Nuprl Lemma : approx-arg-interval_wf
∀l:ℝ. ∀r:{r:ℝ| l < r} . ∀f,f':[l, r] ⟶ℝ.
((∀x,y:{t:ℝ| t ∈ [l, r]} . ((x = y) ⇒ (f'[x] = f'[y])))
⇒ d(f[x])/dx = λx.f'[x] on [l, r]
⇒ (∀B:ℕ
((∀x:{x:ℝ| x ∈ [l, r]} . (|f'[x]| ≤ r(B)))
⇒ (∀x:{x:ℝ| x ∈ [l, r]} . (approx-arg-interval(f;l;r;B;x) ∈ {y:ℝ| y = (f x)} )))))
Proof
Definitions occuring in Statement :
approx-arg-interval: approx-arg-interval(f;l;u;B;x),
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iproper: iproper(I),
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
nat: ℕ,
approx-arg-interval: approx-arg-interval(f;l;u;B;x),
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
int_nzero: ℤ-o,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
has-value: (a)↓,
true: True,
less_than': less_than'(a;b),
less_than: a < b,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nequal: a ≠ b ∈ T ,
real: ℝ,
cand: A c∧ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
guard: {T},
subtype_rel: A ⊆r B,
approx-in-interval: approx-in-interval(l;u;x;n),
and: P ∧ Q,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat_plus: ℕ+,
accelerate: accelerate(k;f),
regular-int-seq: k-regular-seq(f),
rational-approx: (x within 1/n),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rsub: x - y,
bdd-diff: bdd-diff(f;g),
req: x = y
Lemmas referenced :
mean-value-for-bounded-derivative,
rccint_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
sq_stable__rless,
i-finite_wf,
int-to-real_wf,
set_wf,
real_wf,
i-member_wf,
all_wf,
rleq_wf,
rabs_wf,
nat_wf,
derivative_wf,
req_wf,
rfun_wf,
rless_wf,
nequal_wf,
int-rdiv_wf,
subtract_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
function-value-type,
regular-int-seq_wf,
top_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
int_subtype_base,
equal-wf-base,
member_rccint_lemma,
int_formula_prop_eq_lemma,
intformeq_wf,
rless-int,
rdiv_wf,
rsub_wf,
rleq_weakening_rless,
subtype_rel_sets,
approx-in-interval_wf,
int-value-type,
set-value-type,
value-type-has-value,
nat_plus_wf,
equal_wf,
less_than_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermMultiply_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
nat_plus_properties,
mul_nat_plus,
rmul_wf,
sq_stable__le,
absval_wf,
rational-approx-property,
rmul_preserves_rleq2,
rleq-int,
sq_stable__less_than,
decidable__le,
less_than'_wf,
radd_wf,
rminus_wf,
rinv_wf2,
false_wf,
rleq_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul_functionality,
req_weakening,
rabs_functionality,
itermMinus_wf,
real_term_value_add_lemma,
real_term_value_minus_lemma,
rmul-rinv,
rmul-int,
radd_functionality,
rminus_functionality,
int-rdiv-req,
rmul-assoc,
rneq_functionality,
rneq-int,
equal-wf-T-base,
minus-one-mul,
true_wf,
req_functionality,
rabs-rmul,
rabs-of-nonneg,
req_inversion,
rinv_functionality2,
rinv-of-rmul,
int-rinv-cancel,
squash_wf,
rminus-int,
rsub_functionality,
uimplies_transitivity,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rabs-difference-symmetry,
rmul_comm,
rmul_over_rminus,
rmul-distrib,
uiff_transitivity,
set_subtype_base,
le_wf,
absval_pos,
mul-non-neg1,
decidable__equal_int,
absval-non-neg,
iff_weakening_equal,
rabs-int,
zero-rleq-rabs,
rmul_functionality_wrt_rleq2,
mul_bounds_1a,
multiply_nat_wf,
nat_plus_subtype_nat,
radd-int,
rleq_weakening,
rmul-rinv3,
rsub-int,
uiff_transitivity2,
req-iff-bdd-diff,
accelerate_wf,
rational-approx_wf,
rmul-rdiv-cancel2,
rmul-rdiv-cancel,
rmul-ac,
rmul_preserves_req,
rdiv_functionality,
radd-rdiv,
rleq-int-fractions,
uiff_transitivity3,
radd_comm,
int_term_value_subtract_lemma,
int_term_value_minus_lemma,
req-int,
rmul-zero-both,
rmul_preserves_rleq,
multiply-is-int-iff,
bdd-diff_inversion,
bdd-diff_weakening,
bdd-diff_functionality,
accelerate-bdd-diff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
lambdaEquality,
setEquality,
applyEquality,
dependent_set_memberEquality,
functionEquality,
closedConclusion,
baseApply,
cumulativity,
instantiate,
promote_hyp,
callbyvalueReduce,
sqequalAxiom,
isect_memberFormation,
lessCases,
equalityElimination,
applyLambdaEquality,
productElimination,
inrFormation,
productEquality,
divideEquality,
equalitySymmetry,
equalityTransitivity,
computeAll,
independent_pairFormation,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
multiplyEquality,
natural_numberEquality,
addEquality,
functionExtensionality,
independent_pairEquality,
minusEquality,
axiomEquality,
addLevel,
universeEquality,
inlFormation,
pointwiseFunctionality
Latex:
\mforall{}l:\mBbbR{}. \mforall{}r:\{r:\mBbbR{}| l < r\} . \mforall{}f,f':[l, r] {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} [l, r]\} . ((x = y) {}\mRightarrow{} (f'[x] = f'[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on [l, r]
{}\mRightarrow{} (\mforall{}B:\mBbbN{}
((\mforall{}x:\{x:\mBbbR{}| x \mmember{} [l, r]\} . (|f'[x]| \mleq{} r(B)))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} [l, r]\} . (approx-arg-interval(f;l;r;B;x) \mmember{} \{y:\mBbbR{}| y = (f x)\} )))))
Date html generated:
2017_10_03-PM-00_25_55
Last ObjectModification:
2017_07_28-AM-08_41_15
Theory : reals
Home
Index