Nuprl Lemma : partition-sum-constant
∀I:Interval
(icompact(I)
⇒ (∀c:ℝ. ∀p:partition(I). ∀y:partition-choice(full-partition(I;p)). (S(λx.c;full-partition(I;p)) = (c * |I|))))
Proof
Definitions occuring in Statement :
partition-sum: S(f;p),
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
i-length: |I|,
interval: Interval,
req: x = y,
rmul: a * b,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
partition-sum: S(f;p),
prop: ℙ,
so_lambda: λ2x.t[x],
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],
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
icompact: icompact(I),
full-partition: full-partition(I;p),
partition: partition(I),
ge: i ≥ j ,
less_than': less_than'(a;b),
int_upper: {i...},
sq_type: SQType(T),
last: last(L),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
bfalse: ff,
nat_plus: ℕ+,
true: True,
select: L[n],
i-length: |I|,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
full-partition_wf,
list_wf,
real_wf,
equal_wf,
partition-choice_wf,
partition_wf,
icompact_wf,
interval_wf,
rsum_wf,
subtract_wf,
length_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,
le_wf,
req_functionality,
rsum_functionality2,
rmul-rsub-distrib,
req_weakening,
length_of_cons_lemma,
length-append,
length_of_nil_lemma,
non_neg_length,
less_than_wf,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
last_wf,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
rsum-telescopes,
req_inversion,
req_wf,
add_nat_plus,
length_wf_nat,
append_wf,
cons_wf,
right-endpoint_wf,
nil_wf,
nat_plus_wf,
nat_plus_properties,
squash_wf,
true_wf,
last-full-partition,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
lambdaEquality,
because_Cache,
addEquality,
setElimination,
rename,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
instantiate,
cumulativity,
hypothesis_subsumption,
imageMemberEquality,
applyEquality,
universeEquality
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}c:\mBbbR{}. \mforall{}p:partition(I). \mforall{}y:partition-choice(full-partition(I;p)).
(S(\mlambda{}x.c;full-partition(I;p)) = (c * |I|))))
Date html generated:
2017_10_03-PM-00_53_31
Last ObjectModification:
2017_07_28-AM-08_47_28
Theory : reals_2
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