Nuprl Lemma : cosine-exists-ext1
∀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,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
multiply: n * m,
minus: -n,
natural_number: $n,
fastexp: i^n,
fact: (n)!
Definitions unfolded in proof :
decidable__int_equal,
decidable__equal_int,
fact-greater-exp,
expfact-property,
rdiv-factorial-limit-zero,
subsequence-converges,
accelerate: accelerate(k;f),
req_functionality,
rleq_functionality,
rmul_preserves_rleq,
r-archimedean,
r-archimedean-rabs,
converges-iff-cauchy-ext,
label: ...$L... t,
so_apply: x[s],
so_lambda: λ2x.t[x],
alternating-series-converges,
iff_weakening_equal,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
or: P ∨ Q,
guard: {T},
prop: ℙ,
has-value: (a)↓,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
strict4: strict4(F),
uimplies: b supposing a,
top: Top,
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
cosine-exists,
member: t ∈ T
Lemmas referenced :
decidable__int_equal,
decidable__equal_int,
fact-greater-exp,
expfact-property,
rdiv-factorial-limit-zero,
subsequence-converges,
req_functionality,
rleq_functionality,
rmul_preserves_rleq,
r-archimedean,
r-archimedean-rabs,
converges-iff-cauchy-ext,
alternating-series-converges,
iff_weakening_equal,
cosine-exists
Rules used in proof :
decideExceptionCases,
independent_functionElimination,
dependent_functionElimination,
equalityEquality,
sqleReflexivity,
unionElimination,
unionEquality,
equalitySymmetry,
equalityTransitivity,
callbyvalueDecide,
because_Cache,
inlFormation,
exceptionSqequal,
imageElimination,
imageMemberEquality,
inrFormation,
applyExceptionCases,
hypothesisEquality,
closedConclusion,
baseApply,
callbyvalueApply,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}x:\mBbbR{}. \{a:\mBbbR{}| \mSigma{}i.-1\^{}i * (x\^{}2 * i)/(2 * i)! = a\}
Date html generated:
2016_07_08-PM-05_55_03
Last ObjectModification:
2016_07_05-PM-03_07_28
Theory : reals
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