author | haftmann |
Sat, 27 Aug 2011 09:44:45 +0200 | |
changeset 44558 | cc878a312673 |
parent 44555 | da75ffe3d988 |
child 44563 | 01b2732cf4ad |
permissions | -rw-r--r-- |
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(* Author: Florian Haftmann, TU Muenchen *) |
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header {* A dedicated set type which is executable on its finite part *} |
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theory Cset |
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imports More_Set More_List |
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begin |
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subsection {* Lifting *} |
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typedef (open) 'a set = "UNIV :: 'a set set" |
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morphisms set_of Set by rule+ |
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hide_type (open) set |
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lemma set_of_Set [simp]: |
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"set_of (Set A) = A" |
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by (rule Set_inverse) rule |
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lemma Set_set_of [simp]: |
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"Set (set_of A) = A" |
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by (fact set_of_inverse) |
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definition member :: "'a Cset.set \<Rightarrow> 'a \<Rightarrow> bool" where |
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"member A x \<longleftrightarrow> x \<in> set_of A" |
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lemma member_set_of: |
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"set_of = member" |
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by (rule ext)+ (simp add: member_def mem_def) |
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lemma member_Set [simp]: |
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"member (Set A) x \<longleftrightarrow> x \<in> A" |
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by (simp add: member_def) |
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lemma Set_inject [simp]: |
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"Set A = Set B \<longleftrightarrow> A = B" |
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by (simp add: Set_inject) |
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lemma set_eq_iff: |
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"A = B \<longleftrightarrow> member A = member B" |
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by (auto simp add: fun_eq_iff set_of_inject [symmetric] member_def mem_def) |
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hide_fact (open) set_eq_iff |
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lemma set_eqI: |
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"member A = member B \<Longrightarrow> A = B" |
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by (simp add: Cset.set_eq_iff) |
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hide_fact (open) set_eqI |
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subsection {* Lattice instantiation *} |
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instantiation Cset.set :: (type) boolean_algebra |
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begin |
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definition less_eq_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where |
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[simp]: "A \<le> B \<longleftrightarrow> set_of A \<subseteq> set_of B" |
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definition less_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where |
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[simp]: "A < B \<longleftrightarrow> set_of A \<subset> set_of B" |
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definition inf_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where |
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[simp]: "inf A B = Set (set_of A \<inter> set_of B)" |
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definition sup_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where |
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[simp]: "sup A B = Set (set_of A \<union> set_of B)" |
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definition bot_set :: "'a Cset.set" where |
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[simp]: "bot = Set {}" |
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definition top_set :: "'a Cset.set" where |
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[simp]: "top = Set UNIV" |
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definition uminus_set :: "'a Cset.set \<Rightarrow> 'a Cset.set" where |
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[simp]: "- A = Set (- (set_of A))" |
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definition minus_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where |
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[simp]: "A - B = Set (set_of A - set_of B)" |
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instance proof |
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qed (auto intro!: Cset.set_eqI simp add: member_def mem_def) |
