Nuprl Lemma : Legendre-rminus
∀n:ℕ. ∀x:ℝ. (Legendre(n;-(x)) = (r(-1)^n * Legendre(n;x)))
Proof
Definitions occuring in Statement :
Legendre: Legendre(n;x),
rnexp: x^k1,
req: x = y,
rmul: a * b,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
Legendre: Legendre(n;x),
le: A ≤ B,
less_than': less_than'(a;b),
nequal: a ≠ b ∈ T ,
int_upper: {i...},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
int_nzero: ℤ-o,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
nat_plus: ℕ+,
rdiv: (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,
rnexp_zero_lemma,
real_wf,
Legendre_1_lemma,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
Legendre_wf,
rminus_wf,
rmul_wf,
rnexp_wf,
int-to-real_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
itermMultiply_wf,
req-iff-rsub-is-0,
rminus-as-rmul,
int-rdiv_wf,
subtype_rel_sets_simple,
le_wf,
nequal_wf,
rsub_wf,
int-rmul_wf,
int_upper_properties,
rdiv_wf,
rless-int,
rless_wf,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
req_functionality,
req_weakening,
rmul_functionality,
rnexp1,
int-rdiv_functionality,
rsub_functionality,
int-rmul_functionality,
int-rdiv-req,
rnexp_step,
rdiv_functionality,
rmul_preserves_req,
radd_wf,
rinv_wf2,
itermMinus_wf,
int-rmul-req,
req_transitivity,
radd_functionality,
rmul-rinv3,
rminus_functionality,
req_inversion,
rsub-int,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
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,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
cumulativity,
intEquality,
equalityElimination,
equalityIstype,
promote_hyp,
minusEquality,
addEquality,
baseClosed,
sqequalBase,
multiplyEquality,
closedConclusion,
inrFormation_alt
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x:\mBbbR{}. (Legendre(n;-(x)) = (r(-1)\^{}n * Legendre(n;x)))
Date html generated:
2019_10_30-AM-11_33_36
Last ObjectModification:
2019_01_04-PM-02_55_08
Theory : reals_2
Home
Index