(*  Title:      HOL/Lifting.thy

     Author:     Brian Huffman and Ondrej Kuncar

     Author:     Cezary Kaliszyk and Christian Urban

 *)

 header {* Lifting package *}

 theory Lifting

 imports Plain Equiv_Relations Transfer

 keywords

   "print_quotmaps" "print_quotients" :: diag and

   "lift_definition" :: thy_goal and

   "setup_lifting" :: thy_decl

 uses

   ("Tools/Lifting/lifting_info.ML")

   ("Tools/Lifting/lifting_term.ML")

   ("Tools/Lifting/lifting_def.ML")

   ("Tools/Lifting/lifting_setup.ML")

 begin

 subsection {* Function map *}

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

 lemma map_fun_id:

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

   by (simp add: fun_eq_iff)

 subsection {* Quotient Predicate *}

 definition

   "Quotient R Abs Rep T \<longleftrightarrow>

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

      (\<forall>a. R (Rep a) (Rep a)) \<and>

      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>

      T = (\<lambda>x y. R x x \<and> Abs x = y)"

 lemma QuotientI:

   assumes "\<And>a. Abs (Rep a) = a"

     and "\<And>a. R (Rep a) (Rep a)"

     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"

     and "T = (\<lambda>x y. R x x \<and> Abs x = y)"

   shows "Quotient R Abs Rep T"

   using assms unfolding Quotient_def by blast

 lemma Quotient_abs_rep:

   assumes a: "Quotient R Abs Rep T"

   shows "Abs (Rep a) = a"

   using a

   unfolding Quotient_def

   by simp

 lemma Quotient_rep_reflp:

   assumes a: "Quotient R Abs Rep T"

   shows "R (Rep a) (Rep a)"

   using a

   unfolding Quotient_def

   by blast

 lemma Quotient_rel:

   assumes a: "Quotient R Abs Rep T"

   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}

   using a

   unfolding Quotient_def

   by blast

 lemma Quotient_cr_rel:

   assumes a: "Quotient R Abs Rep T"

   shows "T = (\<lambda>x y. R x x \<and> Abs x = y)"

   using a

   unfolding Quotient_def

   by blast

 lemma Quotient_refl1: 

   assumes a: "Quotient R Abs Rep T" 

   shows "R r s \<Longrightarrow> R r r"

   using a unfolding Quotient_def 

   by fast

 lemma Quotient_refl2: 

   assumes a: "Quotient R Abs Rep T" 

   shows "R r s \<Longrightarrow> R s s"

   using a unfolding Quotient_def 

   by fast

 lemma Quotient_rel_rep:

   assumes a: "Quotient R Abs Rep T"

   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"

   using a

   unfolding Quotient_def

   by metis

 lemma Quotient_rep_abs:

   assumes a: "Quotient R Abs Rep T"

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

   using a unfolding Quotient_def

   by blast

 lemma Quotient_rel_abs:

   assumes a: "Quotient R Abs Rep T"

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

   using a unfolding Quotient_def

   by blast

 lemma Quotient_symp:

   assumes a: "Quotient R Abs Rep T"

   shows "symp R"

   using a unfolding Quotient_def using sympI by (metis (full_types))

 lemma Quotient_transp:

   assumes a: "Quotient R Abs Rep T"

   shows "transp R"

   using a unfolding Quotient_def using transpI by (metis (full_types))

 lemma Quotient_part_equivp:

   assumes a: "Quotient R Abs Rep T"

   shows "part_equivp R"

 by (metis Quotient_rep_reflp Quotient_symp Quotient_transp a part_equivpI)

 lemma identity_quotient: "Quotient (op =) id id (op =)"

 unfolding Quotient_def by simp 

 lemma Quotient_alt_def:

   "Quotient R Abs Rep T \<longleftrightarrow>

     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>

     (\<forall>b. T (Rep b) b) \<and>

     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"

 apply safe

 apply (simp (no_asm_use) only: Quotient_def, fast)

 apply (simp (no_asm_use) only: Quotient_def, fast)

 apply (simp (no_asm_use) only: Quotient_def, fast)

 apply (simp (no_asm_use) only: Quotient_def, fast)

 apply (simp (no_asm_use) only: Quotient_def, fast)

 apply (simp (no_asm_use) only: Quotient_def, fast)

 apply (rule QuotientI)

 apply simp

 apply metis

 apply simp

 apply (rule ext, rule ext, metis)

 done

 lemma Quotient_alt_def2:

