src/HOL/Quotient.thy
author Cezary Kaliszyk <kaliszyk@in.tum.de>
Fri, 19 Feb 2010 13:54:19 +0100
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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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theory Quotient
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imports Plain ATP_Linkup
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uses
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  ("~~/src/HOL/Tools/Quotient/quotient_info.ML")
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  ("~~/src/HOL/Tools/Quotient/quotient_typ.ML")
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  ("~~/src/HOL/Tools/Quotient/quotient_def.ML")
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  ("~~/src/HOL/Tools/Quotient/quotient_term.ML")
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  ("~~/src/HOL/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 expand_fun_eq
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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: not yet used anywhere *}
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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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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: expand_fun_eq)
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section {* 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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section {* 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_map_id:
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  shows "(id ---> id) = id"
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  by (simp add: expand_fun_eq 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: expand_fun_eq)
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lemma fun_rel_id:
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  assumes a: "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
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  shows "(R1 ===> R2) f g"
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  using a by simp
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lemma fun_rel_id_asm:
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  assumes a: "\<And>x y. R1 x y \<Longrightarrow> (A \<longrightarrow> R2 (f x) (g y))"
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  shows "A \<longrightarrow> (R1 ===> R2) f g"
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  using a by auto
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section {* Quotient Predicate *}
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definition
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  "Quotient E Abs Rep \<equiv>
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   131
     (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and>
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   132
     (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   133
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   134
lemma Quotient_abs_rep:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   135
  assumes a: "Quotient E Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   136
  shows "Abs (Rep a) = a"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   137
  using a
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   138
  unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   139
  by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   140
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   141
lemma Quotient_rep_reflp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   142
  assumes a: "Quotient E Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   143
  shows "E (Rep a) (Rep a)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   144
  using a
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   145
  unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   146
  by blast
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   147
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   148
lemma Quotient_rel:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   149
  assumes a: "Quotient E Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   150
  shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   151
  using a
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   152
  unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   153
  by blast
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   154
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   155
lemma Quotient_rel_rep:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   156
  assumes a: "Quotient R Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   157
  shows "R (Rep a) (Rep b) = (a = b)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   158
  using a
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   159
  unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   160
  by metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   161
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   162
lemma Quotient_rep_abs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   163
  assumes a: "Quotient R Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   164
  shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   165
  using a unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   166
  by blast
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   167
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   168
lemma Quotient_rel_abs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   169
  assumes a: "Quotient E Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   170
  shows "E r s \<Longrightarrow> Abs r = Abs s"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   171
  using a unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   172
  by blast
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   173
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   174
lemma Quotient_symp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   175
  assumes a: "Quotient E Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   176
  shows "symp E"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   177
  using a unfolding Quotient_def symp_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   178
  by metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   179
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   180
lemma Quotient_transp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   181
  assumes a: "Quotient E Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   182
  shows "transp E"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   183
  using a unfolding Quotient_def transp_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   184
  by metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   185
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   186
lemma identity_quotient:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   187
  shows "Quotient (op =) id id"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   188
  unfolding Quotient_def id_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   189
  by blast
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   190
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   191
lemma fun_quotient:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   192
  assumes q1: "Quotient R1 abs1 rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   193
  and     q2: "Quotient R2 abs2 rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   194
  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   195
proof -
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   196
  have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   197
    using q1 q2
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   198
    unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   199
    unfolding expand_fun_eq
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   200
    by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   201
  moreover
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   202
  have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   203
    using q1 q2
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   204
    unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   205
    by (simp (no_asm)) (metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   206
  moreover
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   207
  have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   208
        (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   209
    unfolding expand_fun_eq
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   210
    apply(auto)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   211
    using q1 q2 unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   212
    apply(metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   213
    using q1 q2 unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   214
    apply(metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   215
    using q1 q2 unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   216
    apply(metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   217
    using q1 q2 unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   218
    apply(metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   219
    done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   220
  ultimately
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   221
  show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   222
    unfolding Quotient_def by blast
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   223
qed
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   224
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   225
lemma abs_o_rep:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   226
  assumes a: "Quotient R Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   227
  shows "Abs o Rep = id"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   228
  unfolding expand_fun_eq
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   229
  by (simp add: Quotient_abs_rep[OF a])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   230
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   231
lemma equals_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   232
  assumes q: "Quotient R Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   233
  and     a: "R xa xb" "R ya yb"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   234
  shows "R xa ya = R xb yb"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   235
  using a Quotient_symp[OF q] Quotient_transp[OF q]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   236
  unfolding symp_def transp_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   237
  by blast
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   238
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   239
lemma lambda_prs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   240
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   241
  and     q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   242
  shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   243
  unfolding expand_fun_eq
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   244
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   245
  by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   246
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   247
lemma lambda_prs1:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   248
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   249
  and     q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   250
  shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   251
  unfolding expand_fun_eq
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   252
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   253
  by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   254
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   255
lemma rep_abs_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   256
  assumes q: "Quotient R Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   257
  and     a: "R x1 x2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   258
  shows "R x1 (Rep (Abs x2))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   259
  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   260
  by metis
4f1fba00f66d Initial version of HOL quotient package.
