Nuprl Lemma : Riemann-sum_wf
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ]. ∀[k:ℕ+]. (Riemann-sum(f;a;b;k) ∈ ℝ)
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Riemann-sum: Riemann-sum(f;a;b;k),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
has-value: (a)↓,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
prop: ℙ
Lemmas referenced :
rccint-icompact,
uniform-partition_wf,
rccint_wf,
partition_wf,
value-type-has-value,
nat_plus_wf,
set-value-type,
less_than_wf,
int-value-type,
valueall-type-has-valueall,
list_wf,
real_wf,
list-valueall-type,
real-valueall-type,
full-partition_wf,
evalall-reduce,
valueall-type-real-list,
partition-sum_wf,
default-partition-choice_wf,
rfun_wf,
set_wf,
rleq_wf,
full-partition-non-dec
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
isectElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
callbyvalueReduce,
intEquality,
lambdaEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
equalityEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:[a, b] {}\mrightarrow{}\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. (Riemann-sum(f;a;b;k) \mmember{} \mBbbR{})
Date html generated:
2016_05_18-AM-10_39_38
Last ObjectModification:
2015_12_27-PM-10_50_01
Theory : reals
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