Nuprl Lemma : derivative-implies-strictly-decreasing-closed
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f,f':[a, b] ⟶ℝ.
(d(f[x])/dx = λx.f'[x] on [a, b]
⇒ ifun(λx.f'[x];[a, b])
⇒ (∀x:{x:ℝ| x ∈ [a, b]} . (f'[x] ≤ r0))
⇒ (∀x:{x:ℝ| x ∈ (a, b)} . (f'[x] < r0))
⇒ f[x] strictly-decreasing for x ∈ [a, b])
Proof
Definitions occuring in Statement :
strictly-decreasing-on-interval: f[x] strictly-decreasing for x ∈ I,
derivative: d(f[x])/dx = λz.g[z] on I,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rooint: (l, u),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
top: Top,
rfun: I ⟶ℝ,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
label: ...$L... t,
subinterval: I ⊆ J ,
cand: A c∧ B,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
i-member: r ∈ I,
rccint: [l, u],
rooint: (l, u),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
strictly-decreasing-on-interval: f[x] strictly-decreasing for x ∈ I,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
i-member_wf,
rooint_wf,
rless_wf,
member_rccint_lemma,
istype-void,
int-to-real_wf,
rccint_wf,
rleq_wf,
ifun_wf,
rccint-icompact,
derivative_wf,
rfun_wf,
real_wf,
derivative-implies-strictly-increasing-closed,
rminus_wf,
member_rooint_lemma,
rleq_weakening_rless,
radd-preserves-rleq,
radd-preserves-rless,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
itermMinus_wf,
subtype_rel_sets_simple,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma,
rless_functionality,
derivative-minus,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_wf,
req_weakening,
req_functionality,
rminus_functionality,
sq_stable__rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
functionIsType,
setIsType,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesis,
applyEquality,
because_Cache,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
closedConclusion,
natural_numberEquality,
lambdaEquality_alt,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
productIsType,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a < b\} . \mforall{}f,f':[a, b] {}\mrightarrow{}\mBbbR{}.
(d(f[x])/dx = \mlambda{}x.f'[x] on [a, b]
{}\mRightarrow{} ifun(\mlambda{}x.f'[x];[a, b])
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . (f'[x] \mleq{} r0))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} (a, b)\} . (f'[x] < r0))
{}\mRightarrow{} f[x] strictly-decreasing for x \mmember{} [a, b])
Date html generated:
2019_10_30-AM-09_08_37
Last ObjectModification:
2018_11_12-AM-11_40_28
Theory : reals
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