Nuprl Lemma : approx-arg_wf
∀f,f':(-∞, ∞) ⟶ℝ.
((∀x,y:ℝ. ((x = y) ⇒ (f'[x] = f'[y])))
⇒ d(f[x])/dx = λx.f'[x] on (-∞, ∞)
⇒ (∀B:ℕ. ((∀x:ℝ. (|f'[x]| ≤ r(B))) ⇒ (∀x:ℝ. (approx-arg(f;B;x) ∈ {y:ℝ| y = (f x)} )))))
Proof
Definitions occuring in Statement :
approx-arg: approx-arg(f;B;x),
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
riiint: (-∞, ∞),
rleq: 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,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
rfun: I ⟶ℝ,
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
approx-arg: approx-arg(f;B;x),
true: True,
label: ...$L... t,
regular-int-seq: k-regular-seq(f),
has-value: (a)↓,
real: ℝ,
nat_plus: ℕ+,
rational-approx: (x within 1/n),
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
less_than': less_than'(a;b),
rdiv: (x/y),
less_than: a < b,
squash: ↓T,
sq_type: SQType(T),
rsub: x - y,
cand: A c∧ B,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
bdd-diff: bdd-diff(f;g),
sq_stable: SqStable(P)
Lemmas referenced :
mean-value-for-bounded-derivative,
riiint_wf,
iproper-riiint,
req_wf,
set_wf,
real_wf,
i-member_wf,
int-to-real_wf,
rleq_functionality_wrt_implies,
rabs_wf,
rleq_weakening_equal,
rleq_weakening,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
all_wf,
rleq_wf,
member_riiint_lemma,
true_wf,
nat_wf,
derivative_wf,
rfun_wf,
value-type-has-value,
int-value-type,
nat_plus_wf,
rational-approx_wf,
rational-approx-property,
rsub_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rmul_wf,
equal_wf,
rmul_preserves_rleq2,
int-rdiv_wf,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
equal-wf-base,
nequal_wf,
rleq-int,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
less_than'_wf,
int_subtype_base,
radd_wf,
rminus_wf,
rinv_wf2,
false_wf,
rleq_functionality,
req_transitivity,
real_term_value_mul_lemma,
rmul_functionality,
req_weakening,
rabs_functionality,
itermAdd_wf,
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,
subtype_base_sq,
req_functionality,
rabs-rmul,
rabs-of-nonneg,
req_inversion,
rinv_functionality2,
rinv-of-rmul,
int-rinv-cancel,
squash_wf,
rminus-int,
rsub_functionality,
rabs-difference-symmetry,
radd_functionality_wrt_rleq,
r-triangle-inequality2,
uimplies_transitivity,
rmul_comm,
rmul_over_rminus,
rmul-distrib,
uiff_transitivity,
rabs-int,
iff_weakening_equal,
absval-non-neg,
decidable__equal_int,
mul-non-neg1,
absval_pos,
le_wf,
set_subtype_base,
rmul_functionality_wrt_rleq2,
zero-rleq-rabs,
nat_plus_subtype_nat,
multiply_nat_wf,
mul_bounds_1a,
radd-int,
rmul-one-both,
rmul-rdiv-cancel,
radd_comm,
rmul-ac,
int_term_value_add_lemma,
rsub-int,
uiff_transitivity2,
absval_wf,
subtract_wf,
regular-int-seq_wf,
less_than_wf,
le-add-cancel,
zero-add,
add-zero,
add-associates,
add_functionality_wrt_le,
not-lt-2,
accelerate_wf,
req-iff-bdd-diff,
rmul-rdiv-cancel2,
req-int,
rmul_preserves_req,
rdiv_functionality,
radd-rdiv,
mul_nat_plus,
rleq-int-fractions,
uiff_transitivity3,
int_term_value_subtract_lemma,
int_term_value_minus_lemma,
rmul-zero-both,
rmul_preserves_rleq,
rleq_weakening_rless,
multiply-is-int-iff,
sq_stable__less_than,
bdd-diff_inversion,
bdd-diff_weakening,
bdd-diff_functionality,
accelerate-bdd-diff
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
hypothesis,
independent_functionElimination,
lambdaFormation,
hypothesisEquality,
isectElimination,
setElimination,
rename,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
setEquality,
functionEquality,
callbyvalueReduce,
multiplyEquality,
inrFormation,
unionElimination,
dependent_pairFormation,
independent_pairFormation,
isect_memberFormation,
baseApply,
closedConclusion,
baseClosed,
independent_pairEquality,
minusEquality,
axiomEquality,
imageMemberEquality,
addLevel,
instantiate,
cumulativity,
imageElimination,
addEquality,
universeEquality,
productEquality,
inlFormation,
functionExtensionality,
promote_hyp,
pointwiseFunctionality
Latex:
\mforall{}f,f':(-\minfty{}, \minfty{}) {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f'[x] = f'[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on (-\minfty{}, \minfty{})
{}\mRightarrow{} (\mforall{}B:\mBbbN{}. ((\mforall{}x:\mBbbR{}. (|f'[x]| \mleq{} r(B))) {}\mRightarrow{} (\mforall{}x:\mBbbR{}. (approx-arg(f;B;x) \mmember{} \{y:\mBbbR{}| y = (f x)\} )))))
Date html generated:
2017_10_03-PM-00_23_26
Last ObjectModification:
2017_07_28-AM-08_40_43
Theory : reals
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