Nuprl Lemma : constant-partition-sum
∀I:Interval
(icompact(I) ⇒ (∀p:partition(I). ∀f:I ⟶ℝ. ∀z:ℝ. ((z ∈ I) ⇒ (S(f;full-partition(I;p)) = ((f z) * |I|)))))
Proof
Definitions occuring in Statement :
partition-sum: S(f;p),
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
rfun: I ⟶ℝ,
i-member: r ∈ I,
i-length: |I|,
interval: Interval,
req: x = y,
rmul: a * b,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
partition-sum: S(f;p),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
so_apply: x[s],
full-partition: full-partition(I;p),
nat: ℕ,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
partition: partition(I),
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
int_upper: {i...},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
select: L[n],
cons: [a / b],
rev_uimplies: rev_uimplies(P;Q),
icompact: icompact(I),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
i-length: |I|
Lemmas referenced :
i-member_wf,
real_wf,
rfun_wf,
partition_wf,
icompact_wf,
interval_wf,
rsum_wf,
subtract_wf,
length_wf,
full-partition_wf,
rmul_wf,
rsub_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
add-is-int-iff,
subtract-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
int_seg_wf,
i-length_wf,
req_weakening,
length_of_cons_lemma,
add_nat_wf,
append_wf,
cons_wf,
right-endpoint_wf,
length_nil,
non_neg_length,
nil_wf,
length_cons,
length_append,
subtype_rel_list,
top_wf,
length-append,
length_of_nil_lemma,
le_wf,
nat_wf,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
add_functionality_wrt_eq,
iff_weakening_equal,
length-singleton,
req_functionality,
rsum_linearity2,
rmul_functionality,
rsum-telescopes,
select-cons-tl,
select-append,
squash_wf,
true_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_subtype_base,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
natural_numberEquality,
lambdaEquality,
applyEquality,
because_Cache,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyLambdaEquality,
independent_functionElimination,
imageMemberEquality,
universeEquality,
equalityElimination,
instantiate,
cumulativity
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}p:partition(I). \mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}z:\mBbbR{}. ((z \mmember{} I) {}\mRightarrow{} (S(f;full-partition(I;p)) = ((f z) * |I|)))))
Date html generated:
2017_10_03-AM-09_45_27
Last ObjectModification:
2017_07_28-AM-07_59_08
Theory : reals
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