   "Quotient R Abs Rep T \<longleftrightarrow>

     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>

     (\<forall>b. T (Rep b) b) \<and>

     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"

   unfolding Quotient_alt_def by (safe, metis+)

 lemma fun_quotient:

   assumes 1: "Quotient R1 abs1 rep1 T1"

   assumes 2: "Quotient R2 abs2 rep2 T2"

   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"

   using assms unfolding Quotient_alt_def2

   unfolding fun_rel_def fun_eq_iff map_fun_apply

   by (safe, metis+)

 lemma apply_rsp:

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

   assumes q: "Quotient R1 Abs1 Rep1 T1"

   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)

 lemma apply_rsp'':

   assumes "Quotient R Abs Rep T"

   and "(R ===> S) f f"

   shows "S (f (Rep x)) (f (Rep x))"

 proof -

   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)

   then show ?thesis using assms(2) by (auto intro: apply_rsp')

 qed

 subsection {* Quotient composition *}

 lemma Quotient_compose:

   assumes 1: "Quotient R1 Abs1 Rep1 T1"

   assumes 2: "Quotient R2 Abs2 Rep2 T2"

   shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"

 proof -

   from 1 have Abs1: "\<And>a b. T1 a b \<Longrightarrow> Abs1 a = b"

     unfolding Quotient_alt_def by simp

   from 1 have Rep1: "\<And>b. T1 (Rep1 b) b"

     unfolding Quotient_alt_def by simp

   from 2 have Abs2: "\<And>a b. T2 a b \<Longrightarrow> Abs2 a = b"

     unfolding Quotient_alt_def by simp

   from 2 have Rep2: "\<And>b. T2 (Rep2 b) b"

     unfolding Quotient_alt_def by simp

   from 2 have R2:

     "\<And>x y. R2 x y \<longleftrightarrow> T2 x (Abs2 x) \<and> T2 y (Abs2 y) \<and> Abs2 x = Abs2 y"

     unfolding Quotient_alt_def by simp

   show ?thesis

     unfolding Quotient_alt_def

     apply simp

     apply safe

     apply (drule Abs1, simp)

     apply (erule Abs2)

     apply (rule pred_compI)

     apply (rule Rep1)

     apply (rule Rep2)

     apply (rule pred_compI, assumption)

     apply (drule Abs1, simp)

     apply (clarsimp simp add: R2)

     apply (rule pred_compI, assumption)

     apply (drule Abs1, simp)+

     apply (clarsimp simp add: R2)

     apply (drule Abs1, simp)+

     apply (clarsimp simp add: R2)

     apply (rule pred_compI, assumption)

     apply (rule pred_compI [rotated])

     apply (erule conversepI)

     apply (drule Abs1, simp)+

     apply (simp add: R2)

     done

 qed

 subsection {* Invariant *}

 definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 

   where "invariant R = (\<lambda>x y. R x \<and> x = y)"

 lemma invariant_to_eq:

   assumes "invariant P x y"

   shows "x = y"

 using assms by (simp add: invariant_def)

 lemma fun_rel_eq_invariant:

   shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"

 by (auto simp add: invariant_def fun_rel_def)

 lemma invariant_same_args:

   shows "invariant P x x \<equiv> P x"

 using assms by (auto simp add: invariant_def)

 lemma UNIV_typedef_to_Quotient:

   assumes "type_definition Rep Abs UNIV"

   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"

   shows "Quotient (op =) Abs Rep T"

 proof -

   interpret type_definition Rep Abs UNIV by fact

   from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis 

     by (fastforce intro!: QuotientI fun_eq_iff)

 qed

 lemma UNIV_typedef_to_equivp:

   fixes Abs :: "'a \<Rightarrow> 'b"

   and Rep :: "'b \<Rightarrow> 'a"

   assumes "type_definition Rep Abs (UNIV::'a set)"

   shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"

 by (rule identity_equivp)

 lemma typedef_to_Quotient:

   assumes "type_definition Rep Abs S"

   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"

   shows "Quotient (Lifting.invariant (\<lambda>x. x \<in> S)) Abs Rep T"

 proof -

   interpret type_definition Rep Abs S by fact

   from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis

     by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)

 qed

 lemma typedef_to_part_equivp:

   assumes "type_definition Rep Abs S"

   shows "part_equivp (Lifting.invariant (\<lambda>x. x \<in> S))"

 proof (intro part_equivpI)

   interpret type_definition Rep Abs S by fact

   show "\<exists>x. Lifting.invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)

 next

   show "symp (Lifting.invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)

 next

   show "transp (Lifting.invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)

 qed

 lemma open_typedef_to_Quotient:

   assumes "type_definition Rep Abs {x. P x}"

