Nuprl Lemma : Legendre-orthogonality
∀[n,m:ℕ]. (r(-1)_∫-r1 Legendre(n;x) * Legendre(m;x) dx = if (n =z m) then (r(2)/r((2 * n) + 1)) else r0 fi )
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
Legendre: Legendre(n;x),
rdiv: (x/y),
req: x = y,
rmul: a * b,
int-to-real: r(n),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
rfun: I ⟶ℝ,
prop: ℙ,
ifun: ifun(f;I),
top: Top,
real-fun: real-fun(f;a;b),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
doublefact: doublefact(n),
lt_int: i <z j,
true: True,
subtract: n - m,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
nequal: a ≠ b ∈ T
Lemmas referenced :
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
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,
istype-void,
right_endpoint_rccint_lemma,
req_functionality,
rmul_functionality,
Legendre_functionality,
req_weakening,
req_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
rdiv_wf,
rless-int,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rless_wf,
istype-nat,
int_subtype_base,
Legendre-rpolynomial,
Legendre-rpolynomial-same-degree,
member_rccint_lemma,
rmul_comm,
fact_wf,
doublefact_wf,
nat_plus_properties,
istype-le,
istype-less_than,
subtract_wf,
decidable__le,
decidable__equal_int,
nat_plus_wf,
set_subtype_base,
less_than_wf,
fact0_redex_lemma,
rinv_wf2,
itermSubtract_wf,
req_transitivity,
rinv1,
rmul-identity1,
rmul-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
req-int,
lt_int_wf,
assert_of_lt_int,
bool_wf,
iff_weakening_uiff,
assert_wf,
mul-commutes,
add-associates,
add-swap,
add-commutes,
rleq-int,
rleq_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rdiv_functionality,
req_inversion,
rmul_preserves_req,
rmul-rinv3,
rinv-mul-as-rdiv,
Legendre-annihilates-rpolynomial,
rpolynomial_wf,
int_seg_wf,
int_seg_properties,
integral_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
sqequalRule,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
setIsType,
universeIsType,
minusEquality,
natural_numberEquality,
isect_memberEquality_alt,
closedConclusion,
addEquality,
multiplyEquality,
inrFormation_alt,
approximateComputation,
int_eqEquality,
independent_pairFormation,
isectIsTypeImplies,
cumulativity,
intEquality,
applyEquality,
applyLambdaEquality,
productIsType,
imageMemberEquality,
baseClosed,
functionIsType
Latex:
\mforall{}[n,m:\mBbbN{}].
(r(-1)\_\mint{}\msupminus{}r1 Legendre(n;x) * Legendre(m;x) dx = if (n =\msubz{} m) then (r(2)/r((2 * n) + 1)) else r0 fi )
Date html generated:
2019_10_31-AM-06_19_00
Last ObjectModification:
2019_01_07-AM-11_11_36
Theory : reals_2
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