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diff changeset
   261
4f1fba00f66d Initial version of HOL quotient package.
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parents:
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   262
lemma rep_abs_rsp_left:
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parents:
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   263
  assumes q: "Quotient R Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   264
  and     a: "R x1 x2"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   265
  shows "R (Rep (Abs x1)) x2"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   266
  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   267
  by metis
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   268
4f1fba00f66d Initial version of HOL quotient package.
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   269
text{*
4f1fba00f66d Initial version of HOL quotient package.
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   270
  In the following theorem R1 can be instantiated with anything,
4f1fba00f66d Initial version of HOL quotient package.
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   271
  but we know some of the types of the Rep and Abs functions;
4f1fba00f66d Initial version of HOL quotient package.
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   272
  so by solving Quotient assumptions we can get a unique R1 that
4f1fba00f66d Initial version of HOL quotient package.
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   273
  will be provable; which is why we need to use apply_rsp and
4f1fba00f66d Initial version of HOL quotient package.
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   274
  not the primed version *}
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parents:
diff changeset
   275
4f1fba00f66d Initial version of HOL quotient package.
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   276
lemma apply_rsp:
4f1fba00f66d Initial version of HOL quotient package.
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   277
  fixes f g::"'a \<Rightarrow> 'c"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   278
  assumes q: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
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diff changeset
   279
  and     a: "(R1 ===> R2) f g" "R1 x y"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   280
  shows "R2 (f x) (g y)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   281
  using a by simp
4f1fba00f66d Initial version of HOL quotient package.
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diff changeset
   282
4f1fba00f66d Initial version of HOL quotient package.
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   283
lemma apply_rsp':
4f1fba00f66d Initial version of HOL quotient package.
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   284
  assumes a: "(R1 ===> R2) f g" "R1 x y"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   285
  shows "R2 (f x) (g y)"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   286
  using a by simp
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   287
4f1fba00f66d Initial version of HOL quotient package.
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   288
section {* lemmas for regularisation of ball and bex *}
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   289
4f1fba00f66d Initial version of HOL quotient package.
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   290
lemma ball_reg_eqv:
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   291
  fixes P :: "'a \<Rightarrow> bool"
4f1fba00f66d Initial version of HOL quotient package.
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   292
  assumes a: "equivp R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   293
  shows "Ball (Respects R) P = (All P)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   294
  using a
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   295
  unfolding equivp_def
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   296
  by (auto simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   297
4f1fba00f66d Initial version of HOL quotient package.
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   298
lemma bex_reg_eqv:
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parents:
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   299
  fixes P :: "'a \<Rightarrow> bool"
4f1fba00f66d Initial version of HOL quotient package.
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diff changeset
   300
  assumes a: "equivp R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   301
  shows "Bex (Respects R) P = (Ex P)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   302
  using a
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   303
  unfolding equivp_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   304
  by (auto simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   305
4f1fba00f66d Initial version of HOL quotient package.
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   306
lemma ball_reg_right:
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parents:
diff changeset
   307
  assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   308
  shows "All P \<longrightarrow> Ball R Q"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   309
  using a by (metis COMBC_def Collect_def Collect_mem_eq)
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   310
4f1fba00f66d Initial version of HOL quotient package.
