Nuprl Lemma : equal-functions-by-Taylor
∀F,G:ℕ ⟶ ℝ ⟶ ℝ.
((∀k:ℕ. ∀x,y:ℝ. ((x = y) ⇒ (F[k;x] = F[k;y])))
⇒ (∀k:ℕ. ∀x,y:ℝ. ((x = y) ⇒ (G[k;x] = G[k;y])))
⇒ infinite-deriv-seq((-∞, ∞);i,x.F[i;x])
⇒ infinite-deriv-seq((-∞, ∞);i,x.G[i;x])
⇒ (∀m:ℕ. ∃c:ℝ. ∃N:ℕ. ∀k:{N...}. ∀x:{x:ℝ| |x| ≤ r(m)} . (|F[k;x]| ≤ c))
⇒ (∀m:ℕ. ∃c:ℝ. ∃N:ℕ. ∀k:{N...}. ∀x:{x:ℝ| |x| ≤ r(m)} . (|G[k;x]| ≤ c))
⇒ (∀n:ℕ. (F[n;r0] = G[n;r0]))
⇒ (∀x:ℝ. (F[0;x] = G[0;x])))
Proof
Definitions occuring in Statement :
infinite-deriv-seq: infinite-deriv-seq(I;i,x.F[i; x]),
riiint: (-∞, ∞),
rleq: x ≤ y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
member: t ∈ T,
so_apply: x[s1;s2],
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
label: ...$L... t,
rfun: I ⟶ℝ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
true: True,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
Taylor-series-bounded-converges-everywhere,
nat_wf,
real_wf,
all_wf,
req_wf,
int-to-real_wf,
exists_wf,
int_upper_wf,
rleq_wf,
rabs_wf,
int_upper_subtype_nat,
infinite-deriv-seq_wf,
riiint_wf,
i-member_wf,
fun-converges-to-pointwise,
rsum_wf,
rmul_wf,
rdiv_wf,
int_seg_subtype_nat,
false_wf,
fact_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,
int_seg_wf,
member_riiint_lemma,
req_weakening,
converges-to_functionality,
rsum_functionality,
req_functionality,
rmul_functionality,
rdiv_functionality,
unique-limit
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
isectElimination,
natural_numberEquality,
because_Cache,
setElimination,
rename,
setEquality,
functionEquality,
addEquality,
independent_isectElimination,
independent_pairFormation,
inrFormation,
productElimination,
dependent_set_memberEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
Error :applyLambdaEquality,
voidElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}F,G:\mBbbN{} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}k:\mBbbN{}. \mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (F[k;x] = F[k;y])))
{}\mRightarrow{} (\mforall{}k:\mBbbN{}. \mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (G[k;x] = G[k;y])))
{}\mRightarrow{} infinite-deriv-seq((-\minfty{}, \minfty{});i,x.F[i;x])
{}\mRightarrow{} infinite-deriv-seq((-\minfty{}, \minfty{});i,x.G[i;x])
{}\mRightarrow{} (\mforall{}m:\mBbbN{}. \mexists{}c:\mBbbR{}. \mexists{}N:\mBbbN{}. \mforall{}k:\{N...\}. \mforall{}x:\{x:\mBbbR{}| |x| \mleq{} r(m)\} . (|F[k;x]| \mleq{} c))
{}\mRightarrow{} (\mforall{}m:\mBbbN{}. \mexists{}c:\mBbbR{}. \mexists{}N:\mBbbN{}. \mforall{}k:\{N...\}. \mforall{}x:\{x:\mBbbR{}| |x| \mleq{} r(m)\} . (|G[k;x]| \mleq{} c))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. (F[n;r0] = G[n;r0]))
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. (F[0;x] = G[0;x])))
Date html generated:
2016_10_26-AM-11_52_07
Last ObjectModification:
2016_09_05-AM-10_28_06
Theory : reals
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