Nuprl Lemma : rsum-positive-implies
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ((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},
rless: x < y,
rabs: |x|,
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],
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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
and: P ∧ Q,
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
false: False,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
uimplies: b supposing a,
top: Top,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
nequal: a ≠ b ∈ T ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rneq: x ≠ y,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2
Lemmas referenced :
small-reciprocal-real,
rsum_wf,
int_seg_wf,
rless_wf,
int-to-real_wf,
real_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformless_wf,
satisfiable-full-omega-tt,
nat_plus_properties,
rsum-empty,
rabs_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
less_than_wf,
int_formula_prop_not_lemma,
intformnot_wf,
mul_nat_plus,
rless-int-fractions2,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
intformle_wf,
itermSubtract_wf,
equal-wf-T-base,
int_subtype_base,
equal-wf-base,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
intformand_wf,
int_seg_properties,
int_entire_a,
rneq-int,
subtract_wf,
rdiv_wf,
rless-cases,
all_wf,
rleq_weakening_rless,
rsum_functionality_wrt_rleq2,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
le_wf,
lelt_wf,
rmul_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert_wf,
bnot_wf,
not_wf,
rleq_functionality,
req_weakening,
rsum-constant2,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
add-commutes,
nat_plus_wf,
rmul_preserves_req,
rless-int,
req_wf,
mul_bounds_1b,
rneq_functionality,
rmul-int,
rinv_wf2,
uiff_transitivity,
req_functionality,
rmul_functionality,
rdiv_functionality,
req_inversion,
rinv-of-rmul,
req_transitivity,
real_term_polynomial,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-rinv,
rmul-rinv3,
rabs-rsum,
rabs-of-nonneg,
rless_irreflexivity,
rless_transitivity1,
not-all-int_seg2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
addEquality,
natural_numberEquality,
hypothesis,
productElimination,
functionEquality,
intEquality,
unionElimination,
computeAll,
int_eqEquality,
dependent_pairFormation,
imageElimination,
rename,
setElimination,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
closedConclusion,
baseApply,
baseClosed,
independent_pairFormation,
independent_functionElimination,
because_Cache,
multiplyEquality,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
inrFormation,
applyLambdaEquality,
inlFormation
Latex:
\mforall{}n,m:\mBbbZ{}. \mforall{}x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}. ((r0 < \mSigma{}\{x[i] | n\mleq{}i\mleq{}m\}) {}\mRightarrow{} (\mexists{}i:\{n..m + 1\msupminus{}\}. (r0 < |x[i]|)))
Date html generated:
2017_10_03-AM-09_00_22
Last ObjectModification:
2017_07_28-AM-07_39_28
Theory : reals
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