Nuprl Lemma : poly-approx-property
∀[k:ℕ]. ∀[a:ℕ ⟶ ℝ]. ∀[x:ℝ]. ∀[N:ℕ+]. ((|x| ≤ (r1/r(4))) ⇒ 1-approx(Σ{(a i) * x^i | 0≤i≤k};N;poly-approx(a;x;k;N)))
Proof
Definitions occuring in Statement :
poly-approx: poly-approx(a;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,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
poly-approx: poly-approx(a;x;k;N),
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
real: ℝ,
rneq: x ≠ y,
guard: {T},
less_than: a < b,
squash: ↓T,
ireal-approx: j-approx(x;M;z),
rleq: x ≤ y,
rnonneg: rnonneg(x),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
sq_type: SQType(T),
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
rge: x ≥ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
int_nzero: ℤ-o
Lemmas referenced :
poly-approx-aux-property,
mul_nat_plus,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
value-type-has-value,
nat_plus_wf,
set-value-type,
int-value-type,
ireal-approx-1,
ireal-approx_wf,
le_wf,
equal_wf,
rleq_wf,
rabs_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
less_than'_wf,
rsub_wf,
nat_plus_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rsum_wf,
rmul_wf,
nat_wf,
int_seg_subtype_nat,
rnexp_wf,
int_seg_wf,
poly-approx_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
real_wf,
rsum_functionality,
int_seg_properties,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
req_weakening,
ireal-approx_functionality,
poly-approx-aux_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
req-int-fractions,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rleq_functionality,
rleq_functionality_wrt_implies,
equal-wf-base,
radd_wf,
rleq_weakening_equal,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rmul_preserves_rleq,
rleq-int,
rless_functionality,
rabs-of-nonneg,
rinv_wf2,
rneq_functionality,
rmul-int,
rneq-int,
equal-wf-T-base,
itermSubtract_wf,
req-iff-rsub-is-0,
rmul-one,
req_functionality,
rmul_functionality,
req_transitivity,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
rmul-rinv,
rmul-rinv3,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rabs-rmul,
mul_bounds_1b,
int_entire_a,
rminus_wf,
itermMinus_wf,
req-int,
rsub_functionality,
radd_functionality,
rminus_functionality,
rminus-int,
radd-int,
real_term_value_add_lemma,
real_term_value_minus_lemma,
mul-commutes,
div_rem_sum2,
subtype_rel_sets,
nequal_wf,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
rabs_functionality,
squash_wf,
true_wf,
rabs-int,
iff_weakening_equal,
absval_wf,
rem_bounds_absval_le,
le_functionality,
le_weakening,
absval_pos,
nat_plus_subtype_nat,
rleq-int-fractions,
radd-rdiv,
rdiv_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
because_Cache,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
productElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
callbyvalueReduce,
equalityTransitivity,
equalitySymmetry,
inrFormation,
imageMemberEquality,
baseClosed,
independent_pairEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
functionExtensionality,
multiplyEquality,
axiomEquality,
functionEquality,
applyLambdaEquality,
promote_hyp,
instantiate,
cumulativity,
divideEquality,
remainderEquality,
setEquality,
imageElimination,
universeEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[a:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}]. \mforall{}[N:\mBbbN{}\msupplus{}].
((|x| \mleq{} (r1/r(4))) {}\mRightarrow{} 1-approx(\mSigma{}\{(a i) * x\^{}i | 0\mleq{}i\mleq{}k\};N;poly-approx(a;x;k;N)))
Date html generated:
2018_05_22-PM-02_01_49
Last ObjectModification:
2017_10_25-PM-05_14_16
Theory : reals
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