src/HOL/Quotient.thy
author Cezary Kaliszyk <kaliszyk@in.tum.de>
Tue Oct 19 11:44:42 2010 +0900 (2010-10-19)
changeset 40031 2671cce4d25d
parent 39956 132b79985660
child 40466 c6587375088e
permissions -rw-r--r--
Quotient package: partial equivalence introduction
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(*  Title:      Quotient.thy
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    Author:     Cezary Kaliszyk and Christian Urban
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*)
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header {* Definition of Quotient Types *}
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theory Quotient
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imports Plain Hilbert_Choice
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uses
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  ("Tools/Quotient/quotient_info.ML")
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  ("Tools/Quotient/quotient_typ.ML")
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  ("Tools/Quotient/quotient_def.ML")
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  ("Tools/Quotient/quotient_term.ML")
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  ("Tools/Quotient/quotient_tacs.ML")
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begin
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text {*
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  Basic definition for equivalence relations
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  that are represented by predicates.
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*}
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definition
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  "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)"
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definition
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  "reflp E \<equiv> \<forall>x. E x x"
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definition
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  "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"
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definition
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  "transp E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"
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lemma equivp_reflp_symp_transp:
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  shows "equivp E = (reflp E \<and> symp E \<and> transp E)"
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  unfolding equivp_def reflp_def symp_def transp_def fun_eq_iff
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  by blast
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lemma equivp_reflp:
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  shows "equivp E \<Longrightarrow> E x x"
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  by (simp only: equivp_reflp_symp_transp reflp_def)
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lemma equivp_symp:
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  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y x"
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  by (metis equivp_reflp_symp_transp symp_def)
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lemma equivp_transp:
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  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y z \<Longrightarrow> E x z"
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  by (metis equivp_reflp_symp_transp transp_def)
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lemma equivpI:
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  assumes "reflp R" "symp R" "transp R"
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  shows "equivp R"
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  using assms by (simp add: equivp_reflp_symp_transp)
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lemma identity_equivp:
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  shows "equivp (op =)"
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  unfolding equivp_def
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  by auto
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text {* Partial equivalences *}
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definition
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  "part_equivp E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"
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lemma equivp_implies_part_equivp:
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  assumes a: "equivp E"
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  shows "part_equivp E"
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  using a
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  unfolding equivp_def part_equivp_def
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  by auto
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lemma part_equivp_symp:
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  assumes e: "part_equivp R"
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  and a: "R x y"
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  shows "R y x"
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  using e[simplified part_equivp_def] a
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  by (metis)
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lemma part_equivp_typedef:
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  shows "part_equivp R \<Longrightarrow> \<exists>d. d \<in> (\<lambda>c. \<exists>x. R x x \<and> c = R x)"
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  unfolding part_equivp_def mem_def
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  apply clarify
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  apply (intro exI)
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  apply (rule conjI)
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  apply assumption
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  apply (rule refl)
