(*  Title:      Quotient.thy

    Author:     Cezary Kaliszyk and Christian Urban

*)

header {* Definition of Quotient Types *}

theory Quotient

imports Plain Hilbert_Choice Equiv_Relations

uses

  ("Tools/Quotient/quotient_info.ML")

  ("Tools/Quotient/quotient_typ.ML")

  ("Tools/Quotient/quotient_def.ML")

  ("Tools/Quotient/quotient_term.ML")

  ("Tools/Quotient/quotient_tacs.ML")

begin

text {*

  Basic definition for equivalence relations

  that are represented by predicates.

*}

definition

  "reflp E \<longleftrightarrow> (\<forall>x. E x x)"

lemma refl_reflp:

  "refl A \<longleftrightarrow> reflp (\<lambda>x y. (x, y) \<in> A)"

  by (simp add: refl_on_def reflp_def)

definition

  "symp E \<longleftrightarrow> (\<forall>x y. E x y \<longrightarrow> E y x)"

lemma sym_symp:

  "sym A \<longleftrightarrow> symp (\<lambda>x y. (x, y) \<in> A)"

  by (simp add: sym_def symp_def)

definition

  "transp E \<longleftrightarrow> (\<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z)"

lemma trans_transp:

  "trans A \<longleftrightarrow> transp (\<lambda>x y. (x, y) \<in> A)"

  by (auto simp add: trans_def transp_def)

definition

  "equivp E \<longleftrightarrow> (\<forall>x y. E x y = (E x = E y))"

lemma equivp_reflp_symp_transp:

  shows "equivp E = (reflp E \<and> symp E \<and> transp E)"

  unfolding equivp_def reflp_def symp_def transp_def fun_eq_iff

  by blast

lemma equiv_equivp:

  "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"

  by (simp add: equiv_def equivp_reflp_symp_transp refl_reflp sym_symp trans_transp)

lemma equivp_reflp:

  shows "equivp E \<Longrightarrow> E x x"

  by (simp only: equivp_reflp_symp_transp reflp_def)

lemma equivp_symp:

  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y x"

  by (simp add: equivp_def)

lemma equivp_transp:

  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y z \<Longrightarrow> E x z"

  by (simp add: equivp_def)

lemma equivpI:

  assumes "reflp R" "symp R" "transp R"

  shows "equivp R"

  using assms by (simp add: equivp_reflp_symp_transp)

lemma identity_equivp:

  shows "equivp (op =)"

  unfolding equivp_def

  by auto

text {* Partial equivalences *}

definition

  "part_equivp E \<longleftrightarrow> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"

lemma equivp_implies_part_equivp:

  assumes a: "equivp E"

  shows "part_equivp E"

  using a

  unfolding equivp_def part_equivp_def

  by auto

lemma part_equivp_symp:

  assumes e: "part_equivp R"

  and a: "R x y"

  shows "R y x"

  using e[simplified part_equivp_def] a

  by (metis)

lemma part_equivp_typedef:

  shows "part_equivp R \<Longrightarrow> \<exists>d. d \<in> (\<lambda>c. \<exists>x. R x x \<and> c = R x)"

  unfolding part_equivp_def mem_def

  apply clarify

  apply (intro exI)

  apply (rule conjI)

  apply assumption

  apply (rule refl)

  done

lemma part_equivp_refl_symp_transp:

  shows "part_equivp E \<longleftrightarrow> ((\<exists>x. E x x) \<and> symp E \<and> transp E)"

proof

  assume "part_equivp E"

  then show "(\<exists>x. E x x) \<and> symp E \<and> transp E"

  unfolding part_equivp_def symp_def transp_def

  by metis

next

  assume a: "(\<exists>x. E x x) \<and> symp E \<and> transp E"

  then have b: "(\<forall>x y. E x y \<longrightarrow> E y x)" and c: "(\<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z)"

    unfolding symp_def transp_def by (metis, metis)

  have "(\<forall>x y. E x y = (E x x \<and> E y y \<and> E x = E y))"

  proof (intro allI iffI conjI)

    fix x y

    assume d: "E x y"

    then show "E x x" using b c by metis

    show "E y y" using b c d by metis

    show "E x = E y" unfolding fun_eq_iff using b c d by metis

  next

    fix x y

    assume "E x x \<and> E y y \<and> E x = E y"

