src/HOL/Lifting.thy
author Andreas Lochbihler
Thu Sep 26 15:50:33 2013 +0200 (2013-09-26)
changeset 53927 abe2b313f0e5
parent 53651 ee90c67502c9
child 53944 50c8f7f21327
permissions -rw-r--r--
add lemmas
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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 Equiv_Relations Transfer
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keywords
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  "parametric" and
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  "print_quot_maps" "print_quotients" :: diag and
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  "lift_definition" :: thy_goal and
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  "setup_lifting" "lifting_forget" "lifting_update" :: thy_decl
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begin
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subsection {* Function map *}
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context
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begin
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interpretation lifting_syntax .
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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 {* Other predicates on relations *}
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definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
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  where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
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lemma left_totalI:
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  "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
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unfolding left_total_def by blast
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lemma left_totalE:
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  assumes "left_total R"
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  obtains "(\<And>x. \<exists>y. R x y)"
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using assms unfolding left_total_def by blast
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lemma bi_total_conv_left_right: "bi_total R \<longleftrightarrow> left_total R \<and> right_total R"
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by(simp add: left_total_def right_total_def bi_total_def)
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definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
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  where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
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lemma bi_unique_conv_left_right: "bi_unique R \<longleftrightarrow> left_unique R \<and> right_unique R"
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by(auto simp add: left_unique_def right_unique_def bi_unique_def)
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lemma left_uniqueI: "(\<And>x y z. \<lbrakk> A x z; A y z \<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> left_unique A"
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unfolding left_unique_def by blast
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lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y"
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unfolding left_unique_def by blast
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lemma left_total_fun:
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  "\<lbrakk>left_unique A; left_total B\<rbrakk> \<Longrightarrow> left_total (A ===> B)"
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  unfolding left_total_def fun_rel_def
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  apply (rule allI, rename_tac f)
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  apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
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  apply clarify
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  apply (subgoal_tac "(THE x. A x y) = x", simp)
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  apply (rule someI_ex)
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  apply (simp)
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  apply (rule the_equality)
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  apply assumption
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  apply (simp add: left_unique_def)
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  done
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lemma left_unique_fun:
