Nuprl Lemma : rpolynomial-composition1
∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ. ∀b,c,d:ℝ. ∃a':ℕn + 1 ⟶ ℝ. ∀x:ℝ. ((Σi≤n. a'_i * x^i) = ((Σi≤n. a_i * ((c - b) * x) + b^i) - d))
Proof
Definitions occuring in Statement :
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: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
rpolynomial: (Σi≤n. a_i * x^i),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
prop: ℙ,
so_apply: x[s],
ge: i ≥ j ,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
subtype_rel: A ⊆r B,
cand: A c∧ B,
subtract: n - m,
true: True,
nat_plus: ℕ+,
guard: {T},
int_iseg: {i...j},
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}. \mforall{}b,c,d:\mBbbR{}.
\mexists{}a':\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}. \mforall{}x:\mBbbR{}. ((\mSigma{}i\mleq{}n. a'\_i * x\^{}i) = ((\mSigma{}i\mleq{}n. a\_i * ((c - b) * x) + b\^{}i) - d))
Date html generated:
2020_05_20-AM-11_13_55
Last ObjectModification:
2020_01_02-PM-02_01_05
Theory : reals
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