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  done
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lemma part_equivp_refl_symp_transp:
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  shows "part_equivp E \<longleftrightarrow> ((\<exists>x. E x x) \<and> symp E \<and> transp E)"
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proof
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  assume "part_equivp E"
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  then show "(\<exists>x. E x x) \<and> symp E \<and> transp E"
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  unfolding part_equivp_def symp_def transp_def
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  by metis
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next
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  assume a: "(\<exists>x. E x x) \<and> symp E \<and> transp E"
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  then have b: "(\<forall>x y. E x y \<longrightarrow> E y x)" and c: "(\<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z)"
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    unfolding symp_def transp_def by (metis, metis)
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  have "(\<forall>x y. E x y = (E x x \<and> E y y \<and> E x = E y))"
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  proof (intro allI iffI conjI)
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    fix x y
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    assume d: "E x y"
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    then show "E x x" using b c by metis
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    show "E y y" using b c d by metis
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    show "E x = E y" unfolding fun_eq_iff using b c d by metis
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  next
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    fix x y
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    assume "E x x \<and> E y y \<and> E x = E y"
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    then show "E x y" using b c by metis
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  qed
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  then show "part_equivp E" unfolding part_equivp_def using a by metis
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qed
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text {* Composition of Relations *}
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abbreviation
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  rel_conj (infixr "OOO" 75)
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where
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  "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
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lemma eq_comp_r:
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  shows "((op =) OOO R) = R"
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  by (auto simp add: fun_eq_iff)
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subsection {* Respects predicate *}
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definition
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  Respects
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where
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  "Respects R x \<equiv> R x x"
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lemma in_respects:
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  shows "(x \<in> Respects R) = R x x"
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  unfolding mem_def Respects_def
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  by simp
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subsection {* Function map and function relation *}
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definition
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  fun_map (infixr "--->" 55)
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where
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[simp]: "fun_map f g h x = g (h (f x))"
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definition
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  fun_rel (infixr "===>" 55)
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where
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[simp]: "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"
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lemma fun_relI [intro]:
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  assumes "\<And>a b. P a b \<Longrightarrow> Q (x a) (y b)"
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  shows "(P ===> Q) x y"
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  using assms by (simp add: fun_rel_def)
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lemma fun_map_id:
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  shows "(id ---> id) = id"
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  by (simp add: fun_eq_iff id_def)
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lemma fun_rel_eq:
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  shows "((op =) ===> (op =)) = (op =)"
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  by (simp add: fun_eq_iff)
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subsection {* Quotient Predicate *}
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definition
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  "Quotient E Abs Rep \<equiv>
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     (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and>
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     (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
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lemma Quotient_abs_rep:
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  assumes a: "Quotient E Abs Rep"
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  shows "Abs (Rep a) = a"
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  using a