    then show "E x y" using b c by metis

  qed

  then show "part_equivp E" unfolding part_equivp_def using a by metis

qed

lemma part_equivpI:

  assumes "\<exists>x. R x x" "symp R" "transp R"

  shows "part_equivp R"

  using assms by (simp add: part_equivp_refl_symp_transp)

text {* Composition of Relations *}

abbreviation

  rel_conj (infixr "OOO" 75)

where

  "r1 OOO r2 \<equiv> r1 OO r2 OO r1"

lemma eq_comp_r:

  shows "((op =) OOO R) = R"

  by (auto simp add: fun_eq_iff)

subsection {* Respects predicate *}

definition

  Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"

where

  "Respects R x = R x x"

lemma in_respects:

  shows "x \<in> Respects R \<longleftrightarrow> R x x"

  unfolding mem_def Respects_def

  by simp

subsection {* Function map and function relation *}

notation map_fun (infixr "--->" 55)

lemma map_fun_id:

  "(id ---> id) = id"

  by (simp add: fun_eq_iff)

definition

  fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" (infixr "===>" 55)

where

  "fun_rel E1 E2 = (\<lambda>f g. \<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"

lemma fun_relI [intro]:

  assumes "\<And>x y. E1 x y \<Longrightarrow> E2 (f x) (g y)"

  shows "(E1 ===> E2) f g"

  using assms by (simp add: fun_rel_def)

lemma fun_relE:

  assumes "(E1 ===> E2) f g" and "E1 x y"

  obtains "E2 (f x) (g y)"

  using assms by (simp add: fun_rel_def)

lemma fun_rel_eq:

  shows "((op =) ===> (op =)) = (op =)"

  by (auto simp add: fun_eq_iff elim: fun_relE)

subsection {* Quotient Predicate *}

definition

  "Quotient E Abs Rep \<longleftrightarrow>

     (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and>

     (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"

lemma Quotient_abs_rep:

  assumes a: "Quotient E Abs Rep"

  shows "Abs (Rep a) = a"

  using a

  unfolding Quotient_def

  by simp

lemma Quotient_rep_reflp:

  assumes a: "Quotient E Abs Rep"

  shows "E (Rep a) (Rep a)"

  using a

  unfolding Quotient_def

  by blast

lemma Quotient_rel:

  assumes a: "Quotient E Abs Rep"

  shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"

  using a

  unfolding Quotient_def

  by blast

lemma Quotient_rel_rep:

  assumes a: "Quotient R Abs Rep"

  shows "R (Rep a) (Rep b) = (a = b)"

  using a

  unfolding Quotient_def

  by metis

lemma Quotient_rep_abs:

  assumes a: "Quotient R Abs Rep"

  shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"

  using a unfolding Quotient_def

  by blast

lemma Quotient_rel_abs:

  assumes a: "Quotient E Abs Rep"

  shows "E r s \<Longrightarrow> Abs r = Abs s"

  using a unfolding Quotient_def

  by blast

lemma Quotient_symp:

  assumes a: "Quotient E Abs Rep"

  shows "symp E"

  using a unfolding Quotient_def symp_def

  by metis

lemma Quotient_transp:

  assumes a: "Quotient E Abs Rep"

  shows "transp E"

  using a unfolding Quotient_def transp_def

  by metis

lemma identity_quotient:

  shows "Quotient (op =) id id"

  unfolding Quotient_def id_def

  by blast

lemma fun_quotient:

  assumes q1: "Quotient R1 abs1 rep1"

  and     q2: "Quotient R2 abs2 rep2"

  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"

proof -

  have "\<And>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"

    using q1 q2 by (simp add: Quotient_def fun_eq_iff)

  moreover

  have "\<And>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"

    by (rule fun_relI)

      (insert q1 q2 Quotient_rel_abs [of R1 abs1 rep1] Quotient_rel_rep [of R2 abs2 rep2],

        simp (no_asm) add: Quotient_def, simp)

  moreover

  have "\<And>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>

        (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"

    apply(auto simp add: fun_rel_def fun_eq_iff)

    using q1 q2 unfolding Quotient_def

    apply(metis)

    using q1 q2 unfolding Quotient_def

    apply(metis)

    using q1 q2 unfolding Quotient_def

    apply(metis)

    using q1 q2 unfolding Quotient_def

    apply(metis)

    done

  ultimately

  show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"

    unfolding Quotient_def by blast

qed

lemma abs_o_rep:

