src/HOL/Quotient_Examples/Cset.thy

added UNION

(*  Title:      HOL/Quotient_Examples/Cset.thy
    Author:     Florian Haftmann, Alexander Krauss, TU Muenchen
*)

header {* A variant of theory Cset from Library, defined as a quotient *}

theory Cset
imports "~~/src/HOL/Library/More_Set" "~~/src/HOL/Library/More_List" "~~/src/HOL/Library/Quotient_Syntax"
begin

subsection {* Lifting *}

(*FIXME: quotient package requires a dedicated constant*)
definition set_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where [simp]: "set_eq \<equiv> op ="

quotient_type 'a set = "'a Set.set" / "set_eq"
by (simp add: identity_equivp)

hide_type (open) set

subsection {* Operations *}

lemma [quot_respect]:
  "(op = ===> set_eq ===> op =) (op \<in>) (op \<in>)"
  "(op = ===> set_eq) Collect Collect"
  "(set_eq ===> op =) More_Set.is_empty More_Set.is_empty"
  "(op = ===> set_eq ===> set_eq) Set.insert Set.insert"
  "(op = ===> set_eq ===> set_eq) More_Set.remove More_Set.remove"
  "(op = ===> set_eq ===> set_eq) image image"
  "(op = ===> set_eq ===> set_eq) More_Set.project More_Set.project"
  "(set_eq ===> op =) Ball Ball"
  "(set_eq ===> op =) Bex Bex"
  "(set_eq ===> op =) Finite_Set.card Finite_Set.card"
  "(set_eq ===> set_eq ===> op =) (op \<subseteq>) (op \<subseteq>)"
  "(set_eq ===> set_eq ===> op =) (op \<subset>) (op \<subset>)"
  "(set_eq ===> set_eq ===> set_eq) (op \<inter>) (op \<inter>)"
  "(set_eq ===> set_eq ===> set_eq) (op \<union>) (op \<union>)"
  "set_eq {} {}"
  "set_eq UNIV UNIV"
  "(set_eq ===> set_eq) uminus uminus"
  "(set_eq ===> set_eq ===> set_eq) minus minus"
  "((set_eq ===> op =) ===> set_eq) Inf Inf"
  "((set_eq ===> op =) ===> set_eq) Sup Sup"
  "(op = ===> set_eq) List.set List.set"
  "(set_eq ===> (op = ===> set_eq) ===> set_eq) UNION UNION"
by (auto simp: fun_rel_eq)

quotient_definition "member :: 'a => 'a Cset.set => bool"
is "op \<in>"
quotient_definition "Set :: ('a => bool) => 'a Cset.set"
is Collect
quotient_definition is_empty where "is_empty :: 'a Cset.set \<Rightarrow> bool"
is More_Set.is_empty
quotient_definition insert where "insert :: 'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set"
is Set.insert
quotient_definition remove where "remove :: 'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set"
is More_Set.remove
quotient_definition map where "map :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a Cset.set \<Rightarrow> 'b Cset.set"
is image
quotient_definition filter where "filter :: ('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set"
is More_Set.project
quotient_definition "forall :: 'a Cset.set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
is Ball
quotient_definition "exists :: 'a Cset.set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
is Bex
quotient_definition card where "card :: 'a Cset.set \<Rightarrow> nat"
is Finite_Set.card
quotient_definition set where "set :: 'a list => 'a Cset.set"
is List.set
quotient_definition subset where "subset :: 'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> bool"
is "op \<subseteq> :: 'a set \<Rightarrow> 'a set \<Rightarrow> bool"
quotient_definition psubset where "psubset :: 'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> bool"
is "op \<subset> :: 'a set \<Rightarrow> 'a set \<Rightarrow> bool"
quotient_definition inter where "inter :: 'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set"
is "op \<inter> :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set"
quotient_definition union where "union :: 'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set"
is "op \<union> :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set"
quotient_definition empty where "empty :: 'a Cset.set"
is "{} :: 'a set"
quotient_definition UNIV where "UNIV :: 'a Cset.set"
is "Set.UNIV :: 'a set"
quotient_definition uminus where "uminus :: 'a Cset.set \<Rightarrow> 'a Cset.set"
is "uminus_class.uminus :: 'a set \<Rightarrow> 'a set"
quotient_definition minus where "minus :: 'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set"
is "(op -) :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set"
quotient_definition Inf where "Inf :: 'a Cset.set set \<Rightarrow> 'a Cset.set"
is "Inf_class.Inf :: 'a set set \<Rightarrow> 'a set"
quotient_definition Sup where "Sup :: 'a Cset.set set \<Rightarrow> 'a Cset.set"
is "Sup_class.Sup :: 'a set set \<Rightarrow> 'a set"
quotient_definition UNION where "UNION :: 'a Cset.set \<Rightarrow> ('a \<Rightarrow> 'b Cset.set) \<Rightarrow> 'b Cset.set"
is "Complete_Lattice.UNION :: 'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"


context complete_lattice
begin

(* FIXME: Would like to use 

quotient_definition "Infimum :: 'a Cset.set \<Rightarrow> 'a"
is "Inf"

but it fails, presumably because @{term "Inf"} is a Free.
*)

definition Infimum :: "'a Cset.set \<Rightarrow> 'a" where
  [simp]: "Infimum A = Inf (\<lambda>x. member x A)"

definition Supremum :: "'a Cset.set \<Rightarrow> 'a" where
  [simp]: "Supremum A = Sup (\<lambda>x. member x A)"

end

hide_const (open) is_empty insert remove map filter forall exists card
  set subset psubset inter union empty UNIV uminus minus Inf Sup UNION

hide_fact (open) is_empty_def insert_def remove_def map_def filter_def
  forall_def exists_def card_def set_def subset_def psubset_def
  inter_def union_def empty_def UNIV_def uminus_def minus_def Inf_def Sup_def
  UNION_def


end