src/HOL/Lifting.thy
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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_iff: "bi_total A = (right_total A \<and> left_total A)"
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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 left_unique_transfer [transfer_rule]:
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  assumes "right_total A"
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  assumes "right_total B"
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  assumes "bi_unique A"
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  shows "((A ===> B ===> op=) ===> implies) left_unique left_unique"
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using assms unfolding left_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
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by metis
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lemma bi_unique_iff: "bi_unique A = (right_unique A \<and> left_unique A)"
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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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lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y"
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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 rel_fun_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 rel_fun_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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lemma [simp]:
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  shows left_unique_conversep: "left_unique A\<inverse>\<inverse> \<longleftrightarrow> right_unique A"
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  and right_unique_conversep: "right_unique A\<inverse>\<inverse> \<longleftrightarrow> left_unique A"
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by(auto simp add: left_unique_def right_unique_def)
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lemma [simp]:
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  shows left_total_conversep: "left_total A\<inverse>\<inverse> \<longleftrightarrow> right_total A"
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  and right_total_conversep: "right_total A\<inverse>\<inverse> \<longleftrightarrow> left_total A"
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by(simp_all add: left_total_def right_total_def)
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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_eq: "R t t \<Longrightarrow> R \<le> op= \<Longrightarrow> Rep (Abs t) = t"
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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 rel_fun_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: rel_funE)
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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: rel_funE)
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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 rel_fun_eq_invariant: "(op= ===> Lifting.invariant P) = Lifting.invariant (\<lambda>f. \<forall>x. P(f x))"
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unfolding invariant_def rel_fun_def by auto
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lemma rel_fun_invariant_rel:
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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 rel_fun_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 invariant_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A"
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   312
  shows "((A ===> op=) ===> A ===> A ===> op=) Lifting.invariant Lifting.invariant"
b2781a3ce958 new parametricity rules and useful lemmas
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parents: 53944
diff changeset
   313
unfolding invariant_def[abs_def] by transfer_prover
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parents: 53944
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   314
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   315
lemma UNIV_typedef_to_Quotient:
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  assumes "type_definition Rep Abs UNIV"
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   317
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
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parents:
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   318
  shows "Quotient (op =) Abs Rep T"
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parents:
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   319
proof -
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parents:
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   320
  interpret type_definition Rep Abs UNIV by fact
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   321
  from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis 
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   322
    by (fastforce intro!: QuotientI fun_eq_iff)
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parents:
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   323
qed
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parents:
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   324
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   325
lemma UNIV_typedef_to_equivp:
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parents:
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   326
  fixes Abs :: "'a \<Rightarrow> 'b"
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parents:
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   327
  and Rep :: "'b \<Rightarrow> 'a"
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parents:
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   328
  assumes "type_definition Rep Abs (UNIV::'a set)"
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kuncar
parents:
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   329
  shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
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parents:
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   330
by (rule identity_equivp)
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parents:
diff changeset
   331
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   332
lemma typedef_to_Quotient:
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   333
  assumes "type_definition Rep Abs S"
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parents: 47354
diff changeset
   334
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
47501
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kuncar
parents: 47436
diff changeset
   335
  shows "Quotient (invariant (\<lambda>x. x \<in> S)) Abs Rep T"
47361
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kuncar
parents: 47354
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   336
proof -
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parents: 47354
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   337
  interpret type_definition Rep Abs S by fact
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diff changeset
   338
  from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
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   339
    by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)
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parents: 47354
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   340
qed
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   341
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
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   342
lemma typedef_to_part_equivp:
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parents: 47354
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   343
  assumes "type_definition Rep Abs S"
47501
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parents: 47436
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   344
  shows "part_equivp (invariant (\<lambda>x. x \<in> S))"
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parents: 47354
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   345
proof (intro part_equivpI)
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
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   346
  interpret type_definition Rep Abs S by fact
47501
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parents: 47436
diff changeset
   347
  show "\<exists>x. invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)
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parents: 47354
diff changeset
   348
next
47501
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parents: 47436
diff changeset
   349
  show "symp (invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)
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kuncar
parents: 47354
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   350
next
47501
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kuncar
parents: 47436
diff changeset
   351
  show "transp (invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)
47361
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kuncar
parents: 47354
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   352
qed
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   353
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
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   354
lemma open_typedef_to_Quotient:
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parents:
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   355