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  "\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)"
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  unfolding left_total_def left_unique_def fun_rel_def
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  by (clarify, rule ext, fast)
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lemma left_total_eq: "left_total op=" unfolding left_total_def by blast
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lemma left_unique_eq: "left_unique op=" unfolding left_unique_def by blast
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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_rep_abs_fold_unmap: 
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  assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'" 
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  shows "R (Rep' x') x"
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proof -
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  have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
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  then show ?thesis using assms(3) by simp
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qed
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lemma Quotient_Rep_eq:
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  assumes "x' \<equiv> Abs x" 
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  shows "Rep x' \<equiv> Rep x'"
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by simp
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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_rel_abs2:
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  assumes "R (Rep x) y"
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  shows "x = Abs y"
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proof -
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  from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
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  then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
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qed
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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 fastforce
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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)"
kuncar@47308
   321
  shows "Quotient (invariant P) Abs Rep T"
huffman@47651
   322
  using typedef_to_Quotient [OF assms] by simp
kuncar@47308
   323
kuncar@47361
   324
lemma open_typedef_to_part_equivp:
kuncar@47308
   325
  assumes "type_definition Rep Abs {x. P x}"
kuncar@47308
   326
  shows "part_equivp (invariant P)"
huffman@47651
   327
  using typedef_to_part_equivp [OF assms] by simp
kuncar@47308
   328
huffman@47376
   329
text {* Generating transfer rules for quotients. *}
huffman@47376
   330
huffman@47537
   331
context
huffman@47537
   332
  fixes R Abs Rep T
huffman@47537
   333
  assumes 1: "Quotient R Abs Rep T"
huffman@47537
   334
begin
huffman@47376
   335
huffman@47537
   336
lemma Quotient_right_unique: "right_unique T"
huffman@47537
   337
  using 1 unfolding Quotient_alt_def right_unique_def by metis
huffman@47537
   338
huffman@47537
   339
lemma Quotient_right_total: "right_total T"
huffman@47537
   340
  using 1 unfolding Quotient_alt_def right_total_def by metis
huffman@47537
   341
huffman@47537
   342
lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
huffman@47537
   343
  using 1 unfolding Quotient_alt_def fun_rel_def by simp
huffman@47376
   344
huffman@47538
   345
lemma Quotient_abs_induct:
huffman@47538
   346
  assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
huffman@47538
   347
  using 1 assms unfolding Quotient_def by metis
huffman@47538
   348
huffman@47537
   349
end
huffman@47537
   350
huffman@47537
   351
text {* Generating transfer rules for total quotients. *}
huffman@47376
   352
huffman@47537
   353
context
huffman@47537
   354
  fixes R Abs Rep T
huffman@47537
   355
  assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
huffman@47537
   356
begin
huffman@47376
   357
huffman@47537
   358
lemma Quotient_bi_total: "bi_total T"
huffman@47537
   359
  using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
huffman@47537
   360
huffman@47537
   361
lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
huffman@47537
   362
  using 1 2 unfolding Quotient_alt_def reflp_def fun_rel_def by simp
huffman@47537
   363
huffman@47575
   364
lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
huffman@47575
   365
  using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
huffman@47575
