src/HOL/Lifting.thy
author huffman
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lifting package produces abs_eq_iff rules for total quotients
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(*  Title:      HOL/Lifting.thy
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    Author:     Brian Huffman and Ondrej Kuncar
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    Author:     Cezary Kaliszyk and Christian Urban
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*)
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header {* Lifting package *}
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theory Lifting
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imports Plain Equiv_Relations Transfer
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keywords
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  "print_quotmaps" "print_quotients" :: diag and
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  "lift_definition" :: thy_goal and
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  "setup_lifting" :: thy_decl
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uses
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  ("Tools/Lifting/lifting_util.ML")
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  ("Tools/Lifting/lifting_info.ML")
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  ("Tools/Lifting/lifting_term.ML")
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  ("Tools/Lifting/lifting_def.ML")
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  ("Tools/Lifting/lifting_setup.ML")
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begin
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subsection {* Function map *}
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notation map_fun (infixr "--->" 55)
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lemma map_fun_id:
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  "(id ---> id) = id"
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  by (simp add: fun_eq_iff)
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subsection {* Quotient Predicate *}
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definition
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  "Quotient R Abs Rep T \<longleftrightarrow>
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     (\<forall>a. Abs (Rep a) = a) \<and> 
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     (\<forall>a. R (Rep a) (Rep a)) \<and>
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     (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
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     T = (\<lambda>x y. R x x \<and> Abs x = y)"
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lemma QuotientI:
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  assumes "\<And>a. Abs (Rep a) = a"
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    and "\<And>a. R (Rep a) (Rep a)"
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    and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
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    and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
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  shows "Quotient R Abs Rep T"
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  using assms unfolding Quotient_def by blast
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context
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  fixes R Abs Rep T
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  assumes a: "Quotient R Abs Rep T"
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begin
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lemma Quotient_abs_rep: "Abs (Rep a) = a"
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  using a unfolding Quotient_def
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  by simp
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lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rel:
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  "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
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  using a unfolding Quotient_def
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  by fast
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lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
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  using a unfolding Quotient_def
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  by fast
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lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
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  using a unfolding Quotient_def
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  by metis
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lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_symp: "symp R"
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  using a unfolding Quotient_def using sympI by (metis (full_types))
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lemma Quotient_transp: "transp R"
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  using a unfolding Quotient_def using transpI by (metis (full_types))
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lemma Quotient_part_equivp: "part_equivp R"
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by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
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end
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lemma identity_quotient: "Quotient (op =) id id (op =)"
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unfolding Quotient_def by simp 
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text {* TODO: Use one of these alternatives as the real definition. *}
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lemma Quotient_alt_def:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
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    (\<forall>b. T (Rep b) b) \<and>
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    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
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apply safe
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (rule QuotientI)
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apply simp
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apply metis
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apply simp
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apply (rule ext, rule ext, metis)
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done
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lemma Quotient_alt_def2:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
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    (\<forall>b. T (Rep b) b) \<and>
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    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
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  unfolding Quotient_alt_def by (safe, metis+)
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lemma Quotient_alt_def3:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and>
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    (\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
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  unfolding Quotient_alt_def2 by (safe, metis+)
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lemma Quotient_alt_def4:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
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  unfolding Quotient_alt_def3 fun_eq_iff by auto
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lemma fun_quotient:
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  assumes 1: "Quotient R1 abs1 rep1 T1"
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  assumes 2: "Quotient R2 abs2 rep2 T2"
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  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
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  using assms unfolding Quotient_alt_def2
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  unfolding fun_rel_def fun_eq_iff map_fun_apply
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  by (safe, metis+)
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lemma apply_rsp:
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  fixes f g::"'a \<Rightarrow> 'c"
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  assumes q: "Quotient R1 Abs1 Rep1 T1"
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  and     a: "(R1 ===> R2) f g" "R1 x y"
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  shows "R2 (f x) (g y)"
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  using a by (auto elim: fun_relE)
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lemma apply_rsp':
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  assumes a: "(R1 ===> R2) f g" "R1 x y"
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  shows "R2 (f x) (g y)"
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  using a by (auto elim: fun_relE)
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lemma apply_rsp'':
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  assumes "Quotient R Abs Rep T"
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  and "(R ===> S) f f"
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  shows "S (f (Rep x)) (f (Rep x))"
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proof -
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  from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
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  then show ?thesis using assms(2) by (auto intro: apply_rsp')
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qed
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subsection {* Quotient composition *}
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lemma Quotient_compose:
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  assumes 1: "Quotient R1 Abs1 Rep1 T1"
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  assumes 2: "Quotient R2 Abs2 Rep2 T2"
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  shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
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  using assms unfolding Quotient_alt_def4 by (auto intro!: ext)
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lemma equivp_reflp2:
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  "equivp R \<Longrightarrow> reflp R"
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  by (erule equivpE)
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subsection {* Respects predicate *}
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definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
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  where "Respects R = {x. R x x}"
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lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
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  unfolding Respects_def by simp
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subsection {* Invariant *}
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definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
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  where "invariant R = (\<lambda>x y. R x \<and> x = y)"
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lemma invariant_to_eq:
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  assumes "invariant P x y"
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  shows "x = y"
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using assms by (simp add: invariant_def)
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lemma fun_rel_eq_invariant:
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  shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
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by (auto simp add: invariant_def fun_rel_def)
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lemma invariant_same_args:
