src/HOL/Quotient_Examples/Lift_Set.thy

adjusted to set/pred distinction by means of type constructor `set`

(*  Title:      HOL/Quotient_Examples/Lift_Set.thy
    Author:     Lukas Bulwahn and Ondrej Kuncar
*)

header {* Example of lifting definitions with the quotient infrastructure *}

theory Lift_Set
imports Main
begin

definition set where "set = (UNIV :: ('a \<Rightarrow> bool) set)"

typedef (open) 'a set = "set :: ('a \<Rightarrow> bool) set"
  morphisms member Set
  unfolding set_def by auto

text {* Here is some ML setup that should eventually be incorporated in the typedef command. *}

local_setup {* fn lthy =>
  let
    val quotients =
      {qtyp = @{typ "'a set"}, rtyp = @{typ "'a => bool"},
        equiv_rel = @{term "dummy"}, equiv_thm = @{thm refl}}
    val qty_full_name = @{type_name "set"}

    fun qinfo phi = Quotient_Info.transform_quotients phi quotients
  in
    lthy
    |> Local_Theory.declaration {syntax = false, pervasive = true}
        (fn phi =>
          Quotient_Info.update_quotients qty_full_name (qinfo phi) #>
          Quotient_Info.update_abs_rep qty_full_name
            (Quotient_Info.transform_abs_rep phi {abs = @{term "Set"}, rep = @{term "member"}}))
  end
*}

text {* Now, we can employ quotient_definition to lift definitions. *}

quotient_definition empty where "empty :: 'a set"
is "bot :: 'a \<Rightarrow> bool"

term "Lift_Set.empty"
thm Lift_Set.empty_def

definition insertp :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" where
  "insertp x P y \<longleftrightarrow> y = x \<or> P y"

quotient_definition insert where "insert :: 'a => 'a set => 'a set"
is insertp

term "Lift_Set.insert"
thm Lift_Set.insert_def

end