Nuprl Lemma : Legendre-roots-ext
∀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 :
member: t ∈ T,
subtract: n - m,
Legendre-roots-sq,
decidable__equal_int,
decidable__int_equal,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a
Lemmas referenced :
Legendre-roots-sq,
lifting-strict-int_eq,
istype-void,
strict4-decide,
decidable__equal_int,
decidable__int_equal
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
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_57
Last ObjectModification:
2019_01_18-PM-10_10_11
Theory : reals_2
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