Nuprl Lemma : rpolynomial-complete-factors-ordered
∀n:ℕ+. ∀a:ℕn + 1 ⟶ ℝ. ∀z:ℕn ⟶ ℝ.
((∀j:ℕn - 1. ((z j) < (z (j + 1))))
⇒ ∀[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),
rless: 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],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
uimplies: b supposing a,
false: False,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
subtract: n - m,
nat: ℕ,
ge: i ≥ j ,
guard: {T},
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
rneq: x ≠ y,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
rpolynomial-complete-factors,
istype-int,
set_subtype_base,
lelt_wf,
int_subtype_base,
istype-void,
int_seg_wf,
subtract_wf,
rless_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
add-member-int_seg2,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
real_wf,
nat_plus_wf,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
primrec-wf2,
nat_wf,
less_than_wf,
istype-nat,
zero-add,
sq_stable__less_than,
squash_wf,
true_wf,
subtract-add-cancel,
subtype_base_sq,
add-associates,
equal_wf,
istype-universe,
add_functionality_wrt_eq,
add-comm,
subtype_rel_self,
iff_weakening_equal,
add-swap,
add-commutes,
rless_transitivity2,
rleq_weakening_rless,
minus-one-mul,
add-mul-special,
zero-mul,
add-zero,
int_seg_subtype_nat,
istype-false,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
functionIsType,
equalityIstype,
applyEquality,
isectElimination,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
inhabitedIsType,
universeIsType,
dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
because_Cache,
closedConclusion,
addEquality,
setIsType,
functionEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
hyp_replacement,
applyLambdaEquality,
instantiate,
cumulativity,
universeEquality,
multiplyEquality,
inrFormation_alt,
inlFormation_alt
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}. \mforall{}z:\mBbbN{}n {}\mrightarrow{} \mBbbR{}.
((\mforall{}j:\mBbbN{}n - 1. ((z j) < (z (j + 1))))
{}\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_55
Last ObjectModification:
2019_01_14-PM-11_49_41
Theory : reals
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