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  shows "invariant P x x \<equiv> P x"
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using assms by (auto simp add: invariant_def)
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lemma UNIV_typedef_to_Quotient:
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  assumes "type_definition Rep Abs UNIV"
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  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
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  shows "Quotient (op =) Abs Rep T"
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proof -
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  interpret type_definition Rep Abs UNIV by fact
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  from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis 
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    by (fastforce intro!: QuotientI fun_eq_iff)
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qed
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lemma UNIV_typedef_to_equivp:
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  fixes Abs :: "'a \<Rightarrow> 'b"
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  and Rep :: "'b \<Rightarrow> 'a"
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  assumes "type_definition Rep Abs (UNIV::'a set)"
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  shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
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by (rule identity_equivp)
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lemma typedef_to_Quotient:
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  assumes "type_definition Rep Abs S"
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  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
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  shows "Quotient (invariant (\<lambda>x. x \<in> S)) Abs Rep T"
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proof -
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  interpret type_definition Rep Abs S by fact
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  from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
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    by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)
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qed
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lemma typedef_to_part_equivp:
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  assumes "type_definition Rep Abs S"
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  shows "part_equivp (invariant (\<lambda>x. x \<in> S))"
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proof (intro part_equivpI)
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  interpret type_definition Rep Abs S by fact
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  show "\<exists>x. invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)
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next
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  show "symp (invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)
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next
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  show "transp (invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)
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qed
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lemma open_typedef_to_Quotient:
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  assumes "type_definition Rep Abs {x. P x}"
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  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
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  shows "Quotient (invariant P) Abs Rep T"
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  using typedef_to_Quotient [OF assms] by simp
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lemma open_typedef_to_part_equivp:
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  assumes "type_definition Rep Abs {x. P x}"
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  shows "part_equivp (invariant P)"
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  using typedef_to_part_equivp [OF assms] by simp
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text {* Generating transfer rules for quotients. *}
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context
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  fixes R Abs Rep T
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  assumes 1: "Quotient R Abs Rep T"
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begin
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lemma Quotient_right_unique: "right_unique T"
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  using 1 unfolding Quotient_alt_def right_unique_def by metis
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lemma Quotient_right_total: "right_total T"
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  using 1 unfolding Quotient_alt_def right_total_def by metis
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lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
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  using 1 unfolding Quotient_alt_def fun_rel_def by simp
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lemma Quotient_abs_induct:
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  assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
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  using 1 assms unfolding Quotient_def by metis
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lemma Quotient_All_transfer:
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  "((T ===> op =) ===> op =) (Ball (Respects R)) All"
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  unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
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  by (auto, metis Quotient_abs_induct)
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lemma Quotient_Ex_transfer:
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  "((T ===> op =) ===> op =) (Bex (Respects R)) Ex"
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  unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
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  by (auto, metis Quotient_rep_reflp [OF 1] Quotient_abs_rep [OF 1])
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lemma Quotient_forall_transfer:
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  "((T ===> op =) ===> op =) (transfer_bforall (\<lambda>x. R x x)) transfer_forall"
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  using Quotient_All_transfer
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  unfolding transfer_forall_def transfer_bforall_def
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    Ball_def [abs_def] in_respects .
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end
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text {* Generating transfer rules for total quotients. *}
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context
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  fixes R Abs Rep T
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  assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
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begin
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lemma Quotient_bi_total: "bi_total T"
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  using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
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lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
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  using 1 2 unfolding Quotient_alt_def reflp_def fun_rel_def by simp
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lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
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  using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
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lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
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  using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
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end
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text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
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context
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  fixes Rep Abs A T
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  assumes type: "type_definition Rep Abs A"
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  assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
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begin
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lemma typedef_bi_unique: "bi_unique T"
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  unfolding bi_unique_def T_def
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  by (simp add: type_definition.Rep_inject [OF type])
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lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
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  unfolding fun_rel_def T_def by simp
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lemma typedef_All_transfer: "((T ===> op =) ===> op =) (Ball A) All"
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  unfolding T_def fun_rel_def
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  by (metis type_definition.Rep [OF type]
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    type_definition.Abs_inverse [OF type])
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lemma typedef_Ex_transfer: "((T ===> op =) ===> op =) (Bex A) Ex"
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  unfolding T_def fun_rel_def
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  by (metis type_definition.Rep [OF type]
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    type_definition.Abs_inverse [OF type])
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lemma typedef_forall_transfer:
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  "((T ===> op =) ===> op =)
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    (transfer_bforall (\<lambda>x. x \<in> A)) transfer_forall"
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  unfolding transfer_bforall_def transfer_forall_def Ball_def [symmetric]
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  by (rule typedef_All_transfer)
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end
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text {* Generating the correspondence rule for a constant defined with
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  @{text "lift_definition"}. *}
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lemma Quotient_to_transfer:
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  assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
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  shows "T c c'"
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  using assms by (auto dest: Quotient_cr_rel)
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subsection {* ML setup *}
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use "Tools/Lifting/lifting_util.ML"
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use "Tools/Lifting/lifting_info.ML"
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setup Lifting_Info.setup
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declare fun_quotient[quot_map]
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use "Tools/Lifting/lifting_term.ML"
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use "Tools/Lifting/lifting_def.ML"
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use "Tools/Lifting/lifting_setup.ML"
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hide_const (open) invariant
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end