Nuprl Lemma : Taylor-approx_wf
∀[I:Interval]. ∀[n:ℕ]. ∀[F:ℕn + 1 ⟶ I ⟶ℝ]. ∀[a,b:{a:ℝ| a ∈ I} ]. (Taylor-approx(n;a;b;i,x.F[i;x]) ∈ ℝ)
Proof
Definitions occuring in Statement :
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
lelt: i ≤ j < k,
ge: i ≥ j ,
int_seg: {i..j-},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
nat: ℕ,
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rsum_wf,
rmul_wf,
rdiv_wf,
i-member_wf,
int-to-real_wf,
fact_wf,
int_seg_subtype_nat,
false_wf,
nat_plus_wf,
rless-int,
int_seg_properties,
nat_properties,
decidable__lt,
le_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
rless_wf,
rnexp_wf,
rsub_wf,
int_seg_wf,
set_wf,
real_wf,
rfun_wf,
nat_wf,
interval_wf
Rules used in proof :
functionEquality,
axiomEquality,
computeAll,
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
voidElimination,
equalitySymmetry,
equalityTransitivity,
unionElimination,
independent_functionElimination,
productElimination,
dependent_functionElimination,
inrFormation,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
addEquality,
dependent_set_memberEquality,
because_Cache,
applyEquality,
lambdaEquality,
hypothesis,
hypothesisEquality,
natural_numberEquality,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
sqequalRule,
rename,
thin,
setElimination,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[I:Interval]. \mforall{}[n:\mBbbN{}]. \mforall{}[F:\mBbbN{}n + 1 {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}]. \mforall{}[a,b:\{a:\mBbbR{}| a \mmember{} I\} ].
(Taylor-approx(n;a;b;i,x.F[i;x]) \mmember{} \mBbbR{})
Date html generated:
2016_05_18-AM-10_28_36
Last ObjectModification:
2016_01_17-AM-00_26_40
Theory : reals
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