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Step * 1 1 2 1 1 of Lemma bag-map-union2


1. A : Type
2. B : Type
3. g : A ⟶ B
4. x : bag(bag(A))
5. bag-map(λa.{g a};bag-union(x)) = bag-union(bag-map(λb.bag-map(λa.{g a};b);x)) ∈ bag(bag(B))
6. bag-union(bag-map(λa.{g a};bag-union(x))) = bag-union(bag-union(bag-map(λb.bag-map(λa.{g a};b);x))) ∈ bag(B)
⊢ bag-map(g;bag-union(x)) = bag-union(bag-map(λa.{g a};bag-union(x))) ∈ bag(B)
BY
{ (Fold `bag-combine` 0 THEN (RWO "bag-combine-single-right-as-map" 0 THENA Auto) THEN EqCDA THEN Auto) }

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Latex:

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Latex:

1.  A  :  Type
2.  B  :  Type
3.  g  :  A  {}\mrightarrow{}  B
4.  x  :  bag(bag(A))
5.  bag-map(\mlambda{}a.\{g  a\};bag-union(x))  =  bag-union(bag-map(\mlambda{}b.bag-map(\mlambda{}a.\{g  a\};b);x))
6.  bag-union(bag-map(\mlambda{}a.\{g  a\};bag-union(x)))
=  bag-union(bag-union(bag-map(\mlambda{}b.bag-map(\mlambda{}a.\{g  a\};b);x)))
\mvdash{}  bag-map(g;bag-union(x))  =  bag-union(bag-map(\mlambda{}a.\{g  a\};bag-union(x)))


By

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Latex:
(Fold  `bag-combine`  0
  THEN  (RWO  "bag-combine-single-right-as-map"  0  THENA  Auto)
  THEN  EqCDA
  THEN  Auto)

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