Nuprl Lemma : rpolynomial-complete-factors
∀n:ℕ+. ∀a:ℕn + 1 ⟶ ℝ. ∀z:ℕn ⟶ ℝ.
((∀i,j:ℕn. ((¬(i = j ∈ ℤ)) ⇒ z i ≠ z j))
⇒ ∀[x:ℝ]. ((Σi≤n. a_i * x^i) = ((a n) * rprod(0;n - 1;j.x - z j))) supposing ∀j:ℕn. ((Σi≤n. a_i * z j^i) = r0))
Proof
Definitions occuring in Statement :
rprod: rprod(n;m;k.x[k]),
rpolynomial: (Σi≤n. a_i * x^i),
rneq: x ≠ y,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rprod: rprod(n;m;k.x[k]),
rpolynomial: (Σi≤n. a_i * x^i),
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
bfalse: ff,
le: A ≤ B,
less_than': less_than'(a;b),
btrue: tt,
req_int_terms: t1 ≡ t2,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
rneq: x ≠ y,
guard: {T},
sq_type: SQType(T),
rless: x < y,
sq_exists: ∃x:A [B[x]],
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
req_witness,
rpolynomial_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
rmul_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
rprod_wf,
rsub_wf,
subtract-add-cancel,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_seg_wf,
subtract_wf,
real_wf,
req_wf,
int_seg_properties,
int-to-real_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
itermAdd_wf,
int_term_value_add_lemma,
add-subtract-cancel,
rneq_wf,
primrec-wf-nat-plus,
all_wf,
not_wf,
equal-wf-base,
nat_plus_subtype_nat,
uall_wf,
nat_plus_wf,
rpolynomial-linear-factor,
itermSubtract_wf,
int_term_value_subtract_lemma,
req_functionality,
req_weakening,
rsum_wf,
rnexp_wf,
int_seg_subtype_nat,
istype-false,
rnexp_zero_lemma,
rmul_functionality,
rsum-single,
itermMultiply_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
subtype_rel_function,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
subtype_rel_self,
subtype_base_sq,
radd-preserves-rless,
rless_wf,
radd_wf,
rless_functionality,
real_term_value_add_lemma,
rmul_preserves_req,
req_inversion,
le-add-cancel2,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
isect_memberFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
natural_numberEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
applyEquality,
independent_pairFormation,
productIsType,
closedConclusion,
productElimination,
because_Cache,
int_eqEquality,
addEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
equalityIstype,
intEquality,
sqequalBase,
equalitySymmetry,
isectIsType,
functionEquality,
isectEquality,
minusEquality,
setIsType,
baseApply,
baseClosed,
multiplyEquality,
instantiate,
equalityTransitivity,
inlFormation_alt,
inrFormation_alt,
equalityElimination,
promote_hyp,
cumulativity
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}. \mforall{}z:\mBbbN{}n {}\mrightarrow{} \mBbbR{}.
((\mforall{}i,j:\mBbbN{}n. ((\mneg{}(i = j)) {}\mRightarrow{} z i \mneq{} z j))
{}\mRightarrow{} \mforall{}[x:\mBbbR{}]. ((\mSigma{}i\mleq{}n. a\_i * x\^{}i) = ((a n) * rprod(0;n - 1;j.x - z j)))
supposing \mforall{}j:\mBbbN{}n. ((\mSigma{}i\mleq{}n. a\_i * z j\^{}i) = r0))
Date html generated:
2019_10_29-AM-10_20_39
Last ObjectModification:
2019_01_14-PM-11_17_37
Theory : reals
Home
Index