Nuprl Lemma : partition-sum-rleq
∀I:Interval
(icompact(I)
⇒ (∀f,g:I ⟶ℝ.
∀p:partition(I). ∀y:partition-choice(full-partition(I;p)).
(S(f;full-partition(I;p)) ≤ S(g;full-partition(I;p)))
supposing ∀x:ℝ. ((x ∈ I) ⇒ ((f x) ≤ (g x)))))
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),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
partition-sum: S(f;p),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
partition: partition(I),
full-partition: full-partition(I;p),
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
less_than: a < b,
guard: {T},
so_apply: x[s],
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rleq: x ≤ y,
rnonneg: rnonneg(x),
real: ℝ,
icompact: icompact(I),
sq_stable: SqStable(P),
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
frs-non-dec: frs-non-dec(L),
rsub: x - y
Lemmas referenced :
partition-choice-indep-funtype,
int_seg_wf,
length_wf,
real_wf,
i-member_wf,
length_of_cons_lemma,
length_nil,
non_neg_length,
nil_wf,
length_cons,
right-endpoint_wf,
cons_wf,
append_wf,
length_append,
subtype_rel_list,
top_wf,
length-append,
length_of_nil_lemma,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rsum_functionality_wrt_rleq,
subtract_wf,
full-partition_wf,
rmul_wf,
decidable__lt,
add-is-int-iff,
intformand_wf,
intformless_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
lelt_wf,
rsub_wf,
select_wf,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_wf,
equal_wf,
partition-choice_wf,
partition_wf,
less_than'_wf,
partition-sum_wf,
nat_plus_wf,
all_wf,
rleq_wf,
rfun_wf,
icompact_wf,
interval_wf,
set_wf,
sq_stable__rleq,
full-partition-non-dec,
radd-preserves-rleq,
radd_wf,
int-to-real_wf,
rminus_wf,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd-ac,
req_weakening,
radd_functionality,
radd-rminus-both,
radd-zero-both,
rmul_preserves_rleq2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
independent_isectElimination,
hypothesis,
sqequalRule,
functionEquality,
natural_numberEquality,
addEquality,
setElimination,
rename,
setEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
lambdaEquality,
unionElimination,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
equalityTransitivity,
equalitySymmetry,
computeAll,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
independent_pairEquality,
minusEquality,
axiomEquality,
imageMemberEquality,
imageElimination
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}f,g:I {}\mrightarrow{}\mBbbR{}.
\mforall{}p:partition(I). \mforall{}y:partition-choice(full-partition(I;p)).
(S(f;full-partition(I;p)) \mleq{} S(g;full-partition(I;p)))
supposing \mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} ((f x) \mleq{} (g x)))))
Date html generated:
2017_10_03-PM-00_53_35
Last ObjectModification:
2017_07_28-AM-08_47_28
Theory : reals_2
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