Nuprl Lemma : Riemann-sum-rsub
∀[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)) = Riemann-sum(λx.((f x) - g x);a;b;k))
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
rsub: x - y,
req: x = y,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x]
Definitions unfolded in proof :
partition-sum: partition-sum(f;x;p),
default-partition-choice: default-partition-choice(p),
nat_plus: ℕ+,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
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),
rfun: I ⟶ℝ,
prop: ℙ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
has-value: (a)↓,
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
top: Top,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
less_than: a < b,
not: ¬A,
false: False,
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-},
rev_uimplies: rev_uimplies(P;Q),
le: A ≤ B,
rsub: x - y
Lemmas referenced :
rmul_functionality,
rminus-as-rmul,
radd_comm,
radd-ac,
radd-assoc,
rminus-rminus,
rminus-radd,
rmul_over_rminus,
rmul-distrib,
req_transitivity,
rminus_functionality,
radd_functionality,
uiff_transitivity,
req_wf,
radd_wf,
int-to-real_wf,
rminus_wf,
req_weakening,
rsum_linearity-rsub,
req_inversion,
req_functionality,
rsum_wf,
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,
rsum_functionality,
le_wf,
list_set_type,
full-partition_wf,
full-partition-point-member,
member_rccint_lemma,
uniform-partition_wf,
partition_wf,
evalall-reduce,
sq_stable__req,
rsub_wf,
Riemann-sum_wf,
rleq_wf,
real_wf,
i-member_wf,
rccint_wf,
rccint-icompact,
req_witness,
nat_plus_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,
equalitySymmetry,
equalityTransitivity,
natural_numberEquality,
intEquality,
independent_isectElimination,
isect_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
setEquality,
applyEquality,
lambdaEquality,
sqequalRule,
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,
voidElimination,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
computeAll,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation,
unionElimination,
addEquality,
productEquality,
minusEquality
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) - Riemann-sum(g;a;b;k)) = Riemann-sum(\mlambda{}x.((f x) - g x);a;b;k))
Date html generated:
2016_05_18-AM-10_40_10
Last ObjectModification:
2016_01_17-AM-00_22_18
Theory : reals
Home
Index