Nuprl Lemma : poly-approx-aux-property
∀[k:ℕ]. ∀[a:ℕ ⟶ ℝ]. ∀[x:ℝ]. ∀[xM:ℤ]. ∀[M:ℕ+]. ∀[n:ℕ].
((|x| ≤ (r1/r(4))) ⇒ 1-approx(x;M;xM) ⇒ k + 1-approx(Σ{(a (n + i)) * x^i | 0≤i≤k};M;poly-approx-aux(a;x;xM;M;n;k)))
Proof
Definitions occuring in Statement :
poly-approx-aux: poly-approx-aux(a;x;xM;M;n;k),
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],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
ireal-approx: j-approx(x;M;z),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
true: True,
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
less_than': less_than'(a;b),
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
poly-approx-aux: poly-approx-aux(a;x;xM;M;n;k),
so_apply: x[s],
decidable: Dec(P),
nat_plus: ℕ+,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
real: ℝ,
subtype_rel: A ⊆r B,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rge: x ≥ y
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[a:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}]. \mforall{}[xM:\mBbbZ{}]. \mforall{}[M:\mBbbN{}\msupplus{}]. \mforall{}[n:\mBbbN{}].
((|x| \mleq{} (r1/r(4)))
{}\mRightarrow{} 1-approx(x;M;xM)
{}\mRightarrow{} k + 1-approx(\mSigma{}\{(a (n + i)) * x\^{}i | 0\mleq{}i\mleq{}k\};M;poly-approx-aux(a;x;xM;M;n;k)))
Date html generated:
2020_05_20-AM-11_16_52
Last ObjectModification:
2020_01_03-AM-00_57_58
Theory : reals
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