Step
*
2
2
1
1
of Lemma
mk-DeMorgan-algebra_wf
1. L : BoundedLatticeStructure
2. lattice-axioms(L)
3. bounded-lattice-axioms(L)
4. ∀[a,b,c:Point(L)]. (a ∧ b ∨ c = a ∧ b ∨ a ∧ c ∈ Point(L))
5. n : Point(L) ⟶ Point(L)
6. ∀x:Point(L). ((n (n x)) = x ∈ Point(L))
7. (∀x,y:Point(L). ((n x ∧ y) = n x ∨ n y ∈ Point(L))) ∨ (∀x,y:Point(L). ((n x ∨ y) = n x ∧ n y ∈ Point(L)))
8. L["neg" := n] ∈ BoundedLatticeStructure
9. L["neg" := n] ∈ DeMorganAlgebraStructure
⊢ L = L["neg" := n] ∈ BoundedLatticeStructure
BY
{ ((RecordExt THEN Try (Trivial))
THEN Try ((Reduce 0
THEN Folds ``lattice-point lattice-0 lattice-1`` 0
THEN ((RepeatFor 2 ((FunExt THENA Auto)) THEN Folds ``lattice-meet lattice-join`` 0 THEN Auto)
ORELSE Auto
)))
) }
1
1. L : BoundedLatticeStructure
2. lattice-axioms(L)
3. bounded-lattice-axioms(L)
4. ∀[a,b,c:Point(L)]. (a ∧ b ∨ c = a ∧ b ∨ a ∧ c ∈ Point(L))
5. n : Point(L) ⟶ Point(L)
6. ∀x:Point(L). ((n (n x)) = x ∈ Point(L))
7. (∀x,y:Point(L). ((n x ∧ y) = n x ∨ n y ∈ Point(L))) ∨ (∀x,y:Point(L). ((n x ∨ y) = n x ∧ n y ∈ Point(L)))
8. L["neg" := n] ∈ BoundedLatticeStructure
9. L["neg" := n] ∈ DeMorganAlgebraStructure
10. L ∈ record(x.Top)
11. L["neg" := n] ∈ record(x.Top)
⊢ L = L["neg" := n] ∈ record(x.Top)
Latex:
Latex:
1. L : BoundedLatticeStructure
2. lattice-axioms(L)
3. bounded-lattice-axioms(L)
4. \mforall{}[a,b,c:Point(L)]. (a \mwedge{} b \mvee{} c = a \mwedge{} b \mvee{} a \mwedge{} c)
5. n : Point(L) {}\mrightarrow{} Point(L)
6. \mforall{}x:Point(L). ((n (n x)) = x)
7. (\mforall{}x,y:Point(L). ((n x \mwedge{} y) = n x \mvee{} n y)) \mvee{} (\mforall{}x,y:Point(L). ((n x \mvee{} y) = n x \mwedge{} n y))
8. L["neg" := n] \mmember{} BoundedLatticeStructure
9. L["neg" := n] \mmember{} DeMorganAlgebraStructure
\mvdash{} L = L["neg" := n]
By
Latex:
((RecordExt THEN Try (Trivial))
THEN Try ((Reduce 0
THEN Folds ``lattice-point lattice-0 lattice-1`` 0
THEN ((RepeatFor 2 ((FunExt THENA Auto))
THEN Folds ``lattice-meet lattice-join`` 0
THEN Auto)
ORELSE Auto
)))
)
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