src/HOL/Word/Misc_Typedef.thy

 *)
 
 section \<open>Type Definition Theorems\<close>
 
 theory Misc_Typedef
-imports Main
+  imports Main
 begin
 
 section "More lemmas about normal type definitions"
 
-lemma
-  tdD1: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Rep x \<in> A" and
-  tdD2: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Abs (Rep x) = x" and
-  tdD3: "type_definition Rep Abs A \<Longrightarrow> \<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y"
+lemma tdD1: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Rep x \<in> A"
+  and tdD2: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Abs (Rep x) = x"
+  and tdD3: "type_definition Rep Abs A \<Longrightarrow> \<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y"
   by (auto simp: type_definition_def)
 
-lemma td_nat_int:
-  "type_definition int nat (Collect (op <= 0))"
+lemma td_nat_int: "type_definition int nat (Collect (op \<le> 0))"
   unfolding type_definition_def by auto
 
 context type_definition
 begin
 
 declare Rep [iff] Rep_inverse [simp] Rep_inject [simp]
 
-lemma Abs_eqD: "Abs x = Abs y ==> x \<in> A ==> y \<in> A ==> x = y"
+lemma Abs_eqD: "Abs x = Abs y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y"
   by (simp add: Abs_inject)
 
-lemma Abs_inverse':
-  "r : A ==> Abs r = a ==> Rep a = r"
+lemma Abs_inverse': "r \<in> A \<Longrightarrow> Abs r = a \<Longrightarrow> Rep a = r"
   by (safe elim!: Abs_inverse)
 
-lemma Rep_comp_inverse:
-  "Rep \<circ> f = g ==> Abs \<circ> g = f"
+lemma Rep_comp_inverse: "Rep \<circ> f = g \<Longrightarrow> Abs \<circ> g = f"
   using Rep_inverse by auto
 
-lemma Rep_eqD [elim!]: "Rep x = Rep y ==> x = y"
+lemma Rep_eqD [elim!]: "Rep x = Rep y \<Longrightarrow> x = y"
   by simp
 
-lemma Rep_inverse': "Rep a = r ==> Abs r = a"
+lemma Rep_inverse': "Rep a = r \<Longrightarrow> Abs r = a"
   by (safe intro!: Rep_inverse)
 
-lemma comp_Abs_inverse:
-  "f \<circ> Abs = g ==> g \<circ> Rep = f"
+lemma comp_Abs_inverse: "f \<circ> Abs = g \<Longrightarrow> g \<circ> Rep = f"
   using Rep_inverse by auto
 
-lemma set_Rep:
-  "A = range Rep"
+lemma set_Rep: "A = range Rep"
 proof (rule set_eqI)
-  fix x
-  show "(x \<in> A) = (x \<in> range Rep)"
+  show "x \<in> A \<longleftrightarrow> x \<in> range Rep" for x
     by (auto dest: Abs_inverse [of x, symmetric])
 qed
 
 lemma set_Rep_Abs: "A = range (Rep \<circ> Abs)"
 proof (rule set_eqI)
-  fix x
-  show "(x \<in> A) = (x \<in> range (Rep \<circ> Abs))"
+  show "x \<in> A \<longleftrightarrow> x \<in> range (Rep \<circ> Abs)" for x
     by (auto dest: Abs_inverse [of x, symmetric])
 qed
 
 lemma Abs_inj_on: "inj_on Abs A"
   unfolding inj_on_def

 lemma image: "Abs ` A = UNIV"
   by (auto intro!: image_eqI)
 
 lemmas td_thm = type_definition_axioms
 
-lemma fns1:
-  "Rep \<circ> fa = fr \<circ> Rep | fa \<circ> Abs = Abs \<circ> fr ==> Abs \<circ> fr \<circ> Rep = fa"
+lemma fns1: "Rep \<circ> fa = fr \<circ> Rep \<or> fa \<circ> Abs = Abs \<circ> fr \<Longrightarrow> Abs \<circ> fr \<circ> Rep = fa"
   by (auto dest: Rep_comp_inverse elim: comp_Abs_inverse simp: o_assoc)
 
 lemmas fns1a = disjI1 [THEN fns1]
 lemmas fns1b = disjI2 [THEN fns1]
 
-lemma fns4:
-  "Rep \<circ> fa \<circ> Abs = fr ==>
-   Rep \<circ> fa = fr \<circ> Rep & fa \<circ> Abs = Abs \<circ> fr"
+lemma fns4: "Rep \<circ> fa \<circ> Abs = fr \<Longrightarrow> Rep \<circ> fa = fr \<circ> Rep \<and> fa \<circ> Abs = Abs \<circ> fr"
   by auto
 
 end
 
-interpretation nat_int: type_definition int nat "Collect (op <= 0)"
+interpretation nat_int: type_definition int nat "Collect (op \<le> 0)"
   by (rule td_nat_int)
 
 declare
   nat_int.Rep_cases [cases del]
   nat_int.Abs_cases [cases del]

 
 
 subsection "Extended form of type definition predicate"
 
 lemma td_conds:
-  "norm \<circ> norm = norm ==> (fr \<circ> norm = norm \<circ> fr) =
-    (norm \<circ> fr \<circ> norm = fr \<circ> norm & norm \<circ> fr \<circ> norm = norm \<circ> fr)"
+  "norm \<circ> norm = norm \<Longrightarrow>
+    fr \<circ> norm = norm \<circ> fr \<longleftrightarrow> norm \<circ> fr \<circ> norm = fr \<circ> norm \<and> norm \<circ> fr \<circ> norm = norm \<circ> fr"
   apply safe
     apply (simp_all add: comp_assoc)
    apply (simp_all add: o_assoc)
   done
 
