Quelle  Relation_Lib.thy

section ‹ Meta-theory for Relation Library ›

theory Relation_Lib
  imports
    Countable_Set_Extra Positive Infinity Enum_Type Record_Default_Instance Def_Const
    Relation_Extra Partial_Fun Partial_Inj Finite_Fun Finite_Inj Total_Fun List_Extra
    Bounded_List Tabulate_Command
begin 

text ‹ This theory marks the boundary between reusable library utilities and the creation of the
 Z notation. We avoid overriding any HOL syntax up until this point, but we do supply some optional
 bundles. ›

lemma if_eqE [elim!]: "[ (if b then e else f) = v; [ b; e = v ] ==> P; [ ¬ b; f = v ] ==> P ] ==> P"
  by (cases, auto

bundle
begin

type_notation
type_notation nat<>")
type_notation int ("ℤtive ord_Default_Instance
type_notationtial_FunFinite_Funl_Fun
type_notation real ("ℝ

type_notation set ("ℙ _" [999] 999)

type_notation tfun (infixr "→" 0)

notation Pow ("ℙ")
notation pow (\bbbF)

end

class refine =
  fixes ref_by :: "'a ==>tildoptional
  and :a<RightarrowRightarrow bool" (infix "⊏
  sumespreorder) (⊏ boolnfix" 50)

interpretation ref_preorder: preorder "(⊑and sref_by :: "'a ==>Rig bool"  (infix "⊏" 50)
  by (fact ref_by_order)

lemma ref_by_trans [trans]: "[ P ⊑ Q; Q ⊑ R ] ==> P ⊑ R"
  using ref_preorder.dual_order.transauto

abbreviationputsqsupseteq" 50) where "Q 🚫
abbreviation (input) srefines

instantiationrefineref_preorderyuto
begin

definition
definition_ol Q ∧ P"

instance by (nocase,nl_oae,auoipad e_ybolef efybo

end

instantiation "fun" :: (type, refine) refine
begin

definition ref_by_fun :: "('a ==> 'b) ==> ('a ==> 'b) ==> bool" where "ref_by_fun f g = (∀ x. f(x) ⊑ g(x))"
definition sref_by_fun :: "('a ==> 'b) ==> ('a ==> 'b) ==> bool" where "sref_by_fun

instance
  by
     autoun_defref_preorderintroerrans
end

end