Nuprl Lemma : nearby-increasing-partition-avoids
∀I:Interval
((icompact(I) ∧ iproper(I))
⇒ (∀p:partition(I)
(frs-increasing(p)
⇒ (∀L:ℝ List. ∀e:{e:ℝ| r0 < e} .
∃q:partition(I). (frs-increasing(q) ∧ nearby-partitions(e;p;q) ∧ frs-separated(q;L))))))
Proof
Definitions occuring in Statement :
nearby-partitions: nearby-partitions(e;p;q),
partition: partition(I),
frs-separated: frs-separated(p;q),
frs-increasing: frs-increasing(p),
icompact: icompact(I),
iproper: iproper(I),
interval: Interval,
rless: x < y,
int-to-real: r(n),
real: ℝ,
list: T List,
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,
partition: partition(I),
member: t ∈ T,
sq_stable: SqStable(P),
squash: ↓T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
l_all: (∀x∈L.P[x]),
frs-separated: frs-separated(p;q),
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
nearby-partitions: nearby-partitions(e;p;q),
cand: A c∧ B,
exists: ∃x:A. B[x],
frs-increasing: frs-increasing(p),
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
subtract: n - m,
ge: i ≥ j ,
le: A ≤ B,
less_than: a < b,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
real: ℝ,
true: True,
sq_type: SQType(T),
iff: P ⇐⇒ Q,
partitions: partitions(I;p),
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than': less_than'(a;b),
bfalse: ff,
frs-non-dec: frs-non-dec(L),
icompact: icompact(I),
bool: 𝔹,
unit: Unit,
bnot: ¬bb,
last: last(L),
iproper: iproper(I),
rev_implies: P ⇐ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rsub: x - y,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
nat: ℕ,
rneq: x ≠ y,
rnonneg: rnonneg(x),
rleq: x ≤ y
Lemmas referenced :
sq_stable__partitions,
list_induction,
real_wf,
all_wf,
rless_wf,
int-to-real_wf,
frs-increasing_wf,
partitions_wf,
exists_wf,
partition_wf,
nearby-partitions_wf,
frs-separated_wf,
list_wf,
nil_wf,
set_wf,
cons_wf,
icompact_wf,
iproper_wf,
interval_wf,
int_seg_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
base_wf,
stuck-spread,
length_of_nil_lemma,
add-member-int_seg2,
length_wf,
length_of_cons_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
non_neg_length,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
less_than_wf,
nat_plus_properties,
sq_stable__less_than,
select_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
squash_wf,
le_wf,
add-subtract-cancel,
nat_plus_wf,
le_weakening2,
subtype_base_sq,
int_subtype_base,
true_wf,
select_cons_tl,
iff_weakening_equal,
frs-increasing-non-dec,
rleq_wf,
last_cons,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
false_wf,
right-endpoint_wf,
rleq_transitivity,
left-endpoint_wf,
null_wf3,
subtype_rel_list,
top_wf,
bool_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
length_wf_nat,
nat_wf,
length-singleton,
rless-cases,
radd-preserves-rless,
rsub_wf,
radd_wf,
rless_functionality,
real_term_polynomial,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
avoid-reals-simple,
rmin_wf,
rminus_wf,
req_transitivity,
rmin_functionality,
itermMinus_wf,
real_term_value_minus_lemma,
radd_functionality,
req_weakening,
rmin_strict_ub,
sq_stable__rless,
rless-implies-rless,
rneq_wf,
l_member_wf,
l_all_wf2,
rabs_wf,
rleq_weakening_rless,
rless_transitivity1,
rsub-rmin,
rleq_functionality,
rmax_wf,
rmax_lb,
radd-zero-both,
radd_comm,
rminus-rminus,
rmul-zero-both,
radd-int,
rminus-as-rmul,
rmul_functionality,
rmul-distrib2,
rmul-identity1,
req_inversion,
radd-ac,
rminus-radd,
uiff_transitivity,
rmul_wf,
rabs-difference-bound-rleq,
radd-rminus-assoc,
radd-rminus-both,
radd-assoc,
rless_transitivity2,
radd-preserves-rleq,
sq_stable__rleq,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rminus_functionality,
rmin-rleq,
radd-rmin,
l_all_nil,
l_all_cons,
select-cons-hd,
last_singleton,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
minus-one-mul-top,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
rmul-int,
rmul-rdiv-cancel2,
rmul_preserves_rless,
rless-int,
rdiv_wf,
trivial-rless-radd,
frs-increasing-cons,
rmin_ub,
rmul-rdiv-cancel,
rmul_comm,
rmul-distrib,
less_than'_wf,
rleq-int,
rmul_preserves_rleq2,
rmul_over_rminus,
rmul-one-both,
int_formula_prop_eq_lemma,
intformeq_wf,
and_wf,
last_wf,
assert_wf,
add_functionality_wrt_eq,
decidable__equal_int,
rmin_lb,
rneq-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
setElimination,
rename,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
isectElimination,
lambdaEquality,
setEquality,
natural_numberEquality,
functionEquality,
independent_isectElimination,
productEquality,
because_Cache,
computeAll,
intEquality,
int_eqEquality,
voidEquality,
voidElimination,
isect_memberEquality,
independent_pairFormation,
dependent_set_memberEquality,
dependent_pairFormation,
unionElimination,
addEquality,
applyEquality,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
cumulativity,
universeEquality,
instantiate,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
applyLambdaEquality,
minusEquality,
levelHypothesis,
addLevel,
multiplyEquality,
inrFormation,
axiomEquality,
independent_pairEquality,
isect_memberFormation,
impliesFunctionality,
inlFormation
Latex:
\mforall{}I:Interval
((icompact(I) \mwedge{} iproper(I))
{}\mRightarrow{} (\mforall{}p:partition(I)
(frs-increasing(p)
{}\mRightarrow{} (\mforall{}L:\mBbbR{} List. \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}q:partition(I)
(frs-increasing(q) \mwedge{} nearby-partitions(e;p;q) \mwedge{} frs-separated(q;L))))))
Date html generated:
2017_10_03-AM-09_39_57
Last ObjectModification:
2017_07_28-AM-07_55_30
Theory : reals
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