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

∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)].  (/\(λx.free-dl-inc(x)"(s)) = {s} ∈ Point(free-dist-lattice(T; eq)))
BY
{ ((Auto
    THEN (Assert deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice(T; eq))) BY
                (RWO "free-dl-point" 0 THEN Auto))
    THEN (Assert {s} ∈ Point(free-dist-lattice(T; eq)) BY
                (RWO  "free-dl-point" 0 THEN Auto)))
   THEN (InstLemma `lattice-fset-meet-is-glb` [⌜free-dist-lattice(T; eq)⌝;⌜deq-fset(deq-fset(eq))⌝]⋅ THENA Auto)
   THEN D -1
   THEN (InstHyp [⌜λx.free-dl-inc(x)"(s)⌝;⌜{s}⌝] (-1)⋅
         THENA (Auto THEN (RWO "member-fset-image-iff" (-1)  THENA 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 = ((λx.free-dl-inc(x)) x1) ∈ Point(free-dist-lattice(T; eq))))
⊢ {s} ≤ x

2
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))
⊢ /\(λx.free-dl-inc(x)"(s)) = {s} ∈ Point(free-dist-lattice(T; eq))

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Latex:

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Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[s:fset(T)].    (/\mbackslash{}(\mlambda{}x.free-dl-inc(x)"(s))  =  \{s\})


By

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Latex:
((Auto
    THEN  (Assert  deq-fset(deq-fset(eq))  \mmember{}  EqDecider(Point(free-dist-lattice(T;  eq)))  BY
                            (RWO  "free-dl-point"  0  THEN  Auto))
    THEN  (Assert  \{s\}  \mmember{}  Point(free-dist-lattice(T;  eq))  BY
                            (RWO    "free-dl-point"  0  THEN  Auto)))
  THEN  (InstLemma  `lattice-fset-meet-is-glb`  [\mkleeneopen{}free-dist-lattice(T;  eq)\mkleeneclose{};\mkleeneopen{}deq-fset(deq-fset(eq))\mkleeneclose{}]\mcdot{}
              THENA  Auto
              )
  THEN  D  -1
  THEN  (InstHyp  [\mkleeneopen{}\mlambda{}x.free-dl-inc(x)"(s)\mkleeneclose{};\mkleeneopen{}\{s\}\mkleeneclose{}]  (-1)\mcdot{}
              THENA  (Auto  THEN  (RWO  "member-fset-image-iff"  (-1)    THENA  Auto))
              ))

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