Nuprl Lemma : bag-summation-filter
∀[T,R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[p:T ⟶ 𝔹]. ∀[f:T ⟶ R].
  Σ(x∈[x∈b|p[x]]). f[x] = Σ(x∈b). if p[x] then f[x] else zero fi  ∈ R supposing IsMonoid(R;add;zero) ∧ Comm(R;add)
Proof
Definitions occuring in Statement : 
bag-summation: Σ(x∈b). f[x]
, 
bag-filter: [x∈b|p[x]]
, 
bag: bag(T)
, 
comm: Comm(T;op)
, 
ifthenelse: if b then t else f fi 
, 
bool: 𝔹
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
and: P ∧ Q
, 
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
, 
prop: ℙ
, 
squash: ↓T
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
all: ∀x:A. B[x]
, 
cand: A c∧ B
, 
true: True
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
infix_ap: x f y
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
ifthenelse: if b then t else f fi 
, 
uiff: uiff(P;Q)
, 
bfalse: ff
, 
bnot: ¬bb
, 
not: ¬A
, 
false: False
, 
exists: ∃x:A. B[x]
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
assert: ↑b
, 
monoid_p: IsMonoid(T;op;id)
, 
ident: Ident(T;op;id)
Lemmas referenced : 
bag-summation-split, 
monoid_p_wf, 
comm_wf, 
bool_wf, 
bag_wf, 
equal_wf, 
squash_wf, 
true_wf, 
bag-summation_wf, 
assert_wf, 
bag-filter_wf, 
ifthenelse_wf, 
iff_weakening_equal, 
eqtt_to_assert, 
bag-summation-is-zero, 
bnot_wf, 
assert_elim, 
bfalse_wf, 
and_wf, 
btrue_neq_bfalse, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
bag-member_wf, 
set_wf, 
assoc_wf, 
not_assert_elim
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
productElimination, 
productEquality, 
cumulativity, 
functionExtensionality, 
applyEquality, 
sqequalRule, 
isect_memberEquality, 
axiomEquality, 
because_Cache, 
equalityTransitivity, 
equalitySymmetry, 
functionEquality, 
universeEquality, 
lambdaEquality, 
imageElimination, 
setEquality, 
lambdaFormation, 
setElimination, 
rename, 
independent_isectElimination, 
independent_pairFormation, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
independent_functionElimination, 
unionElimination, 
equalityElimination, 
dependent_functionElimination, 
addLevel, 
levelHypothesis, 
dependent_set_memberEquality, 
applyLambdaEquality, 
voidElimination, 
dependent_pairFormation, 
promote_hyp, 
instantiate
Latex:
\mforall{}[T,R:Type].  \mforall{}[add:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].  \mforall{}[b:bag(T)].  \mforall{}[p:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[f:T  {}\mrightarrow{}  R].
    \mSigma{}(x\mmember{}[x\mmember{}b|p[x]]).  f[x]  =  \mSigma{}(x\mmember{}b).  if  p[x]  then  f[x]  else  zero  fi   
    supposing  IsMonoid(R;add;zero)  \mwedge{}  Comm(R;add)
Date html generated:
2017_10_01-AM-09_02_04
Last ObjectModification:
2017_07_26-PM-04_43_22
Theory : bags
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