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Step * 1 1 1 1 2 1 of Lemma lattice-meet-fset-join-distrib


1. l : BoundedDistributiveLattice
2. eq : EqDecider(Point(l))
3. ∀[a,b,c:Point(l)].  (a ∧ b ∨ c = a ∧ b ∨ a ∧ c ∈ Point(l))
4. u : Point(l)@i
5. v : Point(l) List@i
6. ∀bs:Point(l) List. (\/(v) ∧ \/(bs) = \/(f-union(eq;eq;v;a.λb.a ∧ b"(bs))) ∈ Point(l))
7. bs : Point(l) List@i
8. \/(v) ∧ \/(bs) = \/(f-union(eq;eq;v;a.λb.a ∧ b"(bs))) ∈ Point(l)
⊢ u ∨ \/(v) ∧ \/(bs) = \/(f-union(eq;eq;[u / v];a.λb.a ∧ b"(bs))) ∈ Point(l)
BY
{ (Subst ⌜u ∨ \/(v) ∧ \/(bs) = u ∧ \/(bs) ∨ \/(v) ∧ \/(bs) ∈ Point(l)⌝ 0⋅ THEN Auto) }

1
1. l : BoundedDistributiveLattice
2. eq : EqDecider(Point(l))
3. ∀[a,b,c:Point(l)].  (a ∧ b ∨ c = a ∧ b ∨ a ∧ c ∈ Point(l))
4. u : Point(l)@i
5. v : Point(l) List@i
6. ∀bs:Point(l) List. (\/(v) ∧ \/(bs) = \/(f-union(eq;eq;v;a.λb.a ∧ b"(bs))) ∈ Point(l))
7. bs : Point(l) List@i
8. \/(v) ∧ \/(bs) = \/(f-union(eq;eq;v;a.λb.a ∧ b"(bs))) ∈ Point(l)
⊢ u ∧ \/(bs) ∨ \/(v) ∧ \/(bs) = \/(f-union(eq;eq;[u / v];a.λb.a ∧ b"(bs))) ∈ Point(l)

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Latex:

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Latex:

1.  l  :  BoundedDistributiveLattice
2.  eq  :  EqDecider(Point(l))
3.  \mforall{}[a,b,c:Point(l)].    (a  \mwedge{}  b  \mvee{}  c  =  a  \mwedge{}  b  \mvee{}  a  \mwedge{}  c)
4.  u  :  Point(l)@i
5.  v  :  Point(l)  List@i
6.  \mforall{}bs:Point(l)  List.  (\mbackslash{}/(v)  \mwedge{}  \mbackslash{}/(bs)  =  \mbackslash{}/(f-union(eq;eq;v;a.\mlambda{}b.a  \mwedge{}  b"(bs))))
7.  bs  :  Point(l)  List@i
8.  \mbackslash{}/(v)  \mwedge{}  \mbackslash{}/(bs)  =  \mbackslash{}/(f-union(eq;eq;v;a.\mlambda{}b.a  \mwedge{}  b"(bs)))
\mvdash{}  u  \mvee{}  \mbackslash{}/(v)  \mwedge{}  \mbackslash{}/(bs)  =  \mbackslash{}/(f-union(eq;eq;[u  /  v];a.\mlambda{}b.a  \mwedge{}  b"(bs)))


By

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Latex:
(Subst  \mkleeneopen{}u  \mvee{}  \mbackslash{}/(v)  \mwedge{}  \mbackslash{}/(bs)  =  u  \mwedge{}  \mbackslash{}/(bs)  \mvee{}  \mbackslash{}/(v)  \mwedge{}  \mbackslash{}/(bs)\mkleeneclose{}  0\mcdot{}  THEN  Auto)

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