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parents:
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   311
lemma bex_reg_left:
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   312
  assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   313
  shows "Bex R Q \<longrightarrow> Ex P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   314
  using a by (metis COMBC_def Collect_def Collect_mem_eq)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   315
4f1fba00f66d Initial version of HOL quotient package.
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parents:
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   316
lemma ball_reg_left:
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parents:
diff changeset
   317
  assumes a: "equivp R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   318
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   319
  using a by (metis equivp_reflp in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   320
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   321
lemma bex_reg_right:
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parents:
diff changeset
   322
  assumes a: "equivp R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   323
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   324
  using a by (metis equivp_reflp in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   325
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   326
lemma ball_reg_eqv_range:
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parents:
diff changeset
   327
  fixes P::"'a \<Rightarrow> bool"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   328
  and x::"'a"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   329
  assumes a: "equivp R2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   330
  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   331
  apply(rule iffI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   332
  apply(rule allI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   333
  apply(drule_tac x="\<lambda>y. f x" in bspec)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   334
  apply(simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   335
  apply(rule impI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   336
  using a equivp_reflp_symp_transp[of "R2"]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   337
  apply(simp add: reflp_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   338
  apply(simp)
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   339
  apply(simp)
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   340
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   341
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   342
lemma bex_reg_eqv_range:
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   343
  assumes a: "equivp R2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   344
  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   345
  apply(auto)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   346
  apply(rule_tac x="\<lambda>y. f x" in bexI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   347
  apply(simp)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   348
  apply(simp add: Respects_def in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   349
  apply(rule impI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   350
  using a equivp_reflp_symp_transp[of "R2"]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   351
  apply(simp add: reflp_def)
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   352
  done
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   353
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   354
(* Next four lemmas are unused *)
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   355
lemma all_reg:
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   356
  assumes a: "!x :: 'a. (P x --> Q x)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   357
  and     b: "All P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   358
  shows "All Q"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   359
  using a b by (metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   360
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   361
lemma ex_reg:
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   362
  assumes a: "!x :: 'a. (P x --> Q x)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   363
  and     b: "Ex P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   364
  shows "Ex Q"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   365
  using a b by metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   366
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   367
lemma ball_reg:
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   368
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   369
  and     b: "Ball R P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   370
  shows "Ball R Q"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   371
  using a b by (metis COMBC_def Collect_def Collect_mem_eq)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   372
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   373
lemma bex_reg:
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   374
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   375
  and     b: "Bex R P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   376
  shows "Bex R Q"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   377
  using a b by (metis COMBC_def Collect_def Collect_mem_eq)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   378
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   379
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   380
lemma ball_all_comm:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   381
  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   382
  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   383
  using assms by auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   384
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   385
lemma bex_ex_comm:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   386
  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   387
  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   388
  using assms by auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   389
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   390
section {* Bounded abstraction *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   391
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   392
definition
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   393
  Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   394
where
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   395
  "x \<in> p \<Longrightarrow> Babs p m x = m x"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   396
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   397
lemma babs_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   398
  assumes q: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   399
  and     a: "(R1 ===> R2) f g"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   400
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   401
  apply (auto simp add: Babs_def in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   402
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   403
  using a apply (simp add: Babs_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   404
  apply (simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   405
  using Quotient_rel[OF q]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   406
  by metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   407
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   408
lemma babs_prs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   409
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   410
  and     q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   411
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   412
  apply (rule ext)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   413
  apply (simp)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   414
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   415
  apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   416
  apply (simp add: in_respects Quotient_rel_rep[OF q1])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   417
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   418
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   419
lemma babs_simp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   420
  assumes q: "Quotient R1 Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   421
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   422
  apply(rule iffI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   423
  apply(simp_all only: babs_rsp[OF q])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   424
  apply(auto simp add: Babs_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   425
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   426
  apply(metis Babs_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   427
  apply (simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   428
  using Quotient_rel[OF q]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   429
  by metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   430
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   431
(* If a user proves that a particular functional relation
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   432
   is an equivalence this may be useful in regularising *)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   433
