Nuprl Lemma : cosine-exists-ext
∀x:ℝ. ∃a:ℝ. Σi.-1^i * (x^2 * i)/(2 * i)! = 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,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m,
accelerate: accelerate(k;f),
cosine-exists,
iff_weakening_equal,
r-archimedean-rabs,
alternating-series-converges,
rleq_functionality,
subsequence-converges,
decidable__le,
any: any x,
req_functionality,
rdiv-factorial-limit-zero-from-bound,
r-archimedean,
rmax_lb,
converges-iff-cauchy-ext,
expfact-property,
canonical-bound-property,
fact-greater-exp,
decidable__equal_int,
decidable__and,
decidable__not,
decidable__less_than',
decidable__int_equal,
decidable__implies,
decidable__false,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
uimplies: b supposing a
Lemmas referenced :
cosine-exists,
lifting-strict-spread,
istype-void,
strict4-spread,
lifting-strict-decide,
lifting-strict-int_eq,
strict4-decide,
iff_weakening_equal,
r-archimedean-rabs,
alternating-series-converges,
rleq_functionality,
subsequence-converges,
decidable__le,
req_functionality,
rdiv-factorial-limit-zero-from-bound,
r-archimedean,
rmax_lb,
converges-iff-cauchy-ext,
expfact-property,
canonical-bound-property,
fact-greater-exp,
decidable__equal_int,
decidable__and,
decidable__not,
decidable__less_than',
decidable__int_equal,
decidable__implies,
decidable__false
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}x:\mBbbR{}. \mexists{}a:\mBbbR{}. \mSigma{}i.-1\^{}i * (x\^{}2 * i)/(2 * i)! = a
Date html generated:
2019_10_29-AM-10_35_27
Last ObjectModification:
2019_04_02-AM-10_10_15
Theory : reals
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