Nuprl Lemma : item-rleq-rsum-of-nonneg
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ((∀i:{n..m + 1-}. (r0 ≤ x[i])) ⇒ (∀i:{n..m + 1-}. (x[i] ≤ Σ{x[i] | n≤i≤m})))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
int_seg: {i..j-},
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
lelt: i ≤ j < k,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
le: A ≤ B,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2
Lemmas referenced :
rsum-split,
int_seg_wf,
rleq_wf,
int-to-real_wf,
real_wf,
istype-int,
rsum_wf,
radd_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
istype-le,
istype-less_than,
rsum_nonneg,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
rleq_weakening,
req-iff-rsub-is-0,
rleq_functionality,
req_weakening,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
radd_functionality,
rsum-split-last,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
functionIsType,
universeIsType,
addEquality,
natural_numberEquality,
applyEquality,
inhabitedIsType,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}n,m:\mBbbZ{}. \mforall{}x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}.
((\mforall{}i:\{n..m + 1\msupminus{}\}. (r0 \mleq{} x[i])) {}\mRightarrow{} (\mforall{}i:\{n..m + 1\msupminus{}\}. (x[i] \mleq{} \mSigma{}\{x[i] | n\mleq{}i\mleq{}m\})))
Date html generated:
2019_10_29-AM-10_12_51
Last ObjectModification:
2019_04_11-PM-06_35_46
Theory : reals
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