  assumes a: "Quotient R Abs Rep"

  shows "Abs o Rep = id"

  unfolding fun_eq_iff

  by (simp add: Quotient_abs_rep[OF a])

lemma equals_rsp:

  assumes q: "Quotient R Abs Rep"

  and     a: "R xa xb" "R ya yb"

  shows "R xa ya = R xb yb"

  using a Quotient_symp[OF q] Quotient_transp[OF q]

  unfolding symp_def transp_def

  by blast

lemma lambda_prs:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"

  unfolding fun_eq_iff

  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]

  by (simp add:)

lemma lambda_prs1:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"

  unfolding fun_eq_iff

  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]

  by (simp add:)

lemma rep_abs_rsp:

  assumes q: "Quotient R Abs Rep"

  and     a: "R x1 x2"

  shows "R x1 (Rep (Abs x2))"

  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]

  by metis

lemma rep_abs_rsp_left:

  assumes q: "Quotient R Abs Rep"

  and     a: "R x1 x2"

  shows "R (Rep (Abs x1)) x2"

  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]

  by metis

text{*

  In the following theorem R1 can be instantiated with anything,

  but we know some of the types of the Rep and Abs functions;

  so by solving Quotient assumptions we can get a unique R1 that

  will be provable; which is why we need to use @{text apply_rsp} and

  not the primed version *}

lemma apply_rsp:

  fixes f g::"'a \<Rightarrow> 'c"

  assumes q: "Quotient R1 Abs1 Rep1"

  and     a: "(R1 ===> R2) f g" "R1 x y"

  shows "R2 (f x) (g y)"

  using a by (auto elim: fun_relE)

lemma apply_rsp':

  assumes a: "(R1 ===> R2) f g" "R1 x y"

  shows "R2 (f x) (g y)"

  using a by (auto elim: fun_relE)

subsection {* lemmas for regularisation of ball and bex *}

lemma ball_reg_eqv:

  fixes P :: "'a \<Rightarrow> bool"

  assumes a: "equivp R"

  shows "Ball (Respects R) P = (All P)"

  using a

  unfolding equivp_def

  by (auto simp add: in_respects)

lemma bex_reg_eqv:

  fixes P :: "'a \<Rightarrow> bool"

  assumes a: "equivp R"

  shows "Bex (Respects R) P = (Ex P)"

  using a

  unfolding equivp_def

  by (auto simp add: in_respects)

lemma ball_reg_right:

  assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"

  shows "All P \<longrightarrow> Ball R Q"

  using a by (metis Collect_def Collect_mem_eq)

lemma bex_reg_left:

  assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"

  shows "Bex R Q \<longrightarrow> Ex P"

  using a by (metis Collect_def Collect_mem_eq)

lemma ball_reg_left:

  assumes a: "equivp R"

  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"

  using a by (metis equivp_reflp in_respects)

lemma bex_reg_right:

  assumes a: "equivp R"

  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"

  using a by (metis equivp_reflp in_respects)

lemma ball_reg_eqv_range:

  fixes P::"'a \<Rightarrow> bool"

  and x::"'a"

  assumes a: "equivp R2"

  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"

  apply(rule iffI)

  apply(rule allI)

  apply(drule_tac x="\<lambda>y. f x" in bspec)

  apply(simp add: in_respects fun_rel_def)

  apply(rule impI)

  using a equivp_reflp_symp_transp[of "R2"]

  apply(simp add: reflp_def)

  apply(simp)

  apply(simp)

  done

lemma bex_reg_eqv_range:

  assumes a: "equivp R2"

  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"

  apply(auto)

  apply(rule_tac x="\<lambda>y. f x" in bexI)

  apply(simp)

  apply(simp add: Respects_def in_respects fun_rel_def)

  apply(rule impI)

  using a equivp_reflp_symp_transp[of "R2"]

  apply(simp add: reflp_def)

  done

(* Next four lemmas are unused *)

lemma all_reg:

  assumes a: "!x :: 'a. (P x --> Q x)"

  and     b: "All P"

  shows "All Q"

  using a b by (metis)

lemma ex_reg:

  assumes a: "!x :: 'a. (P x --> Q x)"

  and     b: "Ex P"

  shows "Ex Q"

  using a b by metis

lemma ball_reg:

  assumes a: "!x :: 'a. (R x --> P x --> Q x)"

  and     b: "Ball R P"

  shows "Ball R Q"

  using a b by (metis Collect_def Collect_mem_eq)

lemma bex_reg:

  assumes a: "!x :: 'a. (R x --> P x --> Q x)"

  and     b: "Bex R P"

  shows "Bex R Q"

  using a b by (metis Collect_def Collect_mem_eq)

lemma ball_all_comm:

