diff -r e72018e0dd75 -r 2b002c6b0f7d src/HOL/Quotient_Examples/List_Quotient_Set.thy --- a/src/HOL/Quotient_Examples/List_Quotient_Set.thy Mon Oct 31 19:12:41 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,191 +0,0 @@ -(* Title: HOL/Quotient_Examples/List_Quotient_Set.thy - Author: Florian Haftmann, Alexander Krauss, TU Muenchen -*) - -header {* Implementation of type Quotient_Set.set based on lists. Code equations obtained via quotient lifting. *} - -theory List_Quotient_Set -imports Quotient_Set -begin - -lemma [quot_respect]: "((op = ===> set_eq ===> set_eq) ===> op = ===> set_eq ===> set_eq) - foldr foldr" -by (simp add: fun_rel_eq) - -lemma [quot_preserve]: "((id ---> abs_set ---> rep_set) ---> id ---> rep_set ---> abs_set) foldr = foldr" -apply (rule ext)+ -by (induct_tac xa) (auto simp: Quotient_abs_rep[OF Quotient_set]) - - -subsection {* Relationship to lists *} - -(*FIXME: maybe define on sets first and then lift -> more canonical*) -definition coset :: "'a list \ 'a Quotient_Set.set" where - "coset xs = Quotient_Set.uminus (Quotient_Set.set xs)" - -code_datatype Quotient_Set.set List_Quotient_Set.coset - -lemma member_code [code]: - "member x (Quotient_Set.set xs) \ List.member xs x" - "member x (coset xs) \ \ List.member xs x" -unfolding coset_def -apply (lifting in_set_member) -by descending (simp add: in_set_member) - -definition (in term_syntax) - setify :: "'a\typerep list \ (unit \ Code_Evaluation.term) - \ 'a Quotient_Set.set \ (unit \ Code_Evaluation.term)" where - [code_unfold]: "setify xs = Code_Evaluation.valtermify Quotient_Set.set {\} xs" - -notation fcomp (infixl "\>" 60) -notation scomp (infixl "\\" 60) - -instantiation Quotient_Set.set :: (random) random -begin - -definition - "Quickcheck.random i = Quickcheck.random i \\ (\xs. Pair (setify xs))" - -instance .. - -end - -no_notation fcomp (infixl "\>" 60) -no_notation scomp (infixl "\\" 60) - -subsection {* Basic operations *} - -lemma is_empty_set [code]: - "Quotient_Set.is_empty (Quotient_Set.set xs) \ List.null xs" - by (lifting is_empty_set) -hide_fact (open) is_empty_set - -lemma empty_set [code]: - "Quotient_Set.empty = Quotient_Set.set []" - by (lifting set.simps(1)[symmetric]) -hide_fact (open) empty_set - -lemma UNIV_set [code]: - "Quotient_Set.UNIV = coset []" - unfolding coset_def by descending simp -hide_fact (open) UNIV_set - -lemma remove_set [code]: - "Quotient_Set.remove x (Quotient_Set.set xs) = Quotient_Set.set (removeAll x xs)" - "Quotient_Set.remove x (coset xs) = coset (List.insert x xs)" -unfolding coset_def -apply descending -apply (simp add: More_Set.remove_def) -apply descending -by (simp add: remove_set_compl) - -lemma insert_set [code]: - "Quotient_Set.insert x (Quotient_Set.set xs) = Quotient_Set.set (List.insert x xs)" - "Quotient_Set.insert x (coset xs) = coset (removeAll x xs)" -unfolding coset_def -apply (lifting set_insert[symmetric]) -by descending simp - -lemma map_set [code]: - "Quotient_Set.map f (Quotient_Set.set xs) = Quotient_Set.set (remdups (List.map f xs))" -by descending simp - -lemma filter_set [code]: - "Quotient_Set.filter P (Quotient_Set.set