Nuprl Lemma : Legendre-roots-unique
∀[n:ℕ]. ∀[z:ℕn ⟶ ℝ].
(∀[i:ℕn]. ((z i) = Legendre-root(n;i))) supposing
((∀i:ℕn. (Legendre(n;z i) = r0)) and
(∀i:ℕn - 1. ((z i) < (z (i + 1)))))
Proof
Definitions occuring in Statement :
Legendre-root: Legendre-root(n;i),
Legendre: Legendre(n;x),
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
top: Top,
implies: P ⇒ Q,
nat: ℕ,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
uiff: uiff(P;Q),
subtract: n - m,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
Legendre-rpolynomial,
req_witness,
Legendre-root_wf,
member_rooint_lemma,
istype-void,
int_seg_wf,
req_wf,
Legendre_wf,
int-to-real_wf,
subtract_wf,
rless_wf,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
add-member-int_seg2,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
real_wf,
istype-nat,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
rdiv_wf,
doublefact_wf,
fact_wf,
rless-int,
nat_plus_properties,
rless-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
rless_functionality,
req_weakening,
rpolynomial_wf,
rpolynomial-complete-roots-unique,
iff_weakening_uiff,
req_functionality,
Legendre-roots-rless,
sq_stable__less_than,
sq_stable__req,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
applyEquality,
hypothesis,
lambdaEquality_alt,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
natural_numberEquality,
because_Cache,
isectIsTypeImplies,
functionIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
productIsType,
closedConclusion,
addEquality,
multiplyEquality,
inrFormation_alt,
applyLambdaEquality,
lambdaFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIstype
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[z:\mBbbN{}n {}\mrightarrow{} \mBbbR{}].
(\mforall{}[i:\mBbbN{}n]. ((z i) = Legendre-root(n;i))) supposing
((\mforall{}i:\mBbbN{}n. (Legendre(n;z i) = r0)) and
(\mforall{}i:\mBbbN{}n - 1. ((z i) < (z (i + 1)))))
Date html generated:
2019_10_31-AM-06_20_23
Last ObjectModification:
2019_01_19-PM-01_10_28
Theory : reals_2
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