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  unfolding Quotient_def
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  by simp
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lemma Quotient_rep_reflp:
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  assumes a: "Quotient E Abs Rep"
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  shows "E (Rep a) (Rep a)"
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  using a
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  unfolding Quotient_def
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  by blast
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lemma Quotient_rel:
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  assumes a: "Quotient E Abs Rep"
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  shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
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  using a
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  unfolding Quotient_def
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  by blast
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lemma Quotient_rel_rep:
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  assumes a: "Quotient R Abs Rep"
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  shows "R (Rep a) (Rep b) = (a = b)"
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  using a
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  unfolding Quotient_def
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  by metis
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lemma Quotient_rep_abs:
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  assumes a: "Quotient R Abs Rep"
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  shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rel_abs:
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  assumes a: "Quotient E Abs Rep"
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  shows "E r s \<Longrightarrow> Abs r = Abs s"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_symp:
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  assumes a: "Quotient E Abs Rep"
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  shows "symp E"
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  using a unfolding Quotient_def symp_def
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  by metis
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lemma Quotient_transp:
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  assumes a: "Quotient E Abs Rep"
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  shows "transp E"
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  using a unfolding Quotient_def transp_def
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  by metis
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lemma identity_quotient:
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  shows "Quotient (op =) id id"
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  unfolding Quotient_def id_def
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  by blast
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lemma fun_quotient:
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  assumes q1: "Quotient R1 abs1 rep1"
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  and     q2: "Quotient R2 abs2 rep2"
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  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
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proof -
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  have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
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    using q1 q2
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    unfolding Quotient_def
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    unfolding fun_eq_iff
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    by simp
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  moreover
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  have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
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    using q1 q2
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    unfolding Quotient_def
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    by (simp (no_asm)) (metis)
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  moreover
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  have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
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        (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
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    unfolding fun_eq_iff
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    apply(auto)
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    using q1 q2 unfolding Quotient_def
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    apply(metis)
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    using q1 q2 unfolding Quotient_def
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    apply(metis)
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    using q1 q2 unfolding Quotient_def
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    apply(metis)
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    using q1 q2 unfolding Quotient_def
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    apply(metis)
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    done
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  ultimately
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  show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
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    unfolding Quotient_def by blast
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qed
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lemma abs_o_rep:
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  assumes a: "Quotient R Abs Rep"
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  shows "Abs o Rep = id"
