Nuprl Lemma : Legendre-differential-equation
∀n:ℕ
∃f,g:(-∞, ∞) ⟶ℝ
((∀x,y:ℝ. ((x = y) ⇒ ((f x) = (f y))))
∧ (∀x,y:ℝ. ((x = y) ⇒ ((g x) = (g y))))
∧ d(g[x])/dx = λx.f[x] on (-∞, ∞)
∧ d(Legendre(n;x))/dx = λx.g[x] on (-∞, ∞)
∧ (∀x:ℝ. (((((r1 - x * x) * (f x)) - (r(2) * x) * (g x)) + (r(n * (n + 1)) * Legendre(n;x))) = r0))
∧ (0 < n ⇒ (∀x:ℝ. (((r1 - x * x) * (g x)) = ((r(n) * Legendre(n - 1;x)) - (r(n) * x) * Legendre(n;x))))))
Proof
Definitions occuring in Statement :
Legendre: Legendre(n;x),
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
riiint: (-∞, ∞),
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
less_than: a < b,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
rfun: I ⟶ℝ,
cand: A c∧ B,
subtract: n - m,
true: True,
label: ...$L... t,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
Legendre: Legendre(n;x),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bnot: ¬bb,
assert: ↑b,
int_upper: {i...},
rneq: x ≠ y,
rdiv: (x/y),
rfun-eq: rfun-eq(I;f;g),
int-to-real: r(n),
r-ap: f(x),
eq_int: (i =z j),
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+
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,
int-to-real_wf,
real_wf,
i-member_wf,
riiint_wf,
req_weakening,
req_wf,
derivative-const,
Legendre_1_lemma,
derivative-id,
Legendre_0_lemma,
member_riiint_lemma,
true_wf,
derivative_wf,
Legendre_wf,
nat_properties,
radd_wf,
rsub_wf,
rmul_wf,
guard_wf,
exists_wf,
rfun_wf,
all_wf,
int_seg_subtype_nat,
istype-false,
less_than_wf,
primrec-wf2,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
itermMultiply_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
int-rdiv_wf,
nequal_wf,
int-rmul_wf,
int_term_value_mul_lemma,
derivative-int-rdiv,
derivative-sub,
derivative-int-rmul,
derivative-add,
derivative-mul-x,
req_functionality,
rsub_functionality,
int-rmul_functionality,
radd_functionality,
rmul_functionality,
int-rdiv_functionality,
Legendre_functionality,
bool_wf,
bool_subtype_base,
equal_wf,
squash_wf,
istype-universe,
eq_int_eq_false,
bfalse_wf,
iff_weakening_equal,
subtract-add-cancel,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
int-rmul-req,
req_transitivity,
req_inversion,
rmul-int,
rsub-int,
radd-int,
req-implies-req,
subtype_rel_sets_simple,
le_wf,
rdiv_wf,
rless-int,
int_upper_properties,
rless_wf,
int-rdiv-req,
rmul_preserves_req,
rinv_wf2,
rmul-rinv3,
derivative_unique,
iproper-riiint,
rdiv_functionality,
int-rinv-cancel,
rminus_wf,
itermMinus_wf,
rless_functionality,
rminus-int,
real_term_value_minus_lemma,
nat_plus_properties,
radd-preserves-req,
rmul-rinv,
rmul-zero,
rminus_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
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,
setIsType,
cumulativity,
intEquality,
functionIsType,
inhabitedIsType,
closedConclusion,
productEquality,
functionEquality,
multiplyEquality,
addEquality,
imageElimination,
equalityIstype,
baseApply,
baseClosed,
sqequalBase,
universeEquality,
imageMemberEquality,
equalityElimination,
promote_hyp,
inrFormation_alt,
minusEquality
Latex:
\mforall{}n:\mBbbN{}
\mexists{}f,g:(-\minfty{}, \minfty{}) {}\mrightarrow{}\mBbbR{}
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} ((f x) = (f y))))
\mwedge{} (\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} ((g x) = (g y))))
\mwedge{} d(g[x])/dx = \mlambda{}x.f[x] on (-\minfty{}, \minfty{})
\mwedge{} d(Legendre(n;x))/dx = \mlambda{}x.g[x] on (-\minfty{}, \minfty{})
\mwedge{} (\mforall{}x:\mBbbR{}
(((((r1 - x * x) * (f x)) - (r(2) * x) * (g x)) + (r(n * (n + 1)) * Legendre(n;x))) = r0))
\mwedge{} (0 < n
{}\mRightarrow{} (\mforall{}x:\mBbbR{}
(((r1 - x * x) * (g x)) = ((r(n) * Legendre(n - 1;x)) - (r(n) * x) * Legendre(n;x))))))
Date html generated:
2019_10_30-AM-11_33_24
Last ObjectModification:
2019_01_04-PM-01_34_24
Theory : reals_2
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