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


1. T : Type
2. eq : EqDecider(T)
3. s : fset(T)
4. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice(T; eq)))
5. {s} ∈ Point(free-dist-lattice(T; eq))
6. ∀[s:fset(Point(free-dist-lattice(T; eq)))]. ∀[x:Point(free-dist-lattice(T; eq))].  /\(s) ≤ x supposing x ∈ s
7. ∀[s:fset(Point(free-dist-lattice(T; eq)))]. ∀[v:Point(free-dist-lattice(T; eq))].
     ((∀x:Point(free-dist-lattice(T; eq)). (x ∈ s ⇒ v ≤ x)) ⇒ v ≤ /\(s))
8. x : Point(free-dist-lattice(T; eq))
9. ↓∃x1:T. (x1 ∈ s ∧ (x = ((λx.free-dl-inc(x)) x1) ∈ Point(free-dist-lattice(T; eq))))
⊢ {s} ≤ x
BY
{ (Reduce (-1) THEN RepUR ``free-dl-inc`` -1 THEN RWO "free-dl-le" 0 THEN Auto) }

1
1. T : Type
2. eq : EqDecider(T)
3. s : fset(T)
4. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice(T; eq)))
5. {s} ∈ Point(free-dist-lattice(T; eq))
6. ∀[s:fset(Point(free-dist-lattice(T; eq)))]. ∀[x:Point(free-dist-lattice(T; eq))].  /\(s) ≤ x supposing x ∈ s
7. ∀[s:fset(Point(free-dist-lattice(T; eq)))]. ∀[v:Point(free-dist-lattice(T; eq))].
     ((∀x:Point(free-dist-lattice(T; eq)). (x ∈ s ⇒ v ≤ x)) ⇒ v ≤ /\(s))
8. x : Point(free-dist-lattice(T; eq))
9. ↓∃x1:T. (x1 ∈ s ∧ (x = {{x1}} ∈ Point(free-dist-lattice(T; eq))))
⊢ fset-ac-le(eq;{s};x)

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Latex:

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Latex:

1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  s  :  fset(T)
4.  deq-fset(deq-fset(eq))  \mmember{}  EqDecider(Point(free-dist-lattice(T;  eq)))
5.  \{s\}  \mmember{}  Point(free-dist-lattice(T;  eq))
6.  \mforall{}[s:fset(Point(free-dist-lattice(T;  eq)))].  \mforall{}[x:Point(free-dist-lattice(T;  eq))].
          /\mbackslash{}(s)  \mleq{}  x  supposing  x  \mmember{}  s
7.  \mforall{}[s:fset(Point(free-dist-lattice(T;  eq)))].  \mforall{}[v:Point(free-dist-lattice(T;  eq))].
          ((\mforall{}x:Point(free-dist-lattice(T;  eq)).  (x  \mmember{}  s  {}\mRightarrow{}  v  \mleq{}  x))  {}\mRightarrow{}  v  \mleq{}  /\mbackslash{}(s))
8.  x  :  Point(free-dist-lattice(T;  eq))
9.  \mdownarrow{}\mexists{}x1:T.  (x1  \mmember{}  s  \mwedge{}  (x  =  ((\mlambda{}x.free-dl-inc(x))  x1)))
\mvdash{}  \{s\}  \mleq{}  x


By

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

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