(*  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