Nuprl Lemma : ddr_wf
∀x:ℝ. ∀n:ℕ+. (ddr(x;n) ∈ {y:ℝ| |x - y| ≤ (r1/r(5 * 10^n - 1))} )
Proof
Definitions occuring in Statement :
ddr: ddr(x;n),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
exp: i^n,
nat_plus: ℕ+,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
ddr: ddr(x;n),
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
le: A ≤ B,
int_upper: {i...},
has-value: (a)↓,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
uiff: uiff(P;Q),
real: ℝ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
subtract: n - m,
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rational-approx: (x within 1/n),
divides: b | a
Lemmas referenced :
rational-approx-property,
mul_nat_plus,
exp_wf_nat_plus,
subtract_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
nat_plus_wf,
real_wf,
exp-fastexp,
nat_plus_subtype_nat,
exp_wf2,
less_than_wf,
mul_preserves_le,
false_wf,
exp_step,
itermMultiply_wf,
int_term_value_mul_lemma,
int_upper_wf,
int_upper_properties,
value-type-has-value,
int-value-type,
div_rem_sum,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
rem_bounds_1,
int_upper_subtype_nat,
add-is-int-iff,
multiply-is-int-iff,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
rat-to-real_wf,
subtype_rel_sets,
equal_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rneq-int,
int_entire_a,
set_subtype_base,
exp_wf4,
nat_wf,
div-cancel2,
nat_properties,
decidable__equal_int,
rational-approx_wf,
le_antisymmetry_iff,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
mul-associates,
add-associates,
add-swap,
rless-int,
rless_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
div_rem_sum2,
divides_iff_rem_zero,
equal-wf-T-base,
rat-to-real-req,
int-rdiv_wf,
rleq_functionality,
req_weakening,
rabs_functionality,
rsub_functionality,
int-rdiv-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
because_Cache,
dependent_set_memberEquality,
setElimination,
rename,
hypothesis,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
productElimination,
callbyvalueReduce,
addLevel,
instantiate,
cumulativity,
independent_functionElimination,
divideEquality,
imageElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
setEquality,
multiplyEquality,
addEquality,
minusEquality,
inrFormation
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. (ddr(x;n) \mmember{} \{y:\mBbbR{}| |x - y| \mleq{} (r1/r(5 * 10\^{}n - 1))\} )
Date html generated:
2017_10_03-AM-08_45_16
Last ObjectModification:
2017_07_28-AM-07_31_58
Theory : reals
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