Nuprl Lemma : bag-summation-partitions-primes
∀[h:ℕ+ ⟶ ℕ+ ⟶ ℤ]. ∀[b:bag(Prime)].
(Σ(p∈bag-partitions(IntDeq;b)). h[Π(fst(p));Π(snd(p))] = Σ i|Π(b). h[i;Π(b) ÷ i] ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-partitions: bag-partitions(eq;bs),
divisors-sum: Σ i|n. f[i] ,
Prime: Prime,
int-bag-product: Π(b),
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
int-deq: IntDeq,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
lambda: λx.A[x],
function: x:A ⟶ B[x],
divide: 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,
subtype_rel: A ⊆r B,
integ_dom: IntegDom{i},
int_ring: ℤ-rng,
rng_car: |r|,
pi1: fst(t),
rng_zero: 0,
pi2: snd(t),
rng_plus: +r,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
let: let,
squash: ↓T,
prop: ℙ,
nat_plus: ℕ+,
Prime: Prime,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtract: n - m,
so_apply: x[s]
Lemmas referenced :
bag-summation-partitions-primes-general,
int_ring_wf,
integ_dom_wf,
nat_plus_wf,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
gen-divisors-sum-int-ring,
int-bag-product_wf,
subtype_rel_bag,
Prime_wf,
bag-product-primes,
less_than_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equal-wf-base,
div-positive-1,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
zero-add,
le-add-cancel2,
int_seg_wf,
divisors-sum_wf,
iff_weakening_equal,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
functionExtensionality,
functionEquality,
intEquality,
instantiate,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageElimination,
universeEquality,
dependent_set_memberEquality,
because_Cache,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
productElimination,
divideEquality,
addEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
baseClosed,
minusEquality,
imageMemberEquality,
axiomEquality
Latex:
\mforall{}[h:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. \mforall{}[b:bag(Prime)].
(\mSigma{}(p\mmember{}bag-partitions(IntDeq;b)). h[\mPi{}(fst(p));\mPi{}(snd(p))] = \mSigma{} i|\mPi{}(b). h[i;\mPi{}(b) \mdiv{} i] )
Date html generated:
2018_05_21-PM-09_50_28
Last ObjectModification:
2017_07_26-PM-06_31_21
Theory : bags_2
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