Nuprl Lemma : length-open_box-le-3
∀[X:CubicalSet]. ∀[I:Cname List].
∀J:nameset(I) List
∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[box:open_box(X;I;J;x;i)]. ((¬↑null(J)) ⇒ (||box|| ≤ 3) ⇒ (∀j'∈J.j' = hd(J) ∈ Cname))
Proof
Definitions occuring in Statement :
open_box: open_box(X;I;J;x;i),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
l_all: (∀x∈L.P[x]),
hd: hd(l),
length: ||as||,
null: null(as),
list: T List,
int_seg: {i..j-},
assert: ↑b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
open_box: open_box(X;I;J;x;i),
subtype_rel: A ⊆r B,
top: Top,
l_all: (∀x∈L.P[x]),
nameset: nameset(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
uiff: uiff(P;Q),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
cons: [a / b],
bfalse: ff,
nat: ℕ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
listp: A List+,
deq: EqDecider(T),
cand: A c∧ B,
l_member: (x ∈ l),
ge: i ≥ j
Lemmas referenced :
length-open_box,
subtype_base_sq,
int_subtype_base,
le_wf,
length_wf,
I-face_wf,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
nameset_wf,
top_wf,
int_seg_wf,
open_box_wf,
coordinate_name_wf,
list_wf,
cubical-set_wf,
cname_deq_wf,
strong-subtype-deq-subtype,
l_member_wf,
strong-subtype-set2,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
remove-repeats_wf,
add-is-int-iff,
multiply-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
false_wf,
l_all_iff,
equal_wf,
hd_wf,
listp_properties,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
length_wf_nat,
nat_wf,
list_subtype_base,
nameset_subtype_base,
decidable__lt,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
remove_repeats_cons_lemma,
remove_repeats_nil_lemma,
subtype_rel_self,
subtype_rel_dep_function,
iff_wf,
all_wf,
bool_wf,
subtype_rel_set,
bnot_wf,
filter_wf5,
member-remove-repeats,
nat_properties,
select_wf,
set_subtype_base,
non_neg_length,
int_formula_prop_eq_lemma,
intformeq_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
length_one,
member_singleton,
l_member_subtype,
hd_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setElimination,
rename,
natural_numberEquality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
sqequalRule,
axiomEquality,
setEquality,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
hypothesis_subsumption,
addEquality,
minusEquality,
dependent_set_memberEquality,
functionExtensionality,
functionEquality,
applyLambdaEquality,
computeAll
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)].
((\mneg{}\muparrow{}null(J)) {}\mRightarrow{} (||box|| \mleq{} 3) {}\mRightarrow{} (\mforall{}j'\mmember{}J.j' = hd(J)))
Date html generated:
2017_10_05-AM-10_20_37
Last ObjectModification:
2017_07_28-AM-11_20_59
Theory : cubical!sets
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