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Step * 1 1 1 2 1 1 1 1 2 of Lemma free-dlwc-basis


1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : fset(fset(T))
5. ↑fset-antichain(eq;x)
6. fset-all(x;a.fset-contains-none(eq;a;x.Cs[x]))
7. x ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
8. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
9. ∀s:fset(T). (s ∈ x ⇒ ({s} ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))))
10. λs.{s}"(x) ∈ fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
11. ∀[x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]. x@0 ≤ \/(λs.{s}"(x)) supposing x@0 ∈ λs.{s}"(x)
12. ∀[u:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]
      ((∀x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])). (x@0 ∈ λs.{s}"(x) ⇒ x@0 ≤ u))
      ⇒ \/(λs.{s}"(x)) ≤ u)
13. \/(λs.{s}"(x)) ≤ x
14. \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
15. \/(λs.{s}"(x)) ∈ {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 
16. z : fset(T)
17. z ∈ x
18. ∀[P:fset(T) ⟶ 𝔹]. ∀[s:fset(fset(T))].  uiff({x ∈ s | P[x]} = {} ∈ fset(fset(T));¬(∃x:fset(T). (x ∈ s ∧ (↑P[x]))))
19. z ∈ \/(λs.{s}"(x))
⊢ z ⊆ z
BY
{ EAuto 1 }

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Latex:

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Latex:

1.  T  :  Type
2.  eq  :  EqDecider(T)
3.  Cs  :  T  {}\mrightarrow{}  fset(fset(T))
4.  x  :  fset(fset(T))
5.  \muparrow{}fset-antichain(eq;x)
6.  fset-all(x;a.fset-contains-none(eq;a;x.Cs[x]))
7.  x  \mmember{}  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
8.  deq-fset(deq-fset(eq))  \mmember{}  EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
9.  \mforall{}s:fset(T).  (s  \mmember{}  x  {}\mRightarrow{}  (\{s\}  \mmember{}  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))))
10.  \mlambda{}s.\{s\}"(x)  \mmember{}  fset(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
11.  \mforall{}[x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]
            x@0  \mleq{}  \mbackslash{}/(\mlambda{}s.\{s\}"(x))  supposing  x@0  \mmember{}  \mlambda{}s.\{s\}"(x)
12.  \mforall{}[u:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))]
            ((\mforall{}x@0:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])).  (x@0  \mmember{}  \mlambda{}s.\{s\}"(x)  {}\mRightarrow{}  x@0  \mleq{}  u))
            {}\mRightarrow{}  \mbackslash{}/(\mlambda{}s.\{s\}"(x))  \mleq{}  u)
13.  \mbackslash{}/(\mlambda{}s.\{s\}"(x))  \mleq{}  x
14.  \mbackslash{}/(\mlambda{}s.\{s\}"(x))  \mmember{}  Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
15.  \mbackslash{}/(\mlambda{}s.\{s\}"(x))  \mmember{}  \{ac:fset(fset(T))| 
                                            (\muparrow{}fset-antichain(eq;ac))  \mwedge{}  fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))\} 
16.  z  :  fset(T)
17.  z  \mmember{}  x
18.  \mforall{}[P:fset(T)  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[s:fset(fset(T))].
            uiff(\{x  \mmember{}  s  |  P[x]\}  =  \{\};\mneg{}(\mexists{}x:fset(T).  (x  \mmember{}  s  \mwedge{}  (\muparrow{}P[x]))))
19.  z  \mmember{}  \mbackslash{}/(\mlambda{}s.\{s\}"(x))
\mvdash{}  z  \msubseteq{}  z


By

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Latex:
EAuto  1

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