(*  Title:      HOL/Lifting.thy

     Author:     Brian Huffman and Ondrej Kuncar

     Author:     Cezary Kaliszyk and Christian Urban

 *)

 header {* Lifting package *}

 theory Lifting

 imports Equiv_Relations Transfer

 keywords

   "parametric" and

   "print_quot_maps" "print_quotients" :: diag and

   "lift_definition" :: thy_goal and

   "setup_lifting" "lifting_forget" "lifting_update" :: thy_decl

 begin

 subsection {* Function map *}

 context

 begin

 interpretation lifting_syntax .

 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

 context

   fixes R Abs Rep T

   assumes a: "Quotient R Abs Rep T"

 begin

 lemma Quotient_abs_rep: "Abs (Rep a) = a"

   using a unfolding Quotient_def

   by simp

 lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"

   using a unfolding Quotient_def

   by blast

 lemma Quotient_rel:

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

   using a unfolding Quotient_def

   by blast

 lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"

   using a unfolding Quotient_def

   by fast

 lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"

   using a unfolding Quotient_def

   by fast

 lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"

   using a unfolding Quotient_def

   by metis

 lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"

   using a unfolding Quotient_def

   by blast

 lemma Quotient_rep_abs_eq: "R t t \<Longrightarrow> R \<le> op= \<Longrightarrow> Rep (Abs t) = t"

   using a unfolding Quotient_def

   by blast

 lemma Quotient_rep_abs_fold_unmap: 

   assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'" 

   shows "R (Rep' x') x"

 proof -

   have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto

   then show ?thesis using assms(3) by simp

 qed

 lemma Quotient_Rep_eq:

   assumes "x' \<equiv> Abs x" 

   shows "Rep x' \<equiv> Rep x'"

 by simp

 lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"

   using a unfolding Quotient_def

   by blast

 lemma Quotient_rel_abs2:

   assumes "R (Rep x) y"

   shows "x = Abs y"

 proof -

   from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)

   then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)

 qed

 lemma Quotient_symp: "symp R"

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

 lemma Quotient_transp: "transp R"

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

 lemma Quotient_part_equivp: "part_equivp R"

 by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)

 end

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

 unfolding Quotient_def by simp 

 text {* TODO: Use one of these alternatives as the real definition. *}

 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 Quotient_alt_def3:

   "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> (\<exists>z. T x z \<and> T y z))"

   unfolding Quotient_alt_def2 by (safe, metis+)

 lemma Quotient_alt_def4:

   "Quotient R Abs Rep T \<longleftrightarrow>

     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"

   unfolding Quotient_alt_def3 fun_eq_iff by auto

 lemma Quotient_alt_def5:

   "Quotient R Abs Rep T \<longleftrightarrow>

     T \<le> BNF_Util.Grp UNIV Abs \<and> BNF_Util.Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>"

   unfolding Quotient_alt_def4 Grp_def by blast

 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 rel_fun_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: rel_funE)

 lemma apply_rsp':

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

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

   using a by (auto elim: rel_funE)

 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)"

   using assms unfolding Quotient_alt_def4 by fastforce

 lemma equivp_reflp2:

   "equivp R \<Longrightarrow> reflp R"

   by (erule equivpE)

 subsection {* Respects predicate *}

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

   where "Respects R = {x. R x x}"

 lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"

   unfolding Respects_def by simp

 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 (eq_onp (\<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: eq_onp_def fun_eq_iff)

 qed

 lemma typedef_to_part_equivp:

   assumes "type_definition Rep Abs S"

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

 proof (intro part_equivpI)

   interpret type_definition Rep Abs S by fact

   show "\<exists>x. eq_onp (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: eq_onp_def)

 next

   show "symp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: eq_onp_def)

 next

   show "transp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: eq_onp_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 (eq_onp P) Abs Rep T"

   using typedef_to_Quotient [OF assms] by simp

 lemma open_typedef_to_part_equivp:

