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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