Nuprl Lemma : length-open_box-ge-3
∀[X:CubicalSet]. ∀[I:Cname List].
∀J:nameset(I) List
∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[box:open_box(X;I;J;x;i)]. (3 < ||box|| ⇒ (∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ 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_exists: (∃x∈L. P[x]),
length: ||as||,
list: T List,
int_seg: {i..j-},
less_than: a < b,
uall: ∀[x:A]. B[x],
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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
open_box: open_box(X;I;J;x;i),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
int_seg: {i..j-},
coordinate_name: Cname,
int_upper: {i...},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
false: False,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons: [a / b],
less_than': less_than'(a;b),
cand: A c∧ B,
l_exists: (∃x∈L. P[x]),
le: A ≤ B,
nat_plus: ℕ+,
true: True,
select: L[n],
sq_stable: SqStable(P),
l_member: (x ∈ l),
nat: ℕ,
ge: i ≥ j
Lemmas referenced :
length-open_box,
less_than_wf,
length_wf,
I-face_wf,
open_box_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
cubical-set_wf,
cname_deq_wf,
strong-subtype-deq-subtype,
l_member_wf,
strong-subtype-set2,
int_seg_properties,
decidable__lt,
remove-repeats_wf,
add-is-int-iff,
multiply-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
false_wf,
decidable__l_exists_better-extract,
l_exists_wf,
not_wf,
equal_wf,
decidable__not,
decidable__equal-coordinate_name,
subtype_base_sq,
int_subtype_base,
remove-repeats-length-one,
list-cases,
product_subtype_list,
remove_repeats_nil_lemma,
length_of_nil_lemma,
cons_member,
cons_wf,
all_wf,
isect_wf,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
lelt_wf,
select_wf,
sq_stable__l_member,
sq_stable__le,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
nameset_subtype_base,
nat_properties,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
setElimination,
rename,
applyEquality,
independent_isectElimination,
lambdaEquality,
because_Cache,
sqequalRule,
setEquality,
productElimination,
unionElimination,
pointwiseFunctionality,
equalityTransitivity,
promote_hyp,
imageElimination,
baseClosed,
baseApply,
closedConclusion,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
instantiate,
cumulativity,
hypothesis_subsumption,
inlFormation,
productEquality,
dependent_set_memberEquality,
imageMemberEquality,
addEquality
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)].
(3 < ||box|| {}\mRightarrow{} (\mexists{}j1\mmember{}J. (\mexists{}j2\mmember{}J. \mneg{}(j1 = j2))))
Date html generated:
2017_10_05-AM-10_20_28
Last ObjectModification:
2017_07_28-AM-11_20_55
Theory : cubical!sets
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