Theory Quotient_Type

(*  Title:      HOL/Library/Quotient_Type.thy
    Author:     Markus Wenzel, TU Muenchen

section ‹Quotient types›

theory Quotient_Type
imports Main

text ‹We introduce the notion of quotient types over equivalence relations
  via type classes.›

subsection ‹Equivalence relations and quotient types›

text ‹Type class equiv› models equivalence relations
  ∼ :: 'a ⇒ 'a ⇒ bool›.›

class eqv =
  fixes eqv :: "'a  'a  bool"  (infixl "" 50)

class equiv = eqv +
  assumes equiv_refl [intro]: "x  x"
    and equiv_trans [trans]: "x  y  y  z  x  z"
    and equiv_sym [sym]: "x  y  y  x"

lemma equiv_not_sym [sym]: "¬ x  y  ¬ y  x"
proof -
  assume "¬ x  y"
  then show "¬ y  x" by (rule contrapos_nn) (rule equiv_sym)

lemma not_equiv_trans1 [trans]: "¬ x  y  y  z  ¬ x  z"
proof -
  assume "¬ x  y" and "y  z"
  show "¬ x  z"
    assume "x  z"
    also from y  z have "z  y" ..
    finally have "x  y" .
    with ¬ x  y show False by contradiction

lemma not_equiv_trans2 [trans]: "x  y  ¬ y  z  ¬ x  z"
proof -
  assume "¬ y  z"
  then have "¬ z  y" ..
  assume "x  y"
  then have "y  x" ..
  finally have "¬ z  x" .
  then show "¬ x  z" ..


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

definition (in eqv) "quot = {{x. a  x} | a. True}"

typedef (overloaded) 'a quot = "quot :: 'a::eqv set set"
  unfolding quot_def by blast

lemma quotI [intro]: "{x. a  x}  quot"
  unfolding quot_def by blast

lemma quotE [elim]:
  assumes "R  quot"
  obtains a where "R = {x. a  x}"
  using assms unfolding quot_def by blast

text ‹Abstracted equivalence classes are the canonical representation of
  elements of a quotient type.›

definition "class" :: "'a::equiv  'a quot"  ("_")
  where "a = Abs_quot {x. a  x}"

theorem quot_exhaust: "a. A = a"
proof (cases A)
  fix R
  assume R: "A = Abs_quot R"
  assume "R  quot"
  then have "a. R = {x. a  x}" by blast
  with R have "a. A = Abs_quot {x. a  x}" by blast
  then show ?thesis unfolding class_def .

lemma quot_cases [cases type: quot]:
  obtains a where "A = a"
  using quot_exhaust by blast

subsection ‹Equality on quotients›

text ‹Equality of canonical quotient elements coincides with the original

theorem quot_equality [iff?]: "a = b  a  b"
  assume eq: "a = b"
  show "a  b"
  proof -
    from eq have "{x. a  x} = {x. b  x}"
      by (simp only: class_def Abs_quot_inject quotI)
    moreover have "a  a" ..
    ultimately have "a  {x. b  x}" by blast
    then have "b  a" by blast
    then show ?thesis ..
  assume ab: "a  b"
  show "a = b"
  proof -
    have "{x. a  x} = {x. b  x}"
    proof (rule Collect_cong)
      fix x show "(a  x) = (b  x)"
        from ab have "b  a" ..
        also assume "a  x"
        finally show "b  x" .
        note ab
        also assume "b  x"
        finally show "a  x" .
    then show ?thesis by (simp only: class_def)

subsection ‹Picking representing elements›

definition pick :: "'a::equiv quot  'a"
  where "pick A = (SOME a. A = a)"

theorem pick_equiv [intro]: "pick a  a"
proof (unfold pick_def)
  show "(SOME x. a = x)  a"
  proof (rule someI2)
    show "a = a" ..
    fix x assume "a = x"
    then have "a  x" ..
    then show "x  a" ..

theorem pick_inverse [intro]: "pick A = A"
proof (cases A)
  fix a assume a: "A = a"
  then have "pick A  a" by (simp only: pick_equiv)
  then have "pick A = a" ..
  with a show ?thesis by simp

text ‹The following rules support canonical function definitions on quotient
  types (with up to two arguments). Note that the stripped-down version
  without additional conditions is sufficient most of the time.›

theorem quot_cond_function:
  assumes eq: "X Y. P X Y  f X Y  g (pick X) (pick Y)"
    and cong: "x x' y y'. x = x'  y = y'
       P x y  P x' y'  g x y = g x' y'"
    and P: "P a b"
  shows "f a b = g a b"
proof -
  from eq and P have "f a b = g (pick a) (pick b)" by (simp only:)
  also have "... = g a b"
  proof (rule cong)
    show "pick a = a" ..
    show "pick b = b" ..
    show "P a b" by (rule P)
    ultimately show "P pick a pick b" by (simp only:)
  finally show ?thesis .

theorem quot_function:
  assumes "X Y. f X Y  g (pick X) (pick Y)"
    and "x x' y y'. x = x'  y = y'  g x y = g x' y'"
  shows "f a b = g a b"
  using assms and TrueI
  by (rule quot_cond_function)

theorem quot_function':
  "(X Y. f X Y  g (pick X) (pick Y)) 
    (x x' y y'. x  x'  y  y'  g x y = g x' y') 
    f a b = g a b"
  by (rule quot_function) (simp_all only: quot_equality)