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Step * 8 of Lemma free-dl_wf


1. X : Type
2. EquivRel(X List List;as,bs.dlattice-eq(X;as;bs))
3. ∀[a,b:free-dl-type(X)].  (free-dl-meet(a;b) = free-dl-meet(b;a) ∈ free-dl-type(X))
4. ∀[a,b:free-dl-type(X)].  (free-dl-join(a;b) = free-dl-join(b;a) ∈ free-dl-type(X))
5. ∀[a,b,c:free-dl-type(X)].  (free-dl-meet(a;free-dl-meet(b;c)) = free-dl-meet(free-dl-meet(a;b);c) ∈ free-dl-type(X))
6. ∀[a,b,c:free-dl-type(X)].  (free-dl-join(a;free-dl-join(b;c)) = free-dl-join(free-dl-join(a;b);c) ∈ free-dl-type(X))
7. ∀[a,b:free-dl-type(X)].  (free-dl-join(a;free-dl-meet(a;b)) = a ∈ free-dl-type(X))
8. a : free-dl-type(X)
9. b : free-dl-type(X)
⊢ free-dl-meet(a;free-dl-join(a;b)) = a ∈ free-dl-type(X)
BY
{ ((OnVar `a' newQuotientElim THENA Auto)
   THEN (OnVar `b' newQuotientElim THENA Auto)
   THEN EqTypeCD
   THEN Auto
   THEN RepUR ``free-dl-join`` 0
   THEN Auto
   THEN (Assert Trans(X List List;as,bs.dlattice-eq(X;as;bs)) BY
               Auto)
   THEN UseTrans ⌜as⌝⋅
   THEN All Thin
   THEN RenameVar `bs' (-1)
   THEN D 0
   THEN RWO  "dlattice-order-iff" 0
   THEN Auto) }

1
1. X : Type
2. as : X List List
3. bs : X List List
4. x : X List
5. (x ∈ as)
⊢ ∃y:X List. ((y ∈ free-dl-meet(as;as @ bs)) ∧ l_subset(X;y;x))

2
1. X : Type
2. as : X List List
3. bs : X List List
4. x : X List
5. (x ∈ free-dl-meet(as;as @ bs))
⊢ ∃y:X List. ((y ∈ as) ∧ l_subset(X;y;x))

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Latex:

---

Latex:

1.  X  :  Type
2.  EquivRel(X  List  List;as,bs.dlattice-eq(X;as;bs))
3.  \mforall{}[a,b:free-dl-type(X)].    (free-dl-meet(a;b)  =  free-dl-meet(b;a))
4.  \mforall{}[a,b:free-dl-type(X)].    (free-dl-join(a;b)  =  free-dl-join(b;a))
5.  \mforall{}[a,b,c:free-dl-type(X)].
          (free-dl-meet(a;free-dl-meet(b;c))  =  free-dl-meet(free-dl-meet(a;b);c))
6.  \mforall{}[a,b,c:free-dl-type(X)].
          (free-dl-join(a;free-dl-join(b;c))  =  free-dl-join(free-dl-join(a;b);c))
7.  \mforall{}[a,b:free-dl-type(X)].    (free-dl-join(a;free-dl-meet(a;b))  =  a)
8.  a  :  free-dl-type(X)
9.  b  :  free-dl-type(X)
\mvdash{}  free-dl-meet(a;free-dl-join(a;b))  =  a


By

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Latex:
((OnVar  `a'  newQuotientElim  THENA  Auto)
  THEN  (OnVar  `b'  newQuotientElim  THENA  Auto)
  THEN  EqTypeCD
  THEN  Auto
  THEN  RepUR  ``free-dl-join``  0
  THEN  Auto
  THEN  (Assert  Trans(X  List  List;as,bs.dlattice-eq(X;as;bs))  BY
                          Auto)
  THEN  UseTrans  \mkleeneopen{}as\mkleeneclose{}\mcdot{}
  THEN  All  Thin
  THEN  RenameVar  `bs'  (-1)
  THEN  D  0
  THEN  RWO    "dlattice-order-iff"  0
  THEN  Auto)

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