src/HOL/ex/PER.thy

no longer declare .psimps rules as [simp].

This regularly caused confusion (e.g., they show up in simp traces
when the regular simp rules are disabled). In the few places where the
rules are used, explicitly mentioning them actually clarifies the
proof text.

(*  Title:      HOL/ex/PER.thy
    Author:     Oscar Slotosch and Markus Wenzel, TU Muenchen
*)

header {* Partial equivalence relations *}

theory PER imports Main begin

text {*
  Higher-order quotients are defined over partial equivalence
  relations (PERs) instead of total ones.  We provide axiomatic type
  classes @{text "equiv < partial_equiv"} and a type constructor
  @{text "'a quot"} with basic operations.  This development is based
  on:

  Oscar Slotosch: \emph{Higher Order Quotients and their
  Implementation in Isabelle HOL.}  Elsa L. Gunter and Amy Felty,
  editors, Theorem Proving in Higher Order Logics: TPHOLs '97,
  Springer LNCS 1275, 1997.
*}


subsection {* Partial equivalence *}

text {*
  Type class @{text partial_equiv} models partial equivalence
  relations (PERs) using the polymorphic @{text "\<sim> :: 'a => 'a =>
  bool"} relation, which is required to be symmetric and transitive,
  but not necessarily reflexive.
*}

class partial_equiv =
  fixes eqv :: "'a => 'a => bool"    (infixl "\<sim>" 50)
  assumes partial_equiv_sym [elim?]: "x \<sim> y ==> y \<sim> x"
  assumes partial_equiv_trans [trans]: "x \<sim> y ==> y \<sim> z ==> x \<sim> z"

text {*
  \medskip The domain of a partial equivalence relation is the set of
  reflexive elements.  Due to symmetry and transitivity this
  characterizes exactly those elements that are connected with
  \emph{any} other one.
*}

definition
  "domain" :: "'a::partial_equiv set" where
  "domain = {x. x \<sim> x}"

lemma domainI [intro]: "x \<sim> x ==> x \<in> domain"
  unfolding domain_def by blast

lemma domainD [dest]: "x \<in> domain ==> x \<sim> x"
  unfolding domain_def by blast

theorem domainI' [elim?]: "x \<sim> y ==> x \<in> domain"
proof
  assume xy: "x \<sim> y"
  also from xy have "y \<sim> x" ..
  finally show "x \<sim> x" .
qed


subsection {* Equivalence on function spaces *}

text {*
  The @{text \<sim>} relation is lifted to function spaces.  It is
  important to note that this is \emph{not} the direct product, but a
  structural one corresponding to the congruence property.
*}

instantiation "fun" :: (partial_equiv, partial_equiv) partial_equiv
begin

definition
  eqv_fun_def: "f \<sim> g == \<forall>x \<in> domain. \<forall>y \<in> domain. x \<sim> y --> f x \<sim> g y"

lemma partial_equiv_funI [intro?]:
    "(!!x y. x \<in> domain ==> y \<in> domain ==> x \<sim> y ==> f x \<sim> g y) ==> f \<sim> g"
  unfolding eqv_fun_def by blast

lemma partial_equiv_funD [dest?]:
    "f \<sim> g ==> x \<in> domain ==> y \<in> domain ==> x \<sim> y ==> f x \<sim> g y"
  unfolding eqv_fun_def by blast

text {*
  The class of partial equivalence relations is closed under function
  spaces (in \emph{both} argument positions).
*}

instance proof
  fix f g h :: "'a::partial_equiv => 'b::partial_equiv"
  assume fg: "f \<sim> g"
  show "g \<sim> f"
  proof
    fix x y :: 'a
    assume x: "x \<in> domain" and y: "y \<in> domain"
    assume "x \<sim> y" then have "y \<sim> x" ..
    with fg y x have "f y \<sim> g x" ..
    then show "g x \<sim> f y" ..
  qed
  assume gh: "g \<sim> h"
  show "f \<sim> h"
  proof
    fix x y :: 'a
    assume x: "x \<in> domain" and y: "y \<in> domain" and "x \<sim> y"
    with fg have "f x \<sim> g y" ..
    also from y have "y \<sim> y" ..
    with gh y y have "g y \<sim> h y" ..
    finally show "f x \<sim> h y" .
  qed
qed

end


subsection {* Total equivalence *}

text {*
  The class of total equivalence relations on top of PERs.  It
  coincides with the standard notion of equivalence, i.e.\ @{text "\<sim>
  :: 'a => 'a => bool"} is required to be reflexive, transitive and
  symmetric.
*}

class equiv =
  assumes eqv_refl [intro]: "x \<sim> x"

text {*
  On total equivalences all elements are reflexive, and congruence
  holds unconditionally.
*}

theorem equiv_domain [intro]: "(x::'a::equiv) \<in> domain"
proof
  show "x \<sim> x" ..
qed

