Nuprl Lemma : power-sum-split
∀[n:ℕ]. ∀[k:ℕn + 1]. ∀[x:ℤ]. ∀[a:ℕn ⟶ ℤ]. (Σi<n.a[i]*x^i = (Σi<k.a[i]*x^i + (x^k * Σi<n - k.a[k + i]*x^i)) ∈ ℤ)
Proof
Definitions occuring in Statement :
power-sum: Σi<n.a[i]*x^i,
exp: i^n,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
power-sum: Σi<n.a[i]*x^i,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
sq_type: SQType(T)
Lemmas referenced :
sum_split,
exp_wf2,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
int_seg_wf,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
sum_scalar_mult,
subtract_wf,
itermSubtract_wf,
intformless_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
decidable__lt,
istype-less_than,
sum_wf,
subtype_rel_self,
iff_weakening_equal,
mul-commutes,
mul-swap,
add-commutes,
exp_add,
int_seg_subtype_nat,
istype-false,
mul_assoc,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
multiplyEquality,
applyEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesis,
productElimination,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
addEquality,
productIsType,
imageMemberEquality,
baseClosed,
functionIsType,
lambdaFormation_alt,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[k:\mBbbN{}n + 1]. \mforall{}[x:\mBbbZ{}]. \mforall{}[a:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}].
(\mSigma{}i<n.a[i]*x\^{}i = (\mSigma{}i<k.a[i]*x\^{}i + (x\^{}k * \mSigma{}i<n - k.a[k + i]*x\^{}i)))
Date html generated:
2020_05_20-AM-08_16_09
Last ObjectModification:
2019_12_26-PM-04_06_00
Theory : general
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