xs) = Quotient_Set.set (List.filter P xs)" -by descending (simp add: project_set) - -lemma forall_set [code]: - "Quotient_Set.forall (Quotient_Set.set xs) P \ list_all P xs" -(* FIXME: why does (lifting Ball_set_list_all) fail? *) -by descending (fact Ball_set_list_all) - -lemma exists_set [code]: - "Quotient_Set.exists (Quotient_Set.set xs) P \ list_ex P xs" -by descending (fact Bex_set_list_ex) - -lemma card_set [code]: - "Quotient_Set.card (Quotient_Set.set xs) = length (remdups xs)" -by (lifting length_remdups_card_conv[symmetric]) - -lemma compl_set [simp, code]: - "Quotient_Set.uminus (Quotient_Set.set xs) = coset xs" -unfolding coset_def by descending simp - -lemma compl_coset [simp, code]: - "Quotient_Set.uminus (coset xs) = Quotient_Set.set xs" -unfolding coset_def by descending simp - -lemma Inf_inf [code]: - "Quotient_Set.Inf (Quotient_Set.set (xs\'a\complete_lattice list)) = foldr inf xs top" - "Quotient_Set.Inf (coset ([]\'a\complete_lattice list)) = bot" - unfolding List_Quotient_Set.UNIV_set[symmetric] - by (lifting Inf_set_foldr Inf_UNIV) - -lemma Sup_sup [code]: - "Quotient_Set.Sup (Quotient_Set.set (xs\'a\complete_lattice list)) = foldr sup xs bot" - "Quotient_Set.Sup (coset ([]\'a\complete_lattice list)) = top" - unfolding List_Quotient_Set.UNIV_set[symmetric] - by (lifting Sup_set_foldr Sup_UNIV) - -subsection {* Derived operations *} - -lemma subset_eq_forall [code]: - "Quotient_Set.subset A B \ Quotient_Set.forall A (\x. member x B)" -by descending blast - -lemma subset_subset_eq [code]: - "Quotient_Set.psubset A B \ Quotient_Set.subset A B \ \ Quotient_Set.subset B A" -by descending blast - -instantiation Quotient_Set.set :: (type) equal -begin - -definition [code]: - "HOL.equal A B \ Quotient_Set.subset A B \ Quotient_Set.subset B A" - -instance -apply intro_classes -unfolding equal_set_def -by descending auto - -end - -lemma [code nbe]: - "HOL.equal (A :: 'a Quotient_Set.set) A \ True" - by (fact equal_refl) - - -subsection {* Functorial operations *} - -lemma inter_project [code]: - "Quotient_Set.inter A (Quotient_Set.set xs) = Quotient_Set.set (List.filter (\x. Quotient_Set.member x A) xs)" - "Quotient_Set.inter A (coset xs) = foldr Quotient_Set.remove xs A" -apply descending -apply auto -unfolding coset_def -apply descending -apply simp -by (metis diff_eq minus_set_foldr) - -lemma subtract_remove [code]: - "Quotient_Set.minus A (Quotient_Set.set xs) = foldr Quotient_Set.remove xs A" - "Quotient_Set.minus A (coset xs) = Quotient_Set.set (List.filter (\x. member x A) xs)" -unfolding coset_def -apply (lifting minus_set_foldr) -by descending auto - -lemma union_insert [code]: - "Quotient_Set.union (Quotient_Set.set xs) A = foldr Quotient_Set.insert xs A" - "Quotient_Set.union (coset xs) A = coset (List.filter (\x. \ member x A) xs)" -unfolding coset_def -apply (lifting union_set_foldr) -by descending auto - -lemma UNION_code [code]: - "Quotient_Set.UNION (Quotient_Set.set []) f = Quotient_Set.set []" - "Quotient_Set.UNION (Quotient_Set.set (x#xs)) f = - Quotient_Set.union (f x) (Quotient_Set.UNION (Quotient_Set.set xs) f)" - by (descending, simp)+ - - -end