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end |
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instantiation Cset.set :: (type) complete_lattice |
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begin |
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definition Inf_set :: "'a Cset.set set \<Rightarrow> 'a Cset.set" where |
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[simp]: "Inf_set As = Set (Inf (image set_of As))" |
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definition Sup_set :: "'a Cset.set set \<Rightarrow> 'a Cset.set" where |
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[simp]: "Sup_set As = Set (Sup (image set_of As))" |
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instance proof |
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qed (auto simp add: le_fun_def) |
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end |
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instance Cset.set :: (type) complete_boolean_algebra proof |
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qed (unfold INF_def SUP_def, auto) |
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subsection {* Basic operations *} |
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abbreviation empty :: "'a Cset.set" where "empty \<equiv> bot" |
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hide_const (open) empty |
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abbreviation UNIV :: "'a Cset.set" where "UNIV \<equiv> top" |
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hide_const (open) UNIV |
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definition is_empty :: "'a Cset.set \<Rightarrow> bool" where |
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[simp]: "is_empty A \<longleftrightarrow> More_Set.is_empty (set_of A)" |
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definition insert :: "'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where |
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[simp]: "insert x A = Set (Set.insert x (set_of A))" |
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definition remove :: "'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where |
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[simp]: "remove x A = Set (More_Set.remove x (set_of A))" |
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definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a Cset.set \<Rightarrow> 'b Cset.set" where |
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[simp]: "map f A = Set (image f (set_of A))" |
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enriched_type map: map |
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by (simp_all add: fun_eq_iff image_compose) |
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definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where |
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[simp]: "filter P A = Set (More_Set.project P (set_of A))" |
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definition forall :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where |
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[simp]: "forall P A \<longleftrightarrow> Ball (set_of A) P" |
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definition exists :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where |
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[simp]: "exists P A \<longleftrightarrow> Bex (set_of A) P" |
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definition card :: "'a Cset.set \<Rightarrow> nat" where |
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[simp]: "card A = Finite_Set.card (set_of A)" |
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context complete_lattice |
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begin |
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definition Infimum :: "'a Cset.set \<Rightarrow> 'a" where |
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[simp]: "Infimum A = Inf (set_of A)" |
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definition Supremum :: "'a Cset.set \<Rightarrow> 'a" where |
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[simp]: "Supremum A = Sup (set_of A)" |
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end |
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subsection {* More operations *} |
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text {* conversion from @{typ "'a list"} *} |
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definition set :: "'a list \<Rightarrow> 'a Cset.set" where |
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"set xs = Set (List.set xs)" |
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hide_const (open) set |
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definition coset :: "'a list \<Rightarrow> 'a Cset.set" where |
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"coset xs = Set (- List.set xs)" |
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hide_const (open) coset |
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text {* conversion from @{typ "'a Predicate.pred"} *} |
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definition pred_of_cset :: "'a Cset.set \<Rightarrow> 'a Predicate.pred" where |
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[code del]: "pred_of_cset = Predicate.Pred \<circ> Cset.member" |
