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Step * 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 : Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
11. ↓∃x1:T
      (x1 ∈ s ∧ (x = ((λx.free-dlwc-inc(eq;a.Cs[a];x)) x1) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))))
⊢ {s} ≤ x
BY
{ (Reduce (-1) THEN RepUR ``free-dlwc-inc`` -1 THEN RWO "free-dlwc-le" 0 THEN 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 : Point(free-dist-lattice-with-constraints(T;eq;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-ac-le(eq;{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.  x  :  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
11.  \mdownarrow{}\mexists{}x1:T.  (x1  \mmember{}  s  \mwedge{}  (x  =  ((\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x))  x1)))
\mvdash{}  \{s\}  \mleq{}  x


By

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Latex:
(Reduce  (-1)  THEN  RepUR  ``free-dlwc-inc``  -1  THEN  RWO  "free-dlwc-le"  0  THEN  Auto)

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