Nuprl Lemma : nearby-increasing-partition
∀I:Interval
((icompact(I) ∧ iproper(I))
⇒ (∀p:partition(I). ∀e:{e:ℝ| r0 < e} . ∃q:partition(I). (frs-increasing(q) ∧ nearby-partitions(e;p;q))))
Proof
Definitions occuring in Statement :
nearby-partitions: nearby-partitions(e;p;q),
partition: partition(I),
frs-increasing: frs-increasing(p),
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,
exists: ∃x:A. B[x],
partition: partition(I),
iproper: iproper(I),
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
icompact: icompact(I),
subtype_rel: A ⊆r B,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
nearby-partitions: nearby-partitions(e;p;q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y,
int_seg: {i..j-},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
nil: [],
select: L[n],
frs-increasing: frs-increasing(p),
cand: A c∧ B,
squash: ↓T,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
sq_stable: SqStable(P),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
less_than: a < b,
real: ℝ,
rneq: x ≠ y,
true: True,
subtract: n - m,
decidable: Dec(P),
cons: [a / b],
less_than': less_than'(a;b),
ge: i ≥ j ,
partitions: partitions(I;p),
frs-non-dec: frs-non-dec(L),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
last: last(L)
Lemmas referenced :
small-reciprocal-real,
real_wf,
rless_wf,
int-to-real_wf,
partition_wf,
icompact_wf,
iproper_wf,
interval_wf,
null_wf3,
subtype_rel_list,
top_wf,
istype-void,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
list_wf,
nil_wf,
nearby-partitions_wf,
frs-increasing_wf,
le_witness_for_triv,
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,
full-omega-unsat,
nat_plus_properties,
int_seg_properties,
istype-base,
stuck-spread,
length_of_nil_lemma,
iff_weakening_equal,
subtype_rel_self,
squash_wf,
partitions_wf,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
istype-nat,
length_wf_nat,
sq_stable__partitions,
list_ind_cons_lemma,
list_ind_nil_lemma,
cons_wf,
int_term_value_add_lemma,
itermAdd_wf,
false_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract-is-int-iff,
decidable__le,
int_formula_prop_not_lemma,
intformnot_wf,
rless-int,
rdiv_wf,
sq_stable__less_than,
length_wf,
subtract_wf,
exists_wf,
right-endpoint_wf,
istype-assert,
last_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
decidable__lt,
length_of_cons_lemma,
product_subtype_list,
list-cases,
istype-false,
select_wf,
left-endpoint_wf,
or_wf,
not_wf,
list_induction,
last-cons,
non_neg_length,
rless-cases,
istype-less_than,
last_cons,
true_wf,
rleq_wf,
add-subtract-cancel,
istype-universe,
less_than_wf,
add-swap,
istype-le,
add-member-int_seg2,
select_cons_tl,
rleq_transitivity,
add-is-int-iff,
nat_plus_wf,
req-iff-rsub-is-0,
rsub_wf,
rless-implies-rless,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
int_formula_prop_eq_lemma,
intformeq_wf,
imax_nat_plus,
imax_wf,
mul_nat_plus,
int_term_value_mul_lemma,
itermMultiply_wf,
multiply-is-int-iff,
multiply_nat_plus,
rless_transitivity2,
less_than_transitivity1,
rleq-int-fractions,
imax_ub,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
less_than_transitivity2,
nat_plus_subtype_nat,
mul_preserves_le,
int_entire_a,
rneq-int,
mklist_length,
select-mklist,
mklist_wf,
rless_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
rinv_wf2,
rmul_wf,
rmul_preserves_rless,
rless_functionality,
req_transitivity,
rmul-rinv3,
real_term_value_mul_lemma,
frs-increasing-non-dec,
rleq_weakening,
rleq_weakening_rless,
decidable__equal_int,
rleq_functionality,
req_weakening,
rleq-int-fractions2,
rabs_wf,
rabs_functionality,
rabs-of-nonneg,
radd_wf,
radd_functionality_wrt_rleq,
real_term_value_add_lemma,
radd_functionality_wrt_rless1,
itermMinus_wf,
rminus_wf,
real_term_value_minus_lemma,
rabs-rminus,
lt_int_wf,
assert_of_lt_int,
add-member-int_seg1,
nat_properties,
int_seg_subtype_nat,
rmul_functionality,
rmul-rinv,
radd_functionality,
rminus_functionality,
rminus-int,
radd-preserves-rless,
req_inversion,
radd-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
setElimination,
rename,
independent_functionElimination,
setIsType,
universeIsType,
hypothesis,
isectElimination,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
productIsType,
applyEquality,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
because_Cache,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
baseClosed,
productEquality,
applyLambdaEquality,
hyp_replacement,
functionIsTypeImplies,
independent_pairFormation,
int_eqEquality,
approximateComputation,
universeEquality,
imageMemberEquality,
imageElimination,
closedConclusion,
intEquality,
equalityIsType4,
dependent_set_memberEquality_alt,
unionIsType,
equalityIsType3,
functionIsType,
baseApply,
pointwiseFunctionality,
inrFormation_alt,
equalityIsType1,
minusEquality,
addEquality,
hypothesis_subsumption,
functionEquality,
inlFormation_alt,
multiplyEquality,
functionExtensionality
Latex:
\mforall{}I:Interval
((icompact(I) \mwedge{} iproper(I))
{}\mRightarrow{} (\mforall{}p:partition(I). \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}q:partition(I). (frs-increasing(q) \mwedge{} nearby-partitions(e;p;q))))
Date html generated:
2019_10_29-AM-10_48_37
Last ObjectModification:
2018_12_13-PM-01_45_19
Theory : reals
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