Nuprl Lemma : Taylor-approx_functionality
∀[I:Interval]. ∀[n:ℕ]. ∀[F:ℕn + 1 ⟶ I ⟶ℝ].
∀[a1,b1,a2,b2:{a:ℝ| a ∈ I} ].
(Taylor-approx(n;a1;b1;i,x.F[i;x]) = Taylor-approx(n;a2;b2;i,x.F[i;x])) supposing ((a1 = a2) and (b1 = b2))
supposing ∀k:ℕn + 1. ∀x,y:{a:ℝ| a ∈ I} . ((x = y) ⇒ (F[k;x] = F[k;y]))
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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_apply: x[s],
pointwise-req: x[k] = y[k] for k ∈ [n,m],
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ
Lemmas referenced :
rsum_functionality,
rmul_wf,
rdiv_wf,
int-to-real_wf,
fact_wf,
int_seg_subtype_nat,
false_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,
rmul_functionality,
itermAdd_wf,
intformle_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
lelt_wf,
i-member_wf,
rnexp_functionality,
rsub_functionality,
req_witness,
Taylor-approx_wf,
int_seg_wf,
rfun_wf,
real_wf,
req_wf,
set_wf,
all_wf,
nat_wf,
interval_wf,
req_weakening,
rdiv_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality,
addEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
Error :applyLambdaEquality,
voidElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
setEquality,
functionEquality
Latex:
\mforall{}[I:Interval]. \mforall{}[n:\mBbbN{}]. \mforall{}[F:\mBbbN{}n + 1 {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}].
\mforall{}[a1,b1,a2,b2:\{a:\mBbbR{}| a \mmember{} I\} ].
(Taylor-approx(n;a1;b1;i,x.F[i;x]) = Taylor-approx(n;a2;b2;i,x.F[i;x])) supposing
((a1 = a2) and
(b1 = b2))
supposing \mforall{}k:\mBbbN{}n + 1. \mforall{}x,y:\{a:\mBbbR{}| a \mmember{} I\} . ((x = y) {}\mRightarrow{} (F[k;x] = F[k;y]))
Date html generated:
2016_10_26-AM-11_44_27
Last ObjectModification:
2016_08_28-PM-10_37_45
Theory : reals
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