Nuprl Lemma : bag-summation-split
∀[T,R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[p:T ⟶ 𝔹]. ∀[f:T ⟶ R].
Σ(x∈b). f[x] = (Σ(x∈[x∈b|p[x]]). f[x] add Σ(x∈[x∈b|¬bp[x]]). f[x]) ∈ 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),
bnot: ¬bb,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
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,
and: P ∧ Q,
so_apply: x[s],
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
true: True,
subtype_rel: A ⊆r B,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
monoid_p: IsMonoid(T;op;id)
Lemmas referenced :
bag-filter-split,
infix_ap_wf,
bag-summation_wf,
assert_wf,
bag-filter_wf,
bnot_wf,
bag-summation-append,
subtype_rel_bag,
monoid_p_wf,
comm_wf,
bool_wf,
bag_wf,
equal_wf,
squash_wf,
true_wf,
assoc_wf,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
productElimination,
applyEquality,
functionExtensionality,
independent_pairFormation,
hypothesis,
cumulativity,
setEquality,
sqequalRule,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
natural_numberEquality,
productEquality,
functionEquality,
universeEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
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{}b). f[x] = (\mSigma{}(x\mmember{}[x\mmember{}b|p[x]]). f[x] add \mSigma{}(x\mmember{}[x\mmember{}b|\mneg{}\msubb{}p[x]]). f[x])
supposing IsMonoid(R;add;zero) \mwedge{} Comm(R;add)
Date html generated:
2017_10_01-AM-08_51_16
Last ObjectModification:
2017_07_26-PM-04_33_14
Theory : bags
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