Nuprl Lemma : general-partition-sum
∀I:Interval
(icompact(I)
⇒ (∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. ∀e:{e:ℝ| r0 < e} .
∃d:{d:ℝ| r0 < d}
∀p,q:{p:partition(I)| partition-mesh(I;p) ≤ d} . ∀x:partition-choice(full-partition(I;p)).
∀y:partition-choice(full-partition(I;q)).
(|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| ≤ (e * |I|))))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
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),
icompact: icompact(I),
rfun: I ⟶ℝ,
i-length: |I|,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
label: ...$L... t,
less_than': less_than'(a;b),
less_than: a < b,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
itermConstant: "const",
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
sq_exists: ∃x:A [B[x]],
rless: x < y,
or: P ∨ Q,
rneq: x ≠ y,
nat_plus: ℕ+,
rfun: I ⟶ℝ,
so_apply: x[s],
so_lambda: λ2x.t[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],
sq_stable: SqStable(P),
rdiv: (x/y),
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
icompact: icompact(I),
rgt: x > y,
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
real: ℝ
Lemmas referenced :
interval_wf,
rfun_wf,
continuous_wf,
set_wf,
less_than_wf,
req-iff-rsub-is-0,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_term_value_const_lemma,
itermSubtract_wf,
real_term_polynomial,
rleq_weakening,
rleq_weakening_equal,
equal_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
rneq-int,
rleq_functionality_wrt_implies,
subtype_rel_sets,
rless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
real_wf,
rsub_wf,
rabs_wf,
rleq_wf,
i-member_wf,
sq_exists_wf,
all_wf,
iff_weakening_equal,
i-approx-of-compact,
true_wf,
squash_wf,
i-approx_wf,
icompact_wf,
nat_plus_wf,
rmul_preserves_rless,
small-reciprocal-real-ext,
rmul_wf,
itermMultiply_wf,
rinv_wf2,
sq_stable__rless,
rless_functionality,
req_transitivity,
rmul-rinv3,
real_polynomial_null,
istype-int,
istype-void,
real_term_value_mul_lemma,
sq_stable__rleq,
sq_stable__all,
sq_stable__and,
rleq_weakening_rless,
less_than'_wf,
partition_wf,
partition-mesh_wf,
partition-choice_wf,
full-partition_wf,
partition-sum_wf,
i-length_wf,
le_witness_for_triv,
rleq-iff-all-rless,
rless-cases,
Inorm_wf,
rmul-is-positive,
rmul-rinv,
rmul_functionality,
req_weakening,
rinv-mul-as-rdiv,
rless-int-fractions2,
full-omega-unsat,
istype-less_than,
rmul_preserves_rleq2,
Inorm-non-neg,
rleq_functionality,
rabs_functionality,
partition-sum-bound,
rabs-difference-symmetry,
radd_wf,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
itermAdd_wf,
real_term_value_add_lemma,
radd-preserves-rleq,
rminus_wf,
itermMinus_wf,
real_term_value_minus_lemma,
rmul-nonneg-case1,
icompact-length-nonneg,
iproper-length-iff,
nearby-separated-partition-sum,
separated-partition-sum,
rmul_preserves_rleq,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
int_term_value_mul_lemma,
nequal_wf,
int-rinv-cancel,
radd_functionality,
radd_functionality_wrt_rless1,
radd-non-neg,
set_subtype_base,
rleq-int-fractions2,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rmul_functionality_wrt_rleq2
Rules used in proof :
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
inrFormation,
rename,
setElimination,
universeEquality,
functionEquality,
productEquality,
productElimination,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
imageElimination,
lambdaEquality,
dependent_set_memberEquality,
because_Cache,
thin,
isectElimination,
hypothesis,
extract_by_obid,
introduction,
setEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
sqequalHypSubstitution,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inrFormation_alt,
dependent_set_memberEquality_alt,
closedConclusion,
universeIsType,
applyLambdaEquality,
approximateComputation,
lambdaEquality_alt,
isect_memberEquality_alt,
axiomEquality,
independent_pairEquality,
minusEquality,
inhabitedIsType,
lambdaFormation_alt,
dependent_pairFormation_alt,
functionIsType,
setIsType,
equalityIstype,
functionIsTypeImplies,
inlFormation_alt,
productIsType,
hyp_replacement,
instantiate,
cumulativity,
sqequalBase,
addEquality,
multiplyEquality
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} I. \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}d:\{d:\mBbbR{}| r0 < d\}
\mforall{}p,q:\{p:partition(I)| partition-mesh(I;p) \mleq{} d\} . \mforall{}x:partition-choice(full-partition(I;p)).
\mforall{}y:partition-choice(full-partition(I;q)).
(|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| \mleq{} (e * |I|))))
Date html generated:
2019_10_30-AM-11_38_13
Last ObjectModification:
2019_01_27-PM-05_13_50
Theory : reals_2
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