Nuprl Lemma : q-geometric-series-converges
∀a:{a:ℚ| |a| < 1} . ∀e:{e:ℚ| 0 < e ∧ (e ≤ 1)} . ∃n:ℕ. ∀m:ℕ. ((n ≤ m) ⇒ |Σ0 ≤ i < m. a ↑ i - (1/1 - a)| < e)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qsum: Σa ≤ j < b. E[j],
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
rationals: ℚ,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
true: True,
false: False,
qsub: r - s,
guard: {T},
iff: P ⇐⇒ Q,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
qabs: |r|,
qpositive: qpositive(r),
btrue: tt,
lt_int: i <z j,
bor: p ∨bq,
qadd: r + s,
qmul: r * s,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
bnot: ¬bb,
or: P ∨ Q,
exists: ∃x:A. B[x],
qlog-type: qlog-type(q;e),
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_type: SQType(T),
it: ⋅,
unit: Unit,
bool: 𝔹,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
top: Top
Lemmas referenced :
qadd_wf,
qsub_wf,
int-subtype-rationals,
zero-qle-qabs,
sq_stable_from_decidable,
qless_wf,
qabs_wf,
decidable__qless,
qle_wf,
rationals_wf,
set-value-type,
equal_wf,
rationals-value-type,
qmul_wf,
equal-wf-T-base,
squash_wf,
true_wf,
istype-universe,
qadd_ac_1_q,
qadd_comm_q,
subtype_rel_self,
qinverse_q,
mon_ident_q,
iff_weakening_equal,
qmul-positive,
qabs-positive,
qlog_wf,
nat_plus_subtype_nat,
qexp_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
istype-nat,
qsum_wf,
int_seg_subtype_nat,
istype-false,
int_seg_wf,
qdiv_wf,
q-geometric-series,
qabs-zero,
iff_weakening_uiff,
assert_wf,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert-qeq,
eqtt_to_assert,
qeq_wf2,
qmul-preserves-eq,
qmul_over_plus_qrng,
qmul_over_minus_qrng,
qmul-qdiv-cancel,
qadd_assoc,
istype-void,
qabs-qdiv,
qexp-qabs,
qabs-neg,
not_wf,
nat_properties,
qmul_preserves_qless,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
qmul_comm_qrng,
qmul_com,
qexp_preserves_qle,
decidable__qle,
qle_weakening_lt_qorder,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
qexp-one,
qle_witness,
qexp-nonneg,
le_wf,
qmul_preserves_qle2,
qmul_one_qrng,
qexp-add,
subtract-add-cancel,
qless_transitivity_1_qorder
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
applyLambdaEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
equalityIstype,
because_Cache,
natural_numberEquality,
applyEquality,
sqequalRule,
independent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
productIsType,
universeIsType,
productElimination,
closedConclusion,
setEquality,
cutEval,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaEquality_alt,
independent_isectElimination,
setIsType,
minusEquality,
hyp_replacement,
instantiate,
universeEquality,
inlFormation_alt,
dependent_pairFormation_alt,
unionElimination,
approximateComputation,
int_eqEquality,
Error :memTop,
voidElimination,
functionIsType,
equalityIsType3,
cumulativity,
promote_hyp,
equalityIsType1,
equalityElimination,
isect_memberFormation_alt,
isect_memberEquality_alt
Latex:
\mforall{}a:\{a:\mBbbQ{}| |a| < 1\} . \mforall{}e:\{e:\mBbbQ{}| 0 < e \mwedge{} (e \mleq{} 1)\} .
\mexists{}n:\mBbbN{}. \mforall{}m:\mBbbN{}. ((n \mleq{} m) {}\mRightarrow{} |\mSigma{}0 \mleq{} i < m. a \muparrow{} i - (1/1 - a)| < e)
Date html generated:
2020_05_20-AM-09_27_18
Last ObjectModification:
2020_01_05-AM-00_14_32
Theory : rationals
Home
Index