Nuprl Lemma : Legendre-deriv-equation1
∀n:ℕ+
∃g:(-∞, ∞) ⟶ℝ
((∀x,y:ℝ. ((x = y) ⇒ ((g x) = (g y))))
∧ d(Legendre(n;x))/dx = λx.g[x] on (-∞, ∞)
∧ (∀x:ℝ. ((x * Legendre(n;x)) = (Legendre(n - 1;x) + ((x^2 - r1/r(n)) * (g x))))))
Proof
Definitions occuring in Statement :
Legendre: Legendre(n;x),
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
riiint: (-∞, ∞),
rdiv: (x/y),
rnexp: x^k1,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
nat: ℕ,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rfun: I ⟶ℝ,
true: True,
so_lambda: λ2x.t[x],
label: ...$L... t,
so_apply: x[s],
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
Legendre-differential-equation,
nat_plus_subtype_nat,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rmul_preserves_req,
rmul_wf,
Legendre_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
radd_wf,
rdiv_wf,
rsub_wf,
rnexp_wf,
rless-int,
real_wf,
req_wf,
member_riiint_lemma,
true_wf,
derivative_wf,
riiint_wf,
i-member_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
int-to-real_wf,
rless_wf,
nat_plus_wf,
rminus_wf,
itermMultiply_wf,
rinv_wf2,
itermAdd_wf,
itermMinus_wf,
req-implies-req,
req-iff-rsub-is-0,
req_functionality,
req_transitivity,
radd_functionality,
rmul_functionality,
req_weakening,
rmul-rinv3,
rminus_functionality,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
rnexp2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
productElimination,
dependent_pairFormation_alt,
independent_pairFormation,
independent_functionElimination,
isectElimination,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
dependent_set_memberEquality_alt,
because_Cache,
inrFormation_alt,
productIsType,
functionIsType,
setIsType,
closedConclusion
Latex:
\mforall{}n:\mBbbN{}\msupplus{}
\mexists{}g:(-\minfty{}, \minfty{}) {}\mrightarrow{}\mBbbR{}
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} ((g x) = (g y))))
\mwedge{} d(Legendre(n;x))/dx = \mlambda{}x.g[x] on (-\minfty{}, \minfty{})
\mwedge{} (\mforall{}x:\mBbbR{}. ((x * Legendre(n;x)) = (Legendre(n - 1;x) + ((x\^{}2 - r1/r(n)) * (g x))))))
Date html generated:
2019_10_30-AM-11_33_50
Last ObjectModification:
2019_01_04-PM-04_59_54
Theory : reals_2
Home
Index