Nuprl Lemma : length-open_box
∀[X:CubicalSet]. ∀[I:Cname List].
∀J:nameset(I) List
∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[box:open_box(X;I;J;x;i)]. (||box|| = (1 + (||remove-repeats(CnameDeq;J)|| * 2)) ∈ ℤ)
Proof
Definitions occuring in Statement :
open_box: open_box(X;I;J;x;i),
cubical-set: CubicalSet,
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
remove-repeats: remove-repeats(eq;L),
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nameset: nameset(L),
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
open_box: open_box(X;I;J;x;i),
and: P ∧ Q,
cand: A c∧ B,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
sq_stable: SqStable(P),
coordinate_name: Cname,
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_all: (∀x∈L.P[x]),
I-face: I-face(X;I),
face-name: face-name(f),
face-dimension: dimension(f),
pi1: fst(t),
pi2: snd(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
equipollent: A ~ B,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
respects-equality: respects-equality(S;T),
isl: isl(x),
true: True,
subtract: n - m,
l_member: (x ∈ l),
pairwise: (∀x,y∈L. P[x; y]),
surject: Surj(A;B;f),
l_exists: (∃x∈L. P[x])
Lemmas referenced :
cname_deq_wf,
strong-subtype-deq-subtype,
coordinate_name_wf,
l_member_wf,
strong-subtype-set2,
equipollent-nsub,
length_wf_nat,
I-face_wf,
length_wf,
nameset_wf,
remove-repeats_wf,
add_nat_wf,
istype-void,
istype-le,
multiply_nat_wf,
nat_properties,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
add-is-int-iff,
multiply-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
intformeq_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_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
open_box_wf,
subtype_rel_list,
int_seg_wf,
list_wf,
cubical-set_wf,
mul_bounds_1a,
equipollent_functionality_wrt_equipollent2,
equipollent_transitivity,
equipollent_inversion,
equipollent-add,
union_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent-multiply,
product_functionality_wrt_equipollent_left,
nameset-equipollent,
select_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
cons_member,
eq-cname_wf,
eqtt_to_assert,
assert-eq-cname,
istype-less_than,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal_wf,
cons_wf,
biject_wf,
face-dimension_wf,
bnot_wf,
not_wf,
respects-equality-set-trivial2,
istype-assert,
bool_cases,
iff_transitivity,
assert_of_bnot,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
int_subtype_base,
subtract_wf,
respects-equality-product,
respects-equality-trivial,
subtype-base-respects-equality,
istype-base,
set_subtype_base,
le_wf,
lelt_wf,
decidable__equal_int,
int_seg_subtype_special,
int_seg_cases,
face-name_wf,
decidable__equal_int_seg,
select_member,
member_wf,
istype-false,
nameset_subtype_base,
pi1_wf_top,
pi2_wf,
product_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
hypothesis,
applyEquality,
thin,
sqequalHypSubstitution,
isectElimination,
setEquality,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality_alt,
universeIsType,
setElimination,
rename,
productElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
addEquality,
natural_numberEquality,
multiplyEquality,
imageElimination,
independent_pairFormation,
voidElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
equalityIstype,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
functionIsTypeImplies,
unionEquality,
productEquality,
because_Cache,
equalityElimination,
inlEquality_alt,
productIsType,
instantiate,
cumulativity,
inrEquality_alt,
independent_pairEquality,
unionIsType,
functionIsType,
intEquality,
sqequalBase,
hypothesis_subsumption
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[I:Cname List].
\mforall{}J:nameset(I) List
\mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2]. \mforall{}[box:open\_box(X;I;J;x;i)].
(||box|| = (1 + (||remove-repeats(CnameDeq;J)|| * 2)))
Date html generated:
2020_05_21-AM-10_51_40
Last ObjectModification:
2020_02_07-PM-09_03_35
Theory : cubical!sets
Home
Index