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definition of_pred :: "'a Predicate.pred \<Rightarrow> 'a Cset.set" where |
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"of_pred = Cset.Set \<circ> Collect \<circ> Predicate.eval" |
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definition of_seq :: "'a Predicate.seq \<Rightarrow> 'a Cset.set" where |
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"of_seq = of_pred \<circ> Predicate.pred_of_seq" |
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text {* monad operations *} |
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definition single :: "'a \<Rightarrow> 'a Cset.set" where |
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"single a = Set {a}" |
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definition bind :: "'a Cset.set \<Rightarrow> ('a \<Rightarrow> 'b Cset.set) \<Rightarrow> 'b Cset.set" (infixl "\<guillemotright>=" 70) where |
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"A \<guillemotright>= f = (SUP x : set_of A. f x)" |
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subsection {* Simplified simprules *} |
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lemma empty_simp [simp]: "member Cset.empty = bot" |
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by (simp add: fun_eq_iff bot_apply) |
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lemma UNIV_simp [simp]: "member Cset.UNIV = top" |
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by (simp add: fun_eq_iff top_apply) |
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lemma is_empty_simp [simp]: |
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"is_empty A \<longleftrightarrow> set_of A = {}" |
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by (simp add: More_Set.is_empty_def) |
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declare is_empty_def [simp del] |
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lemma remove_simp [simp]: |
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"remove x A = Set (set_of A - {x})" |
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by (simp add: More_Set.remove_def) |
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declare remove_def [simp del] |
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lemma filter_simp [simp]: |
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"filter P A = Set {x \<in> set_of A. P x}" |
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by (simp add: More_Set.project_def) |
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declare filter_def [simp del] |
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lemma set_of_set [simp]: |
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"set_of (Cset.set xs) = set xs" |
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by (simp add: set_def) |
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hide_fact (open) set_def |
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lemma member_set [simp]: |
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"member (Cset.set xs) = (\<lambda>x. x \<in> set xs)" |
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by (simp add: fun_eq_iff member_def) |
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hide_fact (open) member_set |
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lemma set_of_coset [simp]: |
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"set_of (Cset.coset xs) = - set xs" |
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by (simp add: coset_def) |
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hide_fact (open) coset_def |
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lemma member_coset [simp]: |
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"member (Cset.coset xs) = (\<lambda>x. x \<in> - set xs)" |
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by (simp add: fun_eq_iff member_def) |
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hide_fact (open) member_coset |
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lemma set_simps [simp]: |
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"Cset.set [] = Cset.empty" |
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"Cset.set (x # xs) = insert x (Cset.set xs)" |
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by(simp_all add: Cset.set_def) |
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226 |
|
44555 | 227 |
lemma member_SUP [simp]: |
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"member (SUPR A f) = SUPR A (member \<circ> f)" |
44555 | 229 |
by (auto simp add: fun_eq_iff SUP_apply member_def, unfold SUP_def, auto) |
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230 |
|
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lemma member_bind [simp]: |
44555 | 232 |
"member (P \<guillemotright>= f) = SUPR (set_of P) (member \<circ> f)" |
233 |
by (simp add: bind_def Cset.set_eq_iff) |
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234 |
|
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lemma member_single [simp]: |
44555 | 236 |
"member (single a) = (\<lambda>x. x \<in> {a})" |
237 |
by (simp add: single_def fun_eq_iff) |
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238 |
|
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lemma single_sup_simps [simp]: |
