Nuprl Lemma : rsum_functionality_wrt_rleq
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. Σ{x[k] | n≤k≤m} ≤ Σ{y[k] | n≤k≤m} supposing x[k] ≤ 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,
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,
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,
int_seg: {i..j-},
lelt: i ≤ j < k,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
cand: A c∧ B,
top: Top,
nat: ℕ,
less_than: a < b,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
intformless_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
subtract_wf,
int_seg_properties,
select-from-upto,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
not_wf,
bnot_wf,
assert_wf,
lt_int_wf,
select-map,
lelt_wf,
top_wf,
subtype_rel_list,
length-from-upto,
length-map,
length_wf,
nat_wf,
length_wf_nat,
map-length,
radd-list_functionality_wrt_rleq,
valueall-type-real-list,
evalall-reduce,
from-upto_wf,
less_than_wf,
le_wf,
and_wf,
map_wf,
real-valueall-type,
list-valueall-type,
list_wf,
valueall-type-has-valueall,
int-value-type,
value-type-has-value,
pointwise-rleq_wf,
nat_plus_wf,
real_wf,
int_seg_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,
addEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
intEquality,
voidElimination,
independent_isectElimination,
setEquality,
callbyvalueReduce,
voidEquality,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
productEquality,
unionElimination,
instantiate,
cumulativity,
independent_functionElimination,
impliesFunctionality,
dependent_pairFormation,
int_eqEquality,
computeAll
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x,y:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
\mSigma{}\{x[k] | n\mleq{}k\mleq{}m\} \mleq{} \mSigma{}\{y[k] | n\mleq{}k\mleq{}m\} supposing x[k] \mleq{} y[k] for k \mmember{} [n,m]
Date html generated:
2016_05_18-AM-07_45_11
Last ObjectModification:
2016_01_17-AM-02_07_50
Theory : reals
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