  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"

  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"

  using assms by auto

lemma bex_ex_comm:

  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"

  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"

  using assms by auto

subsection {* Bounded abstraction *}

definition

  Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"

where

  "x \<in> p \<Longrightarrow> Babs p m x = m x"

lemma babs_rsp:

  assumes q: "Quotient R1 Abs1 Rep1"

  and     a: "(R1 ===> R2) f g"

  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"

  apply (auto simp add: Babs_def in_respects fun_rel_def)

  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")

  using a apply (simp add: Babs_def fun_rel_def)

  apply (simp add: in_respects fun_rel_def)

  using Quotient_rel[OF q]

  by metis

lemma babs_prs:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"

  apply (rule ext)

  apply (simp add:)

  apply (subgoal_tac "Rep1 x \<in> Respects R1")

  apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

  apply (simp add: in_respects Quotient_rel_rep[OF q1])

  done

lemma babs_simp:

  assumes q: "Quotient R1 Abs Rep"

  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"

  apply(rule iffI)

  apply(simp_all only: babs_rsp[OF q])

  apply(auto simp add: Babs_def fun_rel_def)

  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")

  apply(metis Babs_def)

  apply (simp add: in_respects)

  using Quotient_rel[OF q]

  by metis

(* If a user proves that a particular functional relation

   is an equivalence this may be useful in regularising *)

lemma babs_reg_eqv:

  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"

  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)

(* 3 lemmas needed for proving repabs_inj *)

lemma ball_rsp:

  assumes a: "(R ===> (op =)) f g"

  shows "Ball (Respects R) f = Ball (Respects R) g"

  using a by (auto simp add: Ball_def in_respects elim: fun_relE)

lemma bex_rsp:

  assumes a: "(R ===> (op =)) f g"

  shows "(Bex (Respects R) f = Bex (Respects R) g)"

  using a by (auto simp add: Bex_def in_respects elim: fun_relE)

lemma bex1_rsp:

  assumes a: "(R ===> (op =)) f g"

  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"

  using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 

(* 2 lemmas needed for cleaning of quantifiers *)

lemma all_prs:

  assumes a: "Quotient R absf repf"

  shows "Ball (Respects R) ((absf ---> id) f) = All f"

  using a unfolding Quotient_def Ball_def in_respects id_apply comp_def map_fun_def

  by metis

lemma ex_prs:

  assumes a: "Quotient R absf repf"

  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"

  using a unfolding Quotient_def Bex_def in_respects id_apply comp_def map_fun_def

  by metis

subsection {* @{text Bex1_rel} quantifier *}

definition

  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"

where

  "Bex1_rel R P \<longleftrightarrow> (\<exists>x \<in> Respects R. P x) \<and> (\<forall>x \<in> Respects R. \<forall>y \<in> Respects R. ((P x \<and> P y) \<longrightarrow> (R x y)))"

lemma bex1_rel_aux:

  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"

  unfolding Bex1_rel_def

  apply (erule conjE)+

  apply (erule bexE)

  apply rule

  apply (rule_tac x="xa" in bexI)

  apply metis

  apply metis

  apply rule+

  apply (erule_tac x="xaa" in ballE)

  prefer 2

  apply (metis)

  apply (erule_tac x="ya" in ballE)

  prefer 2

  apply (metis)

  apply (metis in_respects)

  done

lemma bex1_rel_aux2:

  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"

  unfolding Bex1_rel_def

  apply (erule conjE)+

  apply (erule bexE)

  apply rule

  apply (rule_tac x="xa" in bexI)

  apply metis

  apply metis

  apply rule+

  apply (erule_tac x="xaa" in ballE)

  prefer 2

  apply (metis)

  apply (erule_tac x="ya" in ballE)

  prefer 2

  apply (metis)

  apply (metis in_respects)

  done

lemma bex1_rel_rsp:

  assumes a: "Quotient R absf repf"

  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"

  apply (simp add: fun_rel_def)

  apply clarify

  apply rule

  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)

  apply (erule bex1_rel_aux2)

  apply assumption

  done

lemma ex1_prs:

  assumes a: "Quotient R absf repf"