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shows single_sup: "sup (single a) A = insert a A" |
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and sup_single: "sup A (single a) = insert a A" |
44555 | 242 |
by (auto simp add: Cset.set_eq_iff single_def) |
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243 |
|
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lemma single_bind [simp]: |
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"single a \<guillemotright>= B = B a" |
44555 | 246 |
by (simp add: Cset.set_eq_iff SUP_insert single_def) |
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247 |
|
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lemma bind_bind: |
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"(A \<guillemotright>= B) \<guillemotright>= C = A \<guillemotright>= (\<lambda>x. B x \<guillemotright>= C)" |
44555 | 250 |
by (simp add: bind_def, simp only: SUP_def image_image, simp) |
251 |
||
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lemma bind_single [simp]: |
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"A \<guillemotright>= single = A" |
44555 | 254 |
by (simp add: Cset.set_eq_iff SUP_apply fun_eq_iff single_def member_def) |
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255 |
|
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256 |
lemma bind_const: "A \<guillemotright>= (\<lambda>_. B) = (if Cset.is_empty A then Cset.empty else B)" |
44555 | 257 |
by (auto simp add: Cset.set_eq_iff fun_eq_iff) |
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258 |
|
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259 |
lemma empty_bind [simp]: |
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"Cset.empty \<guillemotright>= f = Cset.empty" |
44555 | 261 |
by (simp add: Cset.set_eq_iff fun_eq_iff bot_apply) |
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262 |
|
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263 |
lemma member_of_pred [simp]: |
44555 | 264 |
"member (of_pred P) = (\<lambda>x. x \<in> {x. Predicate.eval P x})" |
265 |
by (simp add: of_pred_def fun_eq_iff) |
|
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266 |
|
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267 |
lemma member_of_seq [simp]: |
44555 | 268 |
"member (of_seq xq) = (\<lambda>x. x \<in> {x. Predicate.member xq x})" |
269 |
by (simp add: of_seq_def eval_member) |
|
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270 |
|
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lemma eval_pred_of_cset [simp]: |
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"Predicate.eval (pred_of_cset A) = Cset.member A" |
44555 | 273 |
by (simp add: pred_of_cset_def) |
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274 |
|
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subsection {* Default implementations *} |
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276 |
|
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277 |
lemma set_code [code]: |
44555 | 278 |
"Cset.set = (\<lambda>xs. fold insert xs Cset.empty)" |
279 |
proof (rule ext, rule Cset.set_eqI) |
|
280 |
fix xs :: "'a list" |
|
281 |
show "member (Cset.set xs) = member (fold insert xs Cset.empty)" |
|
282 |
by (simp add: fold_commute_apply [symmetric, where ?h = Set and ?g = Set.insert] |
|
283 |
fun_eq_iff Cset.set_def union_set [symmetric]) |
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284 |
qed |
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285 |
|
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lemma single_code [code]: |
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"single a = insert a Cset.empty" |
44555 | 288 |
by (simp add: Cset.single_def) |
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289 |
|
44558 | 290 |
lemma compl_set [simp]: |
291 |
"- Cset.set xs = Cset.coset xs" |
|
292 |
by (simp add: Cset.set_def Cset.coset_def) |
|
293 |
||
294 |
lemma compl_coset [simp]: |
|
295 |
"- Cset.coset xs = Cset.set xs" |
|
296 |
by (simp add: Cset.set_def Cset.coset_def) |
|
297 |
||
298 |
lemma member_cset_of: |
|
299 |
"member = set_of" |
|
300 |
by (rule ext)+ (simp add: member_def mem_def) |
|
301 |
||
302 |
lemma inter_project: |
|
303 |
"inf A (Cset.set xs) = Cset.set (List.filter (Cset.member A) xs)" |
|
304 |
"inf A (Cset.coset xs) = foldr Cset.remove xs A" |
|
305 |
proof - |
|
306 |
show "inf A (Cset.set xs) = Cset.set (List.filter (member A) xs)" |
|
307 |
by (simp add: inter project_def Cset.set_def member_def) |
|
308 |
have *: "\<And>x::'a. Cset.remove = (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member)" |
|
309 |
by (simp add: fun_eq_iff More_Set.remove_def member_cset_of) |
|
310 |
have "member \<circ> fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs = |
|
311 |
fold More_Set.remove xs \<circ> member" |
|
312 |
by (rule fold_commute) (simp add: fun_eq_iff mem_def) |
|
313 |
then have "fold More_Set.remove xs (member A) = |
|
314 |
member (fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs A)" |
|
315 |
by (simp add: fun_eq_iff) |
|
316 |
then have "inf A (Cset.coset xs) = fold Cset.remove xs A" |
|
317 |
by (simp add: Diff_eq [symmetric] minus_set * member_cset_of) |
|
318 |
moreover have "\<And>x y :: 'a. Cset.remove y \<circ> Cset.remove x = Cset.remove x \<circ> Cset.remove y" |
|
319 |
by (auto simp add: More_Set.remove_def * member_cset_of) |
|
320 |
ultimately show "inf A (Cset.coset xs) = foldr Cset.remove xs A" |
