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Step * 2 1 1 1 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))
14. a : T
15. a ∈ s
16. (∀[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)
∧ (∀[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)))
⊢ a ∈ x
BY
{ ((InstHyp [⌜λx.free-dlwc-inc(eq;a.Cs[a];x)"(s)⌝;⌜free-dlwc-inc(eq;a.Cs[a];a)⌝] 8⋅
    THENA (Auto THEN (RWO "member-fset-image-iff" 0  THENA Auto) THEN Reduce 0 THEN D 0 THEN Auto)
    )
   THEN Thin (-2)
   ) }

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))
14. a : T
15. a ∈ s
16. /\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s)) ≤ free-dlwc-inc(eq;a.Cs[a];a)
⊢ a ∈ 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))
14.  a  :  T
15.  a  \mmember{}  s
16.  (\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)
\mwedge{}  (\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)))
\mvdash{}  a  \mmember{}  x


By

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Latex:
((InstHyp  [\mkleeneopen{}\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(s)\mkleeneclose{};\mkleeneopen{}free-dlwc-inc(eq;a.Cs[a];a)\mkleeneclose{}]  8\mcdot{}
    THENA  (Auto  THEN  (RWO  "member-fset-image-iff"  0    THENA  Auto)  THEN  Reduce  0  THEN  D  0  THEN  Auto)
    )
  THEN  Thin  (-2)
  )

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