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  unfolding fun_eq_iff
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  by (simp add: Quotient_abs_rep[OF a])
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lemma equals_rsp:
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  assumes q: "Quotient R Abs Rep"
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  and     a: "R xa xb" "R ya yb"
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  shows "R xa ya = R xb yb"
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  using a Quotient_symp[OF q] Quotient_transp[OF q]
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  unfolding symp_def transp_def
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  by blast
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lemma lambda_prs:
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  assumes q1: "Quotient R1 Abs1 Rep1"
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  and     q2: "Quotient R2 Abs2 Rep2"
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  shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
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  unfolding fun_eq_iff
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  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
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  by simp
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lemma lambda_prs1:
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  assumes q1: "Quotient R1 Abs1 Rep1"
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  and     q2: "Quotient R2 Abs2 Rep2"
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  shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
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  unfolding fun_eq_iff
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  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
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  by simp
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lemma rep_abs_rsp:
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  assumes q: "Quotient R Abs Rep"
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  and     a: "R x1 x2"
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  shows "R x1 (Rep (Abs x2))"
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  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
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  by metis
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lemma rep_abs_rsp_left:
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  assumes q: "Quotient R Abs Rep"
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  and     a: "R x1 x2"
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  shows "R (Rep (Abs x1)) x2"
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  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
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  by metis
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text{*
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  In the following theorem R1 can be instantiated with anything,
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  but we know some of the types of the Rep and Abs functions;
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  so by solving Quotient assumptions we can get a unique R1 that
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  will be provable; which is why we need to use @{text apply_rsp} and
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  not the primed version *}
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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"
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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 simp
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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 simp
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subsection {* lemmas for regularisation of ball and bex *}
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lemma ball_reg_eqv:
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  fixes P :: "'a \<Rightarrow> bool"
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  assumes a: "equivp R"
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  shows "Ball (Respects R) P = (All P)"
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  using a
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  unfolding equivp_def
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  by (auto simp add: in_respects)
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lemma bex_reg_eqv:
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  fixes P :: "'a \<Rightarrow> bool"
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  assumes a: "equivp R"
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  shows "Bex (Respects R) P = (Ex P)"
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   341
  using a
kaliszyk@35222
   342
  unfolding equivp_def
kaliszyk@35222
   343
  by (auto simp add: in_respects)
kaliszyk@35222
   344
kaliszyk@35222
   345
lemma ball_reg_right:
kaliszyk@35222
   346
  assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
kaliszyk@35222
   347
  shows "All P \<longrightarrow> Ball R Q"
blanchet@39956
   348
  using a by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   349
kaliszyk@35222
   350
lemma bex_reg_left:
kaliszyk@35222
   351
  assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
kaliszyk@35222
   352
  shows "Bex R Q \<longrightarrow> Ex P"
blanchet@39956
   353
  using a by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   354
kaliszyk@35222
   355
lemma ball_reg_left:
kaliszyk@35222
   356
  assumes a: "equivp R"
kaliszyk@35222
   357
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
kaliszyk@35222
   358
  using a by (metis equivp_reflp in_respects)
kaliszyk@35222
   359
kaliszyk@35222
   360
lemma bex_reg_right:
kaliszyk@35222
   361
  assumes a: "equivp R"
kaliszyk@35222
   362
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