   366
huffman@47889
   367
lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
huffman@47889
   368
  using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
huffman@47889
   369
huffman@47537
   370
end
huffman@47376
   371
huffman@47368
   372
text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
huffman@47368
   373
huffman@47534
   374
context
huffman@47534
   375
  fixes Rep Abs A T
huffman@47368
   376
  assumes type: "type_definition Rep Abs A"
huffman@47534
   377
  assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
huffman@47534
   378
begin
huffman@47534
   379
kuncar@51994
   380
lemma typedef_left_unique: "left_unique T"
kuncar@51994
   381
  unfolding left_unique_def T_def
kuncar@51994
   382
  by (simp add: type_definition.Rep_inject [OF type])
kuncar@51994
   383
huffman@47534
   384
lemma typedef_bi_unique: "bi_unique T"
huffman@47368
   385
  unfolding bi_unique_def T_def
huffman@47368
   386
  by (simp add: type_definition.Rep_inject [OF type])
huffman@47368
   387
kuncar@51374
   388
(* the following two theorems are here only for convinience *)
kuncar@51374
   389
kuncar@51374
   390
lemma typedef_right_unique: "right_unique T"
kuncar@51374
   391
  using T_def type Quotient_right_unique typedef_to_Quotient 
kuncar@51374
   392
  by blast
kuncar@51374
   393
kuncar@51374
   394
lemma typedef_right_total: "right_total T"
kuncar@51374
   395
  using T_def type Quotient_right_total typedef_to_Quotient 
kuncar@51374
   396
  by blast
kuncar@51374
   397
huffman@47535
   398
lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
huffman@47535
   399
  unfolding fun_rel_def T_def by simp
huffman@47535
   400
huffman@47534
   401
end
huffman@47534
   402
huffman@47368
   403
text {* Generating the correspondence rule for a constant defined with
huffman@47368
   404
  @{text "lift_definition"}. *}
huffman@47368
   405
huffman@47351
   406
lemma Quotient_to_transfer:
huffman@47351
   407
  assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
huffman@47351
   408
  shows "T c c'"
huffman@47351
   409
  using assms by (auto dest: Quotient_cr_rel)
huffman@47351
   410
kuncar@47982
   411
text {* Proving reflexivity *}
kuncar@47982
   412
kuncar@51994
   413
definition reflp' :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where "reflp' R \<equiv> reflp R"
kuncar@47982
   414
kuncar@47982
   415
lemma Quotient_to_left_total:
kuncar@47982
   416
  assumes q: "Quotient R Abs Rep T"
kuncar@47982
   417
  and r_R: "reflp R"
kuncar@47982
   418
  shows "left_total T"
kuncar@47982
   419
using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
kuncar@47982
   420
kuncar@47982
   421
lemma reflp_Quotient_composition:
kuncar@51994
   422
  assumes "left_total R"
kuncar@51994
   423
  assumes "reflp T"
kuncar@51994
   424
  shows "reflp (R OO T OO R\<inverse>\<inverse>)"
kuncar@51994
   425
using assms unfolding reflp_def left_total_def by fast
kuncar@51994
   426
kuncar@51994
   427
lemma reflp_fun1:
kuncar@51994
   428
  assumes "is_equality R"
kuncar@51994
   429
  assumes "reflp' S"
kuncar@51994
   430
  shows "reflp (R ===> S)"
kuncar@51994
   431
using assms unfolding is_equality_def reflp'_def reflp_def fun_rel_def by blast
kuncar@51994
   432
kuncar@51994
   433
lemma reflp_fun2:
kuncar@51994
   434
  assumes "is_equality R"
kuncar@51994
   435
  assumes "is_equality S"
kuncar@51994
   436
  shows "reflp (R ===> S)"
kuncar@51994
   437
using assms unfolding is_equality_def reflp_def fun_rel_def by blast
kuncar@51994
   438
kuncar@51994
   439
lemma is_equality_Quotient_composition:
kuncar@51994
   440
  assumes "is_equality T"
kuncar@51994
   441
  assumes "left_total R"
kuncar@51994
   442
  assumes "left_unique R"
kuncar@51994
   443
  shows "is_equality (R OO T OO R\<inverse>\<inverse>)"
kuncar@51994
   444
using assms unfolding is_equality_def left_total_def left_unique_def OO_def conversep_iff
kuncar@51994
   445
by fastforce
kuncar@47982
   446
kuncar@52307
   447
lemma left_total_composition: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
kuncar@52307
   448
unfolding left_total_def OO_def by fast
kuncar@52307
   449
kuncar@52307
   450
lemma left_unique_composition: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
kuncar@52307
   451
unfolding left_unique_def OO_def by fast
kuncar@52307
   452
kuncar@47982
   453
lemma reflp_equality: "reflp (op =)"
kuncar@47982
   454
by (auto intro: reflpI)
kuncar@47982
   455
kuncar@51374
   456
text {* Proving a parametrized correspondence relation *}
kuncar@51374
   457
kuncar@51374
   458
lemma eq_OO: "op= OO R = R"