  assumes "type_definition Rep Abs {x. P x}"
47354
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parents: 47351
diff changeset
   356
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
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parents:
diff changeset
   357
  shows "Quotient (invariant P) Abs Rep T"
47651
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parents: 47575
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   358
  using typedef_to_Quotient [OF assms] by simp
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parents:
diff changeset
   359
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   360
lemma open_typedef_to_part_equivp:
47308
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parents:
diff changeset
   361
  assumes "type_definition Rep Abs {x. P x}"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   362
  shows "part_equivp (invariant P)"
47651
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huffman
parents: 47575
diff changeset
   363
  using typedef_to_part_equivp [OF assms] by simp
47308
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parents:
diff changeset
   364
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   365
text {* Generating transfer rules for quotients. *}
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   366
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   367
context
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parents: 47536
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   368
  fixes R Abs Rep T
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   369
  assumes 1: "Quotient R Abs Rep T"
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   370
begin
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   371
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   372
lemma Quotient_right_unique: "right_unique T"
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   373
  using 1 unfolding Quotient_alt_def right_unique_def by metis
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parents: 47536
diff changeset
   374
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   375
lemma Quotient_right_total: "right_total T"
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parents: 47536
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   376
  using 1 unfolding Quotient_alt_def right_total_def by metis
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parents: 47536
diff changeset
   377
b06be48923a4 more usage of context blocks
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   378
lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
55945
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blanchet
parents: 55737
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   379
  using 1 unfolding Quotient_alt_def rel_fun_def by simp
47376
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parents: 47369
diff changeset
   380
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   381
lemma Quotient_abs_induct:
1f0ec5b8135a add lemma Quotient_abs_induct
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parents: 47537
diff changeset
   382
  assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   383
  using 1 assms unfolding Quotient_def by metis
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   384
47537
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parents: 47536
diff changeset
   385
end
b06be48923a4 more usage of context blocks
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parents: 47536
diff changeset
   386
b06be48923a4 more usage of context blocks
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   387
text {* Generating transfer rules for total quotients. *}
47376
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huffman
parents: 47369
diff changeset
   388
47537
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huffman
parents: 47536
diff changeset
   389
context
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   390
  fixes R Abs Rep T
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   391
  assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
b06be48923a4 more usage of context blocks
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parents: 47536
diff changeset
   392
begin
47376
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huffman
parents: 47369
diff changeset
   393
47537
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diff changeset
   394
lemma Quotient_bi_total: "bi_total T"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   395
  using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   396
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   397
lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   398
  using 1 2 unfolding Quotient_alt_def reflp_def rel_fun_def by simp
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   399
47575
b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   400
lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   401
  using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   402
47889
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   403
lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   404
  using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   405
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   406
end
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   407
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   408
text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   409
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   410
context
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   411
  fixes Rep Abs A T
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   412
  assumes type: "type_definition Rep Abs A"
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   413
  assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   414
begin
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   415
51994
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   416
lemma typedef_left_unique: "left_unique T"
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   417
  unfolding left_unique_def T_def
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   418
  by (simp add: type_definition.Rep_inject [OF type])
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   419
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   420
lemma typedef_bi_unique: "bi_unique T"
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   421
  unfolding bi_unique_def T_def
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   422
  by (simp add: type_definition.Rep_inject [OF type])
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   423
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   424
(* the following two theorems are here only for convinience *)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   425
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   426
lemma typedef_right_unique: "right_unique T"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   427
  using T_def type Quotient_right_unique typedef_to_Quotient 
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   428
  by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   429
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   430
lemma typedef_right_total: "right_total T"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   431
  using T_def type Quotient_right_total typedef_to_Quotient 
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   432
  by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   433
47535
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   434
lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   435
  unfolding rel_fun_def T_def by simp
47535
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   436
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   437
end
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   438
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   439
text {* Generating the correspondence rule for a constant defined with
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   440
  @{text "lift_definition"}. *}
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   441
47351
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   442
lemma Quotient_to_transfer:
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   443
  assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   444
  shows "T c c'"
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   445
  using assms by (auto dest: Quotient_cr_rel)
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   446
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   447
text {* Proving reflexivity *}
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   448
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   449
lemma Quotient_to_left_total:
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   450
  assumes q: "Quotient R Abs Rep T"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   451
  and r_R: "reflp R"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   452
  shows "left_total T"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   453
using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   454
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   455
lemma Quotient_composition_ge_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   456
  assumes "left_total T"
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   457
  assumes "R \<ge> op="