kaliszyk@35222
   363
  using a by (metis equivp_reflp in_respects)
kaliszyk@35222
   364
kaliszyk@35222
   365
lemma ball_reg_eqv_range:
kaliszyk@35222
   366
  fixes P::"'a \<Rightarrow> bool"
kaliszyk@35222
   367
  and x::"'a"
kaliszyk@35222
   368
  assumes a: "equivp R2"
kaliszyk@35222
   369
  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
kaliszyk@35222
   370
  apply(rule iffI)
kaliszyk@35222
   371
  apply(rule allI)
kaliszyk@35222
   372
  apply(drule_tac x="\<lambda>y. f x" in bspec)
kaliszyk@35222
   373
  apply(simp add: in_respects)
kaliszyk@35222
   374
  apply(rule impI)
kaliszyk@35222
   375
  using a equivp_reflp_symp_transp[of "R2"]
kaliszyk@35222
   376
  apply(simp add: reflp_def)
kaliszyk@35222
   377
  apply(simp)
kaliszyk@35222
   378
  apply(simp)
kaliszyk@35222
   379
  done
kaliszyk@35222
   380
kaliszyk@35222
   381
lemma bex_reg_eqv_range:
kaliszyk@35222
   382
  assumes a: "equivp R2"
kaliszyk@35222
   383
  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
kaliszyk@35222
   384
  apply(auto)
kaliszyk@35222
   385
  apply(rule_tac x="\<lambda>y. f x" in bexI)
kaliszyk@35222
   386
  apply(simp)
kaliszyk@35222
   387
  apply(simp add: Respects_def in_respects)
kaliszyk@35222
   388
  apply(rule impI)
kaliszyk@35222
   389
  using a equivp_reflp_symp_transp[of "R2"]
kaliszyk@35222
   390
  apply(simp add: reflp_def)
kaliszyk@35222
   391
  done
kaliszyk@35222
   392
kaliszyk@35222
   393
(* Next four lemmas are unused *)
kaliszyk@35222
   394
lemma all_reg:
kaliszyk@35222
   395
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   396
  and     b: "All P"
kaliszyk@35222
   397
  shows "All Q"
kaliszyk@35222
   398
  using a b by (metis)
kaliszyk@35222
   399
kaliszyk@35222
   400
lemma ex_reg:
kaliszyk@35222
   401
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   402
  and     b: "Ex P"
kaliszyk@35222
   403
  shows "Ex Q"
kaliszyk@35222
   404
  using a b by metis
kaliszyk@35222
   405
kaliszyk@35222
   406
lemma ball_reg:
kaliszyk@35222
   407
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
kaliszyk@35222
   408
  and     b: "Ball R P"
kaliszyk@35222
   409
  shows "Ball R Q"
blanchet@39956
   410
  using a b by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   411
kaliszyk@35222
   412
lemma bex_reg:
kaliszyk@35222
   413
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
kaliszyk@35222
   414
  and     b: "Bex R P"
kaliszyk@35222
   415
  shows "Bex R Q"
blanchet@39956
   416
  using a b by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   417
kaliszyk@35222
   418
kaliszyk@35222
   419
lemma ball_all_comm:
kaliszyk@35222
   420
  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
kaliszyk@35222
   421
  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
kaliszyk@35222
   422
  using assms by auto
kaliszyk@35222
   423
kaliszyk@35222
   424
lemma bex_ex_comm:
kaliszyk@35222
   425
  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
kaliszyk@35222
   426
  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
kaliszyk@35222
   427
  using assms by auto
kaliszyk@35222
   428
huffman@35294
   429
subsection {* Bounded abstraction *}
kaliszyk@35222
   430
kaliszyk@35222
   431
definition
kaliszyk@35222
   432
  Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
kaliszyk@35222
   433
where
kaliszyk@35222
   434
  "x \<in> p \<Longrightarrow> Babs p m x = m x"
kaliszyk@35222
   435
kaliszyk@35222
   436
lemma babs_rsp:
kaliszyk@35222
   437
  assumes q: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   438
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   439
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
kaliszyk@35222
   440
  apply (auto simp add: Babs_def in_respects)
kaliszyk@35222
   441
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   442
  using a apply (simp add: Babs_def)
kaliszyk@35222
   443
  apply (simp add: in_respects)
kaliszyk@35222
   444
  using Quotient_rel[OF q]
kaliszyk@35222
   445
  by metis
kaliszyk@35222
   446
kaliszyk@35222
   447
lemma babs_prs:
kaliszyk@35222
   448
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   449
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   450
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   451
  apply (rule ext)
kaliszyk@35222
   452
  apply (simp)
kaliszyk@35222
   453
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kaliszyk@35222
   454
  apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
kaliszyk@35222
   455
  apply (simp add: in_respects Quotient_rel_rep[OF q1])
kaliszyk@35222
   456
  done
kaliszyk@35222
   457
kaliszyk@35222
   458
lemma babs_simp:
kaliszyk@35222
   459
  assumes q: "Quotient R1 Abs Rep"
kaliszyk@35222
   460
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   461
  apply(rule iffI)
kaliszyk@35222
   462
  apply(simp_all only: babs_rsp[OF q])
kaliszyk@35222
   463
  apply(auto simp add: Babs_def)
kaliszyk@35222
   464
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   465
  apply(metis Babs_def)
kaliszyk@35222
   466
  apply (simp add: in_respects)
kaliszyk@35222
   467
  using Quotient_rel[OF q]
kaliszyk@35222
   468
  by metis
kaliszyk@35222
   469
kaliszyk@35222
   470
(* If a user proves that a particular functional relation
kaliszyk@35222
   471
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   472
lemma babs_reg_eqv:
kaliszyk@35222
   473
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302
   474
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222
   475
kaliszyk@35222
   476
kaliszyk@35222
   477
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   478
lemma ball_rsp:
kaliszyk@35222
   479
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   480
  shows "Ball (Respects R) f = Ball (Respects R) g"
kaliszyk@35222
   481
  using a by (simp add: Ball_def in_respects)
kaliszyk@35222
   482
kaliszyk@35222
   483
lemma bex_rsp:
kaliszyk@35222
   484
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   485
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
kaliszyk@35222
   486
