Nuprl Lemma : shift-rpolynomial
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x:ℝ].
((x * (Σi≤n. a_i * x^i)) = (Σi≤n + 1. λi.if (i =z 0) then r0 else a (i - 1) fi _i * x^i))
Proof
Definitions occuring in Statement :
rpolynomial: (Σi≤n. a_i * x^i),
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
eq_int: (i =z j),
subtract: n - m,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
rpolynomial: (Σi≤n. a_i * x^i),
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
uiff: uiff(P;Q),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
prop: ℙ,
and: P ∧ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
nat_plus: ℕ+
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}].
((x * (\mSigma{}i\mleq{}n. a\_i * x\^{}i)) = (\mSigma{}i\mleq{}n + 1. \mlambda{}i.if (i =\msubz{} 0) then r0 else a (i - 1) fi \_i * x\^{}i))
Date html generated:
2020_05_20-AM-11_12_01
Last ObjectModification:
2019_12_28-AM-11_35_08
Theory : reals
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