Nuprl Lemma : trivial-Taylor-approx
∀[I:Interval]. ∀[n:ℕ]. ∀[F:ℕn + 1 ⟶ I ⟶ℝ]. ∀[a,b:{x:ℝ| x ∈ I} ].
  ((a = b) 
⇒ (Taylor-approx(n;a;b;i,x.F[i;x]) = F[0;a]))
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
, 
req: x = y
, 
real: ℝ
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s1;s2]
, 
implies: P 
⇒ Q
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
, 
add: n + m
, 
natural_number: $n
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x])
, 
prop: ℙ
, 
so_lambda: λ2x y.t[x; y]
, 
label: ...$L... t
, 
rfun: I ⟶ℝ
, 
so_apply: x[s1;s2]
, 
nat: ℕ
, 
subtype_rel: A ⊆r B
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
and: P ∧ Q
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
false: False
, 
not: ¬A
, 
ge: i ≥ j 
, 
all: ∀x:A. B[x]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
rneq: x ≠ y
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
squash: ↓T
, 
fact: (n)!
, 
primrec: primrec(n;b;c)
, 
true: True
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
pointwise-req: x[k] = y[k] for k ∈ [n,m]
, 
subtract: n - m
, 
rsub: x - y
Lemmas referenced : 
req_wf, 
req_witness, 
Taylor-approx_wf, 
int_seg_wf, 
rfun_wf, 
i-member_wf, 
real_wf, 
false_wf, 
nat_properties, 
decidable__lt, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformless_wf, 
itermConstant_wf, 
itermAdd_wf, 
itermVar_wf, 
intformle_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_less_lemma, 
int_term_value_constant_lemma, 
int_term_value_add_lemma, 
int_term_value_var_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_wf, 
lelt_wf, 
set_wf, 
nat_wf, 
interval_wf, 
rsum_wf, 
rmul_wf, 
rdiv_wf, 
int-to-real_wf, 
fact_wf, 
int_seg_subtype_nat, 
rless-int, 
int_seg_properties, 
le_wf, 
nat_plus_properties, 
rless_wf, 
rnexp_wf, 
rsub_wf, 
radd_wf, 
nat_plus_wf, 
decidable__le, 
rneq-int, 
fact-non-zero, 
fact0_redex_lemma, 
rnexp_zero_lemma, 
req_weakening, 
req_functionality, 
rsum-split-first, 
radd_functionality, 
req_transitivity, 
rsum-zero, 
uiff_transitivity, 
rmul-rdiv-cancel2, 
radd_comm, 
radd-zero-both, 
rsum_functionality, 
equal_wf, 
rmul_functionality, 
rnexp_functionality, 
rsub_functionality, 
rminus_wf, 
exp_wf2, 
not-lt-2, 
condition-implies-le, 
add-commutes, 
minus-add, 
minus-zero, 
zero-add, 
add_functionality_wrt_le, 
le-add-cancel, 
less_than_wf, 
rmul-zero-both, 
rmul_comm, 
radd-rminus-both, 
rnexp-int, 
squash_wf, 
true_wf, 
exp-zero
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lambdaFormation, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
setElimination, 
rename, 
hypothesisEquality, 
hypothesis, 
sqequalRule, 
lambdaEquality, 
dependent_functionElimination, 
applyEquality, 
functionExtensionality, 
natural_numberEquality, 
addEquality, 
because_Cache, 
dependent_set_memberEquality, 
setEquality, 
independent_pairFormation, 
unionElimination, 
independent_isectElimination, 
dependent_pairFormation, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
computeAll, 
independent_functionElimination, 
functionEquality, 
inrFormation, 
productElimination, 
equalityTransitivity, 
equalitySymmetry, 
applyLambdaEquality, 
imageMemberEquality, 
baseClosed, 
minusEquality, 
imageElimination
Latex:
\mforall{}[I:Interval].  \mforall{}[n:\mBbbN{}].  \mforall{}[F:\mBbbN{}n  +  1  {}\mrightarrow{}  I  {}\mrightarrow{}\mBbbR{}].  \mforall{}[a,b:\{x:\mBbbR{}|  x  \mmember{}  I\}  ].
    ((a  =  b)  {}\mRightarrow{}  (Taylor-approx(n;a;b;i,x.F[i;x])  =  F[0;a]))
Date html generated:
2017_10_03-PM-00_38_09
Last ObjectModification:
2017_07_28-AM-08_45_03
Theory : reals
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