Nuprl Lemma : Legendre-orthogonal
∀[n,k:ℕ].
r(-1)_∫-r1 x^k * Legendre(n;x) dx = if (k =z n) then (r(2 * (n)!)/r(doublefact((2 * n) + 1))) else r0 fi
supposing k ≤ n
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
Legendre: Legendre(n;x),
rdiv: (x/y),
rnexp: x^k1,
req: x = y,
rmul: a * b,
int-to-real: r(n),
doublefact: doublefact(n),
fact: (n)!,
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,
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
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,
bool: 𝔹,
unit: Unit,
it: ⋅,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
doublefact: doublefact(n),
lt_int: i <z j,
true: True,
bfalse: ff,
nat_plus: ℕ+,
req_int_terms: t1 ≡ t2,
nequal: a ≠ b ∈ T ,
int_upper: {i...},
subtract: n - m,
rat_term_to_real: rat_term_to_real(f;t),
rtermConstant: "const",
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermSubtract: left "-" right,
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
pi2: snd(t),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
assert: ↑b,
bnot: ¬bb,
int_nzero: ℤ-o,
rdiv: (x/y),
primrec: primrec(n;b;c),
fact: (n)!,
Legendre: Legendre(n;x),
rtermVar: rtermVar(var),
rge: x ≥ y
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
req_witness,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
Legendre_0_lemma,
fact0_redex_lemma,
nat_wf,
le_wf,
rnexp_zero_lemma,
rmul_wf,
rnexp_wf,
int-to-real_wf,
real_wf,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rmul_functionality,
rnexp_functionality,
req_weakening,
req_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
rdiv_wf,
doublefact_wf,
rless-int,
rless_wf,
Legendre_1_lemma,
Legendre_wf,
Legendre_functionality,
int_seg_subtype_nat,
istype-false,
fact_wf,
nat_plus_properties,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
rsub_wf,
rleq_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
req-int-fractions2,
integral_functionality,
rmul-int,
integral-const,
req_transitivity,
req_inversion,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
upper_subtype_nat,
nequal-le-implies,
zero-add,
int_upper_properties,
req-implies-req,
riiint_wf,
assert-rat-term-eq2,
rtermSubtract_wf,
rtermMultiply_wf,
rtermConstant_wf,
rtermDivide_wf,
ftc-total-integral,
derivative-const-mul,
derivative-rdiv-const-alt,
real_term_value_var_lemma,
rnexp2,
derivative-rnexp,
derivative_functionality,
rnexp1,
rmul_comm,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
rsub_functionality,
rdiv_functionality,
rnexp-one,
rnexp-minus-one,
nequal_wf,
rinv_wf2,
rmul_preserves_req,
int-rinv-cancel2,
lelt_wf,
iff_weakening_uiff,
integral-rmul-const,
iff_weakening_equal,
bfalse_wf,
eq_int_eq_false,
istype-universe,
true_wf,
squash_wf,
equal_wf,
int-rmul_functionality,
int-rdiv_functionality,
int-rmul_wf,
int-rdiv_wf,
integral-int-rdiv,
int-rdiv-req,
integral-rsub,
integral-int-rmul,
rmul-identity1,
upper_subtype_upper,
int_upper_wf,
less_than_wf,
nat_plus_wf,
radd_wf,
int-rmul-req,
radd_functionality,
rinv1,
real_term_value_add_lemma,
Legendre-deriv-equation1,
member_riiint_lemma,
rnexp_step,
rmul_assoc,
rnexp-add,
itermMinus_wf,
rminus_wf,
rmul-rinv3,
rminus_functionality,
real_term_value_minus_lemma,
istype-true,
integral-radd,
iff_imp_equal_bool,
assert_wf,
equal-wf-base,
istype-assert,
integral-by-parts,
iproper-riiint,
derivative-rdiv-const,
derivative-sub,
derivative-const,
remainder_wfa,
ifthenelse_wf,
btrue_wf,
rem_rec_case,
add-associates,
add-swap,
add-commutes,
rtermVar_wf,
rinv-mul-as-rdiv,
radd-preserves-req,
rmul_preserves_rneq_iff2,
rneq_functionality,
radd-int,
rmul-rinv,
rneq_wf,
add-subtract-cancel,
rmul-int-rdiv,
nat_plus_inc_int_nzero,
int_nzero-rational,
int-subtype-rationals,
equal_functionality_wrt_subtype_rel2,
rationals_wf,
not_functionality_wrt_implies,
rneq-int,
mul-commutes,
assert_of_lt_int,
lt_int_wf,
req-int,
rleq-int,
fact_unroll,
rless_functionality_wrt_implies,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
because_Cache,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
cumulativity,
intEquality,
setIsType,
minusEquality,
closedConclusion,
equalityElimination,
inrFormation_alt,
imageMemberEquality,
baseClosed,
equalityIstype,
multiplyEquality,
addEquality,
promote_hyp,
sqequalBase,
universeEquality,
imageElimination,
baseApply,
functionIsType
Latex:
\mforall{}[n,k:\mBbbN{}].
r(-1)\_\mint{}\msupminus{}r1 x\^{}k * Legendre(n;x) dx
= if (k =\msubz{} n) then (r(2 * (n)!)/r(doublefact((2 * n) + 1))) else r0 fi
supposing k \mleq{} n
Date html generated:
2019_10_31-AM-06_18_17
Last ObjectModification:
2019_04_03-AM-00_26_50
Theory : reals_2
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