Nuprl Lemma : rsum-triangle-inequality2
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. ((Σ{|y[i]| | n≤i≤m} - Σ{|x[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,
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,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
uiff: uiff(P;Q)
Lemmas referenced :
req_weakening,
radd_comm,
rabs_functionality,
rsum_functionality2,
rleq_functionality,
rsum-triangle-inequality1,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
le_wf,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
real_wf,
nat_plus_wf,
int_seg_wf,
radd_wf,
rabs_wf,
rsum_wf,
rsub_wf,
less_than'_wf
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,
hypothesis,
addEquality,
natural_numberEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality,
intEquality,
voidElimination,
dependent_set_memberEquality,
independent_pairFormation,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
voidEquality,
computeAll,
lambdaFormation
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x,y:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
((\mSigma{}\{|y[i]| | n\mleq{}i\mleq{}m\} - \mSigma{}\{|x[i]| | n\mleq{}i\mleq{}m\}) \mleq{} \mSigma{}\{|x[i] + y[i]| | n\mleq{}i\mleq{}m\})
Date html generated:
2016_05_18-AM-07_48_30
Last ObjectModification:
2016_01_17-AM-02_08_57
Theory : reals
Home
Index