Nuprl Lemma : fun-converges-to_functionality
∀I:Interval. ∀f1,f2:ℕ ⟶ I ⟶ℝ.
∀[g:I ⟶ℝ]
((∀n:ℕ. ∀x:{x:ℝ| x ∈ I} . (f1[n;x] = f2[n;x]))
⇒ {lim n→∞.f1[n;x] = λy.g[y] for x ∈ I ⇒ lim n→∞.f2[n;x] = λy.g[y] for x ∈ I})
Proof
Definitions occuring in Statement :
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
member: t ∈ T,
nat_plus: ℕ+,
prop: ℙ,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
le: A ≤ B,
subinterval: I ⊆ J ,
uiff: uiff(P;Q)
Lemmas referenced :
i-approx-is-subinterval,
less_than_wf,
int_upper_wf,
set_wf,
real_wf,
i-member_wf,
i-approx_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
int_upper_subtype_nat,
nat_plus_subtype_nat,
rdiv_wf,
int-to-real_wf,
rless-int,
int_upper_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
nat_plus_wf,
icompact_wf,
fun-converges-to_wf,
nat_wf,
req_wf,
rfun_wf,
interval_wf,
subtype_rel_sets,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
introduction,
extract_by_obid,
dependent_set_memberEquality,
setElimination,
rename,
isectElimination,
natural_numberEquality,
promote_hyp,
productElimination,
dependent_pairFormation,
sqequalRule,
lambdaEquality,
because_Cache,
setEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
inrFormation,
independent_functionElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionEquality
Latex:
\mforall{}I:Interval. \mforall{}f1,f2:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}.
\mforall{}[g:I {}\mrightarrow{}\mBbbR{}]
((\mforall{}n:\mBbbN{}. \mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (f1[n;x] = f2[n;x]))
{}\mRightarrow{} \{lim n\mrightarrow{}\minfty{}.f1[n;x] = \mlambda{}y.g[y] for x \mmember{} I {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f2[n;x] = \mlambda{}y.g[y] for x \mmember{} I\})
Date html generated:
2016_10_26-AM-11_12_47
Last ObjectModification:
2016_08_27-PM-06_59_07
Theory : reals
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