Nuprl Lemma : bag-double-summation2
∀[R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R].
∀[T,A:Type]. ∀[f:T ⟶ bag(A)]. ∀[h:T ⟶ A ⟶ R]. ∀[b:bag(T)].
(Σ(x∈b). Σ(y∈f[x]). h[x;y] = Σ(p∈⋃x∈b.bag-map(λy.<x, y>;f[x])). h[fst(p);snd(p)] ∈ R)
supposing IsMonoid(R;add;zero) ∧ Comm(R;add)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;bs),
bag: bag(T),
comm: Comm(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
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,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
bag-double-summation,
bag_wf,
and_wf,
monoid_p_wf,
comm_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
introduction,
productElimination,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
functionEquality,
universeEquality,
independent_pairFormation,
lambdaEquality,
independent_isectElimination
Latex:
\mforall{}[R:Type]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R].
\mforall{}[T,A:Type]. \mforall{}[f:T {}\mrightarrow{} bag(A)]. \mforall{}[h:T {}\mrightarrow{} A {}\mrightarrow{} R]. \mforall{}[b:bag(T)].
(\mSigma{}(x\mmember{}b). \mSigma{}(y\mmember{}f[x]). h[x;y] = \mSigma{}(p\mmember{}\mcup{}x\mmember{}b.bag-map(\mlambda{}y.<x, y>f[x])). h[fst(p);snd(p)])
supposing IsMonoid(R;add;zero) \mwedge{} Comm(R;add)
Date html generated:
2016_05_15-PM-02_33_02
Last ObjectModification:
2015_12_27-AM-09_47_22
Theory : bags
Home
Index