  using a by (simp add: Bex_def in_respects)
kaliszyk@35222
   487
kaliszyk@35222
   488
lemma bex1_rsp:
kaliszyk@35222
   489
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   490
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
kaliszyk@35222
   491
  using a
kaliszyk@35222
   492
  by (simp add: Ex1_def in_respects) auto
kaliszyk@35222
   493
kaliszyk@35222
   494
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   495
lemma all_prs:
kaliszyk@35222
   496
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   497
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
kaliszyk@35222
   498
  using a unfolding Quotient_def Ball_def in_respects fun_map_def id_apply
kaliszyk@35222
   499
  by metis
kaliszyk@35222
   500
kaliszyk@35222
   501
lemma ex_prs:
kaliszyk@35222
   502
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   503
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
kaliszyk@35222
   504
  using a unfolding Quotient_def Bex_def in_respects fun_map_def id_apply
kaliszyk@35222
   505
  by metis
kaliszyk@35222
   506
huffman@35294
   507
subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222
   508
kaliszyk@35222
   509
definition
kaliszyk@35222
   510
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   511
where
kaliszyk@35222
   512
  "Bex1_rel R P \<longleftrightarrow> (\<exists>x \<in> Respects R. P x) \<and> (\<forall>x \<in> Respects R. \<forall>y \<in> Respects R. ((P x \<and> P y) \<longrightarrow> (R x y)))"
kaliszyk@35222
   513
kaliszyk@35222
   514
lemma bex1_rel_aux:
kaliszyk@35222
   515
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   516
  unfolding Bex1_rel_def
kaliszyk@35222
   517
  apply (erule conjE)+
kaliszyk@35222
   518
  apply (erule bexE)
kaliszyk@35222
   519
  apply rule
kaliszyk@35222
   520
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   521
  apply metis
kaliszyk@35222
   522
  apply metis
kaliszyk@35222
   523
  apply rule+
kaliszyk@35222
   524
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   525
  prefer 2
kaliszyk@35222
   526
  apply (metis)
kaliszyk@35222
   527
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   528
  prefer 2
kaliszyk@35222
   529
  apply (metis)
kaliszyk@35222
   530
  apply (metis in_respects)
kaliszyk@35222
   531
  done
kaliszyk@35222
   532
kaliszyk@35222
   533
lemma bex1_rel_aux2:
kaliszyk@35222
   534
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   535
  unfolding Bex1_rel_def
kaliszyk@35222
   536
  apply (erule conjE)+
kaliszyk@35222
   537
  apply (erule bexE)
kaliszyk@35222
   538
  apply rule
kaliszyk@35222
   539
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   540
  apply metis
kaliszyk@35222
   541
  apply metis
kaliszyk@35222
   542
  apply rule+
kaliszyk@35222
   543
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   544
  prefer 2
kaliszyk@35222
   545
  apply (metis)
kaliszyk@35222
   546
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   547
  prefer 2
kaliszyk@35222
   548
  apply (metis)
kaliszyk@35222
   549
  apply (metis in_respects)
kaliszyk@35222
   550
  done
kaliszyk@35222
   551
kaliszyk@35222
   552
lemma bex1_rel_rsp:
kaliszyk@35222
   553
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   554
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
kaliszyk@35222
   555
  apply simp
kaliszyk@35222
   556
  apply clarify
kaliszyk@35222
   557
  apply rule
kaliszyk@35222
   558
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   559
  apply (erule bex1_rel_aux2)
kaliszyk@35222
   560
  apply assumption
kaliszyk@35222
   561
  done
kaliszyk@35222
   562
kaliszyk@35222
   563
kaliszyk@35222
   564
lemma ex1_prs:
kaliszyk@35222
   565
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   566
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
kaliszyk@35222
   567
apply simp
kaliszyk@35222
   568
apply (subst Bex1_rel_def)
kaliszyk@35222
   569
apply (subst Bex_def)
kaliszyk@35222
   570
apply (subst Ex1_def)
kaliszyk@35222
   571
apply simp
kaliszyk@35222
   572
apply rule
kaliszyk@35222
   573
 apply (erule conjE)+
kaliszyk@35222
   574
 apply (erule_tac exE)
kaliszyk@35222
   575
 apply (erule conjE)
kaliszyk@35222
   576
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222
   577
  apply (rule_tac x="absf x" in exI)
kaliszyk@35222
   578
  apply (simp)
kaliszyk@35222
   579
  apply rule+
kaliszyk@35222
   580
  using a unfolding Quotient_def
kaliszyk@35222
   581
  apply metis
kaliszyk@35222
   582
 apply rule+
kaliszyk@35222
   583
 apply (erule_tac x="x" in ballE)
kaliszyk@35222
   584
  apply (erule_tac x="y" in ballE)
kaliszyk@35222
   585
   apply simp
kaliszyk@35222
   586
  apply (simp add: in_respects)
kaliszyk@35222
   587
 apply (simp add: in_respects)
kaliszyk@35222
   588
apply (erule_tac exE)
kaliszyk@35222
   589
 apply rule
kaliszyk@35222
   590
 apply (rule_tac x="repf x" in exI)
kaliszyk@35222
   591
 apply (simp only: in_respects)
kaliszyk@35222
   592
  apply rule
kaliszyk@35222
   593
 apply (metis Quotient_rel_rep[OF a])
kaliszyk@35222
   594
using a unfolding Quotient_def apply (simp)
kaliszyk@35222
   595
apply rule+
kaliszyk@35222
   596
using a unfolding Quotient_def in_respects
kaliszyk@35222
   597
apply metis
kaliszyk@35222
   598
done
kaliszyk@35222
   599
kaliszyk@38702
   600
lemma bex1_bexeq_reg:
kaliszyk@38702
   601
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222
   602
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222
   603
  apply clarify
kaliszyk@35222
   604
  apply auto
kaliszyk@35222
   605
  apply (rule bexI)
kaliszyk@35222
   606
  apply assumption
kaliszyk@35222
   607
  apply (simp add: in_respects)
kaliszyk@35222
   608
  apply (simp add: in_respects)
kaliszyk@35222
   609
  apply auto
kaliszyk@35222
   610
  done
kaliszyk@35222
   611
kaliszyk@38702
   612
lemma bex1_bexeq_reg_eqv:
kaliszyk@38702
   613
  assumes a: "equivp R"
kaliszyk@38702
   614
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702
   615
  using equivp_reflp[OF a]
kaliszyk@38702
   616
  apply (intro impI)
kaliszyk@38702
   617
  apply (elim ex1E)
