Nuprl Lemma : geometric-series-one-half
We have Σn.(r1/r(2)^n) = r(2) because n = 0 ∈ ℤ is included in the series.⋅
Σn.(r1/r(2)^n) = r(2)
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
rdiv: (x/y),
rnexp: x^k1,
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
so_apply: x[s],
rat_term_to_real: rat_term_to_real(f;t),
rtermConstant: "const",
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermDivide: num "/" denom,
rtermSubtract: left "-" right,
pi2: snd(t),
cand: A c∧ B,
nat_plus: ℕ+,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
uiff: uiff(P;Q),
le: A ≤ B,
subtype_rel: A ⊆r B,
rsub: x - y,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
series-sum_functionality,
rnexp_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
istype-nat,
rsub_wf,
assert-rat-term-eq2,
rtermDivide_wf,
rtermConstant_wf,
rtermSubtract_wf,
istype-int,
geometric-series-converges,
rleq-int-fractions2,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
istype-false,
rless-int-fractions3,
rleq_wf,
rnexp-positive,
rmul_preserves_rless,
radd_wf,
rminus_wf,
rmul_wf,
iff_transitivity,
squash_wf,
true_wf,
real_wf,
rminus-int,
iff_weakening_equal,
rless_functionality,
req_weakening,
radd-int,
radd_functionality,
rminus_functionality,
rmul-rdiv-cancel,
rmul_comm,
rmul-one-both,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rmul-int,
req_functionality,
req_inversion,
rnexp-rdiv,
rnexp-one,
rdiv_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality_alt,
isectElimination,
hypothesisEquality,
closedConclusion,
natural_numberEquality,
hypothesis,
independent_isectElimination,
inrFormation_alt,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
universeIsType,
lambdaFormation_alt,
approximateComputation,
dependent_set_memberEquality_alt,
unionElimination,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
productIsType,
inrFormation,
minusEquality,
addEquality,
multiplyEquality,
addLevel,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaFormation,
levelHypothesis
Latex:
\mSigma{}n.(r1/r(2)\^{}n) = r(2)
Date html generated:
2019_10_29-AM-10_26_25
Last ObjectModification:
2019_04_02-AM-10_00_34
Theory : reals
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