Nuprl Lemma : rneq-iff
∀x,y:ℝ. (x ≠ y ⇐⇒ ∃n:ℕ+. 4 < |(x n) - y n|)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
rneq: x ≠ y,
rless: x < y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
real: ℝ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
decidable: Dec(P),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
nat: ℕ,
le: A ≤ B,
gt: i > j
Lemmas referenced :
absval_unfold,
subtract_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
nat_plus_properties,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__lt,
intformnot_wf,
itermMinus_wf,
int_formula_prop_not_lemma,
int_term_value_minus_lemma,
absval_wf,
or_wf,
sq_exists_wf,
nat_plus_wf,
absval_lbound,
le_wf,
subtract-is-int-iff,
exists_wf,
nat_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
setElimination,
rename,
dependent_pairFormation,
hypothesisEquality,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
hypothesis,
minusEquality,
natural_numberEquality,
equalityElimination,
productElimination,
independent_isectElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
instantiate,
cumulativity,
dependent_set_memberEquality,
addEquality,
inlFormation,
dependent_set_memberFormation,
inrFormation
Latex:
\mforall{}x,y:\mBbbR{}. (x \mneq{} y \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}\msupplus{}. 4 < |(x n) - y n|)
Date html generated:
2017_10_03-AM-08_26_57
Last ObjectModification:
2017_08_30-PM-03_15_44
Theory : reals
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