kaliszyk@38702
   618
  apply (rule mp[OF bex1_bexeq_reg])
kaliszyk@38702
   619
  apply (rule_tac a="x" in ex1I)
kaliszyk@38702
   620
  apply (subst in_respects)
kaliszyk@38702
   621
  apply (rule conjI)
kaliszyk@38702
   622
  apply assumption
kaliszyk@38702
   623
  apply assumption
kaliszyk@38702
   624
  apply clarify
kaliszyk@38702
   625
  apply (erule_tac x="xa" in allE)
kaliszyk@38702
   626
  apply simp
kaliszyk@38702
   627
  done
kaliszyk@38702
   628
huffman@35294
   629
subsection {* Various respects and preserve lemmas *}
kaliszyk@35222
   630
kaliszyk@35222
   631
lemma quot_rel_rsp:
kaliszyk@35222
   632
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   633
  shows "(R ===> R ===> op =) R R"
urbanc@38317
   634
  apply(rule fun_relI)+
kaliszyk@35222
   635
  apply(rule equals_rsp[OF a])
kaliszyk@35222
   636
  apply(assumption)+
kaliszyk@35222
   637
  done
kaliszyk@35222
   638
kaliszyk@35222
   639
lemma o_prs:
kaliszyk@35222
   640
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   641
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   642
  and     q3: "Quotient R3 Abs3 Rep3"
kaliszyk@36215
   643
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215
   644
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kaliszyk@35222
   645
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
nipkow@39302
   646
  unfolding o_def fun_eq_iff by simp_all
kaliszyk@35222
   647
kaliszyk@35222
   648
lemma o_rsp:
kaliszyk@36215
   649
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215
   650
  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
nipkow@39302
   651
  unfolding fun_rel_def o_def fun_eq_iff by auto
kaliszyk@35222
   652
kaliszyk@35222
   653
lemma cond_prs:
kaliszyk@35222
   654
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   655
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kaliszyk@35222
   656
  using a unfolding Quotient_def by auto
kaliszyk@35222
   657
kaliszyk@35222
   658
lemma if_prs:
kaliszyk@35222
   659
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   660
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kaliszyk@36123
   661
  using Quotient_abs_rep[OF q]
nipkow@39302
   662
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   663
kaliszyk@35222
   664
lemma if_rsp:
kaliszyk@35222
   665
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   666
  shows "(op = ===> R ===> R ===> R) If If"
kaliszyk@36123
   667
  by auto
kaliszyk@35222
   668
kaliszyk@35222
   669
lemma let_prs:
kaliszyk@35222
   670
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   671
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@37049
   672
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kaliszyk@37049
   673
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
nipkow@39302
   674
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   675
kaliszyk@35222
   676
lemma let_rsp:
kaliszyk@37049
   677
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
kaliszyk@37049
   678
  by auto
kaliszyk@35222
   679
kaliszyk@38861
   680
lemma mem_rsp:
kaliszyk@38861
   681
  shows "(R1 ===> (R1 ===> R2) ===> R2) op \<in> op \<in>"
kaliszyk@38861
   682
  by (simp add: mem_def)
kaliszyk@38861
   683
kaliszyk@38861
   684
lemma mem_prs:
kaliszyk@38861
   685
  assumes a1: "Quotient R1 Abs1 Rep1"
kaliszyk@38861
   686
  and     a2: "Quotient R2 Abs2 Rep2"
kaliszyk@38861
   687
  shows "(Rep1 ---> (Abs1 ---> Rep2) ---> Abs2) op \<in> = op \<in>"
nipkow@39302
   688
  by (simp add: fun_eq_iff mem_def Quotient_abs_rep[OF a1] Quotient_abs_rep[OF a2])
kaliszyk@38861
   689
kaliszyk@39669
   690
lemma id_rsp:
kaliszyk@39669
   691
  shows "(R ===> R) id id"
kaliszyk@39669
   692
  by simp
kaliszyk@39669
   693
kaliszyk@39669
   694
lemma id_prs:
kaliszyk@39669
   695
  assumes a: "Quotient R Abs Rep"
kaliszyk@39669
   696
  shows "(Rep ---> Abs) id = id"
kaliszyk@39669
   697
  unfolding fun_eq_iff
kaliszyk@39669
   698
  by (simp add: Quotient_abs_rep[OF a])
kaliszyk@39669
   699
kaliszyk@39669
   700
kaliszyk@35222
   701
locale quot_type =
kaliszyk@35222
   702
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@35222
   703
  and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"
kaliszyk@35222
   704
  and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"
kaliszyk@37493
   705
  assumes equivp: "part_equivp R"
kaliszyk@37493
   706
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = R x"
kaliszyk@35222
   707
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@37493
   708
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = R x))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222
   709
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   710
begin
kaliszyk@35222
   711
kaliszyk@35222
   712
definition
kaliszyk@35222
   713
  abs::"'a \<Rightarrow> 'b"
kaliszyk@35222
   714
where
kaliszyk@35222
   715
  "abs x \<equiv> Abs (R x)"
kaliszyk@35222
   716
kaliszyk@35222
   717
definition
kaliszyk@35222
   718
  rep::"'b \<Rightarrow> 'a"
kaliszyk@35222
   719
where
kaliszyk@35222
   720
  "rep a = Eps (Rep a)"
kaliszyk@35222
   721
kaliszyk@37493
   722
lemma homeier5:
kaliszyk@37493
   723
  assumes a: "R r r"
kaliszyk@37493
   724
  shows "Rep (Abs (R r)) = R r"
kaliszyk@37493
   725
  apply (subst abs_inverse)
kaliszyk@37493
   726
  using a by auto
kaliszyk@35222
   727
kaliszyk@37493
   728
theorem homeier6:
kaliszyk@37493
   729
  assumes a: "R r r"
kaliszyk@37493
   730
  and b: "R s s"
kaliszyk@37493
   731
  shows "Abs (R r) = Abs (R s) \<longleftrightarrow> R r = R s"
kaliszyk@37493
   732
  by (metis a b homeier5)
kaliszyk@35222
   733
kaliszyk@37493
   734
theorem homeier8:
kaliszyk@37493
   735
  assumes "R r r"
kaliszyk@37493
   736
  shows "R (Eps (R r)) = R r"
kaliszyk@37493
   737
  using assms equivp[simplified part_equivp_def]
kaliszyk@37493
   738
  apply clarify
kaliszyk@37493
   739
  by (metis assms exE_some)
kaliszyk@35222
   740
kaliszyk@35222
   741
lemma Quotient:
kaliszyk@35222
   742
  shows "Quotient R abs rep"
kaliszyk@37493
