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Step * of Lemma free-dlwc-satisfies-constraints

∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))].
  ((∀x:T. ∀c:fset(T).  (c ∈ Cs[x] ⇒ x ∈ c))
  ⇒ (∀x:T. ∀c:fset(T).
        (c ∈ Cs[x]
        ⇒ (/\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(c)) = 0 ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))))))
BY
{ (Auto
   THEN (Assert deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))) BY
               (RWO  "free-dlwc-point" 0 THEN Auto))
   THEN ((InstLemma `lattice-fset-meet-is-glb` [⌜free-dist-lattice-with-constraints(T;eq;x.Cs[x])⌝;
          ⌜deq-fset(deq-fset(eq))⌝]⋅
          THENA Auto
          )
         THEN D -1
         )
   THEN Thin (-1)
   THEN (InstHyp [⌜λx.free-dlwc-inc(eq;a.Cs[a];x)"(c)⌝] (-1)⋅ THENA Auto)
   THEN Thin (-2)
   THEN ((MoveToConcl (-1) THEN GenConclTerm ⌜/\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(c))⌝⋅) THENA Auto)
   THEN Thin (-1)
   THEN Auto
   THEN ∀h:hyp. (RWO "free-dlwc-point" h THENA Auto) 
   THEN DVar `v'
   THEN (Subst' 0 ~ {} 0 THENA (RW  (SubC (SymbCompC [] 100)) 0 THEN Auto))
   THEN Symmetry
   THEN (RWO "free-dlwc-point" 0 THENA Auto)
   THEN EqTypeCD
   THEN Auto) }

1
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. ∀x:T. ∀c:fset(T).  (c ∈ Cs[x] ⇒ x ∈ c)
5. x : T
6. c : fset(T)
7. c ∈ Cs[x]
8. deq-fset(deq-fset(eq)) ∈ EqDecider({ac:fset(fset(T))| 
                                       (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} )
9. v : fset(fset(T))
10. ↑fset-antichain(eq;v)
11. fset-all(v;a.fset-contains-none(eq;a;x.Cs[x]))
12. ∀[x:{ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} ]
      v ≤ x supposing x ∈ λx.free-dlwc-inc(eq;a.Cs[a];x)"(c)
⊢ {} = v ∈ fset(fset(T))

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Latex:

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Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[Cs:T  {}\mrightarrow{}  fset(fset(T))].
    ((\mforall{}x:T.  \mforall{}c:fset(T).    (c  \mmember{}  Cs[x]  {}\mRightarrow{}  x  \mmember{}  c))
    {}\mRightarrow{}  (\mforall{}x:T.  \mforall{}c:fset(T).    (c  \mmember{}  Cs[x]  {}\mRightarrow{}  (/\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(c))  =  0))))


By

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Latex:
(Auto
  THEN  (Assert  deq-fset(deq-fset(eq))
                            \mmember{}  EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))  BY
                          (RWO    "free-dlwc-point"  0  THEN  Auto))
  THEN  ((InstLemma  `lattice-fset-meet-is-glb`  [\mkleeneopen{}free-dist-lattice-with-constraints(T;eq;x.Cs[x])\mkleeneclose{};
                \mkleeneopen{}deq-fset(deq-fset(eq))\mkleeneclose{}]\mcdot{}
                THENA  Auto
                )
              THEN  D  -1
              )
  THEN  Thin  (-1)
  THEN  (InstHyp  [\mkleeneopen{}\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(c)\mkleeneclose{}]  (-1)\mcdot{}  THENA  Auto)
  THEN  Thin  (-2)
  THEN  ((MoveToConcl  (-1)  THEN  GenConclTerm  \mkleeneopen{}/\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(c))\mkleeneclose{}\mcdot{})  THENA  Auto)
  THEN  Thin  (-1)
  THEN  Auto
  THEN  \mforall{}h:hyp.  (RWO  "free-dlwc-point"  h  THENA  Auto) 
  THEN  DVar  `v'
  THEN  (Subst'  0  \msim{}  \{\}  0  THENA  (RW    (SubC  (SymbCompC  []  100))  0  THEN  Auto))
  THEN  Symmetry
  THEN  (RWO  "free-dlwc-point"  0  THENA  Auto)
  THEN  EqTypeCD
  THEN  Auto)

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