Nuprl Lemma : req-iff-rabs-rleq-bound
∀x,y:ℝ. (x = y ⇐⇒ ∃B:ℕ+. ∀m:ℕ+. (|x - y| ≤ (r(B)/r(m))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_apply: x[s],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
req-iff-rabs-rleq,
req_wf,
nat_plus_wf,
exists_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
real_wf,
less_than_wf,
mul_nat_plus,
mul_bounds_1b,
rleq-int-fractions,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_pairFormation,
productElimination,
independent_functionElimination,
isectElimination,
sqequalRule,
lambdaEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
inrFormation,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
multiplyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x,y:\mBbbR{}. (x = y \mLeftarrow{}{}\mRightarrow{} \mexists{}B:\mBbbN{}\msupplus{}. \mforall{}m:\mBbbN{}\msupplus{}. (|x - y| \mleq{} (r(B)/r(m))))
Date html generated:
2018_05_22-PM-01_57_53
Last ObjectModification:
2017_10_26-PM-02_20_36
Theory : reals
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