   743
  unfolding Quotient_def abs_def rep_def
kaliszyk@37493
   744
  proof (intro conjI allI)
kaliszyk@37493
   745
    fix a r s
kaliszyk@37493
   746
    show "Abs (R (Eps (Rep a))) = a"
kaliszyk@37493
   747
      by (metis equivp exE_some part_equivp_def rep_inverse rep_prop)
kaliszyk@37493
   748
    show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (R r) = Abs (R s))"
kaliszyk@37493
   749
      by (metis homeier6 equivp[simplified part_equivp_def])
kaliszyk@37493
   750
    show "R (Eps (Rep a)) (Eps (Rep a))" proof -
kaliszyk@37493
   751
      obtain x where r: "R x x" and rep: "Rep a = R x" using rep_prop[of a] by auto
kaliszyk@37493
   752
      have "R (Eps (R x)) x" using homeier8 r by simp
kaliszyk@37493
   753
      then have "R x (Eps (R x))" using part_equivp_symp[OF equivp] by fast
kaliszyk@37493
   754
      then have "R (Eps (R x)) (Eps (R x))" using homeier8[OF r] by simp
kaliszyk@37493
   755
      then show "R (Eps (Rep a)) (Eps (Rep a))" using rep by simp
kaliszyk@37493
   756
    qed
kaliszyk@37493
   757
  qed
kaliszyk@35222
   758
kaliszyk@35222
   759
end
kaliszyk@35222
   760
kaliszyk@37493
   761
huffman@35294
   762
subsection {* ML setup *}
kaliszyk@35222
   763
kaliszyk@35222
   764
text {* Auxiliary data for the quotient package *}
kaliszyk@35222
   765
wenzelm@37986
   766
use "Tools/Quotient/quotient_info.ML"
kaliszyk@35222
   767
kaliszyk@35222
   768
declare [[map "fun" = (fun_map, fun_rel)]]
kaliszyk@35222
   769
kaliszyk@35222
   770
lemmas [quot_thm] = fun_quotient
kaliszyk@39669
   771
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp mem_rsp id_rsp
kaliszyk@39669
   772
lemmas [quot_preserve] = if_prs o_prs let_prs mem_prs id_prs
kaliszyk@35222
   773
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   774
kaliszyk@35222
   775
kaliszyk@35222
   776
text {* Lemmas about simplifying id's. *}
kaliszyk@35222
   777
lemmas [id_simps] =
kaliszyk@35222
   778
  id_def[symmetric]
kaliszyk@35222
   779
  fun_map_id
kaliszyk@35222
   780
  id_apply
kaliszyk@35222
   781
  id_o
kaliszyk@35222
   782
  o_id
kaliszyk@35222
   783
  eq_comp_r
kaliszyk@35222
   784
kaliszyk@35222
   785
text {* Translation functions for the lifting process. *}
wenzelm@37986
   786
use "Tools/Quotient/quotient_term.ML"
kaliszyk@35222
   787
kaliszyk@35222
   788
kaliszyk@35222
   789
text {* Definitions of the quotient types. *}
wenzelm@37986
   790
use "Tools/Quotient/quotient_typ.ML"
kaliszyk@35222
   791
kaliszyk@35222
   792
kaliszyk@35222
   793
text {* Definitions for quotient constants. *}
wenzelm@37986
   794
use "Tools/Quotient/quotient_def.ML"
kaliszyk@35222
   795
kaliszyk@35222
   796
kaliszyk@35222
   797
text {*
kaliszyk@35222
   798
  An auxiliary constant for recording some information
kaliszyk@35222
   799
  about the lifted theorem in a tactic.
kaliszyk@35222
   800
*}
kaliszyk@35222
   801
definition
kaliszyk@36116
   802
  "Quot_True (x :: 'a) \<equiv> True"
kaliszyk@35222
   803
kaliszyk@35222
   804
lemma
kaliszyk@35222
   805
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   806
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   807
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   808
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   809
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   810
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   811
kaliszyk@35222
   812
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   813
  by (simp add: Quot_True_def)
kaliszyk@35222
   814
kaliszyk@35222
   815
kaliszyk@35222
   816
text {* Tactics for proving the lifted theorems *}
wenzelm@37986
   817
use "Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222
   818
huffman@35294
   819
subsection {* Methods / Interface *}
kaliszyk@35222
   820
kaliszyk@35222
   821
method_setup lifting =
urbanc@37593
   822
  {* Attrib.thms >> (fn thms => fn ctxt => 
urbanc@38859
   823
       SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt [] thms))) *}
kaliszyk@35222
   824
  {* lifts theorems to quotient types *}
kaliszyk@35222
   825
kaliszyk@35222
   826
method_setup lifting_setup =
urbanc@37593
   827
  {* Attrib.thm >> (fn thm => fn ctxt => 
urbanc@38859
   828
       SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_procedure_tac ctxt [] thm))) *}
kaliszyk@35222
   829
  {* sets up the three goals for the quotient lifting procedure *}
kaliszyk@35222
   830
urbanc@37593
   831
method_setup descending =
urbanc@38859
   832
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_tac ctxt []))) *}
urbanc@37593
   833
  {* decends theorems to the raw level *}
urbanc@37593
   834
urbanc@37593
   835
method_setup descending_setup =
urbanc@38859
   836
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_procedure_tac ctxt []))) *}
urbanc@37593
   837
  {* sets up the three goals for the decending theorems *}
urbanc@37593
   838
kaliszyk@35222
   839
method_setup regularize =
kaliszyk@35222
   840
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}
kaliszyk@35222
   841
  {* proves the regularization goals from the quotient lifting procedure *}
kaliszyk@35222
   842
kaliszyk@35222
   843
method_setup injection =
kaliszyk@35222
   844
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}
kaliszyk@35222
   845
  {* proves the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222
   846
kaliszyk@35222
   847
method_setup cleaning =
kaliszyk@35222
   848
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}
kaliszyk@35222
   849
  {* proves the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222
   850
kaliszyk@35222
   851
attribute_setup quot_lifted =
kaliszyk@35222
   852
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
kaliszyk@35222
   853
  {* lifts theorems to quotient types *}
kaliszyk@35222
   854
kaliszyk@35222
   855
no_notation
kaliszyk@35222
   856
  rel_conj (infixr "OOO" 75) and
kaliszyk@35222
   857
  fun_map (infixr "--->" 55) and
kaliszyk@35222
   858
  fun_rel (infixr "===>" 55)
kaliszyk@35222
   859
kaliszyk@35222
   860
end
kaliszyk@35222
   861