kuncar@51374
   459
unfolding OO_def by metis
kuncar@51374
   460
kuncar@51374
   461
definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
kuncar@51374
   462
"POS A B \<equiv> A \<le> B"
kuncar@51374
   463
kuncar@51374
   464
definition  NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
kuncar@51374
   465
"NEG A B \<equiv> B \<le> A"
kuncar@51374
   466
kuncar@51374
   467
(*
kuncar@51374
   468
  The following two rules are here because we don't have any proper
kuncar@51374
   469
  left-unique ant left-total relations. Left-unique and left-total
kuncar@51374
   470
  assumptions show up in distributivity rules for the function type.
kuncar@51374
   471
*)
kuncar@51374
   472
kuncar@51374
   473
lemma bi_unique_left_unique[transfer_rule]: "bi_unique R \<Longrightarrow> left_unique R"
kuncar@51374
   474
unfolding bi_unique_def left_unique_def by blast
kuncar@51374
   475
kuncar@51374
   476
lemma bi_total_left_total[transfer_rule]: "bi_total R \<Longrightarrow> left_total R"
kuncar@51374
   477
unfolding bi_total_def left_total_def by blast
kuncar@51374
   478
kuncar@51374
   479
lemma pos_OO_eq:
kuncar@51374
   480
  shows "POS (A OO op=) A"
kuncar@51374
   481
unfolding POS_def OO_def by blast
kuncar@51374
   482
kuncar@51374
   483
lemma pos_eq_OO:
kuncar@51374
   484
  shows "POS (op= OO A) A"
kuncar@51374
   485
unfolding POS_def OO_def by blast
kuncar@51374
   486
kuncar@51374
   487
lemma neg_OO_eq:
kuncar@51374
   488
  shows "NEG (A OO op=) A"
kuncar@51374
   489
unfolding NEG_def OO_def by auto
kuncar@51374
   490
kuncar@51374
   491
lemma neg_eq_OO:
kuncar@51374
   492
  shows "NEG (op= OO A) A"
kuncar@51374
   493
unfolding NEG_def OO_def by blast
kuncar@51374
   494
kuncar@51374
   495
lemma POS_trans:
kuncar@51374
   496
  assumes "POS A B"
kuncar@51374
   497
  assumes "POS B C"
kuncar@51374
   498
  shows "POS A C"
kuncar@51374
   499
using assms unfolding POS_def by auto
kuncar@51374
   500
kuncar@51374
   501
lemma NEG_trans:
kuncar@51374
   502
  assumes "NEG A B"
kuncar@51374
   503
  assumes "NEG B C"
kuncar@51374
   504
  shows "NEG A C"
kuncar@51374
   505
using assms unfolding NEG_def by auto
kuncar@51374
   506
kuncar@51374
   507
lemma POS_NEG:
kuncar@51374
   508
  "POS A B \<equiv> NEG B A"
kuncar@51374
   509
  unfolding POS_def NEG_def by auto
kuncar@51374
   510
kuncar@51374
   511
lemma NEG_POS:
kuncar@51374
   512
  "NEG A B \<equiv> POS B A"
kuncar@51374
   513
  unfolding POS_def NEG_def by auto
kuncar@51374
   514
kuncar@51374
   515
lemma POS_pcr_rule:
kuncar@51374
   516
  assumes "POS (A OO B) C"
kuncar@51374
   517
  shows "POS (A OO B OO X) (C OO X)"
kuncar@51374
   518
using assms unfolding POS_def OO_def by blast
kuncar@51374
   519
kuncar@51374
   520
lemma NEG_pcr_rule:
kuncar@51374
   521
  assumes "NEG (A OO B) C"
kuncar@51374
   522
  shows "NEG (A OO B OO X) (C OO X)"
kuncar@51374
   523
using assms unfolding NEG_def OO_def by blast
kuncar@51374
   524
kuncar@51374
   525
lemma POS_apply:
kuncar@51374
   526
  assumes "POS R R'"
kuncar@51374
   527
  assumes "R f g"
kuncar@51374
   528
  shows "R' f g"
kuncar@51374
   529
using assms unfolding POS_def by auto
kuncar@51374
   530
kuncar@51374
   531
text {* Proving a parametrized correspondence relation *}
kuncar@51374
   532
kuncar@51374
   533
lemma fun_mono:
kuncar@51374
   534
  assumes "A \<ge> C"
kuncar@51374
   535
  assumes "B \<le> D"
kuncar@51374
   536
  shows   "(A ===> B) \<le> (C ===> D)"
kuncar@51374
   537
using assms unfolding fun_rel_def by blast
kuncar@51374
   538
kuncar@51374
   539
lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"
kuncar@51374
   540
unfolding OO_def fun_rel_def by blast
kuncar@51374
   541
kuncar@51374
   542
lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"
kuncar@51374
   543
unfolding right_unique_def left_total_def by blast
kuncar@51374
   544
kuncar@51374
   545
lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"
kuncar@51374
   546
unfolding left_unique_def right_total_def by blast
kuncar@51374
   547
kuncar@51374
   548
lemma neg_fun_distr1:
kuncar@51374
   549
assumes 1: "left_unique R" "right_total R"
kuncar@51374
   550
assumes 2: "right_unique R'" "left_total R'"
kuncar@51374
   551
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "
kuncar@51374
   552
  using functional_relation[OF 2] functional_converse_relation[OF 1]
kuncar@51374
   553
  unfolding fun_rel_def OO_def
kuncar@51374
   554
  apply clarify
kuncar@51374
   555
  apply (subst all_comm)
kuncar@51374
   556
  apply (subst all_conj_distrib[symmetric])
