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Step * 2 1 1 1 1 1 of Lemma lattice-extend-meet


1. T : Type
2. eq : EqDecider(T)
3. L : BoundedDistributiveLattice
4. eqL : EqDecider(Point(L))
5. f : T ⟶ Point(L)
6. a : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
7. b : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
8. ∀[s:fset(fset(T))]. (λls./\(ls)"(λxs.f"(xs)"(s)) = λx./\(f"(x))"(s) ∈ fset(Point(L)))
9. a1 : fset(Point(L))
⊢ uiff(↓∃x:fset(T)
         (x ∈ f-union(deq-fset(eq);deq-fset(eq);a;as.λbs.as ⋃ bs"(b))
         ∧ (a1 = f"(x) ∈ fset(Point(L))));↓∃as:fset(Point(L))
                                            (as ∈ λxs.f"(xs)"(a) ∧ a1 ∈ λbs.as ⋃ bs"(λxs.f"(xs)"(b))))
BY
{ ((RWO "member-fset-image-iff member-f-union" 0 THENA Auto) THEN Reduce 0) }

1
1. T : Type
2. eq : EqDecider(T)
3. L : BoundedDistributiveLattice
4. eqL : EqDecider(Point(L))
5. f : T ⟶ Point(L)
6. a : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
7. b : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
8. ∀[s:fset(fset(T))]. (λls./\(ls)"(λxs.f"(xs)"(s)) = λx./\(f"(x))"(s) ∈ fset(Point(L)))
9. a1 : fset(Point(L))
⊢ uiff(↓∃x:fset(T)
         ((↓∃as:fset(T). (as ∈ a ∧ x ∈ λbs.as ⋃ bs"(b))) ∧ (a1 = f"(x) ∈ fset(Point(L))));↓∃as:fset(Point(L))
                                                                                            ((↓∃x:fset(T)
                                                                                                (x ∈ a
                                                                                                ∧ (as
                                                                                                  = f"(x)
                                                                                                  ∈ fset(Point(L)))))
                                                                                            ∧ (↓∃x:fset(Point(L))
                                                                                                 (x ∈ λxs.f"(xs)"(b)
                                                                                                 ∧ (a1
                                                                                                   = as ⋃ x
                                                                                                   ∈ fset(Point(L)))))))

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Latex:

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Latex:

1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  L  :  BoundedDistributiveLattice
4.  eqL  :  EqDecider(Point(L))
5.  f  :  T  {}\mrightarrow{}  Point(L)
6.  a  :  \{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\} 
7.  b  :  \{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\} 
8.  \mforall{}[s:fset(fset(T))].  (\mlambda{}ls./\mbackslash{}(ls)"(\mlambda{}xs.f"(xs)"(s))  =  \mlambda{}x./\mbackslash{}(f"(x))"(s))
9.  a1  :  fset(Point(L))
\mvdash{}  uiff(\mdownarrow{}\mexists{}x:fset(T)
                  (x  \mmember{}  f-union(deq-fset(eq);deq-fset(eq);a;as.\mlambda{}bs.as  \mcup{}  bs"(b))
                  \mwedge{}  (a1  =  f"(x)));\mdownarrow{}\mexists{}as:fset(Point(L))
                                                      (as  \mmember{}  \mlambda{}xs.f"(xs)"(a)  \mwedge{}  a1  \mmember{}  \mlambda{}bs.as  \mcup{}  bs"(\mlambda{}xs.f"(xs)"(b))))


By

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Latex:
((RWO  "member-fset-image-iff  member-f-union"  0  THENA  Auto)  THEN  Reduce  0)

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