Free Boolean Algebra

This theory defines a type constructor representing the free Boolean algebra over a set of generators. Values of type (α)formula represent propositional formulas with uninterpreted variables from type α, ordered by implication. In addition to all the standard Boolean algebra operations, the library also provides a function for building homomorphisms to any other Boolean algebra type. 1 Free Boolean algebras theory Free-Boolean-Algebra imports Main begin 1.1 Free boolean algebra as a set We start by defining the free boolean algebra over type ′a as an inductive set. Here i :: ′a represents a variable; A :: ′a set represents a valuation, assigning a truth value to each variable; and S :: ′a set set represents a formula, as the set of valuations that make the formula true. The set fba contains representatives of formulas built from finite combinations of variables with negation and conjunction. inductive-set fba :: ′a set set set where var : {A. i ∈ A} ∈ fba | Compl : S ∈ fba =⇒ − S ∈ fba | inter : S ∈ fba =⇒ T ∈ fba =⇒ S ∩ T ∈ fba lemma fba-Diff : S ∈ fba =⇒ T ∈ fba =⇒ S − T ∈ fba unfolding Diff-eq by (intro fba.inter fba.Compl) lemma fba-union: S ∈ fba =⇒ T ∈ fba =⇒ S ∪ T ∈ fba proof − assume S ∈ fba and T ∈ fba