Nuprl Lemma : fun-converges_functionality
∀[I:Interval]. ∀f,g:ℕ ⟶ I ⟶ℝ. ((∀n:ℕ. rfun-eq(I;f n;g n)) ⇒ λn.f[n;x]↓ for x ∈ I) ⇒ λn.g[n;x]↓ for x ∈ I))
Proof
Definitions occuring in Statement :
fun-converges: λn.f[n; x]↓ for x ∈ I),
rfun-eq: rfun-eq(I;f;g),
rfun: I ⟶ℝ,
interval: Interval,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
nat_plus: ℕ+,
uimplies: b supposing a,
sq_stable: SqStable(P),
guard: {T},
squash: ↓T,
fun-converges: λn.f[n; x]↓ for x ∈ I),
exists: ∃x:A. B[x],
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
int_upper: {i...},
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
uiff: uiff(P;Q)
Lemmas referenced :
fun-converges_wf,
subtype_rel_self,
real_wf,
i-member_wf,
istype-nat,
rfun-eq_wf,
rfun_wf,
interval_wf,
upper_subtype_nat,
sq_stable__le,
le_weakening2,
istype-int_upper,
i-approx_wf,
istype-less_than,
i-member-approx,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
int_upper_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
int-to-real_wf,
nat_plus_wf,
icompact_wf,
fun-converges-to_wf,
sq_stable__rleq,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
hypothesis,
functionEquality,
setEquality,
setIsType,
functionIsType,
inhabitedIsType,
dependent_functionElimination,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
dependent_set_memberEquality_alt,
productElimination,
dependent_pairFormation_alt,
inrFormation_alt,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
closedConclusion
Latex:
\mforall{}[I:Interval]
\mforall{}f,g:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. ((\mforall{}n:\mBbbN{}. rfun-eq(I;f n;g n)) {}\mRightarrow{} \mlambda{}n.f[n;x]\mdownarrow{} for x \mmember{} I) {}\mRightarrow{} \mlambda{}n.g[n;x]\mdownarrow{} for x \mmember{} I))
Date html generated:
2019_10_30-AM-08_57_35
Last ObjectModification:
2018_11_12-AM-10_57_07
Theory : reals
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