Nuprl Lemma : Legendre-root_wf
∀[n:ℕ]. ∀[i:ℕn].
(Legendre-root(n;i) ∈ {z:ℝ| (z ∈ (r(-1), r1)) ∧ (Legendre(n;z) = r0) ∧ (r0 < (r((-1)^(n - i)) * Legendre(n + 1;z)))} )
Proof
Definitions occuring in Statement :
Legendre-root: Legendre-root(n;i),
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,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
subtract: n - m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Legendre-root: Legendre-root(n;i),
Legendre-roots-ext,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
uimplies: b supposing a,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtract: n - m,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
Legendre-roots-ext,
subtype_rel_self,
sq_exists_wf,
rless_wf,
rmul_wf,
int-to-real_wf,
exp_wf2,
Legendre_wf,
all_wf,
add-member-int_seg2,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
subtract_wf,
istype-less_than,
int_seg_wf,
real_wf,
i-member_wf,
rooint_wf,
req_wf,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
itermAdd_wf,
int_term_value_add_lemma,
member_rooint_lemma,
sq_stable__i-member,
sq_stable__req,
sq_stable__rless,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
introduction,
sqequalHypSubstitution,
isectElimination,
functionEquality,
because_Cache,
setEquality,
productEquality,
lambdaEquality_alt,
setElimination,
rename,
dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
independent_isectElimination,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
functionIsType,
setIsType,
minusEquality,
addEquality,
inhabitedIsType,
lambdaFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[i:\mBbbN{}n].
(Legendre-root(n;i) \mmember{} \{z:\mBbbR{}|
(z \mmember{} (r(-1), r1))
\mwedge{} (Legendre(n;z) = r0)
\mwedge{} (r0 < (r((-1)\^{}(n - i)) * Legendre(n + 1;z)))\} )
Date html generated:
2019_10_31-AM-06_20_05
Last ObjectModification:
2019_01_18-PM-10_11_57
Theory : reals_2
Home
Index