Nuprl Lemma : rsum-of-nonneg-positive-iff
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ((∀i:{n..m + 1-}. (r0 ≤ x[i])) ⇒ (r0 < Σ{x[i] | n≤i≤m} ⇐⇒ ∃i:{n..m + 1-}. (r0 < x[i])))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
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],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
guard: {T},
uimplies: b supposing a,
int_seg: {i..j-},
rless: x < y,
sq_exists: ∃x:{A| B[x]},
subtype_rel: A ⊆r B,
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rge: x ≥ y,
uiff: uiff(P;Q)
Lemmas referenced :
rless_wf,
int-to-real_wf,
rsum_wf,
int_seg_wf,
exists_wf,
all_wf,
rleq_wf,
real_wf,
rsum-positive-implies,
rabs_wf,
rless_functionality,
req_weakening,
rabs-of-nonneg,
sq_stable__less_than,
nat_plus_properties,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
radd_wf,
decidable__lt,
lelt_wf,
rsum-split,
rsum_nonneg,
le_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
subtract_wf,
subtract-add-cancel,
radd_functionality,
rsum-split-last,
trivial-rless-radd,
itermSubtract_wf,
int_term_value_subtract_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
addEquality,
functionEquality,
intEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
dependent_pairFormation,
because_Cache,
independent_isectElimination,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
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{} (r0 < \mSigma{}\{x[i] | n\mleq{}i\mleq{}m\} \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\{n..m + 1\msupminus{}\}. (r0 < x[i])))
Date html generated:
2016_10_26-AM-09_17_28
Last ObjectModification:
2016_09_28-PM-06_09_00
Theory : reals
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