Nuprl Lemma : Riemann-sum-rmul-constant
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ]. ∀[c:ℝ]. ∀[k:ℕ+].
(Riemann-sum(λx.(c * (f x));a;b;k) = (c * Riemann-sum(λx.(f 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,
req: x = y,
rmul: a * b,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rfun: I ⟶ℝ,
prop: ℙ,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
Riemann-sum: Riemann-sum(f;a;b;k),
let: let,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
true: True,
guard: {T},
rev_implies: P ⇐ Q
Lemmas referenced :
sq_stable__req,
Riemann-sum_wf,
rmul_wf,
i-member_wf,
rccint_wf,
real_wf,
subtype_rel_self,
rfun_wf,
req_witness,
nat_plus_wf,
set_wf,
rleq_wf,
rccint-icompact,
partition-sum_wf,
uniform-partition_wf,
default-partition-choice_wf,
full-partition_wf,
full-partition-non-dec,
req_functionality,
partition-sum-rmul-const,
req_weakening,
equal_wf,
squash_wf,
true_wf,
eta_conv,
iff_weakening_equal,
req_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_set_memberEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
dependent_functionElimination,
productElimination,
independent_isectElimination,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:[a, b] {}\mrightarrow{}\mBbbR{}]. \mforall{}[c:\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}].
(Riemann-sum(\mlambda{}x.(c * (f x));a;b;k) = (c * Riemann-sum(\mlambda{}x.(f x);a;b;k)))
Date html generated:
2017_10_03-PM-00_55_26
Last ObjectModification:
2017_07_28-AM-08_47_37
Theory : reals_2
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