Nuprl Lemma : Riemann-sum-rleq
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f,g:[a, b] ⟶ℝ]. ∀[k:ℕ+].
Riemann-sum(f;a;b;k) ≤ Riemann-sum(g;a;b;k) supposing ∀x:ℝ. ((x ∈ [a, b]) ⇒ ((f x) ≤ (g x)))
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
partition-sum: partition-sum(f;x;p),
default-partition-choice: default-partition-choice(p),
nat_plus: ℕ+,
so_apply: x[s],
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
real: ℝ,
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
squash: ↓T,
Riemann-sum: Riemann-sum(f;a;b;k),
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
sq_stable: SqStable(P),
prop: ℙ,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
has-value: (a)↓,
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
top: Top,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
uiff: uiff(P;Q),
less_than: a < b,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
rccint: [l, u],
i-member: r ∈ I,
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
frs-non-dec: frs-non-dec(L)
Lemmas referenced :
rmul_preserves_rleq2,
lelt_wf,
subtype_rel_list,
full-partition-non-dec,
radd-zero-both,
radd-rminus-both,
radd_functionality,
req_weakening,
radd-ac,
radd_comm,
rleq_functionality,
uiff_transitivity,
radd-preserves-rleq,
radd_wf,
int-to-real_wf,
rminus_wf,
equal_wf,
rsum_functionality_wrt_rleq,
subtract_wf,
length_wf,
rmul_wf,
select_wf,
int_seg_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
decidable__lt,
add-is-int-iff,
subtract-is-int-iff,
intformless_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
false_wf,
int_seg_wf,
le_wf,
list_set_type,
full-partition_wf,
full-partition-point-member,
member_rccint_lemma,
uniform-partition_wf,
partition_wf,
evalall-reduce,
sq_stable__rleq,
Riemann-sum_wf,
rleq_wf,
rccint-icompact,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
all_wf,
i-member_wf,
rccint_wf,
rfun_wf,
set_wf,
value-type-has-value,
set-value-type,
less_than_wf,
int-value-type,
list_wf,
and_wf,
valueall-type-has-valueall,
list-valueall-type,
set-valueall-type,
real-valueall-type
Rules used in proof :
equalityEquality,
lambdaFormation,
setEquality,
intEquality,
independent_isectElimination,
voidElimination,
isect_memberEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
natural_numberEquality,
minusEquality,
applyEquality,
independent_pairEquality,
lambdaEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
productElimination,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
hypothesis,
dependent_set_memberEquality,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
rename,
thin,
setElimination,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
callbyvalueReduce,
voidEquality,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
computeAll,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation,
unionElimination,
addEquality,
productEquality,
substitution
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f,g:[a, b] {}\mrightarrow{}\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}].
Riemann-sum(f;a;b;k) \mleq{} Riemann-sum(g;a;b;k) supposing \mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} ((f x) \mleq{} (g x)))
Date html generated:
2016_05_18-AM-10_40_28
Last ObjectModification:
2016_01_17-AM-00_22_51
Theory : reals
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