Nuprl Lemma : close-reals-implies
∀[x,y:ℝ]. ∀[m:ℕ+]. |(x m) - y m| ≤ 4 supposing |x - y| ≤ (r1/r(3 * m))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
multiply: n * m,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
le: A ≤ B,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real: ℝ,
subtype_rel: A ⊆r B,
nat: ℕ
Lemmas referenced :
close-reals-iff,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
le_witness_for_triv,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
nat_plus_wf,
real_wf,
decidable__le,
absval_wf,
subtract_wf,
multiply-is-int-iff,
intformle_wf,
int_formula_prop_le_lemma,
false_wf,
mul_preserves_le,
istype-le,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
inrFormation_alt,
isectIsTypeImplies,
inhabitedIsType,
applyEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
baseClosed
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[m:\mBbbN{}\msupplus{}]. |(x m) - y m| \mleq{} 4 supposing |x - y| \mleq{} (r1/r(3 * m))
Date html generated:
2019_10_29-AM-10_03_27
Last ObjectModification:
2019_06_03-PM-02_11_55
Theory : reals
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