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

   shows "part_equivp (eq_onp P)"

   using typedef_to_part_equivp [OF assms] by simp

 text {* Generating transfer rules for quotients. *}

 context

   fixes R Abs Rep T

   assumes 1: "Quotient R Abs Rep T"

 begin

 lemma Quotient_right_unique: "right_unique T"

   using 1 unfolding Quotient_alt_def right_unique_def by metis

 lemma Quotient_right_total: "right_total T"

   using 1 unfolding Quotient_alt_def right_total_def by metis

 lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"

   using 1 unfolding Quotient_alt_def rel_fun_def by simp

 lemma Quotient_abs_induct:

   assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"

   using 1 assms unfolding Quotient_def by metis

 end

 text {* Generating transfer rules for total quotients. *}

 context

   fixes R Abs Rep T

   assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"

 begin

 lemma Quotient_left_total: "left_total T"

   using 1 2 unfolding Quotient_alt_def left_total_def reflp_def by auto

 lemma Quotient_bi_total: "bi_total T"

   using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto

 lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"

   using 1 2 unfolding Quotient_alt_def reflp_def rel_fun_def by simp

 lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"

   using 1 2 assms unfolding Quotient_alt_def reflp_def by metis

 lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"

   using Quotient_rel [OF 1] 2 unfolding reflp_def by simp

 end

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

 context

   fixes Rep Abs A T

   assumes type: "type_definition Rep Abs A"

   assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"

 begin

 lemma typedef_left_unique: "left_unique T"

   unfolding left_unique_def T_def

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

 lemma typedef_bi_unique: "bi_unique T"

   unfolding bi_unique_def T_def

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

 (* the following two theorems are here only for convinience *)

 lemma typedef_right_unique: "right_unique T"

   using T_def type Quotient_right_unique typedef_to_Quotient 

   by blast

 lemma typedef_right_total: "right_total T"

   using T_def type Quotient_right_total typedef_to_Quotient 

   by blast

 lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"

   unfolding rel_fun_def T_def by simp

 end

 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)

 text {* Proving reflexivity *}

 lemma Quotient_to_left_total:

   assumes q: "Quotient R Abs Rep T"

   and r_R: "reflp R"

   shows "left_total T"

 using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)

 lemma Quotient_composition_ge_eq:

   assumes "left_total T"

   assumes "R \<ge> op="

   shows "(T OO R OO T\<inverse>\<inverse>) \<ge> op="

 using assms unfolding left_total_def by fast

 lemma Quotient_composition_le_eq:

   assumes "left_unique T"

   assumes "R \<le> op="

   shows "(T OO R OO T\<inverse>\<inverse>) \<le> op="

 using assms unfolding left_unique_def by blast

 lemma eq_onp_le_eq:

   "eq_onp P \<le> op=" unfolding eq_onp_def by blast

 lemma reflp_ge_eq:

   "reflp R \<Longrightarrow> R \<ge> op=" unfolding reflp_def by blast

 lemma ge_eq_refl:

   "R \<ge> op= \<Longrightarrow> R x x" by blast

 text {* Proving a parametrized correspondence relation *}

 definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where

 "POS A B \<equiv> A \<le> B"

 definition  NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where

 "NEG A B \<equiv> B \<le> A"

 lemma pos_OO_eq:

   shows "POS (A OO op=) A"

 unfolding POS_def OO_def by blast

 lemma pos_eq_OO:

   shows "POS (op= OO A) A"

 unfolding POS_def OO_def by blast

 lemma neg_OO_eq:

   shows "NEG (A OO op=) A"

 unfolding NEG_def OO_def by auto

 lemma neg_eq_OO:

   shows "NEG (op= OO A) A"

 unfolding NEG_def OO_def by blast

 lemma POS_trans:

   assumes "POS A B"

   assumes "POS B C"

   shows "POS A C"

 using assms unfolding POS_def by auto

 lemma NEG_trans:

   assumes "NEG A B"

   assumes "NEG B C"

   shows "NEG A C"

 using assms unfolding NEG_def by auto

 lemma POS_NEG:

   "POS A B \<equiv> NEG B A"

   unfolding POS_def NEG_def by auto

 lemma NEG_POS:

   "NEG A B \<equiv> POS B A"

   unfolding POS_def NEG_def by auto

 lemma POS_pcr_rule:

   assumes "POS (A OO B) C"

   shows "POS (A OO B OO X) (C OO X)"

 using assms unfolding POS_def OO_def by blast

 lemma NEG_pcr_rule:

   assumes "NEG (A OO B) C"

   shows "NEG (A OO B OO X) (C OO X)"

 using assms unfolding NEG_def OO_def by blast

 lemma POS_apply:

   assumes "POS R R'"

   assumes "R f g"

   shows "R' f g"

 using assms unfolding POS_def by auto

 text {* Proving a parametrized correspondence relation *}

 lemma fun_mono:

   assumes "A \<ge> C"

   assumes "B \<le> D"

   shows   "(A ===> B) \<le> (C ===> D)"

 using assms unfolding rel_fun_def by blast

 lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"

 unfolding OO_def rel_fun_def by blast

 lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"

 unfolding right_unique_def left_total_def by blast

 lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"

 unfolding left_unique_def right_total_def by blast

 lemma neg_fun_distr1:

 assumes 1: "left_unique R" "right_total R"

 assumes 2: "right_unique R'" "left_total R'"

 shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "

   using functional_relation[OF 2] functional_converse_relation[OF 1]

   unfolding rel_fun_def OO_def

   apply clarify

   apply (subst all_comm)

   apply (subst all_conj_distrib[symmetric])

   apply (intro choice)

   by metis

 lemma neg_fun_distr2:

 assumes 1: "right_unique R'" "left_total R'"

 assumes 2: "left_unique S'" "right_total S'"

 shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"

   using functional_converse_relation[OF 2] functional_relation[OF 1]

   unfolding rel_fun_def OO_def

   apply clarify

   apply (subst all_comm)

   apply (subst all_conj_distrib[symmetric])

   apply (intro choice)

   by metis

 subsection {* Domains *}

 lemma composed_equiv_rel_eq_onp:

   assumes "left_unique R"

   assumes "(R ===> op=) P P'"

   assumes "Domainp R = P''"

   shows "(R OO eq_onp P' OO R\<inverse>\<inverse>) = eq_onp (inf P'' P)"

 using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def rel_fun_def eq_onp_def

 fun_eq_iff by blast

 lemma composed_equiv_rel_eq_eq_onp:

   assumes "left_unique R"

   assumes "Domainp R = P"

   shows "(R OO op= OO R\<inverse>\<inverse>) = eq_onp P"

 using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def eq_onp_def

 fun_eq_iff is_equality_def by metis

 lemma pcr_Domainp_par_left_total:

   assumes "Domainp B = P"

   assumes "left_total A"

   assumes "(A ===> op=) P' P"

   shows "Domainp (A OO B) = P'"

 using assms

 unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def rel_fun_def 

 by (fast intro: fun_eq_iff)

 lemma pcr_Domainp_par:

 assumes "Domainp B = P2"

 assumes "Domainp A = P1"

 assumes "(A ===> op=) P2' P2"

 shows "Domainp (A OO B) = (inf P1 P2')"

 using assms unfolding rel_fun_def Domainp_iff[abs_def] OO_def

 by (fast intro: fun_eq_iff)

 definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"

 where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y"

 lemma pcr_Domainp:

 assumes "Domainp B = P"

 shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)"

 using assms by blast

 lemma pcr_Domainp_total:

   assumes "left_total B"

   assumes "Domainp A = P"

   shows "Domainp (A OO B) = P"

 using assms unfolding left_total_def 

 by fast

 lemma Quotient_to_Domainp:

   assumes "Quotient R Abs Rep T"

   shows "Domainp T = (\<lambda>x. R x x)"  

 by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])

 lemma eq_onp_to_Domainp:

   assumes "Quotient (eq_onp P) Abs Rep T"

   shows "Domainp T = P"

 by (simp add: eq_onp_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])

 end

 subsection {* ML setup *}

 ML_file "Tools/Lifting/lifting_util.ML"

 ML_file "Tools/Lifting/lifting_info.ML"

 setup Lifting_Info.setup

 (* setup for the function type *)

 declare fun_quotient[quot_map]

 declare fun_mono[relator_mono]

 lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2

 ML_file "Tools/Lifting/lifting_bnf.ML"

 ML_file "Tools/Lifting/lifting_term.ML"

 ML_file "Tools/Lifting/lifting_def.ML"

 ML_file "Tools/Lifting/lifting_setup.ML"

 hide_const (open) POS NEG

 end