Nuprl Lemma : rexp-approx-for-small
∀N:ℕ+. ∀x:{x:ℝ| |x| ≤ (r1/r(4))} . (∃z:ℤ [(|e^x - (r(z)/r(2 * N))| ≤ (r(2)/r(N)))])
Proof
Definitions occuring in Statement :
rexp: e^x,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
set: {x:A| B[x]} ,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
sq_exists: ∃x:A [B[x]],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
le: A ≤ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
rexp-approx-lemma-ext,
rexp-approx_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
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,
rexp_wf,
radd_wf,
exp_wf2,
fact_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
multiply-is-int-iff,
false_wf,
rleq-int-fractions,
istype-less_than,
rmul_preserves_rleq,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermAdd_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rexp-poly-approx,
radd_functionality_wrt_rleq,
rleq_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
rmul-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberFormation_alt,
isectElimination,
hypothesis,
universeIsType,
because_Cache,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
productElimination,
independent_functionElimination,
unionElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
multiplyEquality,
setIsType,
closedConclusion,
imageMemberEquality,
baseClosed,
applyEquality,
dependent_set_memberEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
promote_hyp,
baseApply
Latex:
\mforall{}N:\mBbbN{}\msupplus{}. \mforall{}x:\{x:\mBbbR{}| |x| \mleq{} (r1/r(4))\} . (\mexists{}z:\mBbbZ{} [(|e\^{}x - (r(z)/r(2 * N))| \mleq{} (r(2)/r(N)))])
Date html generated:
2019_10_30-AM-11_40_54
Last ObjectModification:
2019_02_04-AM-10_56_38
Theory : reals_2
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