Nuprl Lemma : rpolydiv-property
∀[n:ℕ+]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[z,x:ℝ].
((Σi≤n. a_i * x^i) = (((x - z) * (Σi≤n - 1. rpolydiv(n;a;z)_i * x^i)) + (Σi≤n. a_i * z^i)))
Proof
Definitions occuring in Statement :
rpolydiv: rpolydiv(n;a;z),
rpolynomial: (Σi≤n. a_i * x^i),
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: 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,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
sq_stable: SqStable(P),
squash: ↓T,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
lelt: i ≤ j < k,
less_than: a < b,
rpolynomial: (Σi≤n. a_i * x^i),
so_lambda: λ2x.t[x],
so_apply: x[s],
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
rpolydiv: rpolydiv(n;a;z),
eq_int: (i =z j),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
top: Top,
ge: i ≥ j
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[z,x:\mBbbR{}].
((\mSigma{}i\mleq{}n. a\_i * x\^{}i) = (((x - z) * (\mSigma{}i\mleq{}n - 1. rpolydiv(n;a;z)\_i * x\^{}i)) + (\mSigma{}i\mleq{}n. a\_i * z\^{}i)))
Date html generated:
2020_05_20-AM-11_12_19
Last ObjectModification:
2020_01_02-PM-01_23_04
Theory : reals
Home
Index