|
321 |
by (simp add: foldr_fold) |
|
322 |
qed |
|
323 |
||
324 |
lemma subtract_remove: |
|
325 |
"A - Cset.set xs = foldr Cset.remove xs A" |
|
326 |
"A - Cset.coset xs = Cset.set (List.filter (member A) xs)" |
|
327 |
by (simp_all only: diff_eq compl_set compl_coset inter_project) |
|
328 |
||
329 |
lemma union_insert: |
|
330 |
"sup (Cset.set xs) A = foldr Cset.insert xs A" |
|
331 |
"sup (Cset.coset xs) A = Cset.coset (List.filter (Not \<circ> member A) xs)" |
|
332 |
proof - |
|
333 |
have *: "\<And>x::'a. Cset.insert = (\<lambda>x. Set \<circ> Set.insert x \<circ> member)" |
|
334 |
by (simp add: fun_eq_iff member_cset_of) |
|
335 |
have "member \<circ> fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs = |
|
336 |
fold Set.insert xs \<circ> member" |
|
337 |
by (rule fold_commute) (simp add: fun_eq_iff mem_def) |
|
338 |
then have "fold Set.insert xs (member A) = |
|
339 |
member (fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs A)" |
|
340 |
by (simp add: fun_eq_iff) |
|
341 |
then have "sup (Cset.set xs) A = fold Cset.insert xs A" |
|
342 |
by (simp add: union_set * member_cset_of) |
|
343 |
moreover have "\<And>x y :: 'a. Cset.insert y \<circ> Cset.insert x = Cset.insert x \<circ> Cset.insert y" |
|
344 |
by (auto simp add: * member_cset_of) |
|
345 |
ultimately show "sup (Cset.set xs) A = foldr Cset.insert xs A" |
|
346 |
by (simp add: foldr_fold) |
|
347 |
show "sup (Cset.coset xs) A = Cset.coset (List.filter (Not \<circ> member A) xs)" |
|
348 |
by (auto simp add: Cset.coset_def member_cset_of mem_def) |
|
349 |
qed |
|
350 |
||
351 |
context complete_lattice |
|
352 |
begin |
|
353 |
||
354 |
lemma Infimum_inf: |
|
355 |
"Infimum (Cset.set As) = foldr inf As top" |
|
356 |
"Infimum (Cset.coset []) = bot" |
|
357 |
by (simp_all add: Inf_set_foldr) |
|
358 |
||
359 |
lemma Supremum_sup: |
|
360 |
"Supremum (Cset.set As) = foldr sup As bot" |
|
361 |
"Supremum (Cset.coset []) = top" |
|
362 |
by (simp_all add: Sup_set_foldr) |
|
363 |
||
364 |
end |
|
365 |
||
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366 |
lemma of_pred_code [code]: |
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367 |
"of_pred (Predicate.Seq f) = (case f () of |
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368 |
Predicate.Empty \<Rightarrow> Cset.empty |
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369 |
| Predicate.Insert x P \<Rightarrow> Cset.insert x (of_pred P) |
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370 |
| Predicate.Join P xq \<Rightarrow> sup (of_pred P) (of_seq xq))" |
44555 | 371 |
apply (auto split: seq.split simp add: Predicate.Seq_def of_pred_def Cset.set_eq_iff sup_apply eval_member [symmetric] member_def [symmetric] Collect_def mem_def member_set_of) |
372 |
apply (unfold Set.insert_def Collect_def sup_apply member_set_of) |
|
373 |
apply simp_all |
|
374 |
done |
|
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375 |
|
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376 |
lemma of_seq_code [code]: |
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377 |
"of_seq Predicate.Empty = Cset.empty" |
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378 |
"of_seq (Predicate.Insert x P) = Cset.insert x (of_pred P)" |
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379 |
"of_seq (Predicate.Join P xq) = sup (of_pred P) (of_seq xq)" |
44555 | 380 |
apply (auto simp add: of_seq_def of_pred_def Cset.set_eq_iff mem_def Collect_def) |
381 |
apply (unfold Set.insert_def Collect_def sup_apply member_set_of) |
|
382 |
apply simp_all |
|
383 |
done |
|
31846 | 384 |
|
44558 | 385 |
lemma bind_set: |
386 |
"Cset.bind (Cset.set xs) f = fold (sup \<circ> f) xs (Cset.set [])" |
|
387 |
by (simp add: Cset.bind_def SUPR_set_fold) |
|
388 |
hide_fact (open) bind_set |
|
389 |
||
390 |
lemma pred_of_cset_set: |
|
391 |
"pred_of_cset (Cset.set xs) = foldr sup (List.map Predicate.single xs) bot" |
|
392 |
proof - |
|
393 |
have "pred_of_cset (Cset.set xs) = Predicate.Pred (\<lambda>x. x \<in> set xs)" |
|
394 |
by (simp add: Cset.pred_of_cset_def member_set) |
|
395 |
moreover have "foldr sup (List.map Predicate.single xs) bot = \<dots>" |
|
396 |
by (induct xs) (auto simp add: bot_pred_def intro: pred_eqI, simp add: mem_def) |
|
397 |
ultimately show ?thesis by simp |
|
398 |
qed |
|
399 |
hide_fact (open) pred_of_cset_set |
|
400 |
||
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401 |
no_notation bind (infixl "\<guillemotright>=" 70) |
31849 | 402 |
|
43241 | 403 |
hide_const (open) is_empty insert remove map filter forall exists card |
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|
404 |
Inter Union bind single of_pred of_seq |
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|
405 |
|
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406 |
hide_fact (open) set_def pred_of_cset_def of_pred_def of_seq_def single_def |
44555 | 407 |
bind_def empty_simp UNIV_simp set_simps member_bind |
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408 |
member_single single_sup_simps single_sup sup_single single_bind |
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409 |
bind_bind bind_single bind_const empty_bind member_of_pred member_of_seq |
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|
410 |
eval_pred_of_cset set_code single_code of_pred_code of_seq_code |
31849 | 411 |
|
31807 | 412 |
end |