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Step * 1 1 1 1 1 1 1 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. z : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} @i
14. ↓∃x1:{s:fset(T)| s ∈ x} . (x1 ∈ x ∧ (z = {x1} ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))))
⊢ fset-ac-le(eq;z;x)
BY
{ (Unfold `fset-ac-le` 0
   THEN Using [`eq',⌜deq-fset(eq)⌝] (BLemma `fset-all-iff`)⋅
   THEN Auto
   THEN ExRepD
   THEN (RWO "free-dlwc-point" (-3) THENA Auto)
   THEN (HypSubst' (-3) (-1) THENA 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. 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. z : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} @i
14. x1 : {s:fset(T)| s ∈ x} 
15. x1 ∈ x
16. z = {x1} ∈ {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))} 
⊢ ¬↑fset-null({y ∈ x | deq-f-subset(eq) y x1})

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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.  z  :  \{ac:fset(fset(T))| 
                  (\muparrow{}fset-antichain(eq;ac))  \mwedge{}  fset-all(ac;a.fset-contains-none(eq;a;x.Cs[x]))\}  @i
14.  \mdownarrow{}\mexists{}x1:\{s:fset(T)|  s  \mmember{}  x\}  .  (x1  \mmember{}  x  \mwedge{}  (z  =  \{x1\}))
\mvdash{}  fset-ac-le(eq;z;x)


By

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Latex:
(Unfold  `fset-ac-le`  0
  THEN  Using  [`eq',\mkleeneopen{}deq-fset(eq)\mkleeneclose{}]  (BLemma  `fset-all-iff`)\mcdot{}
  THEN  Auto
  THEN  ExRepD
  THEN  (RWO  "free-dlwc-point"  (-3)  THENA  Auto)
  THEN  (HypSubst'  (-3)  (-1)  THENA  Auto)
  THEN  (RWO  "member-fset-singleton"  (-1)  THENA  Auto)
  THEN  (HypSubst'  (-1)  0  THENA  Auto)
  THEN  RepeatFor  2  (Thin  (-1)))

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