Nuprl Lemma : Legendre-rpolynomial
∀n:ℕ. ∃a:ℕn + 1 ⟶ ℝ. ((∀x:ℝ. (Legendre(n;x) = (Σi≤n. a_i * x^i))) ∧ ((a n) = (r(doublefact((2 * n) - 1))/r((n)!))))
Proof
Definitions occuring in Statement :
Legendre: Legendre(n;x),
rpolynomial: (Σi≤n. a_i * x^i),
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
real: ℝ,
doublefact: doublefact(n),
fact: (n)!,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
subtract: n - m,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
nat_plus: ℕ+,
fact: (n)!,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
Legendre: Legendre(n;x),
nequal: a ≠ b ∈ T ,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
int_upper: {i...},
int_nzero: ℤ-o,
bnot: ¬bb,
assert: ↑b,
doublefact: doublefact(n),
lt_int: i <z j,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
rtermMinus: rtermMinus(num),
pi1: fst(t),
rtermSubtract: left "-" right,
rtermConstant: "const",
pi2: snd(t)
Lemmas referenced :
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
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,
istype-less_than,
subtype_rel_self,
Legendre_0_lemma,
fact0_redex_lemma,
int-to-real_wf,
real_wf,
req_wf,
rpolynomial_wf,
nat_properties,
rdiv_wf,
doublefact_wf,
rless-int,
rless_wf,
Legendre_1_lemma,
ifthenelse_wf,
eq_int_wf,
nat_plus_wf,
less_than_wf,
fact_wf,
bool_wf,
bool_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
eq_int_eq_false,
bfalse_wf,
iff_weakening_equal,
eqtt_to_assert,
assert_of_eq_int,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
int-rdiv_wf,
subtype_rel_sets_simple,
le_wf,
nequal_wf,
int_upper_properties,
int-rmul_wf,
subtract-add-cancel,
itermAdd_wf,
int_term_value_add_lemma,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
upper_subtype_upper,
rsub_wf,
iff_weakening_uiff,
assert_wf,
rmul_wf,
Legendre_wf,
rneq-int,
fact-non-zero,
guard_wf,
exists_wf,
all_wf,
int_seg_subtype_nat,
primrec-wf2,
nat_plus_properties,
istype-nat,
rnexp_zero_lemma,
req_weakening,
rinv_wf2,
itermMultiply_wf,
req-int,
radd_wf,
rnexp_wf,
req-iff-rsub-is-0,
req_functionality,
rpolynomial_unroll,
req_transitivity,
rmul_functionality,
rinv1,
rmul-identity1,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
radd_functionality,
rnexp1,
real_term_value_add_lemma,
int-rdiv-req,
rdiv_functionality,
rsub_functionality,
int-rmul-req,
add-associates,
add-swap,
add-commutes,
subtype_rel_function,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-one-mul,
add-mul-special,
zero-mul,
add-zero,
minus-add,
minus-minus,
minus-one-mul-top,
le-add-cancel-alt,
le_int_wf,
assert_of_le_int,
req_inversion,
rsub-rdiv,
uiff_transitivity,
shift-rpolynomial,
rmul-rpolynomial,
rdiv-rpolynomial,
subtract-rpolynomials,
rpolynomial_functionality,
rminus_wf,
itermMinus_wf,
assert-rat-term-eq2,
rtermSubtract_wf,
rtermDivide_wf,
rtermMultiply_wf,
rtermVar_wf,
rtermConstant_wf,
rtermMinus_wf,
rsub-int,
rminus-int,
radd-int,
rminus_functionality,
real_term_value_minus_lemma,
rmul_preserves_req,
rmul-rinv,
rmul-int,
mul_bounds_1b,
int_term_value_mul_lemma,
fact_unroll,
rneq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
unionElimination,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
cumulativity,
intEquality,
imageElimination,
functionIsType,
closedConclusion,
minusEquality,
inrFormation_alt,
imageMemberEquality,
baseClosed,
inhabitedIsType,
universeEquality,
equalityElimination,
equalityIstype,
sqequalBase,
addEquality,
promote_hyp,
multiplyEquality,
functionEquality,
productEquality,
setIsType
Latex:
\mforall{}n:\mBbbN{}
\mexists{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}
((\mforall{}x:\mBbbR{}. (Legendre(n;x) = (\mSigma{}i\mleq{}n. a\_i * x\^{}i))) \mwedge{} ((a n) = (r(doublefact((2 * n) - 1))/r((n)!))))
Date html generated:
2019_10_30-AM-11_33_59
Last ObjectModification:
2019_04_09-PM-05_09_40
Theory : reals_2
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