Nuprl Lemma : bag-summation-partition
∀[A:Type]
  ∀[R,T:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[case:T ⟶ A ⟶ 𝔹]. ∀[f:T ⟶ R]. ∀[c:bag(A)].
    Σ(x∈b). f[x] = Σ(z∈c). Σ(x∈[x∈b|case[x;z]]). f[x] ∈ R 
    supposing (IsMonoid(R;add;zero) ∧ Comm(R;add))
    ∧ (∀x:{x:T| x ↓∈ b} . (∃z:{A| (z ↓∈ c ∧ (↑case[x;z]))}))
    ∧ bag-no-repeats(A;c)
    ∧ (∀z1,z2:A. ∀x:T.  ((↑case[x;z1]) 
⇒ (↑case[x;z2]) 
⇒ (z1 = z2 ∈ A))) 
  supposing ∀x,y:A.  Dec(x = y ∈ A)
Proof
Definitions occuring in Statement : 
bag-member: x ↓∈ bs
, 
bag-no-repeats: bag-no-repeats(T;bs)
, 
bag-summation: Σ(x∈b). f[x]
, 
bag-filter: [x∈b|p[x]]
, 
bag: bag(T)
, 
comm: Comm(T;op)
, 
assert: ↑b
, 
bool: 𝔹
, 
decidable: Dec(P)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s1;s2]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
sq_exists: ∃x:{A| B[x]}
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
, 
monoid_p: IsMonoid(T;op;id)
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
and: P ∧ Q
, 
squash: ↓T
, 
exists: ∃x:A. B[x]
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s1;s2]
, 
so_apply: x[s]
, 
implies: P 
⇒ Q
, 
subtype_rel: A ⊆r B
, 
all: ∀x:A. B[x]
, 
sq_exists: ∃x:{A| B[x]}
, 
cand: A c∧ B
, 
monoid_p: IsMonoid(T;op;id)
, 
bag-filter: [x∈b|p[x]]
, 
top: Top
, 
empty-bag: {}
, 
true: True
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
cons-bag: x.b
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
sq_or: a ↓∨ b
, 
or: P ∨ Q
, 
single-bag: {x}
, 
bag-append: as + bs
, 
append: as @ bs
, 
so_lambda: so_lambda(x,y,z.t[x; y; z])
, 
so_apply: x[s1;s2;s3]
, 
infix_ap: x f y
, 
bag-member: x ↓∈ bs
, 
sq_type: SQType(T)
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
Lemmas referenced : 
bag_to_squash_list, 
all_wf, 
bag-member_wf, 
sq_exists_wf, 
assert_wf, 
list_induction, 
list-subtype-bag, 
equal_wf, 
bag-summation_wf, 
bag-filter_wf, 
list_wf, 
nil_wf, 
cons_wf, 
monoid_p_wf, 
comm_wf, 
bag-no-repeats_wf, 
bag_wf, 
bool_wf, 
decidable_wf, 
filter_nil_lemma, 
bag-summation-empty, 
squash_wf, 
true_wf, 
bag-summation-zero, 
iff_weakening_equal, 
bag-summation-cons, 
cons-bag_wf, 
set_wf, 
bag-member-cons, 
infix_ap_wf, 
list_ind_cons_lemma, 
list_ind_nil_lemma, 
bag-summation-add, 
single-bag_wf, 
bag-append_wf, 
assoc_wf, 
bag-filter-append, 
bag-summation-append, 
ifthenelse_wf, 
bag-summation-filter, 
bag-summation-single, 
bag-extensionality-no-repeats, 
subtype_rel_bag, 
bag-filter-no-repeats, 
bag-single-no-repeats, 
bag-member-single, 
bag-member-filter, 
and_wf, 
assert_elim, 
subtype_base_sq, 
bool_subtype_base
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
sqequalHypSubstitution, 
productElimination, 
thin, 
extract_by_obid, 
isectElimination, 
hypothesisEquality, 
imageElimination, 
promote_hyp, 
hypothesis, 
equalitySymmetry, 
hyp_replacement, 
applyLambdaEquality, 
setEquality, 
cumulativity, 
sqequalRule, 
lambdaEquality, 
productEquality, 
applyEquality, 
functionExtensionality, 
setElimination, 
rename, 
functionEquality, 
because_Cache, 
independent_isectElimination, 
lambdaFormation, 
independent_pairFormation, 
independent_functionElimination, 
voidEquality, 
voidElimination, 
dependent_functionElimination, 
isect_memberEquality, 
axiomEquality, 
equalityTransitivity, 
natural_numberEquality, 
universeEquality, 
imageMemberEquality, 
baseClosed, 
dependent_set_memberEquality, 
inrFormation, 
inlFormation, 
instantiate
Latex:
\mforall{}[A:Type]
    \mforall{}[R,T:Type].  \mforall{}[add:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].  \mforall{}[b:bag(T)].  \mforall{}[case:T  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[f:T  {}\mrightarrow{}  R].
    \mforall{}[c:bag(A)].
        \mSigma{}(x\mmember{}b).  f[x]  =  \mSigma{}(z\mmember{}c).  \mSigma{}(x\mmember{}[x\mmember{}b|case[x;z]]).  f[x] 
        supposing  (IsMonoid(R;add;zero)  \mwedge{}  Comm(R;add))
        \mwedge{}  (\mforall{}x:\{x:T|  x  \mdownarrow{}\mmember{}  b\}  .  (\mexists{}z:\{A|  (z  \mdownarrow{}\mmember{}  c  \mwedge{}  (\muparrow{}case[x;z]))\}))
        \mwedge{}  bag-no-repeats(A;c)
        \mwedge{}  (\mforall{}z1,z2:A.  \mforall{}x:T.    ((\muparrow{}case[x;z1])  {}\mRightarrow{}  (\muparrow{}case[x;z2])  {}\mRightarrow{}  (z1  =  z2))) 
    supposing  \mforall{}x,y:A.    Dec(x  =  y)
Date html generated:
2017_10_01-AM-09_02_30
Last ObjectModification:
2017_07_26-PM-04_43_33
Theory : bags
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