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


1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
5. x = \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
6. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7. x ∈ fset(fset(T))
8. y : fset({s:fset(T)| s ∈ x} )
9. x = y ∈ fset({s:fset(T)| s ∈ x} )
10. s : {s:fset(T)| s ∈ x} 
⊢ ↑fset-contains-none(eq;s;x.Cs[x])
BY
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1
1. T : Type
2. eq : EqDecider(T)
3. Cs : T ⟶ fset(fset(T))
4. x : Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
5. x = \/(λs.{s}"(x)) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
6. deq-fset(deq-fset(eq)) ∈ EqDecider(Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
7. x ∈ fset(fset(T))
8. y : fset({s:fset(T)| s ∈ x} )
9. x = y ∈ fset({s:fset(T)| s ∈ x} )
10. s : fset(T)
11. s ∈ x
⊢ ↑fset-contains-none(eq;s;x.Cs[x])

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


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