Nuprl Lemma : limit-shift-iff
∀m:ℕ. ∀X:ℕ ⟶ ℝ. ∀a:ℝ. (lim n→∞.X[n] = a ⇐⇒ lim n→∞.X[n + m] = a)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
add: n + m
Definitions unfolded in proof :
squash: ↓T,
subtype_rel: A ⊆r B,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
sq_stable: SqStable(P),
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
sq_exists: ∃x:{A| B[x]},
converges-to: lim n→∞.x[n] = y,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
subtract-add-cancel,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
all_wf,
nat_plus_wf,
squash_wf,
less_than'_wf,
sq_stable__rleq,
rless_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
rless-int,
int-to-real_wf,
rdiv_wf,
nat_plus_properties,
rsub_wf,
rabs_wf,
rleq_wf,
sq_stable__all,
real_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
nat_wf,
converges-to_wf,
limit-shift
Rules used in proof :
dependent_set_memberFormation,
imageElimination,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
independent_pairEquality,
productElimination,
inrFormation,
because_Cache,
functionEquality,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
isectElimination,
independent_functionElimination,
independent_pairFormation,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}m:\mBbbN{}. \mforall{}X:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a:\mBbbR{}. (lim n\mrightarrow{}\minfty{}.X[n] = a \mLeftarrow{}{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.X[n + m] = a)
Date html generated:
2016_11_08-AM-09_00_14
Last ObjectModification:
2016_11_06-PM-11_34_32
Theory : reals
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