  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"

apply (simp add:)

apply (subst Bex1_rel_def)

apply (subst Bex_def)

apply (subst Ex1_def)

apply simp

apply rule

 apply (erule conjE)+

 apply (erule_tac exE)

 apply (erule conjE)

 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")

  apply (rule_tac x="absf x" in exI)

  apply (simp)

  apply rule+

  using a unfolding Quotient_def

  apply metis

 apply rule+

 apply (erule_tac x="x" in ballE)

  apply (erule_tac x="y" in ballE)

   apply simp

  apply (simp add: in_respects)

 apply (simp add: in_respects)

apply (erule_tac exE)

 apply rule

 apply (rule_tac x="repf x" in exI)

 apply (simp only: in_respects)

  apply rule

 apply (metis Quotient_rel_rep[OF a])

using a unfolding Quotient_def apply (simp)

apply rule+

using a unfolding Quotient_def in_respects

apply metis

done

lemma bex1_bexeq_reg:

  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"

  apply (simp add: Ex1_def Bex1_rel_def in_respects)

  apply clarify

  apply auto

  apply (rule bexI)

  apply assumption

  apply (simp add: in_respects)

  apply (simp add: in_respects)

  apply auto

  done

lemma bex1_bexeq_reg_eqv:

  assumes a: "equivp R"

  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"

  using equivp_reflp[OF a]

  apply (intro impI)

  apply (elim ex1E)

  apply (rule mp[OF bex1_bexeq_reg])

  apply (rule_tac a="x" in ex1I)

  apply (subst in_respects)

  apply (rule conjI)

  apply assumption

  apply assumption

  apply clarify

  apply (erule_tac x="xa" in allE)

  apply simp

  done

subsection {* Various respects and preserve lemmas *}

lemma quot_rel_rsp:

  assumes a: "Quotient R Abs Rep"

  shows "(R ===> R ===> op =) R R"

  apply(rule fun_relI)+

  apply(rule equals_rsp[OF a])

  apply(assumption)+

  done

lemma o_prs:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  and     q3: "Quotient R3 Abs3 Rep3"

  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"

  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"

  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]

  by (simp_all add: fun_eq_iff)

lemma o_rsp:

  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"

  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"

  by (auto intro!: fun_relI elim: fun_relE)

lemma cond_prs:

  assumes a: "Quotient R absf repf"

  shows "absf (if a then repf b else repf c) = (if a then b else c)"

  using a unfolding Quotient_def by auto

lemma if_prs:

  assumes q: "Quotient R Abs Rep"

  shows "(id ---> Rep ---> Rep ---> Abs) If = If"

  using Quotient_abs_rep[OF q]

  by (auto simp add: fun_eq_iff)

lemma if_rsp:

  assumes q: "Quotient R Abs Rep"

  shows "(op = ===> R ===> R ===> R) If If"

  by (auto intro!: fun_relI)

lemma let_prs:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"

  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]

  by (auto simp add: fun_eq_iff)

lemma let_rsp:

  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"

  by (auto intro!: fun_relI elim: fun_relE)

lemma mem_rsp:

  shows "(R1 ===> (R1 ===> R2) ===> R2) op \<in> op \<in>"

  by (auto intro!: fun_relI elim: fun_relE simp add: mem_def)

lemma mem_prs:

  assumes a1: "Quotient R1 Abs1 Rep1"

  and     a2: "Quotient R2 Abs2 Rep2"

  shows "(Rep1 ---> (Abs1 ---> Rep2) ---> Abs2) op \<in> = op \<in>"

  by (simp add: fun_eq_iff mem_def Quotient_abs_rep[OF a1] Quotient_abs_rep[OF a2])

lemma id_rsp:

  shows "(R ===> R) id id"

  by (auto intro: fun_relI)

lemma id_prs:

  assumes a: "Quotient R Abs Rep"

  shows "(Rep ---> Abs) id = id"

  by (simp add: fun_eq_iff Quotient_abs_rep [OF a])

locale quot_type =

  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

  and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"

  and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"

  assumes equivp: "part_equivp R"

  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = R x"

  and     rep_inverse: "\<And>x. Abs (Rep x) = x"

  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = R x))) \<Longrightarrow> (Rep (Abs c)) = c"

