Nuprl Lemma : geometric-series-converges-ext
∀c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)} . Σn.c^n = (r1/r1 - c)
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rnexp: x^k1,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
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),
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x y.t[x; y],
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],
sq_stable__rless,
rless_functionality,
rnexp-converges-ext,
small-reciprocal-real,
geometric-series-converges,
rsub: x - y,
member: t ∈ T
Lemmas referenced :
is-exception_wf,
base_wf,
has-value_wf_base,
strict4-spread,
lifting-strict-callbyvalue,
geometric-series-converges,
sq_stable__rless,
rless_functionality,
rnexp-converges-ext,
small-reciprocal-real
Rules used in proof :
inlFormation,
imageElimination,
imageMemberEquality,
inrFormation,
applyExceptionCases,
hypothesisEquality,
closedConclusion,
baseApply,
callbyvalueApply,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
equalitySymmetry,
equalityTransitivity,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}c:\{c:\mBbbR{}| (r0 \mleq{} c) \mwedge{} (c < r1)\} . \mSigma{}n.c\^{}n = (r1/r1 - c)
Date html generated:
2018_05_22-PM-02_03_25
Last ObjectModification:
2018_05_21-AM-00_16_06
Theory : reals
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