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Step * 1 1 1 of Lemma lattice-fset-meet-free-dlwc-inc


1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. s : fset(T)
5. ↑fset-contains-none(eq;s;x.Cs[x])
6. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7. {s} ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
8. ∀[s:fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))].
   ∀[x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
     /\(s) ≤ x supposing x ∈ s
9. ∀[s:fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))].
   ∀[v:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
     ((∀x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])). (x ∈ s ⇒ v ≤ x)) ⇒ v ≤ /\(s))
10. x : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 
11. ∃x1:T
     (x1 ∈ s
     ∧ (x
       = if fset-null({c ∈ Cs[x1] | deq-f-subset(eq) c {x1}}) then {{x1}} else {} fi 
       ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))))
⊢ ¬↑fset-null({y ∈ x | deq-f-subset(eq) y s})
BY
{ (ExRepD THEN (RWO "free-dlwc-point" (-1) THENA Auto)) }

1
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. s : fset(T)
5. ↑fset-contains-none(eq;s;x.Cs[x])
6. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7. {s} ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
8. ∀[s:fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))].
   ∀[x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
     /\(s) ≤ x supposing x ∈ s
9. ∀[s:fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))].
   ∀[v:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
     ((∀x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])). (x ∈ s ⇒ v ≤ x)) ⇒ v ≤ /\(s))
10. x : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 
11. x1 : T
12. x1 ∈ s
13. x
= if fset-null({c ∈ Cs[x1] | deq-f-subset(eq) c {x1}}) then {{x1}} else {} fi 
∈ {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 
⊢ ¬↑fset-null({y ∈ x | deq-f-subset(eq) y s})

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Latex:

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Latex:

1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  Cs  :  T  {}\mrightarrow{}  fset(fset(T))
4.  s  :  fset(T)
5.  \muparrow{}fset-contains-none(eq;s;x.Cs[x])
6.  deq-fset(deq-fset(eq))  \mmember{}  EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7.  \{s\}  \mmember{}  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
8.  \mforall{}[s:fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))].
      \mforall{}[x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
          /\mbackslash{}(s)  \mleq{}  x  supposing  x  \mmember{}  s
9.  \mforall{}[s:fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))].
      \mforall{}[v:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
          ((\mforall{}x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])).  (x  \mmember{}  s  {}\mRightarrow{}  v  \mleq{}  x))  {}\mRightarrow{}  v  \mleq{}  /\mbackslash{}(s))
10.  x  :  \{ac:fset(fset(T))| 
                  (\muparrow{}fset-antichain(eq;ac))  \mwedge{}  fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))\} 
11.  \mexists{}x1:T
          (x1  \mmember{}  s  \mwedge{}  (x  =  if  fset-null(\{c  \mmember{}  Cs[x1]  |  deq-f-subset(eq)  c  \{x1\}\})  then  \{\{x1\}\}  else  \{\}  fi  ))
\mvdash{}  \mneg{}\muparrow{}fset-null(\{y  \mmember{}  x  |  deq-f-subset(eq)  y  s\})


By

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Latex:
(ExRepD  THEN  (RWO  "free-dlwc-point"  (-1)  THENA  Auto))

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