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Step * 2 1 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. {s} ≤ /\(λx.free-dl-inc(x)"(s))
9. /\(λx.free-dl-inc(x)"(s)) ∈ {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
⊢ /\(λx.free-dl-inc(x)"(s)) ≤ {s}
BY
{ ((RWO "free-dl-le" 0 THENA Auto)
   THEN Unfold `fset-ac-le` 0
   THEN Using [`eq',⌜deq-fset(eq)⌝] (BLemma `fset-all-iff`)⋅
   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. {s} ≤ /\(λx.free-dl-inc(x)"(s))
9. /\(λx.free-dl-inc(x)"(s)) ∈ {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
10. x : fset(T)
11. x ∈ /\(λx.free-dl-inc(x)"(s))
⊢ ¬↑fset-null({y ∈ {s} | deq-f-subset(eq) y 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.  \{s\}  \mleq{}  /\mbackslash{}(\mlambda{}x.free-dl-inc(x)"(s))
9.  /\mbackslash{}(\mlambda{}x.free-dl-inc(x)"(s))  \mmember{}  \{ac:fset(fset(T))|  \muparrow{}fset-antichain(eq;ac)\} 
\mvdash{}  /\mbackslash{}(\mlambda{}x.free-dl-inc(x)"(s))  \mleq{}  \{s\}


By

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Latex:
((RWO  "free-dl-le"  0  THENA  Auto)
  THEN  Unfold  `fset-ac-le`  0
  THEN  Using  [`eq',\mkleeneopen{}deq-fset(eq)\mkleeneclose{}]  (BLemma  `fset-all-iff`)\mcdot{}
  THEN  Auto)

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