src/HOL/Sum_Type.thy
 author nipkow Fri Oct 29 17:25:22 2010 +0200 (2010-10-29) changeset 40271 6014e8252e57 parent 39302 d7728f65b353 child 40610 70776ecfe324 permissions -rw-r--r--
hide Sum_Type.Plus
```     1 (*  Title:      HOL/Sum_Type.thy
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```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     3     Copyright   1992  University of Cambridge
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```     4 *)
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```     5
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```     6 header{*The Disjoint Sum of Two Types*}
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```     7
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```     8 theory Sum_Type
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```     9 imports Typedef Inductive Fun
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```    10 begin
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```    11
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```    12 subsection {* Construction of the sum type and its basic abstract operations *}
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```    13
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```    14 definition Inl_Rep :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool \<Rightarrow> bool" where
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```    15   "Inl_Rep a x y p \<longleftrightarrow> x = a \<and> p"
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```    16
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```    17 definition Inr_Rep :: "'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool \<Rightarrow> bool" where
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```    18   "Inr_Rep b x y p \<longleftrightarrow> y = b \<and> \<not> p"
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```    19
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```    20 typedef ('a, 'b) sum (infixr "+" 10) = "{f. (\<exists>a. f = Inl_Rep (a::'a)) \<or> (\<exists>b. f = Inr_Rep (b::'b))}"
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```    21   by auto
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```    22
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```    23 lemma Inl_RepI: "Inl_Rep a \<in> sum"
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```    24   by (auto simp add: sum_def)
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```    25
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```    26 lemma Inr_RepI: "Inr_Rep b \<in> sum"
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```    27   by (auto simp add: sum_def)
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```    28
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```    29 lemma inj_on_Abs_sum: "A \<subseteq> sum \<Longrightarrow> inj_on Abs_sum A"
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```    30   by (rule inj_on_inverseI, rule Abs_sum_inverse) auto
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```    31
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```    32 lemma Inl_Rep_inject: "inj_on Inl_Rep A"
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```    33 proof (rule inj_onI)
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```    34   show "\<And>a c. Inl_Rep a = Inl_Rep c \<Longrightarrow> a = c"
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```    35     by (auto simp add: Inl_Rep_def fun_eq_iff)
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```    36 qed
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```    37
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```    38 lemma Inr_Rep_inject: "inj_on Inr_Rep A"
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```    39 proof (rule inj_onI)
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```    40   show "\<And>b d. Inr_Rep b = Inr_Rep d \<Longrightarrow> b = d"
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```    41     by (auto simp add: Inr_Rep_def fun_eq_iff)
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```    42 qed
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```    43
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```    44 lemma Inl_Rep_not_Inr_Rep: "Inl_Rep a \<noteq> Inr_Rep b"
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```    45   by (auto simp add: Inl_Rep_def Inr_Rep_def fun_eq_iff)
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```    46
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```    47 definition Inl :: "'a \<Rightarrow> 'a + 'b" where
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```    48   "Inl = Abs_sum \<circ> Inl_Rep"
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```    49
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```    50 definition Inr :: "'b \<Rightarrow> 'a + 'b" where
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```    51   "Inr = Abs_sum \<circ> Inr_Rep"
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```    52
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```    53 lemma inj_Inl [simp]: "inj_on Inl A"
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```    54 by (auto simp add: Inl_def intro!: comp_inj_on Inl_Rep_inject inj_on_Abs_sum Inl_RepI)
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```    55
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```    56 lemma Inl_inject: "Inl x = Inl y \<Longrightarrow> x = y"
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```    57 using inj_Inl by (rule injD)
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```    58
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```    59 lemma inj_Inr [simp]: "inj_on Inr A"
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```    60 by (auto simp add: Inr_def intro!: comp_inj_on Inr_Rep_inject inj_on_Abs_sum Inr_RepI)
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```    61
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```    62 lemma Inr_inject: "Inr x = Inr y \<Longrightarrow> x = y"
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```    63 using inj_Inr by (rule injD)
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```    64
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```    65 lemma Inl_not_Inr: "Inl a \<noteq> Inr b"
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```    66 proof -
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```    67   from Inl_RepI [of a] Inr_RepI [of b] have "{Inl_Rep a, Inr_Rep b} \<subseteq> sum" by auto
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```    68   with inj_on_Abs_sum have "inj_on Abs_sum {Inl_Rep a, Inr_Rep b}" .
