Nuprl Lemma : rsum-triangle-inequality1
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. ((Σ{|x[i]| | n≤i≤m} - Σ{|y[i]| | n≤i≤m}) ≤ Σ{|x[i] + y[i]| | n≤i≤m})
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
implies: P ⇒ Q,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rge: x ≥ y
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x,y:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
((\mSigma{}\{|x[i]| | n\mleq{}i\mleq{}m\} - \mSigma{}\{|y[i]| | n\mleq{}i\mleq{}m\}) \mleq{} \mSigma{}\{|x[i] + y[i]| | n\mleq{}i\mleq{}m\})
Date html generated:
2020_05_20-AM-11_13_14
Last ObjectModification:
2019_12_15-PM-06_48_55
Theory : reals
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