Nuprl Lemma : Legendre-roots-symmetric
∀[n:ℕ]. ∀[x:ℝ]. uiff(Legendre(n;-(x)) = r0;Legendre(n;x) = r0)
Proof
Definitions occuring in Statement :
Legendre: Legendre(n;x),
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
nat: ℕ,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
req_witness,
Legendre_wf,
int-to-real_wf,
req_wf,
rminus_wf,
real_wf,
istype-nat,
rmul_wf,
rnexp_wf,
req_functionality,
Legendre-rminus,
req_weakening,
rmul_preserves_req,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
req-implies-req,
rsub_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
subtype_base_sq,
int_subtype_base,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
rless-int,
rless_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rneq_functionality,
rnexp-minus-one,
rmul-zero,
rmul_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_functionElimination,
universeIsType,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
minusEquality,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
voidElimination,
remainderEquality,
setElimination,
rename,
closedConclusion,
lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
baseClosed,
sqequalBase,
unionElimination,
equalityElimination,
inrFormation_alt,
imageMemberEquality,
dependent_pairFormation_alt,
promote_hyp,
inlFormation_alt
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}]. uiff(Legendre(n;-(x)) = r0;Legendre(n;x) = r0)
Date html generated:
2019_10_30-AM-11_33_47
Last ObjectModification:
2019_01_07-PM-03_17_18
Theory : reals_2
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