Nuprl Lemma : power-sum-product
∀[x:ℤ]. ∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℤ]. ∀[m:ℕ]. ∀[b:ℕm + 1 ⟶ ℤ].
((Σi<n + 1.a[i]*x^i * Σi<m + 1.b[i]*x^i)
= Σi<(n + m) + 1.Σ(if j ≤z n then a[j] else 0 fi * if i - j ≤z m then b[i - j] else 0 fi | j < i + 1)*x^i
∈ ℤ)
Proof
Definitions occuring in Statement :
power-sum: Σi<n.a[i]*x^i,
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
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 :
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
power-sum: Σi<n.a[i]*x^i,
sq_type: SQType(T),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
le_int: i ≤z j,
lt_int: i <z j,
subtract: n - m,
nat_plus: ℕ+,
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_wf,
subtract-1-ge-0,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_wf,
subtype_base_sq,
int_subtype_base,
sum1,
decidable__lt,
intformnot_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
lelt_wf,
exp_wf2,
int_seg_subtype_nat,
false_wf,
exp0_lemma,
mul-one,
equal_wf,
squash_wf,
true_wf,
sum_scalar_mult,
decidable__le,
le_wf,
sum_wf,
subtype_rel_self,
iff_weakening_equal,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_properties,
itermMultiply_wf,
int_term_value_mul_lemma,
singleton_support_sum,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
not_wf,
equal-wf-T-base,
minus-zero,
add-zero,
le-add-cancel,
add-commutes,
add-associates,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
istype-false,
int_seg_subtype,
subtype_rel_function,
add_functionality_wrt_le,
zero-add,
less-iff-le,
not-lt-2,
sum_split1,
mul-distributes-right,
subtract-add-cancel,
general_arith_equation1,
exp_add,
less_than_wf,
add-subtract-cancel,
istype-universe,
istype-le,
iff_weakening_uiff,
assert_wf,
istype-nat,
set_subtype_base,
ifthenelse_wf,
ite_rw_true,
neg_assert_of_eq_int,
assert_of_eq_int,
eq_int_wf,
sum_linear,
mul_add_distrib,
sum_split,
empty_support,
add-member-int_seg2,
sum-is-zero,
le_weakening2,
zero_ann_a
Rules used in proof :
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
axiomEquality,
universeIsType,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation_alt,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
isect_memberFormation,
functionEquality,
addEquality,
intEquality,
isect_memberEquality,
because_Cache,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
multiplyEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
productElimination,
unionElimination,
dependent_pairFormation,
voidEquality,
lambdaFormation,
imageMemberEquality,
baseClosed,
imageElimination,
universeEquality,
equalityElimination,
promote_hyp,
functionIsType,
minusEquality,
closedConclusion,
dependent_set_memberEquality_alt,
Error :memTop,
productIsType,
equalityIstype,
sqequalBase,
equalityIsType1,
baseApply,
equalityIsType4
Latex:
\mforall{}[x:\mBbbZ{}]. \mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbZ{}]. \mforall{}[m:\mBbbN{}]. \mforall{}[b:\mBbbN{}m + 1 {}\mrightarrow{} \mBbbZ{}].
((\mSigma{}i<n + 1.a[i]*x\^{}i * \mSigma{}i<m + 1.b[i]*x\^{}i)
= \mSigma{}i<(n + m) + 1.\mSigma{}(if j \mleq{}z n then a[j] else 0 fi * if i - j \mleq{}z m then b[i - j] else 0 fi | j < i
+ 1)*x\^{}i)
Date html generated:
2020_05_20-AM-08_16_38
Last ObjectModification:
2020_02_01-PM-00_11_15
Theory : general
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