Nuprl Lemma : Legendre-zero-odd
∀n:ℕ. Legendre(n;r0) = r0 supposing (n rem 2) = 1 ∈ ℤ
Proof
Definitions occuring in Statement :
Legendre: Legendre(n;x),
req: x = y,
int-to-real: r(n),
nat: ℕ,
uimplies: b supposing a,
all: ∀x:A. B[x],
remainder: n rem m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
sq_type: SQType(T),
guard: {T},
eq_int: (i =z j),
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),
true: True,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
Legendre-rminus,
int-to-real_wf,
req_witness,
Legendre_wf,
istype-int,
set_subtype_base,
le_wf,
int_subtype_base,
istype-nat,
rminus_wf,
rmul_wf,
rnexp_wf,
eq_int_wf,
ifthenelse_wf,
btrue_wf,
real_wf,
bfalse_wf,
req_functionality,
req_weakening,
rmul_functionality,
rnexp-minus-one,
subtype_base_sq,
Legendre_functionality,
rminus-zero,
rmul_preserves_req,
rless-int,
rless_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-implies-req,
rsub_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
natural_numberEquality,
hypothesis,
independent_functionElimination,
equalityIstype,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
lambdaEquality_alt,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
minusEquality,
equalityTransitivity,
because_Cache,
inhabitedIsType,
unionElimination,
equalityElimination,
productElimination,
instantiate,
cumulativity,
inrFormation_alt,
independent_pairFormation,
imageMemberEquality,
universeIsType,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}n:\mBbbN{}. Legendre(n;r0) = r0 supposing (n rem 2) = 1
Date html generated:
2019_10_30-AM-11_33_43
Last ObjectModification:
2019_01_07-PM-03_12_26
Theory : reals_2
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