lemma babs_reg_eqv:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   434
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   435
  by (simp add: expand_fun_eq Babs_def in_respects equivp_reflp)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   436
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   437
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   438
(* 3 lemmas needed for proving repabs_inj *)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   439
lemma ball_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   440
  assumes a: "(R ===> (op =)) f g"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   441
  shows "Ball (Respects R) f = Ball (Respects R) g"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   442
  using a by (simp add: Ball_def in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   443
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   444
lemma bex_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   445
  assumes a: "(R ===> (op =)) f g"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   446
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   447
  using a by (simp add: Bex_def in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   448
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   449
lemma bex1_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   450
  assumes a: "(R ===> (op =)) f g"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   451
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   452
  using a
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   453
  by (simp add: Ex1_def in_respects) auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   454
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   455
(* 2 lemmas needed for cleaning of quantifiers *)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   456
lemma all_prs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   457
  assumes a: "Quotient R absf repf"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   458
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   459
  using a unfolding Quotient_def Ball_def in_respects fun_map_def id_apply
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   460
  by metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   461
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   462
lemma ex_prs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   463
  assumes a: "Quotient R absf repf"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   464
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   465
  using a unfolding Quotient_def Bex_def in_respects fun_map_def id_apply
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   466
  by metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   467
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   468
section {* Bex1_rel quantifier *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   469
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   470
definition
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   471
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   472
where
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   473
  "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)))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   474
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   475
lemma bex1_rel_aux:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   476
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   477
  unfolding Bex1_rel_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   478
  apply (erule conjE)+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   479
  apply (erule bexE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   480
  apply rule
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   481
  apply (rule_tac x="xa" in bexI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   482
  apply metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   483
  apply metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   484
  apply rule+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   485
  apply (erule_tac x="xaa" in ballE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   486
  prefer 2
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   487
  apply (metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   488
  apply (erule_tac x="ya" in ballE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   489
  prefer 2
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   490
  apply (metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   491
  apply (metis in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   492
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   493
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   494
lemma bex1_rel_aux2:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   495
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   496
  unfolding Bex1_rel_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   497
  apply (erule conjE)+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   498
  apply (erule bexE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   499
  apply rule
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   500
  apply (rule_tac x="xa" in bexI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   501
  apply metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   502
  apply metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   503
  apply rule+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   504
  apply (erule_tac x="xaa" in ballE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   505
  prefer 2
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   506
  apply (metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   507
  apply (erule_tac x="ya" in ballE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   508
  prefer 2
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   509
  apply (metis)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   510
  apply (metis in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   511
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   512
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   513
lemma bex1_rel_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   514
  assumes a: "Quotient R absf repf"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   515
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   516
  apply simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   517
  apply clarify
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   518
  apply rule
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   519
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   520
  apply (erule bex1_rel_aux2)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   521
  apply assumption
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   522
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   523
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   524
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   525
lemma ex1_prs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   526
  assumes a: "Quotient R absf repf"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   527
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   528
apply simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   529
apply (subst Bex1_rel_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   530
apply (subst Bex_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   531
apply (subst Ex1_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   532
apply simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   533
apply rule
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   534
 apply (erule conjE)+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   535
 apply (erule_tac exE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   536
 apply (erule conjE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   537
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   538
  apply (rule_tac x="absf x" in exI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   539
  apply (simp)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   540
  apply rule+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   541
  using a unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   542
  apply metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   543
 apply rule+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   544
 apply (erule_tac x="x" in ballE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   545
  apply (erule_tac x="y" in ballE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   546
   apply simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   547
  apply (simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   548
 apply (simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   549
apply (erule_tac exE)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   550
 apply rule
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   551
 apply (rule_tac x="repf x" in exI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   552
 apply (simp only: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   553
  apply rule
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   554
 apply (metis Quotient_rel_rep[OF a])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   555
using a unfolding Quotient_def apply (simp)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   556
apply rule+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   557
using a unfolding Quotient_def in_respects
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   558
