Nuprl Lemma : derivative-implies-strictly-increasing
∀I:Interval
(iproper(I)
⇒ (∀f,f':I ⟶ℝ.
(d(f[x])/dx = λx.f'[x] on I
⇒ f'[x] continuous for x ∈ I
⇒ (∀x:{x:ℝ| x ∈ I} . (r0 < f'[x]))
⇒ f[x] strictly-increasing for x ∈ I)))
Proof
Definitions occuring in Statement :
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
derivative: d(f[x])/dx = λz.g[z] on I,
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
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]} ,
natural_number: $n
Definitions unfolded in proof :
increasing-on-interval: f[x] increasing for x ∈ I,
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rgt: x > y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
subinterval: I ⊆ J ,
i-member: r ∈ I,
subtype_rel: A ⊆r B,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
rless: x < y,
or: P ∨ Q,
rneq: x ≠ y,
sq_exists: ∃x:A [B[x]],
uimplies: b supposing a,
true: True,
less_than': less_than'(a;b),
less_than: a < b,
nat_plus: ℕ+,
rccint: [l, u],
i-approx: i-approx(I;n),
continuous: f[x] continuous for x ∈ I,
guard: {T},
exists: ∃x:A. B[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
squash: ↓T,
sq_stable: SqStable(P),
cand: A c∧ B,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
i-member-between,
iff_weakening_equal,
subtype_rel_self,
rleq-implies-rleq,
trivial-rleq-radd,
derivative-implies-increasing,
rless_transitivity1,
req_functionality,
rmul_preserves_req,
rsub_functionality,
rdiv_functionality,
rinv-mul-as-rdiv,
rless_functionality_wrt_implies,
rless-int-fractions,
radd-zero,
radd-preserves-rless,
int-rinv-cancel,
rless-implies-rless,
nequal_wf,
true_wf,
equal-wf-base,
int_subtype_base,
subtype_base_sq,
minus-one-mul-top,
rmul-zero-both,
rmul_preserves_rless,
derivative_functionality_wrt_subinterval,
rfun_subtype,
mean-value-theorem,
rless_functionality,
radd_comm,
rleq_transitivity,
radd-rminus-assoc,
radd-rminus-both,
rless_transitivity2,
equal_wf,
ravg_wf,
rless-cases,
ravg-between,
rmul-rinv3,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real_term_value_add_lemma,
rinv-as-rdiv,
rinv-of-rmul,
req_inversion,
rinv_functionality2,
rmul_functionality,
rminus_functionality,
radd_functionality,
req_transitivity,
rleq_functionality,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
itermMinus_wf,
itermAdd_wf,
equal-wf-T-base,
int_formula_prop_eq_lemma,
intformeq_wf,
rneq-int,
req_weakening,
rmul-int,
rneq_functionality,
rinv_wf2,
rmul_wf,
req-iff-rsub-is-0,
itermSubtract_wf,
rminus_wf,
radd_wf,
radd-preserves-rleq,
rabs-difference-bound-rleq,
rleq_weakening,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
member_rccint_lemma,
squash_wf,
nat_plus_wf,
less_than'_wf,
sq_stable__rleq,
sq_stable__all,
sq_stable__rless,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
rsub_wf,
rabs_wf,
i-approx_wf,
sq_stable__and,
mul_nat_plus,
icompact_wf,
rleq_weakening_rless,
rccint-icompact,
less_than_wf,
small-reciprocal-real,
interval_wf,
iproper_wf,
rfun_wf,
derivative_wf,
continuous_wf,
int-to-real_wf,
all_wf,
i-member_wf,
real_wf,
set_wf,
rless_wf,
rccint_wf,
continuous_functionality_wrt_subinterval,
rleq_wf,
sq_stable__i-member,
rcc-subinterval
Rules used in proof :
universeEquality,
cumulativity,
instantiate,
addLevel,
promote_hyp,
productEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
independent_pairEquality,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
inrFormation,
multiplyEquality,
functionEquality,
isect_memberEquality,
independent_isectElimination,
dependent_set_memberEquality,
applyEquality,
natural_numberEquality,
setEquality,
lambdaEquality,
because_Cache,
isectElimination,
independent_pairFormation,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
productElimination,
hypothesis,
rename,
setElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}f,f':I {}\mrightarrow{}\mBbbR{}.
(d(f[x])/dx = \mlambda{}x.f'[x] on I
{}\mRightarrow{} f'[x] continuous for x \mmember{} I
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (r0 < f'[x]))
{}\mRightarrow{} f[x] strictly-increasing for x \mmember{} I)))
Date html generated:
2018_05_22-PM-02_47_15
Last ObjectModification:
2018_05_21-AM-01_12_47
Theory : reals
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