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Step * 1 1 1 2 1 1 1 of Lemma bag-bind-com


1. A : Type
2. B : Type
3. C : Type
4. f : A ⟶ B ⟶ bag(C)
5. ba : bag(A)
6. u : B
7. v : B List
8. bag-union(bag-map(λa.bag-union(bag-map(λb.f[a;b];v));ba))
= bag-union(bag-map(λb.bag-union(bag-map(λa.f[a;b];ba));v))
∈ bag(C)
9. v ∈ bag(B)
⊢ bag-bind(ba;λa.(f[a;u] + bag-bind(v;λb.f[a;b])))
= (bag-bind(ba;λa.f[a;u]) + bag-bind(ba;λa.bag-bind(v;λb.f[a;b])))
∈ bag(C)
BY
{ (Assert ⌜bag-bind(ba;λa.(((λa.f[a;u]) a) + ((λa.bag-bind(v;λb.f[a;b])) a)))
           = (bag-bind(ba;λa.f[a;u]) + bag-bind(ba;λa.bag-bind(v;λb.f[a;b])))
           ∈ bag(C)⌝⋅
   THEN Reduce (-1)
   THEN Auto
   THEN GenConcl ⌜(λa.bag-bind(v;λb.f[a;b])) = F ∈ (A ⟶ bag(C))⌝⋅
   THEN Auto
   THEN GenConcl ⌜(λa.f[a;u]) = G ∈ (A ⟶ bag(C))⌝⋅
   THEN Auto
   THEN All Thin) }

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Latex:

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Latex:

1.  A  :  Type
2.  B  :  Type
3.  C  :  Type
4.  f  :  A  {}\mrightarrow{}  B  {}\mrightarrow{}  bag(C)
5.  ba  :  bag(A)
6.  u  :  B
7.  v  :  B  List
8.  bag-union(bag-map(\mlambda{}a.bag-union(bag-map(\mlambda{}b.f[a;b];v));ba))
=  bag-union(bag-map(\mlambda{}b.bag-union(bag-map(\mlambda{}a.f[a;b];ba));v))
9.  v  \mmember{}  bag(B)
\mvdash{}  bag-bind(ba;\mlambda{}a.(f[a;u]  +  bag-bind(v;\mlambda{}b.f[a;b])))
=  (bag-bind(ba;\mlambda{}a.f[a;u])  +  bag-bind(ba;\mlambda{}a.bag-bind(v;\mlambda{}b.f[a;b])))


By

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Latex:
(Assert  \mkleeneopen{}bag-bind(ba;\mlambda{}a.(((\mlambda{}a.f[a;u])  a)  +  ((\mlambda{}a.bag-bind(v;\mlambda{}b.f[a;b]))  a)))
                  =  (bag-bind(ba;\mlambda{}a.f[a;u])  +  bag-bind(ba;\mlambda{}a.bag-bind(v;\mlambda{}b.f[a;b])))\mkleeneclose{}\mcdot{}
  THEN  Reduce  (-1)
  THEN  Auto
  THEN  GenConcl  \mkleeneopen{}(\mlambda{}a.bag-bind(v;\mlambda{}b.f[a;b]))  =  F\mkleeneclose{}\mcdot{}
  THEN  Auto
  THEN  GenConcl  \mkleeneopen{}(\mlambda{}a.f[a;u])  =  G\mkleeneclose{}\mcdot{}
  THEN  Auto
  THEN  All  Thin)

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