apply metis
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   559
done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   560
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   561
lemma bex1_bexeq_reg: "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   562
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   563
  apply clarify
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   564
  apply auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   565
  apply (rule bexI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   566
  apply assumption
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   567
  apply (simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   568
  apply (simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   569
  apply auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   570
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   571
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   572
section {* Various respects and preserve lemmas *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   573
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   574
lemma quot_rel_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   575
  assumes a: "Quotient R Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   576
  shows "(R ===> R ===> op =) R R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   577
  apply(rule fun_rel_id)+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   578
  apply(rule equals_rsp[OF a])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   579
  apply(assumption)+
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   580
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   581
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   582
lemma o_prs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   583
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   584
  and     q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   585
  and     q3: "Quotient R3 Abs3 Rep3"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   586
  shows "(Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g)) = f o g"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   587
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   588
  unfolding o_def expand_fun_eq by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   589
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   590
lemma o_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   591
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   592
  and     q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   593
  and     q3: "Quotient R3 Abs3 Rep3"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   594
  and     a1: "(R2 ===> R3) f1 f2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   595
  and     a2: "(R1 ===> R2) g1 g2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   596
  shows "(R1 ===> R3) (f1 o g1) (f2 o g2)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   597
  using a1 a2 unfolding o_def expand_fun_eq
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   598
  by (auto)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   599
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   600
lemma cond_prs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   601
  assumes a: "Quotient R absf repf"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   602
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   603
  using a unfolding Quotient_def by auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   604
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   605
lemma if_prs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   606
  assumes q: "Quotient R Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   607
  shows "Abs (If a (Rep b) (Rep c)) = If a b c"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   608
  using Quotient_abs_rep[OF q] by auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   609
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   610
(* q not used *)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   611
lemma if_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   612
  assumes q: "Quotient R Abs Rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   613
  and     a: "a1 = a2" "R b1 b2" "R c1 c2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   614
  shows "R (If a1 b1 c1) (If a2 b2 c2)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   615
  using a by auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   616
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   617
lemma let_prs:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   618
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   619
  and     q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   620
  shows "Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f)) = Let x f"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   621
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] by auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   622
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   623
lemma let_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   624
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   625
  and     a1: "(R1 ===> R2) f g"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   626
  and     a2: "R1 x y"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   627
  shows "R2 ((Let x f)::'c) ((Let y g)::'c)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   628
  using apply_rsp[OF q1 a1] a2 by auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   629
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   630
locale quot_type =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   631
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   632
  and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   633
  and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   634
  assumes equivp: "equivp R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   635
  and     rep_prop: "\<And>y. \<exists>x. Rep y = R x"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   636
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   637
  and     abs_inverse: "\<And>x. (Rep (Abs (R x))) = (R x)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   638
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   639
begin
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   640
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   641
definition
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   642
  abs::"'a \<Rightarrow> 'b"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   643
where
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   644
  "abs x \<equiv> Abs (R x)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   645
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   646
definition
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   647
  rep::"'b \<Rightarrow> 'a"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   648
where
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   649
  "rep a = Eps (Rep a)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   650
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   651
lemma homeier_lem9:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   652
  shows "R (Eps (R x)) = R x"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   653
proof -
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   654
  have a: "R x x" using equivp by (simp add: equivp_reflp_symp_transp reflp_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   655
  then have "R x (Eps (R x))" by (rule someI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   656
  then show "R (Eps (R x)) = R x"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   657
    using equivp unfolding equivp_def by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   658
qed
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   659
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   660
theorem homeier_thm10:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   661
  shows "abs (rep a) = a"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   662
  unfolding abs_def rep_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   663
proof -
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   664
  from rep_prop
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   665
  obtain x where eq: "Rep a = R x" by auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   666
  have "Abs (R (Eps (Rep a))) = Abs (R (Eps (R x)))" using eq by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   667
  also have "\<dots> = Abs (R x)" using homeier_lem9 by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   668
  also have "\<dots> = Abs (Rep a)" using eq by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   669
  also have "\<dots> = a" using rep_inverse by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   670
  finally
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   671
  show "Abs (R (Eps (Rep a))) = a" by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   672
qed
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   673
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   674
lemma homeier_lem7:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   675
  shows "(R x = R y) = (Abs (R x) = Abs (R y))" (is "?LHS = ?RHS")
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   676
proof -
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   677
  have "?RHS = (Rep (Abs (R x)) = Rep (Abs (R y)))" by (simp add: rep_inject)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   678
