Nuprl Lemma : cosine-poly-approx-1
∀[x:{x:ℝ| (r0 ≤ x) ∧ (x ≤ r1)} ]. ∀[k:ℕ].
(|cosine(x) - Σ{-1^i * (x^2 * i)/(2 * i)! | 0≤i≤k}| ≤ (x^(2 * k) + 2/r(((2 * k) + 2)!)))
Proof
Definitions occuring in Statement :
cosine: cosine(x),
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
int-rdiv: (a)/k1,
int-rmul: k1 * a,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
fastexp: i^n,
fact: (n)!,
nat: ℕ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
uiff: uiff(P;Q),
series-sum: Σn.x[n] = a,
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:A [B[x]],
less_than': less_than'(a;b),
rless: x < y,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
real: ℝ,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
top: Top,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
cand: A c∧ B,
subtract: n - m,
true: True,
int_upper: {i...},
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o
Latex:
\mforall{}[x:\{x:\mBbbR{}| (r0 \mleq{} x) \mwedge{} (x \mleq{} r1)\} ]. \mforall{}[k:\mBbbN{}].
(|cosine(x) - \mSigma{}\{-1\^{}i * (x\^{}2 * i)/(2 * i)! | 0\mleq{}i\mleq{}k\}| \mleq{} (x\^{}(2 * k) + 2/r(((2 * k) + 2)!)))
Date html generated:
2020_05_20-AM-11_26_22
Last ObjectModification:
2019_12_14-PM-04_51_41
Theory : reals
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