Nuprl Lemma : accelerate-rational-approx
∀[k:ℕ+]. ∀[x:ℝ]. ∀[a:ℕ+ ⟶ ℤ]. ((∀n:ℕ+. (|x - (r(a n)/r(2 * n))| ≤ (r(k)/r(n)))) ⇒ (accelerate(k;a) ∈ {y:ℝ| y = x} ))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
accelerate: accelerate(k;f),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
nat_plus: ℕ+,
prop: ℙ,
all: ∀x:A. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top
Lemmas referenced :
rational-approx-implies-req,
accelerate_wf,
regular-int-seq_wf,
req_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,
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,
rless_wf,
real_wf,
nat_plus_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
independent_functionElimination,
productElimination,
dependent_set_memberEquality_alt,
universeIsType,
setElimination,
rename,
sqequalRule,
functionIsType,
inhabitedIsType,
applyEquality,
multiplyEquality,
closedConclusion,
natural_numberEquality,
because_Cache,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbR{}]. \mforall{}[a:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}].
((\mforall{}n:\mBbbN{}\msupplus{}. (|x - (r(a n)/r(2 * n))| \mleq{} (r(k)/r(n)))) {}\mRightarrow{} (accelerate(k;a) \mmember{} \{y:\mBbbR{}| y = x\} ))
Date html generated:
2019_10_29-AM-10_21_35
Last ObjectModification:
2019_02_02-AM-10_27_28
Theory : reals
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