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Step * 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. 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)
BY
{ ((RWO "free-dl-point" (-2) THENA Auto)
   THEN Unfold `fset-ac-le` 0
   THEN Using [`eq',⌜deq-fset(eq)⌝] (BLemma `fset-all-iff`)⋅
   THEN Auto
   THEN (RWO "member-fset-singleton" (-1) THENA Auto)
   THEN (HypSubst' (-1) 0 THENA Auto)
   THEN RepeatFor 2 (Thin (-1))) }

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 : {ac:fset(fset(T))| ↑fset-antichain(eq;ac)} 
9. ∃x1:T. (x1 ∈ s ∧ (x = {{x1}} ∈ Point(free-dist-lattice(T; eq))))
⊢ ¬↑fset-null({y ∈ x | deq-f-subset(eq) y s})

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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  =  \{\{x1\}\}))
\mvdash{}  fset-ac-le(eq;\{s\};x)


By

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Latex:
((RWO  "free-dl-point"  (-2)  THENA  Auto)
  THEN  Unfold  `fset-ac-le`  0
  THEN  Using  [`eq',\mkleeneopen{}deq-fset(eq)\mkleeneclose{}]  (BLemma  `fset-all-iff`)\mcdot{}
  THEN  Auto
  THEN  (RWO  "member-fset-singleton"  (-1)  THENA  Auto)
  THEN  (HypSubst'  (-1)  0  THENA  Auto)
  THEN  RepeatFor  2  (Thin  (-1)))

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