Nuprl Lemma : exists-rneq-iff
∀n:ℕ. ∀a,b:ℕn ⟶ ℝ. (∃i:ℕn. a[i] ≠ b[i] ⇐⇒ r0 < Σ{|a[i] - b[i]| | 0≤i≤n - 1})
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+
Lemmas referenced :
subtract-add-cancel,
rsum-of-nonneg-positive-iff,
subtract_wf,
rabs_wf,
rsub_wf,
int_seg_wf,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
zero-rleq-rabs,
nat_plus_properties,
rless_wf,
int-to-real_wf,
rsum_wf,
iff_wf,
exists_wf,
rneq_wf,
real_wf,
nat_wf,
rneq-iff-rabs,
rneq-if-rabs
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
dependent_functionElimination,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
functionEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbN{}n {}\mrightarrow{} \mBbbR{}. (\mexists{}i:\mBbbN{}n. a[i] \mneq{} b[i] \mLeftarrow{}{}\mRightarrow{} r0 < \mSigma{}\{|a[i] - b[i]| | 0\mleq{}i\mleq{}n - 1\})
Date html generated:
2017_10_03-AM-09_00_31
Last ObjectModification:
2017_06_16-AM-11_47_25
Theory : reals
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