kuncar@51374
   557
  apply (intro choice)
kuncar@51374
   558
  by metis
kuncar@51374
   559
kuncar@51374
   560
lemma neg_fun_distr2:
kuncar@51374
   561
assumes 1: "right_unique R'" "left_total R'"
kuncar@51374
   562
assumes 2: "left_unique S'" "right_total S'"
kuncar@51374
   563
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"
kuncar@51374
   564
  using functional_converse_relation[OF 2] functional_relation[OF 1]
kuncar@51374
   565
  unfolding fun_rel_def OO_def
kuncar@51374
   566
  apply clarify
kuncar@51374
   567
  apply (subst all_comm)
kuncar@51374
   568
  apply (subst all_conj_distrib[symmetric])
kuncar@51374
   569
  apply (intro choice)
kuncar@51374
   570
  by metis
kuncar@51374
   571
kuncar@51956
   572
subsection {* Domains *}
kuncar@51956
   573
kuncar@51956
   574
lemma pcr_Domainp_par_left_total:
kuncar@51956
   575
  assumes "Domainp B = P"
kuncar@51956
   576
  assumes "left_total A"
kuncar@51956
   577
  assumes "(A ===> op=) P' P"
kuncar@51956
   578
  shows "Domainp (A OO B) = P'"
kuncar@51956
   579
using assms
kuncar@51956
   580
unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def fun_rel_def 
kuncar@51956
   581
by (fast intro: fun_eq_iff)
kuncar@51956
   582
kuncar@51956
   583
lemma pcr_Domainp_par:
kuncar@51956
   584
assumes "Domainp B = P2"
kuncar@51956
   585
assumes "Domainp A = P1"
kuncar@51956
   586
assumes "(A ===> op=) P2' P2"
kuncar@51956
   587
shows "Domainp (A OO B) = (inf P1 P2')"
kuncar@51956
   588
using assms unfolding fun_rel_def Domainp_iff[abs_def] OO_def
kuncar@51956
   589
by (fast intro: fun_eq_iff)
kuncar@51956
   590
kuncar@53151
   591
definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"
kuncar@51956
   592
where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y"
kuncar@51956
   593
kuncar@51956
   594
lemma pcr_Domainp:
kuncar@51956
   595
assumes "Domainp B = P"
kuncar@53151
   596
shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)"
kuncar@53151
   597
using assms by blast
kuncar@51956
   598
kuncar@51956
   599
lemma pcr_Domainp_total:
kuncar@51956
   600
  assumes "bi_total B"
kuncar@51956
   601
  assumes "Domainp A = P"
kuncar@51956
   602
  shows "Domainp (A OO B) = P"
kuncar@51956
   603
using assms unfolding bi_total_def 
kuncar@51956
   604
by fast
kuncar@51956
   605
kuncar@51956
   606
lemma Quotient_to_Domainp:
kuncar@51956
   607
  assumes "Quotient R Abs Rep T"
kuncar@51956
   608
  shows "Domainp T = (\<lambda>x. R x x)"  
kuncar@51956
   609
by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
kuncar@51956
   610
kuncar@51956
   611
lemma invariant_to_Domainp:
kuncar@51956
   612
  assumes "Quotient (Lifting.invariant P) Abs Rep T"
kuncar@51956
   613
  shows "Domainp T = P"
kuncar@51956
   614
by (simp add: invariant_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
kuncar@51956
   615
kuncar@53011
   616
end
kuncar@53011
   617
kuncar@47308
   618
subsection {* ML setup *}
kuncar@47308
   619
wenzelm@48891
   620
ML_file "Tools/Lifting/lifting_util.ML"
kuncar@47308
   621
wenzelm@48891
   622
ML_file "Tools/Lifting/lifting_info.ML"
kuncar@47308
   623
setup Lifting_Info.setup
kuncar@47308
   624
kuncar@51994
   625
lemmas [reflexivity_rule] = 
kuncar@52036
   626
  reflp_equality reflp_Quotient_composition is_equality_Quotient_composition 
kuncar@52307
   627
  left_total_fun left_unique_fun left_total_eq left_unique_eq left_total_composition
kuncar@52307
   628
  left_unique_composition
kuncar@51994
   629
kuncar@51994
   630
text {* add @{thm reflp_fun1} and @{thm reflp_fun2} manually through ML
kuncar@51994
   631
  because we don't want to get reflp' variant of these theorems *}
kuncar@51994
   632
kuncar@51994
   633
setup{*
kuncar@51994
   634
Context.theory_map 
kuncar@51994
   635
  (fold
kuncar@51994
   636
    (snd oo (Thm.apply_attribute Lifting_Info.add_reflexivity_rule_raw_attribute)) 
kuncar@51994
   637
      [@{thm reflp_fun1}, @{thm reflp_fun2}])
kuncar@51994
   638
*}
kuncar@51374
   639
kuncar@51374
   640
(* setup for the function type *)
kuncar@47777
   641
declare fun_quotient[quot_map]
kuncar@51374
   642
declare fun_mono[relator_mono]
kuncar@51374
   643
lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2
kuncar@47308
   644
wenzelm@48891
   645
ML_file "Tools/Lifting/lifting_term.ML"
kuncar@47308
   646
wenzelm@48891
   647
ML_file "Tools/Lifting/lifting_def.ML"
kuncar@47308
   648
wenzelm@48891
   649
ML_file "Tools/Lifting/lifting_setup.ML"
kuncar@47308
   650
kuncar@51994
   651
hide_const (open) invariant POS NEG reflp'
kuncar@47308
   652
kuncar@47308
   653
end