theory Foldsimports "Regular_Exp"beginsection {* ``Summation'' for regular expressions *}text {* To obtain equational system out of finite set of equivalence classes, a fold operation on finite sets @{text "folds"} is defined. The use of @{text "SOME"} makes @{text "folds"} more robust than the @{text "fold"} in the Isabelle library. The expression @{text "folds f"} makes sense when @{text "f"} is not @{text "associative"} and @{text "commutitive"}, while @{text "fold f"} does not. *}definition folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"where "folds f z S \<equiv> SOME x. fold_graph f z S x"text {* Plus-combination for a set of regular expressions *}abbreviation Setalt :: "'a rexp set \<Rightarrow> 'a rexp" ("\<Uplus>_" [1000] 999) where "\<Uplus>A \<equiv> folds Plus Zero A"text {* For finite sets, @{term Setalt} is preserved under @{term lang}.*}lemma folds_plus_simp [simp]: fixes rs::"('a rexp) set" assumes a: "finite rs" shows "lang (\<Uplus>rs) = \<Union> (lang ` rs)"unfolding folds_defapply(rule set_eqI)apply(rule someI2_ex)apply(rule_tac finite_imp_fold_graph[OF a])apply(erule fold_graph.induct)apply(auto)doneend