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Step * 1 1 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} 
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]))} 
17. {y ∈ x | deq-f-subset(eq) y x1} = {} ∈ fset(fset(T))
⊢ False
BY
{ ((InstLemma `fset-filter-is-empty` [⌜fset(T)⌝;⌜deq-fset(eq)⌝]⋅ THENA Auto)
   THEN (RWO "-1" (-2) THENA Auto)
   THEN D -2
   THEN Reduce 0
   THEN With ⌜x1⌝ (D 0)⋅
   THEN Auto) }

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]))} 
17. ∀[P:fset(T) ⟶ 𝔹]. ∀[s:fset(fset(T))].  uiff({x ∈ s | P[x]} = {} ∈ fset(fset(T));¬(∃x:fset(T). (x ∈ s ∧ (↑P[x]))))
18. x1 ∈ x
⊢ x1 ⊆ 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.  x1  :  \{s:fset(T)|  s  \mmember{}  x\} 
15.  x1  \mmember{}  x
16.  z  =  \{x1\}
17.  \{y  \mmember{}  x  |  deq-f-subset(eq)  y  x1\}  =  \{\}
\mvdash{}  False


By

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Latex:
((InstLemma  `fset-filter-is-empty`  [\mkleeneopen{}fset(T)\mkleeneclose{};\mkleeneopen{}deq-fset(eq)\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (RWO  "-1"  (-2)  THENA  Auto)
  THEN  D  -2
  THEN  Reduce  0
  THEN  With  \mkleeneopen{}x1\mkleeneclose{}  (D  0)\mcdot{}
  THEN  Auto)

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