Nuprl Lemma : Taylor-theorem-for-2
∀I:Interval
(iproper(I)
⇒ (∀f,g,h:I ⟶ℝ.
((∀x,y:{a:ℝ| a ∈ I} . ((x = y) ⇒ (h[x] = h[y])))
⇒ d(f[x])/dx = λx.g[x] on I
⇒ d(g[x])/dx = λx.h[x] on I
⇒ (∀a,b:{a:ℝ| a ∈ I} . ∀e:ℝ.
((r0 < e)
⇒ (∃c:ℝ
((rmin(a;b) ≤ c)
∧ (c ≤ rmax(a;b))
∧ (|f[b] - f[a] + (g[a] * (b - a)) - ((b - c) * h[c]) * (b - a)| ≤ e))))))))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rmin: rmin(x;y),
rmax: rmax(x;y),
rsub: x - y,
req: x = y,
rmul: a * b,
radd: 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: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
uall: ∀[x:A]. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
real: ℝ,
sq_stable: SqStable(P),
so_apply: x[s1;s2],
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
sq_type: SQType(T),
guard: {T},
select: L[n],
cons: [a / b],
subtract: n - m,
so_lambda: λ2x.t[x],
label: ...$L... t,
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
less_than': less_than'(a;b),
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
bnot: ¬bb,
assert: ↑b,
eq_int: (i =z j),
fact: (n)!,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
true: True,
rev_uimplies: rev_uimplies(P;Q),
lt_int: i <z j,
length: ||as||,
list_ind: list_ind,
nil: [],
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
rtermConstant: "const",
pi2: snd(t),
cand: A c∧ B,
subinterval: I ⊆ J ,
nequal: a ≠ b ∈ T ,
i-member: r ∈ I,
rccint: [l, u]
Lemmas referenced :
Taylor-theorem,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
select_wf,
real_wf,
cons_wf,
nil_wf,
int_seg_properties,
sq_stable__less_than,
int-to-real_wf,
decidable__le,
intformand_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_wf,
req_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_subtype_special,
int_seg_cases,
rless_wf,
derivative_wf,
i-member_wf,
rfun_wf,
iproper_wf,
interval_wf,
differentiable-functional2,
rsum_wf,
rmul_wf,
rdiv_wf,
fact_wf,
int_seg_subtype_nat,
istype-false,
rless-int,
istype-le,
rnexp_wf,
rsub_wf,
lt_int_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
assert_of_lt_int,
fact0_redex_lemma,
rnexp_zero_lemma,
radd_wf,
req_functionality,
rsum_unroll,
req_weakening,
radd_functionality,
rsum_single,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermVar_wf,
rtermConstant_wf,
nat_plus_wf,
set_subtype_base,
rmul_functionality,
rnexp1,
rleq_wf,
rabs_wf,
rmin-rmax-subinterval,
sq_stable__i-member,
rabs_functionality,
member_rccint_lemma,
rsub_functionality,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
neg_assert_of_eq_int,
req_inversion,
rleq_transitivity,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
dependent_set_memberEquality_alt,
natural_numberEquality,
setElimination,
rename,
isectElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
because_Cache,
applyEquality,
productElimination,
imageElimination,
addEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
int_eqEquality,
independent_pairFormation,
instantiate,
cumulativity,
intEquality,
hypothesis_subsumption,
setIsType,
functionIsType,
closedConclusion,
inrFormation_alt,
applyLambdaEquality,
equalityElimination,
equalityIstype,
promote_hyp,
productIsType,
int_eqReduceTrueSq,
int_eqReduceFalseSq
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}f,g,h:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{a:\mBbbR{}| a \mmember{} I\} . ((x = y) {}\mRightarrow{} (h[x] = h[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.g[x] on I
{}\mRightarrow{} d(g[x])/dx = \mlambda{}x.h[x] on I
{}\mRightarrow{} (\mforall{}a,b:\{a:\mBbbR{}| a \mmember{} I\} . \mforall{}e:\mBbbR{}.
((r0 < e)
{}\mRightarrow{} (\mexists{}c:\mBbbR{}
((rmin(a;b) \mleq{} c)
\mwedge{} (c \mleq{} rmax(a;b))
\mwedge{} (|f[b] - f[a] + (g[a] * (b - a)) - ((b - c) * h[c]) * (b - a)| \mleq{} e))))))))
Date html generated:
2019_10_30-AM-10_12_14
Last ObjectModification:
2019_04_02-AM-09_41_56
Theory : reals
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