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Step * 1 of Lemma free-dist-lattice_wf


1. T : Type
2. eq : EqDecider(T)
3. Order({ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ;x,y.fset-ac-le(eq;x;y))
4. ∀[a,b:{ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ].
     least-upper-bound({ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ;x,y.fset-ac-le(eq;x;y);a;b;fset-ac-lub(eq;a;b))
5. ∀[a,b:{ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ].
     greatest-lower-bound({ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ;x,y.fset-ac-le(eq;x;y);a;b;fset-ac-glb(eq;a;b))
6. ∀[a:{ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ]. fset-ac-le(eq;a;{{}})
7. a : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
⊢ fset-ac-le(eq;{};a)
BY
{ (RWO "empty-fset-ac-le" 0 THEN Auto) }

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Latex:

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Latex:

1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  Order(\{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\}  ;x,y.fset-ac-le(eq;x;y))
4.  \mforall{}[a,b:\{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\}  ].
          least-upper-bound(\{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\}  ;x,y.fset-ac-le(eq;x;y);
                                              a;b;fset-ac-lub(eq;a;b))
5.  \mforall{}[a,b:\{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\}  ].
          greatest-lower-bound(\{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\}  ;x,y.fset-ac-le(eq;x;y);a;b;fse\000Ct-ac-glb(eq;a;b))
6.  \mforall{}[a:\{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\}  ].  fset-ac-le(eq;a;\{\{\}\})
7.  a  :  \{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\} 
\mvdash{}  fset-ac-le(eq;\{\};a)


By

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Latex:
(RWO  "empty-fset-ac-le"  0  THEN  Auto)

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