Nuprl Lemma : fun-converges-to-derivative
∀I:Interval
(iproper(I)
⇒ (∀f,f':ℕ ⟶ I ⟶ℝ. ∀F,G:I ⟶ℝ.
((∀n:ℕ. ∀x,y:{a:ℝ| a ∈ I} . ((x = y) ⇒ (f'[n;x] = f'[n;y])))
⇒ lim n→∞.f[n;x] = λy.F[y] for x ∈ I
⇒ lim n→∞.f'[n;x] = λy.G[y] for x ∈ I
⇒ (∀n:ℕ. d(f[n;x])/dx = λx.f'[n;x] on I)
⇒ d(F[x])/dx = λx.G[x] on I)))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
req: x = y,
real: ℝ,
nat: ℕ,
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],
implies: P ⇒ Q,
member: t ∈ T,
i-nonvoid: i-nonvoid(I),
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
uimplies: b supposing a,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
integrate: a_∫- f[t] dt,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
pi1: fst(t)
Lemmas referenced :
iproper-nonvoid,
istype-nat,
derivative_wf,
subtype_rel_self,
real_wf,
i-member_wf,
fun-converges-to_wf,
req_wf,
rfun_wf,
iproper_wf,
interval_wf,
fun-converges-to-pointwise,
sq_stable__i-member,
unique-limit,
req_inversion,
req_weakening,
converges-to_functionality,
fun-converges-to-integral,
derivative-of-integral,
fun-converges-to-rsub,
integrate_wf,
antiderivatives-differ-by-constant,
rsub_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
req_functionality,
rsub_functionality,
real_polynomial_null,
int-to-real_wf,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
nat_wf,
fun-converges-to_functionality,
req-implies-req,
derivative-add,
derivative-const,
radd-zero,
derivative_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
sqequalRule,
functionIsType,
universeIsType,
isectElimination,
lambdaEquality_alt,
applyEquality,
functionEquality,
setEquality,
setIsType,
because_Cache,
setElimination,
rename,
inhabitedIsType,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_pairFormation_alt,
natural_numberEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
promote_hyp,
functionExtensionality,
equalityIsType1
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}f,f':\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. \mforall{}F,G:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}n:\mBbbN{}. \mforall{}x,y:\{a:\mBbbR{}| a \mmember{} I\} . ((x = y) {}\mRightarrow{} (f'[n;x] = f'[n;y])))
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f[n;x] = \mlambda{}y.F[y] for x \mmember{} I
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f'[n;x] = \mlambda{}y.G[y] for x \mmember{} I
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. d(f[n;x])/dx = \mlambda{}x.f'[n;x] on I)
{}\mRightarrow{} d(F[x])/dx = \mlambda{}x.G[x] on I)))
Date html generated:
2019_10_30-AM-11_39_43
Last ObjectModification:
2018_11_10-PM-00_54_27
Theory : reals_2
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