  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"

begin

definition

  abs :: "'a \<Rightarrow> 'b"

where

  "abs x = Abs (R x)"

definition

  rep :: "'b \<Rightarrow> 'a"

where

  "rep a = Eps (Rep a)"

lemma homeier5:

  assumes a: "R r r"

  shows "Rep (Abs (R r)) = R r"

  apply (subst abs_inverse)

  using a by auto

theorem homeier6:

  assumes a: "R r r"

  and b: "R s s"

  shows "Abs (R r) = Abs (R s) \<longleftrightarrow> R r = R s"

  by (metis a b homeier5)

theorem homeier8:

  assumes "R r r"

  shows "R (Eps (R r)) = R r"

  using assms equivp[simplified part_equivp_def]

  apply clarify

  by (metis assms exE_some)

lemma Quotient:

  shows "Quotient R abs rep"

  unfolding Quotient_def abs_def rep_def

  proof (intro conjI allI)

    fix a r s

    show "Abs (R (Eps (Rep a))) = a"

      by (metis equivp exE_some part_equivp_def rep_inverse rep_prop)

    show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (R r) = Abs (R s))"

      by (metis homeier6 equivp[simplified part_equivp_def])

    show "R (Eps (Rep a)) (Eps (Rep a))" proof -

      obtain x where r: "R x x" and rep: "Rep a = R x" using rep_prop[of a] by auto

      have "R (Eps (R x)) x" using homeier8 r by simp

      then have "R x (Eps (R x))" using part_equivp_symp[OF equivp] by fast

      then have "R (Eps (R x)) (Eps (R x))" using homeier8[OF r] by simp

      then show "R (Eps (Rep a)) (Eps (Rep a))" using rep by simp

    qed

  qed

end

subsection {* ML setup *}

text {* Auxiliary data for the quotient package *}

use "Tools/Quotient/quotient_info.ML"

declare [[map "fun" = (map_fun, fun_rel)]]

lemmas [quot_thm] = fun_quotient

lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp mem_rsp id_rsp

lemmas [quot_preserve] = if_prs o_prs let_prs mem_prs id_prs

lemmas [quot_equiv] = identity_equivp

text {* Lemmas about simplifying id's. *}

lemmas [id_simps] =

  id_def[symmetric]

  map_fun_id

  id_apply

  id_o

  o_id

  eq_comp_r

text {* Translation functions for the lifting process. *}

use "Tools/Quotient/quotient_term.ML"

text {* Definitions of the quotient types. *}

use "Tools/Quotient/quotient_typ.ML"

text {* Definitions for quotient constants. *}

use "Tools/Quotient/quotient_def.ML"

text {*

  An auxiliary constant for recording some information

  about the lifted theorem in a tactic.

*}

definition

  Quot_True :: "'a \<Rightarrow> bool"

where

  "Quot_True x \<longleftrightarrow> True"

lemma

  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"

  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"

  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"

  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"

  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"

  by (simp_all add: Quot_True_def ext)

lemma QT_imp: "Quot_True a \<equiv> Quot_True b"

  by (simp add: Quot_True_def)

text {* Tactics for proving the lifted theorems *}

use "Tools/Quotient/quotient_tacs.ML"

subsection {* Methods / Interface *}

method_setup lifting =

  {* Attrib.thms >> (fn thms => fn ctxt => 

       SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt [] thms))) *}

  {* lifts theorems to quotient types *}

method_setup lifting_setup =

  {* Attrib.thm >> (fn thm => fn ctxt => 

       SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_procedure_tac ctxt [] thm))) *}

  {* sets up the three goals for the quotient lifting procedure *}

method_setup descending =

  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_tac ctxt []))) *}

  {* decends theorems to the raw level *}

method_setup descending_setup =

  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_procedure_tac ctxt []))) *}

  {* sets up the three goals for the decending theorems *}

method_setup regularize =

  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}

  {* proves the regularization goals from the quotient lifting procedure *}

method_setup injection =

  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}

  {* proves the rep/abs injection goals from the quotient lifting procedure *}

method_setup cleaning =

  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}

  {* proves the cleaning goals from the quotient lifting procedure *}

attribute_setup quot_lifted =

  {* Scan.succeed Quotient_Tacs.lifted_attrib *}

  {* lifts theorems to quotient types *}

no_notation

  rel_conj (infixr "OOO" 75) and

  map_fun (infixr "--->" 55) and

  fun_rel (infixr "===>" 55)

end