Nuprl Lemma : Legendre-roots-lemma
∀n:{2...}. ∀z:ℕn - 1 ⟶ {x:ℝ| x ∈ (r(-1), r1)} .
((∀i:ℕn - 1. (Legendre(n - 1;z i) = r0))
⇒ (∀i:ℕn - 2. ((z i) < (z (i + 1))))
⇒ (∀i:ℕn. ∀v:{v:ℝ| (v ∈ (if i=0 then r(-1) else (z (i - 1)), if i=n - 1 then r1 else (z i))) ∧ (Legendre(n;v) = r0)} \000C.
(r0 < (r(-1)^n - i * Legendre(n + 1;v)))))
Proof
Definitions occuring in Statement :
Legendre: Legendre(n;x),
rooint: (l, u),
i-member: r ∈ I,
rless: x < y,
rnexp: x^k1,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
int_eq: if a=b then c else d,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
int_upper: {i...},
int_seg: {i..j-},
and: P ∧ Q,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
sq_stable: SqStable(P),
Legendre: Legendre(n;x),
squash: ↓T,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
subtract: n - m,
int_nzero: ℤ-o,
le: A ≤ B,
less_than': less_than'(a;b),
rneq: x ≠ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
rless: x < y,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sequence: sequence(T),
trans: Trans(T;x,y.E[x; y]),
seq-item: s[i],
seq-len: ||s||,
pi1: fst(t),
pi2: snd(t),
sorted-seq: sorted-seq(x,y.R[x; y];s),
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
real: ℝ,
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
eq_int: (i =z j),
less_than: a < b
Lemmas referenced :
sq_stable__rless,
int-to-real_wf,
rmul_wf,
rnexp_wf,
subtract_wf,
int_seg_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
Legendre_wf,
itermAdd_wf,
int_term_value_add_lemma,
subtype_base_sq,
bool_wf,
bool_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
eq_int_eq_false,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
bfalse_wf,
subtype_rel_self,
iff_weakening_equal,
add-subtract-cancel,
decidable__equal_int,
real_wf,
i-member_wf,
rooint_wf,
decidable__lt,
istype-less_than,
member_rooint_lemma,
req_wf,
int_seg_wf,
rless_wf,
add-member-int_seg2,
istype-int_upper,
int-rdiv_wf,
nequal_wf,
rsub_wf,
int-rmul_wf,
upper_subtype_nat,
istype-false,
rdiv_wf,
rless-int,
rmul_preserves_rless,
rminus_wf,
itermMultiply_wf,
rinv_wf2,
itermMinus_wf,
rless_functionality,
req_weakening,
rmul_functionality,
int-rdiv_functionality,
rsub_functionality,
int-rmul_functionality,
req_transitivity,
int-rdiv-req,
rdiv_functionality,
int-rmul-req,
rminus_functionality,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
Legendre-rpolynomial,
rpolynomial-complete-factors-ordered,
req_witness,
doublefact_wf,
fact_wf,
nat_plus_properties,
rprod_wf,
subtract-add-cancel,
rpolynomial_wf,
req_functionality,
req_inversion,
rless-implies-rless,
sorted-seq-iff,
rless_transitivity2,
rleq_weakening_rless,
set_subtype_base,
lelt_wf,
rleq_weakening_equal,
rprod-split,
rprod-of-positive,
radd-preserves-rless,
radd_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
sq_stable__less_than,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
real_term_value_add_lemma,
rprod-rsub-symmetry,
le_wf,
rless_transitivity1,
general_arith_equation2,
nat_properties,
int_term_value_mul_lemma,
exp_wf2,
rmul_assoc,
rnexp-add,
rnexp-int,
exp-minusone,
ifthenelse_wf,
mod2-add1,
mod2-2n,
rprod-empty,
rnexp1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
hypothesis,
dependent_set_memberEquality_alt,
because_Cache,
hypothesisEquality,
productElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
minusEquality,
addEquality,
instantiate,
cumulativity,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
universeEquality,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
imageMemberEquality,
intEquality,
setIsType,
productIsType,
functionIsType,
multiplyEquality,
inrFormation_alt,
isect_memberFormation_alt,
functionExtensionality,
applyLambdaEquality,
dependent_pairEquality_alt,
int_eqReduceFalseSq,
equalityElimination,
int_eqReduceTrueSq,
promote_hyp
Latex:
\mforall{}n:\{2...\}. \mforall{}z:\mBbbN{}n - 1 {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} .
((\mforall{}i:\mBbbN{}n - 1. (Legendre(n - 1;z i) = r0))
{}\mRightarrow{} (\mforall{}i:\mBbbN{}n - 2. ((z i) < (z (i + 1))))
{}\mRightarrow{} (\mforall{}i:\mBbbN{}n. \mforall{}v:\{v:\mBbbR{}|
(v \mmember{} (if i=0 then r(-1) else (z (i - 1)), if i=n - 1 then r1 else (z i)))
\mwedge{} (Legendre(n;v) = r0)\} .
(r0 < (r(-1)\^{}n - i * Legendre(n + 1;v)))))
Date html generated:
2019_10_31-AM-06_19_13
Last ObjectModification:
2019_01_18-PM-04_53_40
Theory : reals_2
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