Nuprl Lemma : Legendre-orthogonal-rpolynomial
∀[n,k:ℕ]. ∀[a:ℕk + 1 ⟶ ℝ].
r(-1)_∫-r1 (Σi≤k. a_i * x^i) * Legendre(n;x) dx
= if (k =z n) then (r(2 * (n)!)/r(doublefact((2 * n) + 1))) * (a n) else r0 fi
supposing k ≤ n
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
Legendre: Legendre(n;x),
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,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
pointwise-req: x[k] = y[k] for k ∈ [n,m],
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s],
int_seg: {i..j-},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
rfun: I ⟶ℝ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
guard: {T},
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
bfalse: ff,
rpolynomial: (Σi≤n. a_i * x^i),
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s1;s2],
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
true: True,
sq_type: SQType(T),
bnot: ¬bb,
so_lambda: λ2x y.t[x; y],
nequal: a ≠ b ∈ T ,
subtract: n - m,
req_int_terms: t1 ≡ t2
Lemmas referenced :
rpolynomial_wf,
req_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,
istype-void,
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,
req_witness,
rmul_wf,
Legendre_wf,
real_wf,
i-member_wf,
rccint_wf,
rmin_wf,
int-to-real_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rmul_functionality,
Legendre_functionality,
req_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
rdiv_wf,
fact_wf,
doublefact_wf,
rless-int,
nat_plus_properties,
rless_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_wf,
istype-nat,
rpolynomial_functionality,
rsum_wf,
rnexp_wf,
int_seg_subtype_nat,
istype-false,
rsum_functionality2,
rnexp_functionality,
rleq_wf,
member_rccint_lemma,
left-endpoint_wf,
right-endpoint_wf,
subtype_rel_self,
int_seg_properties,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
integral_functionality,
req_inversion,
rsum_linearity3,
rmul_assoc,
integral-rsum,
integral-rmul-const,
Legendre-orthogonal,
nequal-le-implies,
set_subtype_base,
le_wf,
int_subtype_base,
radd_wf,
subtract_wf,
subtract-add-cancel,
rsum-split-last,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add-associates,
le_antisymmetry_iff,
add_functionality_wrt_le,
le-add-cancel,
ifthenelse_wf,
btrue_wf,
bfalse_wf,
rmul-zero,
itermSubtract_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
radd_functionality,
rsum-zero-req,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma
Rules used in proof :
because_Cache,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
lambdaFormation_alt,
applyEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
addEquality,
productIsType,
isect_memberFormation_alt,
setIsType,
minusEquality,
productElimination,
closedConclusion,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
equalityElimination,
multiplyEquality,
inrFormation_alt,
applyLambdaEquality,
equalityIstype,
isectIsTypeImplies,
functionIsType,
functionEquality,
setEquality,
promote_hyp,
instantiate,
cumulativity,
intEquality,
sqequalBase
Latex:
\mforall{}[n,k:\mBbbN{}]. \mforall{}[a:\mBbbN{}k + 1 {}\mrightarrow{} \mBbbR{}].
r(-1)\_\mint{}\msupminus{}r1 (\mSigma{}i\mleq{}k. a\_i * x\^{}i) * Legendre(n;x) dx
= if (k =\msubz{} n) then (r(2 * (n)!)/r(doublefact((2 * n) + 1))) * (a n) else r0 fi
supposing k \mleq{} n
Date html generated:
2019_10_31-AM-06_18_27
Last ObjectModification:
2019_01_07-AM-09_54_25
Theory : reals_2
Home
Index