-lemma fn_comm_power:
-  "fa \<circ> tr = tr \<circ> fr ==> fa ^^ n \<circ> tr = tr \<circ> fr ^^ n"
+lemma fn_comm_power: "fa \<circ> tr = tr \<circ> fr \<Longrightarrow> fa ^^ n \<circ> tr = tr \<circ> fr ^^ n"
   apply (rule ext)
   apply (induct n)
    apply (auto dest: fun_cong)
   done
 

 locale td_ext = type_definition +
   fixes norm
   assumes eq_norm: "\<And>x. Rep (Abs x) = norm x"
 begin
 
-lemma Abs_norm [simp]:
-  "Abs (norm x) = Abs x"
+lemma Abs_norm [simp]: "Abs (norm x) = Abs x"
   using eq_norm [of x] by (auto elim: Rep_inverse')
 
-lemma td_th:
-  "g \<circ> Abs = f ==> f (Rep x) = g x"
+lemma td_th: "g \<circ> Abs = f \<Longrightarrow> f (Rep x) = g x"
   by (drule comp_Abs_inverse [symmetric]) simp
 
 lemma eq_norm': "Rep \<circ> Abs = norm"
   by (auto simp: eq_norm)
 

 lemmas td = td_thm
 
 lemma set_iff_norm: "w : A \<longleftrightarrow> w = norm w"
   by (auto simp: set_Rep_Abs eq_norm' eq_norm [symmetric])
 
-lemma inverse_norm:
-  "(Abs n = w) = (Rep w = norm n)"
+lemma inverse_norm: "Abs n = w \<longleftrightarrow> Rep w = norm n"
   apply (rule iffI)
    apply (clarsimp simp add: eq_norm)
   apply (simp add: eq_norm' [symmetric])
   done
 
-lemma norm_eq_iff:
-  "(norm x = norm y) = (Abs x = Abs y)"
+lemma norm_eq_iff: "norm x = norm y \<longleftrightarrow> Abs x = Abs y"
   by (simp add: eq_norm' [symmetric])
 
 lemma norm_comps:
   "Abs \<circ> norm = Abs"
   "norm \<circ> Rep = Rep"
   "norm \<circ> norm = norm"
   by (auto simp: eq_norm' [symmetric] o_def)
 
 lemmas norm_norm [simp] = norm_comps
 
-lemma fns5:
-  "Rep \<circ> fa \<circ> Abs = fr ==>
-  fr \<circ> norm = fr & norm \<circ> fr = fr"
+lemma fns5: "Rep \<circ> fa \<circ> Abs = fr \<Longrightarrow> fr \<circ> norm = fr \<and> norm \<circ> fr = fr"
   by (fold eq_norm') auto
 
 (* following give conditions for converses to td_fns1
   the condition (norm \<circ> fr \<circ> norm = fr \<circ> norm) says that
   fr takes normalised arguments to normalised results,

   takes norm-equivalent arguments to norm-equivalent results,
   (fr \<circ> norm = fr) says that fr
   takes norm-equivalent arguments to the same result, and
   (norm \<circ> fr = fr) says that fr takes any argument to a normalised result
   *)
-lemma fns2:
-  "Abs \<circ> fr \<circ> Rep = fa ==>
-   (norm \<circ> fr \<circ> norm = fr \<circ> norm) = (Rep \<circ> fa = fr \<circ> Rep)"
+lemma fns2: "Abs \<circ> fr \<circ> Rep = fa \<Longrightarrow> norm \<circ> fr \<circ> norm = fr \<circ> norm \<longleftrightarrow> Rep \<circ> fa = fr \<circ> Rep"
   apply (fold eq_norm')
   apply safe
    prefer 2
    apply (simp add: o_assoc)
   apply (rule ext)
   apply (drule_tac x="Rep x" in fun_cong)
   apply auto
   done
 
-lemma fns3:
-  "Abs \<circ> fr \<circ> Rep = fa ==>
-   (norm \<circ> fr \<circ> norm = norm \<circ> fr) = (fa \<circ> Abs = Abs \<circ> fr)"
+lemma fns3: "Abs \<circ> fr \<circ> Rep = fa \<Longrightarrow> norm \<circ> fr \<circ> norm = norm \<circ> fr \<longleftrightarrow> fa \<circ> Abs = Abs \<circ> fr"
   apply (fold eq_norm')
   apply safe
    prefer 2
    apply (simp add: comp_assoc)
   apply (rule ext)
   apply (drule_tac f="a \<circ> b" for a b in fun_cong)
   apply simp
   done
 
-lemma fns:
-  "fr \<circ> norm = norm \<circ> fr ==>
-    (fa \<circ> Abs = Abs \<circ> fr) = (Rep \<circ> fa = fr \<circ> Rep)"
+lemma fns: "fr \<circ> norm = norm \<circ> fr \<Longrightarrow> fa \<circ> Abs = Abs \<circ> fr \<longleftrightarrow> Rep \<circ> fa = fr \<circ> Rep"
   apply safe
    apply (frule fns1b)
    prefer 2
    apply (frule fns1a)
    apply (rule fns3 [THEN iffD1])

      apply (rule fns2 [THEN iffD1])
        apply (simp_all add: comp_assoc)
    apply (simp_all add: o_assoc)
   done
 
-lemma range_norm:
-  "range (Rep \<circ> Abs) = A"
+lemma range_norm: "range (Rep \<circ> Abs) = A"
   by (simp add: set_Rep_Abs)
 
 end
 
 lemmas td_ext_def' =