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Step * 2 1 1 2 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 = \/(λs.{s}"(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. x ∈ fset(fset(T))
10. y : fset({s:fset(T)| s ∈ x} )
11. x = y ∈ fset({s:fset(T)| s ∈ x} )
12. s : fset(T)
13. s ∈ x
⊢ ↑fset-contains-none(eq;s;x.Cs[x])
BY
{ ((InstLemma `fset-all-iff` [⌜fset(T)⌝;⌜deq-fset(eq)⌝]⋅ THENA Auto)
   THEN (RWO "-1" 6 THENA Auto)
   THEN BHyp 6
   THEN Auto) }

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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  =  \mbackslash{}/(\mlambda{}s.\{s\}"(x))
8.  deq-fset(deq-fset(eq))  \mmember{}  EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
9.  x  \mmember{}  fset(fset(T))
10.  y  :  fset(\{s:fset(T)|  s  \mmember{}  x\}  )
11.  x  =  y
12.  s  :  fset(T)
13.  s  \mmember{}  x
\mvdash{}  \muparrow{}fset-contains-none(eq;s;x.Cs[x])


By

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Latex:
((InstLemma  `fset-all-iff`  [\mkleeneopen{}fset(T)\mkleeneclose{};\mkleeneopen{}deq-fset(eq)\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (RWO  "-1"  6  THENA  Auto)
  THEN  BHyp  6
  THEN  Auto)

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