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Step * 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)} 
⊢ lattice-extend'(L;eq;eqL;f;f-union(deq-fset(eq);deq-fset(eq);a;as.λbs.as ⋃ bs"(b))) 
  ≤ lattice-extend'(L;eq;eqL;f;fset-ac-glb(eq;a;b))
BY
{ (Unfold `fset-ac-glb` 0
   THEN (GenConclTerm ⌜f-union(deq-fset(eq);deq-fset(eq);a;as.λbs.as ⋃ bs"(b))⌝⋅ THENA Auto)
   THEN Unfold `lattice-extend\'` 0
   THEN (InstLemma `fset-minimals-ac-le` [⌜T⌝;⌜eq⌝;⌜v⌝]⋅ THENA Auto)
   THEN MoveToConcl (-1)
   THEN (GenConclTerm ⌜fset-minimals(xs,ys.f-proper-subset-dec(eq;xs;ys); v)⌝⋅ THENA Auto)
   THEN All Thin
   THEN RenameVar `as' (-2)
   THEN RenameVar `bs' (-1)) }

1
1. T : Type
2. eq : EqDecider(T)
3. L : BoundedDistributiveLattice
4. eqL : EqDecider(Point(L))
5. f : T ⟶ Point(L)
6. as : fset(fset(T))
7. bs : fset(fset(T))
⊢ fset-ac-le(eq;as;bs) ⇒ \/(λxs./\(f"(xs))"(as)) ≤ \/(λxs./\(f"(xs))"(bs))

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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{}  lattice-extend'(L;eq;eqL;f;f-union(deq-fset(eq);deq-fset(eq);a;as.\mlambda{}bs.as  \mcup{}  bs"(b))) 
    \mleq{}  lattice-extend'(L;eq;eqL;f;fset-ac-glb(eq;a;b))


By

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Latex:
(Unfold  `fset-ac-glb`  0
  THEN  (GenConclTerm  \mkleeneopen{}f-union(deq-fset(eq);deq-fset(eq);a;as.\mlambda{}bs.as  \mcup{}  bs"(b))\mkleeneclose{}\mcdot{}  THENA  Auto)
  THEN  Unfold  `lattice-extend\mbackslash{}'`  0
  THEN  (InstLemma  `fset-minimals-ac-le`  [\mkleeneopen{}T\mkleeneclose{};\mkleeneopen{}eq\mkleeneclose{};\mkleeneopen{}v\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  MoveToConcl  (-1)
  THEN  (GenConclTerm  \mkleeneopen{}fset-minimals(xs,ys.f-proper-subset-dec(eq;xs;ys);  v)\mkleeneclose{}\mcdot{}  THENA  Auto)
  THEN  All  Thin
  THEN  RenameVar  `as'  (-2)
  THEN  RenameVar  `bs'  (-1))

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