Nuprl Lemma : rabs-rsum
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (|Σ{x[i] | n≤i≤m}| ≤ Σ{|x[i]| | n≤i≤m})
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rleq: x ≤ y,
rabs: |x|,
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,
rsum: Σ{x[k] | n≤k≤m},
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,
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
top: Top,
compose: f o g,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T}
Lemmas referenced :
less_than'_wf,
rsub_wf,
rsum_wf,
rabs_wf,
int_seg_wf,
real_wf,
nat_plus_wf,
value-type-has-value,
int-value-type,
valueall-type-has-valueall,
list_wf,
list-valueall-type,
real-valueall-type,
map_wf,
le_wf,
less_than_wf,
from-upto_wf,
evalall-reduce,
valueall-type-real-list,
radd-list_wf-bag,
list-subtype-bag,
map-map,
subtype_rel_self,
equal_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
radd-list-rabs
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
addEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality,
intEquality,
voidElimination,
independent_isectElimination,
setEquality,
productEquality,
lambdaFormation,
dependent_set_memberEquality,
callbyvalueReduce,
voidEquality,
independent_functionElimination
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. (|\mSigma{}\{x[i] | n\mleq{}i\mleq{}m\}| \mleq{} \mSigma{}\{|x[i]| | n\mleq{}i\mleq{}m\})
Date html generated:
2017_10_03-AM-09_00_02
Last ObjectModification:
2017_07_28-AM-07_39_21
Theory : reals
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