Nuprl Lemma : decimal-digits_wf
∀d:ℕ+. ∀x:ℝ. ((d digits of x) ∈ {p:ℤ × ℤ| let n,m = p in |x - r(n) + (r(m)/r(10^d))| ≤ (r(2)/r(10^d))} )
Proof
Definitions occuring in Statement :
decimal-digits: (d digits of x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
spread: spread def,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
fastexp: i^n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
decimal-digits: (d digits of x),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
has-value: (a)↓,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
exp: i^n,
primrec: primrec(n;b;c),
subtract: n - m,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
le: A ≤ B,
int_nzero: ℤ-o,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rational-approx: (x within 1/n),
real: ℝ,
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
real_wf,
nat_plus_wf,
nat_plus_subtype_nat,
exp_wf_nat_plus,
less_than_wf,
value-type-has-value,
set-value-type,
int-value-type,
equal_wf,
exp-fastexp,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
nat_plus_properties,
equal-wf-base,
primrec-wf-nat-plus,
exp_wf2,
true_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
false_wf,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
mul-commutes,
div-cancel,
nequal_wf,
squash_wf,
exp_add,
exp1,
iff_weakening_equal,
rational-approx-property,
decidable__lt,
multiply-is-int-iff,
div_rem_sum,
subtype_rel_sets,
rleq_wf,
rabs_wf,
rsub_wf,
int-rdiv_wf,
int-to-real_wf,
rdiv_wf,
rless-int,
rless_wf,
equal-wf-base-T,
radd_wf,
req_functionality,
req_weakening,
rdiv_functionality,
req_inversion,
radd-int,
radd-rdiv,
rmul_wf,
radd_functionality,
rmul-int,
rneq_wf,
rmul_preserves_req,
req_wf,
uiff_transitivity,
rmul-distrib,
radd_comm,
rmul-rdiv-cancel2,
rleq_functionality,
rabs_functionality,
rsub_functionality,
int-rdiv-req,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq-int-fractions
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
hypothesis,
introduction,
extract_by_obid,
natural_numberEquality,
hypothesisEquality,
applyEquality,
sqequalRule,
isectElimination,
thin,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
callbyvalueReduce,
independent_isectElimination,
intEquality,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
instantiate,
cumulativity,
rename,
setElimination,
baseApply,
closedConclusion,
because_Cache,
multiplyEquality,
divideEquality,
addLevel,
voidElimination,
equalityUniverse,
levelHypothesis,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageElimination,
universeEquality,
productElimination,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
setEquality,
remainderEquality,
inrFormation,
addEquality,
independent_pairEquality
Latex:
\mforall{}d:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbR{}.
((d digits of x) \mmember{} \{p:\mBbbZ{} \mtimes{} \mBbbZ{}| let n,m = p in |x - r(n) + (r(m)/r(10\^{}d))| \mleq{} (r(2)/r(10\^{}d))\} )
Date html generated:
2017_10_03-AM-10_35_45
Last ObjectModification:
2017_07_28-AM-08_13_29
Theory : reals
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