Nuprl Lemma : partition-sum_wf
∀I:Interval
(icompact(I)
⇒ (∀p:partition(I). ∀f:I ⟶ℝ. ∀x:partition-choice(full-partition(I;p)).
(partition-sum(f;x;full-partition(I;p)) ∈ ℝ)))
Proof
Definitions occuring in Statement :
partition-sum: partition-sum(f;x;p),
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
rfun: I ⟶ℝ,
interval: Interval,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
partition-sum: partition-sum(f;x;p),
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
false: False,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
prop: ℙ,
guard: {T},
so_apply: x[s]
Lemmas referenced :
interval_wf,
icompact_wf,
partition_wf,
rfun_wf,
partition-choice_wf,
int_seg_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_seg_properties,
select_wf,
rsub_wf,
lelt_wf,
false_wf,
int_formula_prop_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
subtract-is-int-iff,
add-is-int-iff,
decidable__lt,
partition-choice-member,
rmul_wf,
full-partition_wf,
real_wf,
length_wf,
subtract_wf,
rsum_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_isectElimination,
lambdaEquality,
applyEquality,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
productElimination,
independent_pairFormation,
unionElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addEquality
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}p:partition(I). \mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}x:partition-choice(full-partition(I;p)).
(partition-sum(f;x;full-partition(I;p)) \mmember{} \mBbbR{})))
Date html generated:
2016_05_18-AM-09_04_38
Last ObjectModification:
2016_01_17-AM-02_34_19
Theory : reals
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