Nuprl Lemma : rexp-poly-approx
∀[x:{x:ℝ| |x| ≤ (r1/r(4))} ]. ∀[k:ℕ]. ∀[N:ℕ+].
(|e^x - (r(rexp-approx(x;k;N))/r(2 * N))| ≤ ((r1/r(4^k * 3 * (k)!)) + (r1/r(N))))
Proof
Definitions occuring in Statement :
rexp-approx: rexp-approx(x;k;N),
rexp: e^x,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
fact: (n)!,
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
sq_stable: SqStable(P),
ireal-approx: j-approx(x;M;z),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
so_apply: x[s],
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
small-rexp-remainder,
rexp-approx-property,
sq_stable__rleq,
rabs_wf,
rdiv_wf,
rless-int,
rless_wf,
int-to-real_wf,
le_witness_for_triv,
nat_plus_wf,
istype-nat,
real_wf,
rleq_wf,
rsub_wf,
rsum_wf,
int-rdiv_wf,
fact_wf,
int_seg_subtype_nat,
istype-false,
rnexp_wf,
int_seg_wf,
rexp-approx_wf,
nat_plus_properties,
nat_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,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
nat_plus_inc_int_nzero,
rleq_functionality,
rabs_functionality,
rsub_functionality,
rsum_functionality2,
int-rdiv-req,
req_weakening,
rleq_functionality_wrt_implies,
rexp_wf,
radd_wf,
rleq_weakening_equal,
r-triangle-inequality2,
exp_wf2,
multiply_nat_plus,
istype-less_than,
multiply-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
radd_functionality_wrt_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
because_Cache,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
dependent_functionElimination,
productElimination,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeIsType,
imageElimination,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
setIsType,
closedConclusion,
applyEquality,
lambdaFormation_alt,
addEquality,
multiplyEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
dependent_set_memberEquality_alt,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
equalityIstype
Latex:
\mforall{}[x:\{x:\mBbbR{}| |x| \mleq{} (r1/r(4))\} ]. \mforall{}[k:\mBbbN{}]. \mforall{}[N:\mBbbN{}\msupplus{}].
(|e\^{}x - (r(rexp-approx(x;k;N))/r(2 * N))| \mleq{} ((r1/r(4\^{}k * 3 * (k)!)) + (r1/r(N))))
Date html generated:
2019_10_30-AM-11_40_41
Last ObjectModification:
2019_02_04-AM-10_27_40
Theory : reals_2
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