Nuprl Lemma : Legendre-rpolynomial-same-degree
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[f:[r(-1), r1] ⟶ℝ].
r(-1)_∫-r1 f[x] * Legendre(n;x) dx = ((r(2 * (n)!)/r(doublefact((2 * n) + 1))) * (a n))
supposing ∀x:{x:ℝ| x ∈ [r(-1), r1]} . ((f x) = (Σi≤n. a_i * x^i))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
Legendre: Legendre(n;x),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rpolynomial: (Σi≤n. a_i * x^i),
rdiv: (x/y),
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
doublefact: doublefact(n),
fact: (n)!,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
rfun: I ⟶ℝ,
nat: ℕ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
top: Top,
imax: imax(a;b),
imin: imin(a;b),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
cand: A c∧ B,
i-member: r ∈ I,
guard: {T},
pointwise-req: x[k] = y[k] for k ∈ [n,m],
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
sq_stable: SqStable(P),
squash: ↓T,
sq_type: SQType(T)
Lemmas referenced :
real_wf,
i-member_wf,
rccint_wf,
int-to-real_wf,
req_wf,
rpolynomial_wf,
rfun_wf,
int_seg_wf,
istype-nat,
ifun_wf,
rmin_wf,
rmax_wf,
rccint-icompact,
rmin-rleq-rmax,
member_rccint_lemma,
istype-void,
rleq_wf,
imin_wf,
imax_wf,
rmul_wf,
Legendre_wf,
iff_weakening_uiff,
rleq_functionality,
rmin-int,
req_weakening,
rmax-int,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
left-endpoint_wf,
right-endpoint_wf,
req-int,
req_functionality,
rmul_functionality,
subtype_rel_sets_simple,
req_inversion,
rleq_transitivity,
rleq_weakening,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-le,
istype-less_than,
rpolynomial_functionality,
Legendre_functionality,
sq_stable__req,
integral_wf,
rdiv_wf,
fact_wf,
doublefact_wf,
rless-int,
nat_plus_properties,
rless_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
eq_int_eq_true,
btrue_wf,
subtype_rel_self,
iff_weakening_equal,
Legendre-orthogonal-rpolynomial,
integral_functionality
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
cut,
functionIsType,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
minusEquality,
natural_numberEquality,
hypothesisEquality,
applyEquality,
setElimination,
rename,
addEquality,
dependent_set_memberEquality_alt,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_functionElimination,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
because_Cache,
callbyvalueReduce,
sqleReflexivity,
independent_pairFormation,
productIsType,
promote_hyp,
lambdaFormation_alt,
closedConclusion,
productEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
inrFormation_alt,
inhabitedIsType,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity,
universeEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[f:[r(-1), r1] {}\mrightarrow{}\mBbbR{}].
r(-1)\_\mint{}\msupminus{}r1 f[x] * Legendre(n;x) dx = ((r(2 * (n)!)/r(doublefact((2 * n) + 1))) * (a n))
supposing \mforall{}x:\{x:\mBbbR{}| x \mmember{} [r(-1), r1]\} . ((f x) = (\mSigma{}i\mleq{}n. a\_i * x\^{}i))
Date html generated:
2019_10_31-AM-06_18_49
Last ObjectModification:
2019_01_07-AM-10_47_56
Theory : reals_2
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