src/HOL/Word/Misc_Typedef.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 59807 22bc39064290 child 61799 4cf66f21b764 permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
```     1 (*
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```     2   Author:     Jeremy Dawson and Gerwin Klein, NICTA
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```     3
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```     4   Consequences of type definition theorems, and of extended type definition.
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```     5 *)
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```     6
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```     7 section {* Type Definition Theorems *}
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```     8
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```     9 theory Misc_Typedef
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```    10 imports Main
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```    11 begin
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```    12
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```    13 section "More lemmas about normal type definitions"
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```    14
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```    15 lemma
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```    16   tdD1: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Rep x \<in> A" and
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```    17   tdD2: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Abs (Rep x) = x" and
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```    18   tdD3: "type_definition Rep Abs A \<Longrightarrow> \<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y"
```
```    19   by (auto simp: type_definition_def)
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```    20
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```    21 lemma td_nat_int:
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```    22   "type_definition int nat (Collect (op <= 0))"
```
```    23   unfolding type_definition_def by auto
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```    24
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```    25 context type_definition
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```    26 begin
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```    27
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```    28 declare Rep [iff] Rep_inverse [simp] Rep_inject [simp]
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```    29
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```    30 lemma Abs_eqD: "Abs x = Abs y ==> x \<in> A ==> y \<in> A ==> x = y"
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```    31   by (simp add: Abs_inject)
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```    32
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```    33 lemma Abs_inverse':
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```    34   "r : A ==> Abs r = a ==> Rep a = r"
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```    35   by (safe elim!: Abs_inverse)
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```    36
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```    37 lemma Rep_comp_inverse:
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```    38   "Rep o f = g ==> Abs o g = f"
```
```    39   using Rep_inverse by auto
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```    40
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```    41 lemma Rep_eqD [elim!]: "Rep x = Rep y ==> x = y"
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```    42   by simp
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```    43
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```    44 lemma Rep_inverse': "Rep a = r ==> Abs r = a"
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```    45   by (safe intro!: Rep_inverse)
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```    46
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```    47 lemma comp_Abs_inverse:
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```    48   "f o Abs = g ==> g o Rep = f"
```
```    49   using Rep_inverse by auto
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```    50
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```    51 lemma set_Rep:
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```    52   "A = range Rep"
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```    53 proof (rule set_eqI)
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```    54   fix x
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```    55   show "(x \<in> A) = (x \<in> range Rep)"
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```    56     by (auto dest: Abs_inverse [of x, symmetric])
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```    57 qed
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```    58
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```    59 lemma set_Rep_Abs: "A = range (Rep o Abs)"
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```    60 proof (rule set_eqI)
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```    61   fix x
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```    62   show "(x \<in> A) = (x \<in> range (Rep o Abs))"
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```    63     by (auto dest: Abs_inverse [of x, symmetric])
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```    64 qed
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```    65
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```    66 lemma Abs_inj_on: "inj_on Abs A"
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```    67   unfolding inj_on_def
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```    68   by (auto dest: Abs_inject [THEN iffD1])
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```    69
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```    70 lemma image: "Abs ` A = UNIV"
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```    71   by (auto intro!: image_eqI)
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```    72
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```    73 lemmas td_thm = type_definition_axioms
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```    74
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```    75 lemma fns1:
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```    76   "Rep o fa = fr o Rep | fa o Abs = Abs o fr ==> Abs o fr o Rep = fa"
```
```    77   by (auto dest: Rep_comp_inverse elim: comp_Abs_inverse simp: o_assoc)
```
```    78
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```    79 lemmas fns1a = disjI1 [THEN fns1]
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```    80 lemmas fns1b = disjI2 [THEN fns1]
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```    81
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```    82 lemma fns4:
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```    83   "Rep o fa o Abs = fr ==>
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```    84    Rep o fa = fr o Rep & fa o Abs = Abs o fr"
```
```    85   by auto
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```    86
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```    87 end
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```    88
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```    89 interpretation nat_int: type_definition int nat "Collect (op <= 0)"
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```    90   by (rule td_nat_int)
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```    91
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```    92 declare
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```    93   nat_int.Rep_cases [cases del]
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```    94   nat_int.Abs_cases [cases del]
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```    95   nat_int.Rep_induct [induct del]
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```    96   nat_int.Abs_induct [induct del]
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```    97
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```    98
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```    99 subsection "Extended form of type definition predicate"
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```   100
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```   101 lemma td_conds:
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```   102   "norm o norm = norm ==> (fr o norm = norm o fr) =
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```   103     (norm o fr o norm = fr o norm & norm o fr o norm = norm o fr)"
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```   104   apply safe
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```   105     apply (simp_all add: comp_assoc)
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```   106    apply (simp_all add: o_assoc)
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```   107   done
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```   108
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```   109 lemma fn_comm_power:
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```   110   "fa o tr = tr o fr ==> fa ^^ n o tr = tr o fr ^^ n"
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```   111   apply (rule ext)
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```   112   apply (induct n)
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```   113    apply (auto dest: fun_cong)
