src/HOL/Lifting.thy

add lemmas for generating transfer rules for typedefs

(*  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 copy_type_to_Quotient:
  assumes "type_definition Rep Abs UNIV"
  and T_def: "T \<equiv> (\<lambda>x y. Abs x = y)"
  shows "Quotient (op =) Abs Rep T"
proof -
  interpret type_definition Rep Abs UNIV by fact
  from Abs_inject Rep_inverse T_def show ?thesis by (auto intro!: QuotientI)
qed

lemma copy_type_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 {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 invariant_type_to_Quotient:
  assumes "type_definition Rep Abs {x. P x}"
  and T_def: "T \<equiv> (\<lambda>x y. (invariant P) x x \<and> Abs x = 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 T_def show ?thesis by (auto intro!: QuotientI simp: invariant_def)
qed

lemma invariant_type_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 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