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   458
  shows "(T OO R OO T\<inverse>\<inverse>) \<ge> op="
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   459
using assms unfolding left_total_def by fast
51994
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   460
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   461
lemma Quotient_composition_le_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   462
  assumes "left_unique T"
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   463
  assumes "R \<le> op="
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   464
  shows "(T OO R OO T\<inverse>\<inverse>) \<le> op="
55604
42e4e8c2e8dc less flex-flex pairs
noschinl
parents: 55563
diff changeset
   465
using assms unfolding left_unique_def by blast
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   466
52307
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   467
lemma left_total_composition: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   468
unfolding left_total_def OO_def by fast
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   469
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   470
lemma left_unique_composition: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
55604
42e4e8c2e8dc less flex-flex pairs
noschinl
parents: 55563
diff changeset
   471
unfolding left_unique_def OO_def by blast
52307
32c433c38ddd more reflexivity rules (for OO)
kuncar
parents: 52036
diff changeset
   472
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   473
lemma invariant_le_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   474
  "invariant P \<le> op=" unfolding invariant_def by blast
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   475
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   476
lemma reflp_ge_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   477
  "reflp R \<Longrightarrow> R \<ge> op=" unfolding reflp_def by blast
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   478
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   479
lemma ge_eq_refl:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   480
  "R \<ge> op= \<Longrightarrow> R x x" by blast
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   481
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   482
text {* Proving a parametrized correspondence relation *}
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   483
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   484
definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   485
"POS A B \<equiv> A \<le> B"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   486
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   487
definition  NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   488
"NEG A B \<equiv> B \<le> A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   489
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   490
(*
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   491
  The following two rules are here because we don't have any proper
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   492
  left-unique ant left-total relations. Left-unique and left-total
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   493
  assumptions show up in distributivity rules for the function type.
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   494
*)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   495
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   496
lemma bi_unique_left_unique[transfer_rule]: "bi_unique R \<Longrightarrow> left_unique R"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   497
unfolding bi_unique_def left_unique_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   498
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   499
lemma bi_total_left_total[transfer_rule]: "bi_total R \<Longrightarrow> left_total R"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   500
unfolding bi_total_def left_total_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   501
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   502
lemma pos_OO_eq:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   503
  shows "POS (A OO op=) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   504
unfolding POS_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   505
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   506
lemma pos_eq_OO:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   507
  shows "POS (op= OO A) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   508
unfolding POS_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   509
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   510
lemma neg_OO_eq:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   511
  shows "NEG (A OO op=) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   512
unfolding NEG_def OO_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   513
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   514
lemma neg_eq_OO:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   515
  shows "NEG (op= OO A) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   516
unfolding NEG_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   517
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   518
lemma POS_trans:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   519
  assumes "POS A B"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   520
  assumes "POS B C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   521
  shows "POS A C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   522
using assms unfolding POS_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   523
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   524
lemma NEG_trans:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   525
  assumes "NEG A B"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   526
  assumes "NEG B C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   527
  shows "NEG A C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   528
using assms unfolding NEG_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   529
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   530
lemma POS_NEG:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   531
  "POS A B \<equiv> NEG B A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   532
  unfolding POS_def NEG_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   533
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   534
lemma NEG_POS:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   535
  "NEG A B \<equiv> POS B A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   536
  unfolding POS_def NEG_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   537
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   538
lemma POS_pcr_rule:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   539
  assumes "POS (A OO B) C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   540
  shows "POS (A OO B OO X) (C OO X)"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   541
using assms unfolding POS_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   542
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   543
lemma NEG_pcr_rule:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   544
  assumes "NEG (A OO B) C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   545
  shows "NEG (A OO B OO X) (C OO X)"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   546
using assms unfolding NEG_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   547
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   548
lemma POS_apply:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   549
  assumes "POS R R'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   550
  assumes "R f g"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   551
  shows "R' f g"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   552
using assms unfolding POS_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   553
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   554
text {* Proving a parametrized correspondence relation *}
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   555
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   556
lemma fun_mono:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   557
  assumes "A \<ge> C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   558
  assumes "B \<le> D"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   559
  shows   "(A ===> B) \<le> (C ===> D)"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   560
using assms unfolding rel_fun_def by blast
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   561
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   562
lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   563
unfolding OO_def rel_fun_def by blast
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   564
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   565
lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   566
unfolding right_unique_def left_total_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   567
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   568
lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   569
unfolding left_unique_def right_total_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   570
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   571
lemma neg_fun_distr1:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   572
assumes 1: "left_unique R" "right_total R"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   573
