Nuprl Lemma : power-sum_functionality_wrt_le
∀m:ℤ. ∀n,x:ℕ. ∀a,b:ℕn ⟶ ℤ.  ((∀i:ℕn. (a[i] ≤ b[i])) 
⇒ (Σi<n.a[i]*x^i ≤ Σi<n.b[i]*x^i))
Proof
Definitions occuring in Statement : 
power-sum: Σi<n.a[i]*x^i
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
so_apply: x[s]
, 
le: A ≤ B
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
natural_number: $n
, 
int: ℤ
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
power-sum: Σi<n.a[i]*x^i
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
uimplies: b supposing a
, 
le: A ≤ B
, 
and: P ∧ Q
, 
less_than': less_than'(a;b)
, 
false: False
, 
not: ¬A
, 
prop: ℙ
, 
guard: {T}
, 
int_seg: {i..j-}
, 
ge: i ≥ j 
, 
lelt: i ≤ j < k
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
uiff: uiff(P;Q)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
Lemmas referenced : 
int_formula_prop_wf, 
int_term_value_var_lemma, 
int_term_value_mul_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
itermVar_wf, 
itermMultiply_wf, 
intformle_wf, 
intformnot_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
multiply-is-int-iff, 
decidable__le, 
nat_properties, 
int_seg_properties, 
exp_wf4, 
mul_preserves_le, 
nat_wf, 
le_wf, 
all_wf, 
int_seg_wf, 
false_wf, 
int_seg_subtype_nat, 
exp_wf2, 
sum_le
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
multiplyEquality, 
applyEquality, 
natural_numberEquality, 
setElimination, 
rename, 
independent_isectElimination, 
independent_pairFormation, 
hypothesis, 
because_Cache, 
functionEquality, 
intEquality, 
productElimination, 
dependent_functionElimination, 
dependent_set_memberEquality, 
unionElimination, 
pointwiseFunctionality, 
equalityTransitivity, 
equalitySymmetry, 
promote_hyp, 
baseApply, 
closedConclusion, 
baseClosed, 
dependent_pairFormation, 
int_eqEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
computeAll
Latex:
\mforall{}m:\mBbbZ{}.  \mforall{}n,x:\mBbbN{}.  \mforall{}a,b:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}.    ((\mforall{}i:\mBbbN{}n.  (a[i]  \mleq{}  b[i]))  {}\mRightarrow{}  (\mSigma{}i<n.a[i]*x\^{}i  \mleq{}  \mSigma{}i<n.b[i]*x\^{}i))
Date html generated:
2016_05_15-PM-06_28_41
Last ObjectModification:
2016_01_16-AM-09_57_44
Theory : general
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