   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"

   shows "Quotient (invariant P) Abs Rep T"

 proof -

   interpret type_definition Rep Abs "{x. P x}" by fact

   from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis

     by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)

 qed

 lemma open_typedef_to_part_equivp:

   assumes "type_definition Rep Abs {x. P x}"

   shows "part_equivp (invariant P)"

 proof (intro part_equivpI)

   interpret type_definition Rep Abs "{x. P x}" by fact

   show "\<exists>x. invariant P x x" using Rep by (auto simp: invariant_def)

 next

   show "symp (invariant P)" by (auto intro: sympI simp: invariant_def)

 next

   show "transp (invariant P)" by (auto intro: transpI simp: invariant_def)

 qed

 text {* Generating transfer rules for quotients. *}

 lemma Quotient_right_unique:

   assumes "Quotient R Abs Rep T" shows "right_unique T"

   using assms unfolding Quotient_alt_def right_unique_def by metis

 lemma Quotient_right_total:

   assumes "Quotient R Abs Rep T" shows "right_total T"

   using assms unfolding Quotient_alt_def right_total_def by metis

 lemma Quotient_rel_eq_transfer:

   assumes "Quotient R Abs Rep T"

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

   using assms unfolding Quotient_alt_def fun_rel_def by simp

 lemma Quotient_bi_total:

   assumes "Quotient R Abs Rep T" and "reflp R"

   shows "bi_total T"

   using assms unfolding Quotient_alt_def bi_total_def reflp_def by auto

 lemma Quotient_id_abs_transfer:

   assumes "Quotient R Abs Rep T" and "reflp R"

   shows "(op = ===> T) (\<lambda>x. x) Abs"

   using assms unfolding Quotient_alt_def reflp_def fun_rel_def by simp

 text {* Generating transfer rules for a type defined with @{text "typedef"}. *}

 lemma typedef_bi_unique:

   assumes type: "type_definition Rep Abs A"

   assumes T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"

   shows "bi_unique T"

   unfolding bi_unique_def T_def

   by (simp add: type_definition.Rep_inject [OF type])

 lemma typedef_right_total:

   assumes T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"

   shows "right_total T"

   unfolding right_total_def T_def by simp

 lemma copy_type_bi_total:

   assumes type: "type_definition Rep Abs UNIV"

   assumes T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"

   shows "bi_total T"

   unfolding bi_total_def T_def

   by (metis type_definition.Abs_inverse [OF type] UNIV_I)

 lemma typedef_transfer_All:

   assumes type: "type_definition Rep Abs A"

   assumes T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"

   shows "((T ===> op =) ===> op =) (Ball A) All"

   unfolding T_def fun_rel_def

   by (metis type_definition.Rep [OF type]

     type_definition.Abs_inverse [OF type])

 lemma typedef_transfer_Ex:

   assumes type: "type_definition Rep Abs A"

   assumes T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"

   shows "((T ===> op =) ===> op =) (Bex A) Ex"

   unfolding T_def fun_rel_def

   by (metis type_definition.Rep [OF type]

     type_definition.Abs_inverse [OF type])

 lemma typedef_transfer_bforall:

   assumes type: "type_definition Rep Abs A"

   assumes T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"

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

     (transfer_bforall (\<lambda>x. x \<in> A)) transfer_forall"

   unfolding transfer_bforall_def transfer_forall_def Ball_def [symmetric]

   by (rule typedef_transfer_All [OF assms])

 text {* Generating the correspondence rule for a constant defined with

   @{text "lift_definition"}. *}

 lemma Quotient_to_transfer:

   assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"

   shows "T c c'"

   using assms by (auto dest: Quotient_cr_rel)

 subsection {* ML setup *}

 text {* Auxiliary data for the lifting package *}

 use "Tools/Lifting/lifting_info.ML"

 setup Lifting_Info.setup

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

 use "Tools/Lifting/lifting_term.ML"

 use "Tools/Lifting/lifting_def.ML"

 use "Tools/Lifting/lifting_setup.ML"

 hide_const (open) invariant

 end