Nuprl Lemma : Legendre-roots-sq
∀n:ℕ
(∃z:i:ℕn ⟶ {x:ℝ| (x ∈ (r(-1), r1)) ∧ (Legendre(n;x) = r0) ∧ (r0 < (r((-1)^(n - i)) * Legendre(n + 1;x)))}
[(∀i:ℕn - 1. ((z i) < (z (i + 1))))])
Proof
Definitions occuring in Statement :
Legendre: Legendre(n;x),
rooint: (l, u),
i-member: r ∈ I,
rless: x < y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
exp: i^n,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
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],
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
guard: {T},
subtract: n - m,
so_lambda: λ2x.t[x],
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
sq_type: SQType(T),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
rless: x < y,
nat_plus: ℕ+,
real: ℝ,
int-to-real: r(n),
rmul: a * b,
Legendre: Legendre(n;x),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
bfalse: ff,
int-rdiv: (a)/k1,
accelerate: accelerate(k;f),
imax: imax(a;b),
canonical-bound: canonical-bound(r),
absval: |i|,
exp: i^n,
primrec: primrec(n;b;c),
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: ⋅,
nat: ℕ,
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
bool: 𝔹,
unit: Unit,
assert: ↑b,
nequal: a ≠ b ∈ T ,
req_int_terms: t1 ≡ t2
Lemmas referenced :
member_rooint_lemma,
istype-void,
Legendre_0_lemma,
Legendre_1_lemma,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
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,
sq_stable__all,
rless_wf,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
istype-less_than,
add-member-int_seg2,
decidable__le,
sq_stable__rless,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int-to-real_wf,
rless-int,
req_weakening,
set_subtype_base,
lelt_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
sq_stable__less_than,
rmul_wf,
exp_wf2,
Legendre_wf,
req_wf,
real_wf,
i-member_wf,
rooint_wf,
itermAdd_wf,
int_term_value_add_lemma,
primrec-wf2,
sq_exists_wf,
nat_properties,
all_wf,
istype-nat,
rational_fun_zero_wf,
ratLegendre_wf,
nat_plus_wf,
rccint_wf,
req_functionality,
Legendre_functionality,
ratreal_wf,
ratreal-ratLegendre,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
bool_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
subtract-add-cancel,
two-mul,
itermMultiply_wf,
int_term_value_mul_lemma,
rless_functionality,
rmul_functionality,
Legendre-minus-1,
req_transitivity,
rmul-int,
squash_wf,
true_wf,
exp_add,
exp-minusone,
ifthenelse_wf,
istype-universe,
mod2-2n,
subtype_rel_self,
iff_weakening_equal,
exp1,
Legendre-1,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
general_arith_equation2,
rmul_reverses_rless_iff,
rless-implies-rless,
rsub_wf,
req_inversion,
real_term_value_var_lemma,
rmul_preserves_rless,
rmul_assoc,
Legendre-roots-lemma2,
subtype_rel_sets_simple,
sq_stable__req,
rnexp_wf,
rnexp-int,
rleq_weakening_equal,
sq_stable__rleq,
rleq_weakening_rless,
rless_transitivity2,
rless_transitivity1,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
dependent_set_memberFormation_alt,
lambdaEquality_alt,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesis,
isectElimination,
natural_numberEquality,
hypothesisEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
universeIsType,
functionIsType,
because_Cache,
applyEquality,
dependent_set_memberEquality_alt,
unionElimination,
productIsType,
closedConclusion,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity,
intEquality,
minusEquality,
addEquality,
setIsType,
functionEquality,
setEquality,
productEquality,
isect_memberFormation_alt,
equalityElimination,
int_eqReduceTrueSq,
equalityIstype,
promote_hyp,
int_eqReduceFalseSq,
multiplyEquality,
universeEquality,
functionExtensionality,
baseApply,
sqequalBase
Latex:
\mforall{}n:\mBbbN{}
(\mexists{}z:i:\mBbbN{}n {}\mrightarrow{} \{x:\mBbbR{}|
(x \mmember{} (r(-1), r1))
\mwedge{} (Legendre(n;x) = r0)
\mwedge{} (r0 < (r((-1)\^{}(n - i)) * Legendre(n + 1;x)))\} [(\mforall{}i:\mBbbN{}n - 1. ((z i) < (z (i + 1))))]\000C)
Date html generated:
2019_10_31-AM-06_19_41
Last ObjectModification:
2019_01_18-PM-10_09_02
Theory : reals_2
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