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Step * 2 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)} 
⊢ \/(λxs./\(f"(xs))"(a)) ∧ \/(λxs./\(f"(xs))"(b)) 
  ≤ \/(λxs./\(f"(xs))"(f-union(deq-fset(eq);deq-fset(eq);a;as.λbs.as ⋃ bs"(b))))
BY
{ ((InstLemma `fset-image-compose` [⌜fset(T)⌝;⌜fset(Point(L))⌝;⌜Point(L)⌝;⌜deq-fset(eq)⌝;⌜deq-fset(eqL)⌝;⌜eqL⌝; 
    ⌜λxs.f"(xs)⌝
                                        ; ⌜λls./\(ls)⌝]⋅
    THENA Auto
    )
   THEN RepUR ``compose`` -1
   THEN (RWO "-1<" 0 THENA Auto)) }

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)))
⊢ \/(λls./\(ls)"(λxs.f"(xs)"(a))) ∧ \/(λls./\(ls)"(λxs.f"(xs)"(b))) 
  ≤ \/(λls./\(ls)"(λxs.f"(xs)"(f-union(deq-fset(eq);deq-fset(eq);a;as.λbs.as ⋃ bs"(b)))))

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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)\} 
\mvdash{}  \mbackslash{}/(\mlambda{}xs./\mbackslash{}(f"(xs))"(a))  \mwedge{}  \mbackslash{}/(\mlambda{}xs./\mbackslash{}(f"(xs))"(b)) 
    \mleq{}  \mbackslash{}/(\mlambda{}xs./\mbackslash{}(f"(xs))"(f-union(deq-fset(eq);deq-fset(eq);a;as.\mlambda{}bs.as  \mcup{}  bs"(b))))


By

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Latex:
((InstLemma  `fset-image-compose`  [\mkleeneopen{}fset(T)\mkleeneclose{};\mkleeneopen{}fset(Point(L))\mkleeneclose{};\mkleeneopen{}Point(L)\mkleeneclose{};\mkleeneopen{}deq-fset(eq)\mkleeneclose{};
    \mkleeneopen{}deq-fset(eqL)\mkleeneclose{};\mkleeneopen{}eqL\mkleeneclose{};  \mkleeneopen{}\mlambda{}xs.f"(xs)\mkleeneclose{}
                                                                            ;  \mkleeneopen{}\mlambda{}ls./\mbackslash{}(ls)\mkleeneclose{}]\mcdot{}
    THENA  Auto
    )
  THEN  RepUR  ``compose``  -1
  THEN  (RWO  "-1<"  0  THENA  Auto))

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