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Step * 2 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. {s} ≤ /\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))
11. /\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))
    ∈ {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 
12. x : fset(T)
13. x ∈ /\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))
⊢ ¬↑fset-null({y ∈ {s} | deq-f-subset(eq) y x})
BY
{ (RepUR ``fset-singleton fset-filter`` 0
   THEN (SplitOnConclITE THENA Auto)
   THEN ((RepUR ``fset-null`` 0 THEN Complete (Auto)) ORELSE D -1)) }

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. {s} ≤ /\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))
11. /\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))
    ∈ {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 
12. x : fset(T)
13. x ∈ /\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))
⊢ s ⊆ x

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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.  \{s\}  \mleq{}  /\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(s))
11.  /\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(s))  \mmember{}  \{ac:fset(fset(T))| 
                                                                                            (\muparrow{}fset-antichain(eq;ac))
                                                                                            \mwedge{}  fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))\} 
12.  x  :  fset(T)
13.  x  \mmember{}  /\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(s))
\mvdash{}  \mneg{}\muparrow{}fset-null(\{y  \mmember{}  \{s\}  |  deq-f-subset(eq)  y  x\})


By

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Latex:
(RepUR  ``fset-singleton  fset-filter``  0
  THEN  (SplitOnConclITE  THENA  Auto)
  THEN  ((RepUR  ``fset-null``  0  THEN  Complete  (Auto))  ORELSE  D  -1))

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