author | haftmann |
Sat, 24 Dec 2011 15:53:10 +0100 | |
changeset 45970 | b6d0cff57d96 |
parent 44563 | 01b2732cf4ad |
child 47232 | e2f0176149d0 |
permissions | -rw-r--r-- |
43146 | 1 |
(* Author: Florian Haftmann, TU Muenchen *) |
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header {* Canonical implementation of sets by distinct lists *} |
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theory Dlist_Cset |
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imports Dlist Cset |
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begin |
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definition Set :: "'a dlist \<Rightarrow> 'a Cset.set" where |
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parents:
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"Set dxs = Cset.set (list_of_dlist dxs)" |
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definition Coset :: "'a dlist \<Rightarrow> 'a Cset.set" where |
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"Coset dxs = Cset.coset (list_of_dlist dxs)" |
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code_datatype Set Coset |
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lemma Set_Dlist [simp]: |
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"Set (Dlist xs) = Cset.set xs" |
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by (rule Cset.set_eqI) (simp add: Set_def) |
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lemma Coset_Dlist [simp]: |
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"Coset (Dlist xs) = Cset.coset xs" |
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by (rule Cset.set_eqI) (simp add: Coset_def) |
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lemma member_Set [simp]: |
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"Cset.member (Set dxs) = List.member (list_of_dlist dxs)" |
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by (simp add: Set_def fun_eq_iff List.member_def) |
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lemma member_Coset [simp]: |
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"Cset.member (Coset dxs) = Not \<circ> List.member (list_of_dlist dxs)" |
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by (simp add: Coset_def fun_eq_iff List.member_def) |
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lemma Set_dlist_of_list [code]: |
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"Cset.set xs = Set (dlist_of_list xs)" |
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by (rule Cset.set_eqI) simp |
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lemma Coset_dlist_of_list [code]: |
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"Cset.coset xs = Coset (dlist_of_list xs)" |
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by (rule Cset.set_eqI) simp |
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lemma is_empty_Set [code]: |
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"Cset.is_empty (Set dxs) \<longleftrightarrow> Dlist.null dxs" |
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by (simp add: Dlist.null_def List.null_def Set_def) |
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lemma bot_code [code]: |
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"bot = Set Dlist.empty" |
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by (simp add: empty_def) |
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lemma top_code [code]: |
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"top = Coset Dlist.empty" |
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by (simp add: empty_def Cset.coset_def) |
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lemma insert_code [code]: |
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"Cset.insert x (Set dxs) = Set (Dlist.insert x dxs)" |
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"Cset.insert x (Coset dxs) = Coset (Dlist.remove x dxs)" |
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by (simp_all add: Dlist.insert_def Dlist.remove_def Cset.set_def Cset.coset_def Set_def Coset_def) |
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lemma remove_code [code]: |
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"Cset.remove x (Set dxs) = Set (Dlist.remove x dxs)" |
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"Cset.remove x (Coset dxs) = Coset (Dlist.insert x dxs)" |
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by (simp_all add: Dlist.insert_def Dlist.remove_def Cset.set_def Cset.coset_def Set_def Coset_def Compl_insert) |
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lemma member_code [code]: |
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"Cset.member (Set dxs) = Dlist.member dxs" |
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"Cset.member (Coset dxs) = Not \<circ> Dlist.member dxs" |
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by (simp_all add: List.member_def member_def fun_eq_iff Dlist.member_def) |
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lemma compl_code [code]: |
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"- Set dxs = Coset dxs" |
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"- Coset dxs = Set dxs" |
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by (rule Cset.set_eqI, simp add: fun_eq_iff List.member_def Set_def Coset_def)+ |
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lemma map_code [code]: |
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"Cset.map f (Set dxs) = Set (Dlist.map f dxs)" |
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by (rule Cset.set_eqI) (simp add: fun_eq_iff List.member_def Set_def) |
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lemma filter_code [code]: |
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"Cset.filter f (Set dxs) = Set (Dlist.filter f dxs)" |
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by (rule Cset.set_eqI) (simp add: fun_eq_iff List.member_def Set_def) |
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lemma forall_Set [code]: |
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"Cset.forall P (Set xs) \<longleftrightarrow> list_all P (list_of_dlist xs)" |
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by (simp add: Set_def list_all_iff) |
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lemma exists_Set [code]: |
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"Cset.exists P (Set xs) \<longleftrightarrow> list_ex P (list_of_dlist xs)" |
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by (simp add: Set_def list_ex_iff) |
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lemma card_code [code]: |
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"Cset.card (Set dxs) = Dlist.length dxs" |
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by (simp add: length_def Set_def distinct_card) |
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lemma inter_code [code]: |
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"inf A (Set xs) = Set (Dlist.filter (Cset.member A) xs)" |
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"inf A (Coset xs) = Dlist.foldr Cset.remove xs A" |
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by (simp_all only: Set_def Coset_def foldr_def inter_project list_of_dlist_filter) |
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lemma subtract_code [code]: |
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"A - Set xs = Dlist.foldr Cset.remove xs A" |
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"A - Coset xs = Set (Dlist.filter (Cset.member A) xs)" |
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by (simp_all only: Set_def Coset_def foldr_def subtract_remove list_of_dlist_filter) |
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lemma union_code [code]: |
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"sup (Set xs) A = Dlist.foldr Cset.insert xs A" |
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"sup (Coset xs) A = Coset (Dlist.filter (Not \<circ> Cset.member A) xs)" |
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by (simp_all only: Set_def Coset_def foldr_def union_insert list_of_dlist_filter) |
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context complete_lattice |
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begin |
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lemma Infimum_code [code]: |
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"Infimum (Set As) = Dlist.foldr inf As top" |
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by (simp only: Set_def Infimum_inf foldr_def inf.commute) |
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lemma Supremum_code [code]: |
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"Supremum (Set As) = Dlist.foldr sup As bot" |
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by (simp only: Set_def Supremum_sup foldr_def sup.commute) |
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end |
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declare Cset.single_code [code] |
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Andreas Lochbihler
parents:
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parents:
43241
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lemma bind_set [code]: |
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"Cset.bind (Dlist_Cset.Set xs) f = fold (sup \<circ> f) (list_of_dlist xs) Cset.empty" |
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by (simp add: Cset.bind_set Set_def) |
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parents:
43241
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hide_fact (open) bind_set |
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parents:
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lemma pred_of_cset_set [code]: |
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"pred_of_cset (Dlist_Cset.Set xs) = foldr sup (map Predicate.single (list_of_dlist xs)) bot" |
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by (simp add: Cset.pred_of_cset_set Dlist_Cset.Set_def) |
43971
892030194015
added operations to Cset with code equations in backing implementations
Andreas Lochbihler
parents:
43241
diff
changeset
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hide_fact (open) pred_of_cset_set |
892030194015
added operations to Cset with code equations in backing implementations
Andreas Lochbihler
parents:
43241
diff
changeset
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end |