction_wf,
iota_wf,
subtype_rel_list,
nameset_wf,
nameset_subtype,
l_subset_right_cons_trivial,
lift-id-faces_wf,
term-A-face_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
istype-void,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
add-is-int-iff,
intformand_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
length_wf,
select_wf,
nat_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
l_member_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
fresh-cname-not-equal2,
A-open-box_wf,
cubical-identity_wf,
I-cube_wf,
int_seg_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
length-map,
A-face_wf,
top_wf,
sq_stable__not,
set-path-name_wf,
list-diff_wf,
cname_deq_wf,
nil_wf,
face-map_wf2,
subtype_rel_sets_simple,
not_wf,
member-list-diff,
le_wf,
lelt_wf,
select-map,
member_singleton,
equal_wf,
squash_wf,
true_wf,
istype-universe,
list-diff-cons,
subtype_rel_self,
iff_weakening_equal,
deq_member_cons_lemma,
deq_member_nil_lemma,
bor_wf,
bfalse_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
deq-member_wf,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
list-diff2,
deq_wf,
list-diff-disjoint,
l_disjoint_singleton,
cons_member,
iota'-identity,
cube-set-restriction-comp,
subtype_rel-equal,
add-remove-fresh-sq,
cube-set-restriction-id,
name-comp_wf,
list_subtype_base,
face-map_wf,
name-morph_wf,
face-maps-commute,
cubical-type-at_wf,
cubical-type-ap-morph_wf,
iota-face-map,
name-morph_subtype,
l_subset_wf,
l_subset_refl,
cubical-type-ap-morph-comp,
cubical-term-at-morph,
subtype_rel_weakening,
ext-eq_weakening,
cubical-term-at_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
independent_pairFormation,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
Error :memTop,
universeIsType,
voidElimination,
productIsType,
lambdaFormation_alt,
isect_memberEquality_alt,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
int_eqEquality,
equalityIstype,
imageMemberEquality,
instantiate,
cumulativity,
independent_pairEquality,
axiomEquality,
isectIsTypeImplies,
functionIsType,
intEquality,
equalityIsType1,
setEquality,
equalityIsType4,
hyp_replacement,
equalityIsType3,
universeEquality,
equalityElimination,
inlFormation_alt
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)].
(cubical-id-box(X;A;a;b;I;alpha;box)
\mmember{} A-open-box(X;A;[fresh-cname(I) / I];iota(fresh-cname(I))(alpha);[fresh-cname(I) / J];x;i))
Date html generated:
2020_05_21-AM-11_13_14
Last ObjectModification:
2019_12_08-PM-07_04_17
Theory : cubical!sets
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