assumes 2: "right_unique R'" "left_total R'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   574
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   575
  using functional_relation[OF 2] functional_converse_relation[OF 1]
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   576
  unfolding rel_fun_def OO_def
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   577
  apply clarify
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   578
  apply (subst all_comm)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   579
  apply (subst all_conj_distrib[symmetric])
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   580
  apply (intro choice)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   581
  by metis
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   582
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   583
lemma neg_fun_distr2:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   584
assumes 1: "right_unique R'" "left_total R'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   585
assumes 2: "left_unique S'" "right_total S'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   586
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   587
  using functional_converse_relation[OF 2] functional_relation[OF 1]
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   588
  unfolding rel_fun_def OO_def
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   589
  apply clarify
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   590
  apply (subst all_comm)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   591
  apply (subst all_conj_distrib[symmetric])
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   592
  apply (intro choice)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   593
  by metis
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   594
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   595
subsection {* Domains *}
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   596
55731
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   597
lemma composed_equiv_rel_invariant:
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   598
  assumes "left_unique R"
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   599
  assumes "(R ===> op=) P P'"
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   600
  assumes "Domainp R = P''"
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   601
  shows "(R OO Lifting.invariant P' OO R\<inverse>\<inverse>) = Lifting.invariant (inf P'' P)"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   602
using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def rel_fun_def invariant_def
55731
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   603
fun_eq_iff by blast
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   604
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   605
lemma composed_equiv_rel_eq_invariant:
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   606
  assumes "left_unique R"
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   607
  assumes "Domainp R = P"
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   608
  shows "(R OO op= OO R\<inverse>\<inverse>) = Lifting.invariant P"
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   609
using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def invariant_def
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   610
fun_eq_iff is_equality_def by metis
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   611
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   612
lemma pcr_Domainp_par_left_total:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   613
  assumes "Domainp B = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   614
  assumes "left_total A"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   615
  assumes "(A ===> op=) P' P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   616
  shows "Domainp (A OO B) = P'"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   617
using assms
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   618
unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def rel_fun_def 
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   619
by (fast intro: fun_eq_iff)
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   620
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   621
lemma pcr_Domainp_par:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   622
assumes "Domainp B = P2"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   623
assumes "Domainp A = P1"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   624
assumes "(A ===> op=) P2' P2"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   625
shows "Domainp (A OO B) = (inf P1 P2')"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   626
using assms unfolding rel_fun_def Domainp_iff[abs_def] OO_def
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   627
by (fast intro: fun_eq_iff)
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   628
53151
fbf4d50dec91 remove OP
kuncar
parents: 53011
diff changeset
   629
definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   630
where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   631
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   632
lemma pcr_Domainp:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   633
assumes "Domainp B = P"
53151
fbf4d50dec91 remove OP
kuncar
parents: 53011
diff changeset
   634
shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)"
fbf4d50dec91 remove OP
kuncar
parents: 53011
diff changeset
   635
using assms by blast
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   636
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   637
lemma pcr_Domainp_total:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   638
  assumes "bi_total B"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   639
  assumes "Domainp A = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   640
  shows "Domainp (A OO B) = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   641
using assms unfolding bi_total_def 
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   642
by fast
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   643
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   644
lemma Quotient_to_Domainp:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   645
  assumes "Quotient R Abs Rep T"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   646
  shows "Domainp T = (\<lambda>x. R x x)"  
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   647
by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   648
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   649
lemma invariant_to_Domainp:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   650
  assumes "Quotient (Lifting.invariant P) Abs Rep T"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   651
  shows "Domainp T = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   652
by (simp add: invariant_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   653
53011
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 52307
diff changeset
   654
end
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 52307
diff changeset
   655
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   656
subsection {* ML setup *}
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   657
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   658
ML_file "Tools/Lifting/lifting_util.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   659
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   660
ML_file "Tools/Lifting/lifting_info.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   661
setup Lifting_Info.setup
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   662
51994
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   663
lemmas [reflexivity_rule] = 
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   664
  order_refl[of "op="] invariant_le_eq Quotient_composition_le_eq
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   665
  Quotient_composition_ge_eq
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   666
  left_total_eq left_unique_eq left_total_composition left_unique_composition
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   667
  left_total_fun left_unique_fun
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   668
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   669
(* setup for the function type *)
47777
f29e7dcd7c40 use a quot_map theorem attribute instead of the complicated map attribute
kuncar
parents: 47698
diff changeset
   670
declare fun_quotient[quot_map]
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   671
declare fun_mono[relator_mono]
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   672
lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   673
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   674
ML_file "Tools/Lifting/lifting_term.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   675
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   676
ML_file "Tools/Lifting/lifting_def.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   677
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   678
ML_file "Tools/Lifting/lifting_setup.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   679
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   680
hide_const (open) invariant POS NEG
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   681
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   682
end