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```   114   done
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```   115
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```   116 lemmas fn_comm_power' =
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```   117   ext [THEN fn_comm_power, THEN fun_cong, unfolded o_def]
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```   118
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```   119
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```   120 locale td_ext = type_definition +
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```   121   fixes norm
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```   122   assumes eq_norm: "\<And>x. Rep (Abs x) = norm x"
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```   123 begin
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```   124
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```   125 lemma Abs_norm [simp]:
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```   126   "Abs (norm x) = Abs x"
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```   127   using eq_norm [of x] by (auto elim: Rep_inverse')
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```   128
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```   129 lemma td_th:
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```   130   "g o Abs = f ==> f (Rep x) = g x"
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```   131   by (drule comp_Abs_inverse [symmetric]) simp
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```   132
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```   133 lemma eq_norm': "Rep o Abs = norm"
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```   134   by (auto simp: eq_norm)
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```   135
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```   136 lemma norm_Rep [simp]: "norm (Rep x) = Rep x"
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```   137   by (auto simp: eq_norm' intro: td_th)
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```   138
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```   139 lemmas td = td_thm
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```   140
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```   141 lemma set_iff_norm: "w : A <-> w = norm w"
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```   142   by (auto simp: set_Rep_Abs eq_norm' eq_norm [symmetric])
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```   143
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```   144 lemma inverse_norm:
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```   145   "(Abs n = w) = (Rep w = norm n)"
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```   146   apply (rule iffI)
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```   147    apply (clarsimp simp add: eq_norm)
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```   148   apply (simp add: eq_norm' [symmetric])
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```   149   done
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```   150
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```   151 lemma norm_eq_iff:
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```   152   "(norm x = norm y) = (Abs x = Abs y)"
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```   153   by (simp add: eq_norm' [symmetric])
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```   154
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```   155 lemma norm_comps:
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```   156   "Abs o norm = Abs"
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```   157   "norm o Rep = Rep"
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```   158   "norm o norm = norm"
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```   159   by (auto simp: eq_norm' [symmetric] o_def)
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```   160
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```   161 lemmas norm_norm [simp] = norm_comps
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```   162
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```   163 lemma fns5:
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```   164   "Rep o fa o Abs = fr ==>
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```   165   fr o norm = fr & norm o fr = fr"
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```   166   by (fold eq_norm') auto
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```   167
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```   168 (* following give conditions for converses to td_fns1
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```   169   the condition (norm o fr o norm = fr o norm) says that
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```   170   fr takes normalised arguments to normalised results,
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```   171   (norm o fr o norm = norm o fr) says that fr
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```   172   takes norm-equivalent arguments to norm-equivalent results,
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```   173   (fr o norm = fr) says that fr
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```   174   takes norm-equivalent arguments to the same result, and
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```   175   (norm o fr = fr) says that fr takes any argument to a normalised result
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```   176   *)
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```   177 lemma fns2:
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```   178   "Abs o fr o Rep = fa ==>
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```   179    (norm o fr o norm = fr o norm) = (Rep o fa = fr o Rep)"
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```   180   apply (fold eq_norm')
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```   181   apply safe
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```   182    prefer 2
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```   183    apply (simp add: o_assoc)
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```   184   apply (rule ext)
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```   185   apply (drule_tac x="Rep x" in fun_cong)
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```   186   apply auto
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```   187   done
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```   188
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```   189 lemma fns3:
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```   190   "Abs o fr o Rep = fa ==>
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```   191    (norm o fr o norm = norm o fr) = (fa o Abs = Abs o fr)"
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```   192   apply (fold eq_norm')
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```   193   apply safe
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```   194    prefer 2
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```   195    apply (simp add: comp_assoc)
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```   196   apply (rule ext)
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```   197   apply (drule_tac f="a o b" for a b in fun_cong)
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```   198   apply simp
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```   199   done
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```   200
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```   201 lemma fns:
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```   202   "fr o norm = norm o fr ==>
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```   203     (fa o Abs = Abs o fr) = (Rep o fa = fr o Rep)"
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```   204   apply safe
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```   205    apply (frule fns1b)
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```   206    prefer 2
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```   207    apply (frule fns1a)
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```   208    apply (rule fns3 [THEN iffD1])
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```   209      prefer 3
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```   210      apply (rule fns2 [THEN iffD1])
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```   211        apply (simp_all add: comp_assoc)
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```   212    apply (simp_all add: o_assoc)
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```   213   done
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```   214
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```   215 lemma range_norm:
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```   216   "range (Rep o Abs) = A"
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```   217   by (simp add: set_Rep_Abs)
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```   218
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```   219 end
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```   220
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```   221 lemmas td_ext_def' =
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```   222   td_ext_def [unfolded type_definition_def td_ext_axioms_def]
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```   223
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```   224 end
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```   225
```