Nuprl Lemma : rexp-approx-property
∀[x:ℝ]. ∀[k:ℕ]. ∀[N:ℕ+]. ((|x| ≤ (r1/r(4))) ⇒ 1-approx(Σ{(x^i)/(i)! | 0≤i≤k};N;rexp-approx(x;k;N)))
Proof
Definitions occuring in Statement :
rexp-approx: rexp-approx(x;k;N),
ireal-approx: j-approx(x;M;z),
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
int-rdiv: (a)/k1,
int-to-real: r(n),
real: ℝ,
fact: (n)!,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
rexp-approx: rexp-approx(x;k;N),
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
nat_plus: ℕ+,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
so_apply: x[s],
pointwise-req: x[k] = y[k] for k ∈ [n,m],
iff: P ⇐⇒ Q,
rneq: x ≠ y,
guard: {T},
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
true: True,
ireal-approx: j-approx(x;M;z),
rleq: x ≤ y,
rnonneg: rnonneg(x),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
poly-approx-property,
int-rdiv_wf,
fact_wf,
nat_plus_inc_int_nzero,
int-to-real_wf,
rsum_functionality,
rmul_wf,
int_seg_properties,
nat_plus_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
rnexp_wf,
int_seg_wf,
ireal-approx_functionality,
rexp-approx_wf,
rsum_wf,
rleq_wf,
rabs_wf,
rdiv_wf,
rless-int,
rless_wf,
le_witness_for_triv,
nat_plus_wf,
istype-nat,
real_wf,
decidable__lt,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
req_weakening,
req_functionality,
rmul_functionality,
int-rdiv-req,
req_transitivity,
rinv-mul-as-rdiv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
hypothesis,
applyEquality,
sqequalRule,
natural_numberEquality,
inhabitedIsType,
independent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
because_Cache,
addEquality,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
inrFormation_alt,
imageMemberEquality,
baseClosed,
functionIsTypeImplies,
isectIsTypeImplies,
applyLambdaEquality
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[k:\mBbbN{}]. \mforall{}[N:\mBbbN{}\msupplus{}].
((|x| \mleq{} (r1/r(4))) {}\mRightarrow{} 1-approx(\mSigma{}\{(x\^{}i)/(i)! | 0\mleq{}i\mleq{}k\};N;rexp-approx(x;k;N)))
Date html generated:
2019_10_29-AM-10_39_09
Last ObjectModification:
2019_02_03-PM-10_05_46
Theory : reals
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