Nuprl Lemma : rpolynomial-linear-factor
∀n:ℕ+. ∀a:ℕn + 1 ⟶ ℝ. ∀z:ℝ.
∃b:ℕn ⟶ ℝ. ((∀[x:ℝ]. ((Σi≤n. a_i * x^i) = ((x - z) * (Σi≤n - 1. b_i * x^i)))) ∧ ((b (n - 1)) = (a n)))
supposing (Σi≤n. a_i * z^i) = r0
Proof
Definitions occuring in Statement :
rpolynomial: (Σi≤n. a_i * x^i),
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],
exists: ∃x:A. B[x],
and: 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],
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,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
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,
int_seg: {i..j-},
lelt: i ≤ j < k,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rpolynomial_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
int-to-real_wf,
rpolydiv_wf,
rmul_wf,
rsub_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_rel_function,
int_seg_wf,
int_seg_subtype,
istype-false,
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,
req_wf,
decidable__lt,
istype-less_than,
itermAdd_wf,
int_term_value_add_lemma,
real_wf,
nat_plus_wf,
rpolydiv-property,
radd_wf,
radd-zero,
req_functionality,
req_transitivity,
radd_functionality,
req_weakening,
rpolydiv-rec
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
because_Cache,
applyEquality,
addEquality,
productElimination,
inhabitedIsType,
minusEquality,
multiplyEquality,
productIsType,
isectIsType,
equalityTransitivity,
equalitySymmetry,
functionIsType,
equalityIstype
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}. \mforall{}z:\mBbbR{}.
\mexists{}b:\mBbbN{}n {}\mrightarrow{} \mBbbR{}
((\mforall{}[x:\mBbbR{}]. ((\mSigma{}i\mleq{}n. a\_i * x\^{}i) = ((x - z) * (\mSigma{}i\mleq{}n - 1. b\_i * x\^{}i)))) \mwedge{} ((b (n - 1)) = (a n)))
supposing (\mSigma{}i\mleq{}n. a\_i * z\^{}i) = r0
Date html generated:
2019_10_29-AM-10_16_01
Last ObjectModification:
2019_01_14-PM-10_25_20
Theory : reals
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