theorem equiv_cong [dest?]: "f \<sim> g ==> x \<sim> y ==> f x \<sim> g (y::'a::equiv)"
proof -
  assume "f \<sim> g"
  moreover have "x \<in> domain" ..
  moreover have "y \<in> domain" ..
  moreover assume "x \<sim> y"
  ultimately show ?thesis ..
qed


subsection {* Quotient types *}

text {*
  The quotient type @{text "'a quot"} consists of all
  \emph{equivalence classes} over elements of the base type @{typ 'a}.
*}

typedef 'a quot = "{{x. a \<sim> x}| a::'a::partial_equiv. True}"
  by blast

lemma quotI [intro]: "{x. a \<sim> x} \<in> quot"
  unfolding quot_def by blast

lemma quotE [elim]: "R \<in> quot ==> (!!a. R = {x. a \<sim> x} ==> C) ==> C"
  unfolding quot_def by blast

text {*
  \medskip Abstracted equivalence classes are the canonical
  representation of elements of a quotient type.
*}

definition
  eqv_class :: "('a::partial_equiv) => 'a quot"    ("\<lfloor>_\<rfloor>") where
  "\<lfloor>a\<rfloor> = Abs_quot {x. a \<sim> x}"

theorem quot_rep: "\<exists>a. A = \<lfloor>a\<rfloor>"
proof (cases A)
  fix R assume R: "A = Abs_quot R"
  assume "R \<in> quot" then have "\<exists>a. R = {x. a \<sim> x}" by blast
  with R have "\<exists>a. A = Abs_quot {x. a \<sim> x}" by blast
  then show ?thesis by (unfold eqv_class_def)
qed

lemma quot_cases [cases type: quot]:
  obtains (rep) a where "A = \<lfloor>a\<rfloor>"
  using quot_rep by blast


subsection {* Equality on quotients *}

text {*
  Equality of canonical quotient elements corresponds to the original
  relation as follows.
*}

theorem eqv_class_eqI [intro]: "a \<sim> b ==> \<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
proof -
  assume ab: "a \<sim> b"
  have "{x. a \<sim> x} = {x. b \<sim> x}"
  proof (rule Collect_cong)
    fix x show "(a \<sim> x) = (b \<sim> x)"
    proof
      from ab have "b \<sim> a" ..
      also assume "a \<sim> x"
      finally show "b \<sim> x" .
    next
      note ab
      also assume "b \<sim> x"
      finally show "a \<sim> x" .
    qed
  qed
  then show ?thesis by (simp only: eqv_class_def)
qed

theorem eqv_class_eqD' [dest?]: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor> ==> a \<in> domain ==> a \<sim> b"
proof (unfold eqv_class_def)
  assume "Abs_quot {x. a \<sim> x} = Abs_quot {x. b \<sim> x}"
  then have "{x. a \<sim> x} = {x. b \<sim> x}" by (simp only: Abs_quot_inject quotI)
  moreover assume "a \<in> domain" then have "a \<sim> a" ..
  ultimately have "a \<in> {x. b \<sim> x}" by blast
  then have "b \<sim> a" by blast
  then show "a \<sim> b" ..
qed

theorem eqv_class_eqD [dest?]: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor> ==> a \<sim> (b::'a::equiv)"
proof (rule eqv_class_eqD')
  show "a \<in> domain" ..
qed

lemma eqv_class_eq' [simp]: "a \<in> domain ==> (\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
  using eqv_class_eqI eqv_class_eqD' by (blast del: eqv_refl)

lemma eqv_class_eq [simp]: "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> (b::'a::equiv))"
  using eqv_class_eqI eqv_class_eqD by blast


subsection {* Picking representing elements *}

definition
  pick :: "'a::partial_equiv quot => 'a" where
  "pick A = (SOME a. A = \<lfloor>a\<rfloor>)"

theorem pick_eqv' [intro?, simp]: "a \<in> domain ==> pick \<lfloor>a\<rfloor> \<sim> a"
proof (unfold pick_def)
  assume a: "a \<in> domain"
  show "(SOME x. \<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>) \<sim> a"
  proof (rule someI2)
    show "\<lfloor>a\<rfloor> = \<lfloor>a\<rfloor>" ..
    fix x assume "\<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>"
    from this and a have "a \<sim> x" ..
    then show "x \<sim> a" ..
  qed
qed

theorem pick_eqv [intro, simp]: "pick \<lfloor>a\<rfloor> \<sim> (a::'a::equiv)"
proof (rule pick_eqv')
  show "a \<in> domain" ..
qed

theorem pick_inverse: "\<lfloor>pick A\<rfloor> = (A::'a::equiv quot)"
proof (cases A)
  fix a assume a: "A = \<lfloor>a\<rfloor>"
  then have "pick A \<sim> a" by simp
  then have "\<lfloor>pick A\<rfloor> = \<lfloor>a\<rfloor>" by simp
  with a show ?thesis by simp
qed

end