Nuprl Lemma : sine-exists
∀x:ℝ. ∃a:ℝ. Σi.-1^i * (x^(2 * i) + 1)/((2 * i) + 1)! = a
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
rnexp: x^k1,
int-rdiv: (a)/k1,
int-rmul: k1 * a,
real: ℝ,
fastexp: i^n,
fact: (n)!,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
real: ℝ,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
false: False,
squash: ↓T,
nat: ℕ,
sq_type: SQType(T),
guard: {T},
ge: i ≥ j ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
true: True,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
series-converges: Σn.x[n]↓,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
top: Top,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rleq: x ≤ y,
rnonneg: rnonneg(x),
cand: A c∧ B,
less_than: a < b,
sq_stable: SqStable(P),
sq_exists: ∃x:A [B[x]],
rless: x < y,
rneq: x ≠ y,
stable: Stable{P},
series-sum: Σn.x[n] = a,
int_seg: {i..j-},
lelt: i ≤ j < k,
rge: x ≥ y,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
modulus: a mod n,
remainder: n rem m,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
subtract: n - m
Latex:
\mforall{}x:\mBbbR{}. \mexists{}a:\mBbbR{}. \mSigma{}i.-1\^{}i * (x\^{}(2 * i) + 1)/((2 * i) + 1)! = a
Date html generated:
2020_05_20-AM-11_24_30
Last ObjectModification:
2020_01_02-PM-02_09_05
Theory : reals
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