Nuprl Lemma : Riemann-sum-constant
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[c:ℝ]. ∀[k:ℕ+]. (Riemann-sum(λx.c;a;b;k) = (c * (b - a)))
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
rleq: x ≤ y,
rsub: x - y,
req: x = y,
rmul: a * b,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
lambda: λx.A[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
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,
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
i-length: |I|,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
sq_stable__req,
Riemann-sum_wf,
rmul_wf,
rsub_wf,
req_witness,
nat_plus_wf,
rleq_wf,
real_wf,
rccint-icompact,
partition-sum_wf,
rccint_wf,
uniform-partition_wf,
istype-top,
default-partition-choice_wf,
full-partition_wf,
full-partition-non-dec,
i-length_wf,
req_weakening,
left_endpoint_rccint_lemma,
istype-void,
right_endpoint_rccint_lemma,
req_functionality,
partition-sum-constant
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_set_memberEquality_alt,
hypothesis,
because_Cache,
lambdaEquality_alt,
applyEquality,
sqequalRule,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
setIsType,
dependent_functionElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
natural_numberEquality,
voidElimination
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[c:\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. (Riemann-sum(\mlambda{}x.c;a;b;k) = (c * (b - a)))
Date html generated:
2019_10_30-AM-11_38_44
Last ObjectModification:
2018_11_08-PM-05_58_29
Theory : reals_2
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