Nuprl Lemma : partition-refinement-sum
∀I:Interval
(icompact(I)
⇒ (∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. ∀q:partition(I). ∀n:ℕ+.
((partition-mesh(I;q) ≤ (mc 1 n))
⇒ frs-increasing(q)
⇒ (∀p:partition(I). ∀x:partition-choice(full-partition(I;p)). ∀y:partition-choice(full-partition(I;q)).
(p refines q ⇒ (|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| ≤ ((r1/r(n)) * |I|)))))))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
partition-refines: P refines Q,
partition-sum: S(f;p),
partition-choice: partition-choice(p),
partition-mesh: partition-mesh(I;p),
full-partition: full-partition(I;p),
partition: partition(I),
frs-increasing: frs-increasing(p),
icompact: icompact(I),
rfun: I ⟶ℝ,
i-length: |I|,
interval: Interval,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
nat_plus: ℕ+,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
continuous: f[x] continuous for x ∈ I,
uall: ∀[x:A]. B[x],
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
nat_plus: ℕ+,
rneq: x ≠ y,
or: P ∨ Q,
rless: x < y,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
label: ...$L... t,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
nat: ℕ,
ge: i ≥ j ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_type: SQType(T),
partition: partition(I),
subinterval: I ⊆ J ,
cons: [a / b],
rbetween: x≤y≤z,
select: L[n],
less_than: a < b,
less_than': less_than'(a;b),
subtract: n - m,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
icompact: icompact(I),
sq_stable: SqStable(P),
i-length: |I|,
partition-sum: S(f;p),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
frs-non-dec: frs-non-dec(L),
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
int_upper: {i...},
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
l_all: (∀x∈L.P[x]),
real: ℝ,
frs-increasing: frs-increasing(p),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
nil: [],
left-endpoint: left-endpoint(I),
endpoints: endpoints(I),
rccint: [l, u],
outl: outl(x),
pi1: fst(t),
right-endpoint: right-endpoint(I),
pi2: snd(t),
isl: isl(x),
i-finite: i-finite(I),
rdiv: (x/y)
Lemmas referenced :
nat_plus_wf,
icompact_wf,
i-approx_wf,
squash_wf,
true_wf,
i-approx-of-compact,
iff_weakening_equal,
subtype_rel_sets_simple,
rless_wf,
int-to-real_wf,
all_wf,
real_wf,
i-member_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rleq_functionality_wrt_implies,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
less_than_wf,
int_subtype_base,
rleq_weakening_equal,
rleq_weakening,
continuous_wf,
rfun_wf,
interval_wf,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
nat_properties,
intformle_wf,
int_formula_prop_le_lemma,
ge_wf,
istype-less_than,
le_witness_for_triv,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
lelt_wf,
int_term_value_subtract_lemma,
decidable__le,
istype-le,
subtype_rel_self,
partition-refines_wf,
partition-choice_wf,
full-partition_wf,
frs-increasing_wf,
partition-mesh_wf,
length_wf,
partition_wf,
subinterval_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
length_wf_nat,
list-cases,
product_subtype_list,
right-endpoint_wf,
left-endpoint_wf,
i-member-compact,
false_wf,
length_of_nil_lemma,
member_rccint_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
i-length_wf,
rmul_wf,
rfun_subtype,
partitions_wf,
nil_wf,
partition-sum_wf,
sq_stable__rleq,
req_weakening,
equal-wf-base,
set_wf,
subtype_rel_sets,
subtype_rel_dep_function,
req_functionality,
rsum-single,
rsum_wf,
rleq_functionality,
rabs_functionality,
rsub_functionality,
subtract-is-int-iff,
add-is-int-iff,
select_wf,
constant-partition-sum,
req_inversion,
partition-choice-member,
rsum_linearity-rsub,
rabs-rsum,
le_wf,
rsum_functionality,
rabs-of-nonneg,
equal_wf,
rmul-rsub-distrib,
rabs-rmul,
rmul_functionality,
radd_wf,
radd-preserves-rleq,
full-partition-non-dec,
real_term_value_add_lemma,
rsum_functionality_wrt_rleq,
sq_stable__i-member,
rccint_wf,
partition-mesh-nil,
i-member-diff-bound,
zero-rleq-rabs,
rmul_functionality_wrt_rleq2,
rleq_transitivity,
rsum_linearity2,
length-nil,
length-singleton,
add_functionality_wrt_eq,
add_nat_plus,
nat_wf,
length-append,
top_wf,
subtype_rel_list,
length_append,
length_cons,
non_neg_length,
length_nil,
cons_wf,
append_wf,
add_nat_wf,
rsum-telescopes,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
select-append,
select-cons-tl,
partition-split-cons-mesh,
partition-point-member,
istype-false,
icompact-endpoints,
subtract_nat_wf,
rccint-icompact,
rcc-subinterval,
le_weakening2,
sq_stable__less_than,
add-subtract-cancel,
list_wf,
zero-add,
add-commutes,
add-swap,
add-associates,
add-member-int_seg2,
base_wf,
stuck-spread,
partition-refines-cons,
select0,
right_endpoint_rccint_lemma,
is-partition-choice_wf,
partition-choice-subtype,
int_seg_subtype,
list_ind_cons_lemma,
left_endpoint_rccint_lemma,
int_seg_subtype_special,
int_seg_cases,
rsum-split-shift,
radd_functionality,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
minus-one-mul-top,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
cons_neq_nil,
req_wf,
rsum-split,
int_seg_subtype_nat,
iff_weakening_uiff,
assert_wf,
rsum-shift,
select-cons-hd,
partition-sum_functionality,
req_transitivity,
and_wf,
subtype_rel-equal,
itermMultiply_wf,
rinv_wf2,
r-triangle-inequality,
radd_functionality_wrt_rleq,
rinv-mul-as-rdiv,
real_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
applyEquality,
functionExtensionality,
hypothesisEquality,
setEquality,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
thin,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
productEquality,
closedConclusion,
functionEquality,
setElimination,
rename,
inrFormation_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIstype,
intEquality,
sqequalBase,
productIsType,
functionIsType,
inhabitedIsType,
setIsType,
intWeakElimination,
functionIsTypeImplies,
instantiate,
cumulativity,
applyLambdaEquality,
hypothesis_subsumption,
addEquality,
promote_hyp,
lambdaEquality,
lambdaFormation,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
inrFormation,
universeEquality,
hyp_replacement,
equalityElimination,
baseApply,
pointwiseFunctionality,
inlFormation,
minusEquality,
equalityIsType1
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} I. \mforall{}q:partition(I). \mforall{}n:\mBbbN{}\msupplus{}.
((partition-mesh(I;q) \mleq{} (mc 1 n))
{}\mRightarrow{} frs-increasing(q)
{}\mRightarrow{} (\mforall{}p:partition(I). \mforall{}x:partition-choice(full-partition(I;p)).
\mforall{}y:partition-choice(full-partition(I;q)).
(p refines q
{}\mRightarrow{} (|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| \mleq{} ((r1/r(n)) * |I|)))))))
Date html generated:
2019_10_30-AM-11_36_30
Last ObjectModification:
2019_04_02-PM-04_17_05
Theory : reals_2
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