Nuprl Lemma : bag-summation-partitions-primes-general
∀[r:CRng]. ∀[h:ℕ+ ⟶ ℕ+ ⟶ |r|]. ∀[b:bag(Prime)].
(Σ(p∈bag-partitions(IntDeq;b)). h[Π(fst(p));Π(snd(p))] = let n = Π(b) in Σ i|n. h[i;n ÷ i] ∈ |r|)
Proof
Definitions occuring in Statement :
bag-partitions: bag-partitions(eq;bs),
gen-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+,
let: let,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
function: x:A ⟶ B[x],
divide: n ÷ m,
equal: s = t ∈ T,
crng: CRng,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
gen-divisors-sum: Σ i|n. f[i],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
Prime: Prime,
so_lambda: λ2x.t[x],
int_upper: {i...},
so_apply: x[s],
let: let,
crng: CRng,
rng: Rng,
top: Top,
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
mapfilter: mapfilter(f;P;L),
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
all: ∀x:A. B[x],
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
implies: P ⇒ Q,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
guard: {T},
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B,
bag-member: x ↓∈ bs,
squash: ↓T,
inject: Inj(A;B;f),
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
less_than: a < b,
divides: b | a,
nat: ℕ,
int_nzero: ℤ-o,
sq_type: SQType(T),
l_member: (x ∈ l),
ge: i ≥ j ,
subtract: n - m,
so_apply: x[s1;s2],
monoid_p: IsMonoid(T;op;id),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
int-deq_wf,
strong-subtype-deq-subtype,
Prime_wf,
strong-subtype-set3,
int_upper_wf,
prime_wf,
le_wf,
strong-subtype-self,
bag_wf,
nat_plus_wf,
rng_car_wf,
crng_wf,
bag-summation-map,
bag-partitions_wf,
set-valueall-type,
int-valueall-type,
bag-product-primes,
from-upto_wf,
int-bag-product_wf,
list-subtype-bag,
less_than_wf,
subtype_rel_bag,
int_seg_wf,
bag-extensionality-no-repeats,
decidable__equal_product,
decidable__equal_nat_plus,
bag-map_wf,
pi1_wf_top,
pi2_wf,
assert_wf,
eq_int_wf,
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,
int_subtype_base,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
div-positive-1,
bag-filter_wf,
bag-member_wf,
bag-map-no-repeats,
equal_wf,
prime-product-injection,
no-repeats-bag-partitions,
assert_of_eq_int,
equal-wf-T-base,
and_wf,
subtype_rel_product,
top_wf,
nat_plus_properties,
sq_stable__assert,
intformless_wf,
int_formula_prop_less_lemma,
iff_weakening_equal,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
set_wf,
bag-no-repeats-subtype,
strong-subtype-set2,
bag-no-repeats-filter,
bag-no-repeats-list,
list_wf,
no_repeats-subtype,
no_repeats_from-upto,
bag-member-map,
sq_stable__bag-member,
bag-member-partitions,
squash_wf,
true_wf,
int-bag-product-append,
divisors_bound,
divides_iff_rem_zero,
nequal_wf,
subtype_base_sq,
product_subtype_base,
set_subtype_base,
bag-member-filter-set,
bag-member-list,
decidable__equal_int_seg,
from-upto-member,
nat_properties,
decidable__equal_int,
length_wf,
select_wf,
divide-exact,
subtype_rel_set,
int_nzero_wf,
subtype_rel_sets,
less_than_transitivity1,
le_weakening,
factors_wf,
div_rem_sum2,
append-factors,
minus-zero,
add-zero,
factors-prime-product,
product-factors,
int_nzero_properties,
bag-subtype-list,
bag-summation-filter,
rng_plus_wf,
rng_zero_wf,
rng_all_properties,
rng_plus_comm2,
bag-summation_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
le-add-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
hypothesis,
applyEquality,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
natural_numberEquality,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
isect_memberEquality,
axiomEquality,
functionEquality,
voidElimination,
voidEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
dependent_functionElimination,
addEquality,
setEquality,
productEquality,
independent_functionElimination,
lambdaFormation,
independent_pairEquality,
dependent_set_memberEquality,
productElimination,
remainderEquality,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
baseClosed,
unionElimination,
divideEquality,
equalityTransitivity,
imageElimination,
imageMemberEquality,
addLevel,
levelHypothesis,
universeEquality,
multiplyEquality,
instantiate,
cumulativity,
functionExtensionality,
equalityElimination,
promote_hyp,
minusEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[h:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbN{}\msupplus{} {}\mrightarrow{} |r|]. \mforall{}[b:bag(Prime)].
(\mSigma{}(p\mmember{}bag-partitions(IntDeq;b)). h[\mPi{}(fst(p));\mPi{}(snd(p))] = let n = \mPi{}(b) in \mSigma{} i|n. h[i;n \mdiv{} i])
Date html generated:
2018_05_21-PM-09_50_25
Last ObjectModification:
2017_07_26-PM-06_31_19
Theory : bags_2
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