Nuprl Lemma : rsum_nonneg
∀[n,m:ℤ]. ∀[y:{n..m + 1-} ⟶ ℝ]. r0 ≤ Σ{y[k] | n≤k≤m} supposing r0 ≤ y[k] for k ∈ [n,m]
Proof
Definitions occuring in Statement :
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rsum: Σ{x[k] | n≤k≤m},
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
less_than'_wf,
rsub_wf,
rsum_wf,
int_seg_wf,
int-to-real_wf,
real_wf,
nat_plus_wf,
pointwise-rleq_wf,
req_weakening,
rsum_functionality_wrt_rleq,
req_functionality,
rsum-zero,
rleq_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
lemma_by_obid,
isectElimination,
applyEquality,
addEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
intEquality,
voidElimination,
independent_isectElimination
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[y:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. r0 \mleq{} \mSigma{}\{y[k] | n\mleq{}k\mleq{}m\} supposing r0 \mleq{} y[k] for k \mmember{} [n,m]
Date html generated:
2016_05_18-AM-07_47_58
Last ObjectModification:
2015_12_28-AM-01_04_02
Theory : reals
Home
Index