Nuprl Lemma : Legendre-roots-lemma2
∀n:ℕ. ∀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_seg: {i..j-},
nat: ℕ,
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],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtract: n - m,
subtype_rel: A ⊆r B,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
int_upper: {i...},
sq_stable: SqStable(P),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
int-to-real: r(n),
rmul: a * b,
rnexp: x^k1,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
bfalse: ff,
canonical-bound: canonical-bound(r),
absval: |i|,
accelerate: accelerate(k;f),
fastexp: i^n,
efficient-exp-ext,
Legendre: Legendre(n;x),
int-rdiv: (a)/k1,
imax: imax(a;b),
rsub: x - y,
radd: a + b,
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
int-rmul: k1 * a,
btrue: tt,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
reg-seq-mul: reg-seq-mul(x;y),
rminus: -(x),
nil: [],
it: ⋅,
true: True,
real: ℝ
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
member_rooint_lemma,
Legendre_1_lemma,
Legendre_0_lemma,
int_seg_subtype_special,
int_seg_cases,
real_wf,
rless_wf,
int-to-real_wf,
decidable__le,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
decidable__lt,
istype-le,
istype-less_than,
nat_plus_properties,
req_wf,
Legendre-roots-lemma,
istype-nat,
sq_stable__rless,
rmul_wf,
rnexp_wf,
Legendre_wf,
rless_functionality,
req_weakening,
rmul_functionality,
Legendre_functionality,
efficient-exp-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
functionIsType,
hypothesis_subsumption,
setIsType,
productIsType,
minusEquality,
applyEquality,
dependent_set_memberEquality_alt,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberFormation_alt,
addEquality,
inhabitedIsType
Latex:
\mforall{}n:\mBbbN{}. \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_22
Last ObjectModification:
2019_01_18-PM-05_06_32
Theory : reals_2
Home
Index