Nuprl Lemma : derivative-rsqrt-function
∀I:Interval
(iproper(I)
⇒ (∀f,f':I ⟶ℝ.
((∀x:{x:ℝ| x ∈ I} . (r0 < f[x]))
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f'[x] = f'[y])))
⇒ (∀a:{a:ℝ| a ∈ I} . ∀b:{b:ℝ| (b ∈ I) ∧ (a ≤ b)} . ∃c:{t:ℝ| t ∈ [a, b]} . ∀x:{t:ℝ| t ∈ [a, b]} . (f[c] ≤ f[x])\000C)
⇒ d(f[x])/dx = λx.f'[x] on I
⇒ d(rsqrt(f[x]))/dx = λx.(f'[x]/r(2) * rsqrt(f[x])) on I)))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rsqrt: rsqrt(x),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
top: Top,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
or: P ∨ Q,
cand: A c∧ B,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
rtermMultiply: left "*" right,
rtermConstant: "const",
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
derivative_wf,
real_wf,
i-member_wf,
rleq_wf,
rccint_wf,
i-member-between,
sq_stable__i-member,
member_rccint_lemma,
istype-void,
sq_stable__rleq,
req_wf,
rless_wf,
int-to-real_wf,
rfun_wf,
iproper_wf,
interval_wf,
chain-rule,
roiint_wf,
iproper-roiint,
derivative-rsqrt,
rsqrt_wf,
member_roiint_lemma,
rleq_weakening_rless,
rdiv_wf,
rmul_wf,
rsqrt-positive-iff,
rless-implies-rless,
rmul-is-positive,
rless-int,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
has-minimum-maps-compact,
continuous-implies-functional,
differentiable-continuous,
proper-continuous-is-continuous,
sq_stable__rless,
subtype_rel_sets_simple,
rless_transitivity1,
rleq_weakening,
req_weakening,
req_functionality,
rdiv_functionality,
rmul_functionality,
rsqrt_functionality,
derivative_functionality,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setIsType,
hypothesis,
because_Cache,
functionIsType,
productIsType,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
inhabitedIsType,
independent_isectElimination,
closedConclusion,
inrFormation_alt,
inlFormation_alt,
independent_pairFormation,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
int_eqReduceTrueSq,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceFalseSq
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}f,f':I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (r0 < f[x]))
{}\mRightarrow{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f'[x] = f'[y])))
{}\mRightarrow{} (\mforall{}a:\{a:\mBbbR{}| a \mmember{} I\} . \mforall{}b:\{b:\mBbbR{}| (b \mmember{} I) \mwedge{} (a \mleq{} b)\} .
\mexists{}c:\{t:\mBbbR{}| t \mmember{} [a, b]\} . \mforall{}x:\{t:\mBbbR{}| t \mmember{} [a, b]\} . (f[c] \mleq{} f[x]))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on I
{}\mRightarrow{} d(rsqrt(f[x]))/dx = \mlambda{}x.(f'[x]/r(2) * rsqrt(f[x])) on I)))
Date html generated:
2019_10_31-AM-06_11_52
Last ObjectModification:
2019_04_03-AM-00_26_55
Theory : reals_2
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