Nuprl Lemma : nearby-partition-sum
∀I:Interval
(icompact(I)
⇒ iproper(I)
⇒ (∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. ∀p:partition(I). ∀x:partition-choice(full-partition(I;p)).
∀alpha:{a:ℝ| r0 < a} .
∃e:{e:ℝ| r0 < e}
∀q:partition(I). ∀y:partition-choice(full-partition(I;q)).
(nearby-partitions(e;p;q)
⇒ (∀i:ℕ||p|| + 1. (|x[i] - y[i]| ≤ e))
⇒ (|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| ≤ alpha))))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
partition-sum: S(f;p),
partition-choice-ap: x[i],
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
nearby-partitions: nearby-partitions(e;p;q),
partition: partition(I),
icompact: icompact(I),
rfun: I ⟶ℝ,
iproper: iproper(I),
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
length: ||as||,
int_seg: {i..j-},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
icompact: icompact(I),
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
label: ...$L... t,
le: A ≤ B,
ge: i ≥ j ,
partition: partition(I),
full-partition: full-partition(I;p),
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
rless: x < y,
or: P ∨ Q,
rneq: x ≠ y,
nat_plus: ℕ+,
rfun: I ⟶ℝ,
so_apply: x[s],
so_lambda: λ2x.t[x],
sq_exists: ∃x:A [B[x]],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ,
uall: ∀[x:A]. B[x],
continuous: f[x] continuous for x ∈ I,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
less_than': less_than'(a;b),
less_than: a < b,
sq_stable: SqStable(P),
rdiv: (x/y),
nat: ℕ,
nearby-partitions: nearby-partitions(e;p;q),
lelt: i ≤ j < k,
int_seg: {i..j-},
rleq: x ≤ y,
rnonneg: rnonneg(x),
partition-sum: S(f;p),
sq_type: SQType(T),
real: ℝ,
partition-choice-ap: x[i],
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
select: L[n],
cons: [a / b],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
absval: |i|,
l_all: (∀x∈L.P[x])
Lemmas referenced :
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermSubtract_wf,
interval_wf,
iproper_wf,
rfun_wf,
continuous_wf,
partition_wf,
full-partition_wf,
partition-choice_wf,
int_term_value_add_lemma,
itermAdd_wf,
length_wf,
decidable__equal_int,
length_of_nil_lemma,
length-append,
top_wf,
subtype_rel_list,
length_append,
append_wf,
cons_wf,
right-endpoint_wf,
length_cons,
nil_wf,
non_neg_length,
length_nil,
length_of_cons_lemma,
rleq_weakening,
rleq_weakening_equal,
int_subtype_base,
less_than_wf,
set_subtype_base,
int_formula_prop_eq_lemma,
intformeq_wf,
rneq-int,
rleq_functionality_wrt_implies,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
rsub_wf,
rabs_wf,
rleq_wf,
i-member_wf,
real_wf,
all_wf,
int-to-real_wf,
rless_wf,
subtype_rel_sets_simple,
iff_weakening_equal,
i-approx-of-compact,
true_wf,
squash_wf,
i-approx_wf,
icompact_wf,
nat_plus_wf,
iproper-length,
false_wf,
add-is-int-iff,
length_wf_nat,
add_nat_plus,
istype-less_than,
mul_bounds_1b,
rmul-is-positive,
i-length_wf,
small-reciprocal-real-ext,
rmul_wf,
rmul_preserves_rless,
itermMultiply_wf,
rinv_wf2,
sq_stable__rless,
rless_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
real_term_value_mul_lemma,
Inorm-non-neg,
real_term_value_minus_lemma,
real_term_value_add_lemma,
rless_functionality_wrt_implies,
itermMinus_wf,
rminus_wf,
radd-preserves-rless,
Inorm_wf,
radd_wf,
rmin_strict_ub,
partition-sum_wf,
istype-le,
le_wf,
partition-choice-ap_wf,
int_seg_wf,
nearby-partitions_wf,
rmin_wf,
le_witness_for_triv,
sq_stable__rleq,
rmin_ub,
subtype_base_sq,
int_term_value_subtract_lemma,
nat_wf,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
partition-choice-indep-funtype,
rsum_wf,
select_wf,
int_seg_properties,
rleq_functionality,
rabs_functionality,
req_inversion,
rsum_linearity-rsub,
rabs-rsum,
rsum_functionality_wrt_rleq,
left-endpoint_wf,
minus-zero,
rleq_weakening_rless,
length-singleton,
subtract_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
rabs-int,
select_cons_tl,
subtype_rel_self,
select-append,
add-subtract-cancel,
uiff_transitivity,
rmul-rsub-distrib,
rabs-rmul,
sq_stable__i-member,
rabs-difference-symmetry,
zero-rleq-rabs,
rmul_functionality_wrt_rleq2,
full-partition-point-member,
i-member-diff-bound,
rmul_comm,
rleq-int-fractions2,
int_term_value_mul_lemma,
rmul_functionality_wrt_rless2,
rmul-rdiv2,
rmul-nonneg-case1,
rmul_preserves_rleq,
rneq_functionality,
rmul-int,
rinv_functionality2,
rinv-of-rmul,
rmul-rinv3,
rleq-int,
istype-false,
Inorm-bound,
r-triangle-inequality,
radd_functionality_wrt_rleq,
radd_functionality,
rmin-rleq,
trivial-rless-radd,
rless_transitivity2,
rmul_preserves_rleq2,
rleq-implies-rleq,
r-triangle-inequality2,
rinv-mul-as-rdiv,
rsum-constant2
Rules used in proof :
setIsType,
addEquality,
voidEquality,
inhabitedIsType,
functionIsType,
productIsType,
intEquality,
equalityIsType4,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
inrFormation_alt,
rename,
setElimination,
functionEquality,
closedConclusion,
productEquality,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
universeIsType,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
thin,
isectElimination,
hypothesis,
extract_by_obid,
introduction,
setEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
sqequalHypSubstitution,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalityIsType1,
baseApply,
promote_hyp,
pointwiseFunctionality,
applyLambdaEquality,
multiplyEquality,
inlFormation_alt,
equalityIstype,
hyp_replacement,
functionIsTypeImplies,
instantiate,
cumulativity,
sqequalBase,
minusEquality,
equalityElimination,
universeEquality
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} iproper(I)
{}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} I. \mforall{}p:partition(I).
\mforall{}x:partition-choice(full-partition(I;p)). \mforall{}alpha:\{a:\mBbbR{}| r0 < a\} .
\mexists{}e:\{e:\mBbbR{}| r0 < e\}
\mforall{}q:partition(I). \mforall{}y:partition-choice(full-partition(I;q)).
(nearby-partitions(e;p;q)
{}\mRightarrow{} (\mforall{}i:\mBbbN{}||p|| + 1. (|x[i] - y[i]| \mleq{} e))
{}\mRightarrow{} (|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| \mleq{} alpha))))
Date html generated:
2019_10_30-AM-11_37_08
Last ObjectModification:
2019_01_27-PM-04_26_24
Theory : reals_2
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