  also have "\<dots> = ?LHS" by (simp add: abs_inverse)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   679
  finally show "?LHS = ?RHS" by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   680
qed
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   681
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   682
theorem homeier_thm11:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   683
  shows "R r r' = (abs r = abs r')"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   684
  unfolding abs_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   685
  by (simp only: equivp[simplified equivp_def] homeier_lem7)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   686
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   687
lemma rep_refl:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   688
  shows "R (rep a) (rep a)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   689
  unfolding rep_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   690
  by (simp add: equivp[simplified equivp_def])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   691
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   692
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   693
lemma rep_abs_rsp:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   694
  shows "R f (rep (abs g)) = R f g"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   695
  and   "R (rep (abs g)) f = R g f"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   696
  by (simp_all add: homeier_thm10 homeier_thm11)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   697
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   698
lemma Quotient:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   699
  shows "Quotient R abs rep"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   700
  unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   701
  apply(simp add: homeier_thm10)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   702
  apply(simp add: rep_refl)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   703
  apply(subst homeier_thm11[symmetric])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   704
  apply(simp add: equivp[simplified equivp_def])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   705
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   706
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   707
end
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   708
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   709
section {* ML setup *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   710
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   711
text {* Auxiliary data for the quotient package *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   712
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   713
use "~~/src/HOL/Tools/Quotient/quotient_info.ML"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   714
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   715
declare [[map "fun" = (fun_map, fun_rel)]]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   716
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   717
lemmas [quot_thm] = fun_quotient
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   718
lemmas [quot_respect] = quot_rel_rsp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   719
lemmas [quot_equiv] = identity_equivp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   720
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   721
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   722
text {* Lemmas about simplifying id's. *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   723
lemmas [id_simps] =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   724
  id_def[symmetric]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   725
  fun_map_id
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   726
  id_apply
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   727
  id_o
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   728
  o_id
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   729
  eq_comp_r
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   730
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   731
text {* Translation functions for the lifting process. *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   732
use "~~/src/HOL/Tools/Quotient/quotient_term.ML"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   733
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   734
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   735
text {* Definitions of the quotient types. *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   736
use "~~/src/HOL/Tools/Quotient/quotient_typ.ML"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   737
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   738
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   739
text {* Definitions for quotient constants. *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   740
use "~~/src/HOL/Tools/Quotient/quotient_def.ML"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   741
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   742
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   743
text {*
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   744
  An auxiliary constant for recording some information
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   745
  about the lifted theorem in a tactic.
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   746
*}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   747
definition
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   748
  "Quot_True x \<equiv> True"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   749
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   750
lemma
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   751
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   752
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   753
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   754
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   755
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   756
  by (simp_all add: Quot_True_def ext)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   757
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   758
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   759
  by (simp add: Quot_True_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   760
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   761
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   762
text {* Tactics for proving the lifted theorems *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   763
use "~~/src/HOL/Tools/Quotient/quotient_tacs.ML"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   764
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   765
section {* Methods / Interface *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   766
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   767
method_setup lifting =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   768
  {* Attrib.thms >> (fn thms => fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt thms))) *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   769
  {* lifts theorems to quotient types *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   770
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   771
method_setup lifting_setup =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   772
  {* Attrib.thm >> (fn thms => fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.procedure_tac ctxt thms))) *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   773
  {* sets up the three goals for the quotient lifting procedure *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   774
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   775
method_setup regularize =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   776
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   777
  {* proves the regularization goals from the quotient lifting procedure *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   778
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   779
method_setup injection =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   780
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   781
  {* proves the rep/abs injection goals from the quotient lifting procedure *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   782
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   783
method_setup cleaning =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   784
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   785
  {* proves the cleaning goals from the quotient lifting procedure *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   786
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   787
attribute_setup quot_lifted =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   788
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   789
  {* lifts theorems to quotient types *}
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   790
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   791
no_notation
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   792
  rel_conj (infixr "OOO" 75) and
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   793
  fun_map (infixr "--->" 55) and
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   794
  fun_rel (infixr "===>" 55)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   795
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   796
end
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   797