Nuprl Lemma : nearby-separated-partitions
∀I:Interval
((icompact(I) ∧ iproper(I))
⇒ (∀p,q:partition(I). ∀e:{e:ℝ| r0 < e} .
∃p',q':partition(I). (separated-partitions(p';q') ∧ nearby-partitions(e;p;p') ∧ nearby-partitions(e;q;q'))))
Proof
Definitions occuring in Statement :
separated-partitions: separated-partitions(P;Q),
nearby-partitions: nearby-partitions(e;p;q),
partition: partition(I),
icompact: icompact(I),
iproper: iproper(I),
interval: Interval,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
cand: A c∧ B,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
partition: partition(I),
so_apply: x[s],
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
separated-partitions: separated-partitions(P;Q),
frs-separated: frs-separated(p;q),
nearby-partitions: nearby-partitions(e;p;q),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
subtype_rel: A ⊆r B
Lemmas referenced :
nearby-increasing-partition,
exists_wf,
partition_wf,
separated-partitions_wf,
nearby-partitions_wf,
set_wf,
real_wf,
rless_wf,
int-to-real_wf,
icompact_wf,
iproper_wf,
interval_wf,
rdiv_wf,
rless-int,
rmul_preserves_rless,
rmul_wf,
rmul-zero-both,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rless_functionality,
req_transitivity,
rmul-rinv3,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
nearby-increasing-partition-avoids,
l_all_iff,
l_all_wf2,
rneq_wf,
l_member_wf,
all_wf,
int_seg_wf,
length_wf,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lelt_wf,
rleq_functionality_wrt_implies,
rabs_wf,
rsub_wf,
select_wf,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
radd_wf,
rleq_weakening_equal,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
itermAdd_wf,
rleq_functionality,
req_weakening,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
dependent_pairFormation,
sqequalRule,
isectElimination,
independent_isectElimination,
lambdaEquality,
productEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
dependent_set_memberEquality,
inrFormation,
imageMemberEquality,
baseClosed,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
allFunctionality,
promote_hyp,
addLevel,
impliesFunctionality,
levelHypothesis,
allLevelFunctionality,
impliesLevelFunctionality,
functionEquality,
unionElimination,
inlFormation,
applyEquality
Latex:
\mforall{}I:Interval
((icompact(I) \mwedge{} iproper(I))
{}\mRightarrow{} (\mforall{}p,q:partition(I). \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}p',q':partition(I)
(separated-partitions(p';q') \mwedge{} nearby-partitions(e;p;p') \mwedge{} nearby-partitions(e;q;q'))))
Date html generated:
2019_10_30-AM-07_51_33
Last ObjectModification:
2018_08_23-PM-00_49_47
Theory : reals
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