```
```    69   with Inl_Rep_not_Inr_Rep [of a b] inj_on_contraD have "Abs_sum (Inl_Rep a) \<noteq> Abs_sum (Inr_Rep b)" by auto
```
```    70   then show ?thesis by (simp add: Inl_def Inr_def)
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```    71 qed
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```    72
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```    73 lemma Inr_not_Inl: "Inr b \<noteq> Inl a"
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```    74   using Inl_not_Inr by (rule not_sym)
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```    75
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```    76 lemma sumE:
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```    77   assumes "\<And>x::'a. s = Inl x \<Longrightarrow> P"
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```    78     and "\<And>y::'b. s = Inr y \<Longrightarrow> P"
```
```    79   shows P
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```    80 proof (rule Abs_sum_cases [of s])
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```    81   fix f
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```    82   assume "s = Abs_sum f" and "f \<in> sum"
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```    83   with assms show P by (auto simp add: sum_def Inl_def Inr_def)
```
```    84 qed
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```    85
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```    86 rep_datatype Inl Inr
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```    87 proof -
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```    88   fix P
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```    89   fix s :: "'a + 'b"
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```    90   assume x: "\<And>x\<Colon>'a. P (Inl x)" and y: "\<And>y\<Colon>'b. P (Inr y)"
```
```    91   then show "P s" by (auto intro: sumE [of s])
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```    92 qed (auto dest: Inl_inject Inr_inject simp add: Inl_not_Inr)
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```    93
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```    94
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```    95 subsection {* Projections *}
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```    96
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```    97 lemma sum_case_KK [simp]: "sum_case (\<lambda>x. a) (\<lambda>x. a) = (\<lambda>x. a)"
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```    98   by (rule ext) (simp split: sum.split)
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```    99
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```   100 lemma surjective_sum: "sum_case (\<lambda>x::'a. f (Inl x)) (\<lambda>y::'b. f (Inr y)) = f"
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```   101 proof
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```   102   fix s :: "'a + 'b"
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```   103   show "(case s of Inl (x\<Colon>'a) \<Rightarrow> f (Inl x) | Inr (y\<Colon>'b) \<Rightarrow> f (Inr y)) = f s"
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```   104     by (cases s) simp_all
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```   105 qed
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```   106
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```   107 lemma sum_case_inject:
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```   108   assumes a: "sum_case f1 f2 = sum_case g1 g2"
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```   109   assumes r: "f1 = g1 \<Longrightarrow> f2 = g2 \<Longrightarrow> P"
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```   110   shows P
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```   111 proof (rule r)
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```   112   show "f1 = g1" proof
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```   113     fix x :: 'a
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```   114     from a have "sum_case f1 f2 (Inl x) = sum_case g1 g2 (Inl x)" by simp
```
```   115     then show "f1 x = g1 x" by simp
```
```   116   qed
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```   117   show "f2 = g2" proof
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```   118     fix y :: 'b
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```   119     from a have "sum_case f1 f2 (Inr y) = sum_case g1 g2 (Inr y)" by simp
```
```   120     then show "f2 y = g2 y" by simp
```
```   121   qed
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```   122 qed
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```   123
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```   124 lemma sum_case_weak_cong:
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```   125   "s = t \<Longrightarrow> sum_case f g s = sum_case f g t"
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```   126   -- {* Prevents simplification of @{text f} and @{text g}: much faster. *}
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```   127   by simp
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```   128
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```   129 primrec Projl :: "'a + 'b \<Rightarrow> 'a" where
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```   130   Projl_Inl: "Projl (Inl x) = x"
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```   131
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```   132 primrec Projr :: "'a + 'b \<Rightarrow> 'b" where
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```   133   Projr_Inr: "Projr (Inr x) = x"
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```   134
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```   135 primrec Suml :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a + 'b \<Rightarrow> 'c" where
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```   136   "Suml f (Inl x) = f x"
```
```   137
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```   138 primrec Sumr :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a + 'b \<Rightarrow> 'c" where
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```   139   "Sumr f (Inr x) = f x"
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```   140
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```   141 lemma Suml_inject:
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```   142   assumes "Suml f = Suml g" shows "f = g"
```
```   143 proof
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```   144   fix x :: 'a
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```   145   let ?s = "Inl x \<Colon> 'a + 'b"
```
```   146   from assms have "Suml f ?s = Suml g ?s" by simp
```
```   147   then show "f x = g x" by simp
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```   148 qed
```
```   149
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```   150 lemma Sumr_inject:
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```   151   assumes "Sumr f = Sumr g" shows "f = g"
```
```   152 proof
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```   153   fix x :: 'b
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```   154   let ?s = "Inr x \<Colon> 'a + 'b"
```
```   155   from assms have "Sumr f ?s = Sumr g ?s" by simp
```
```   156   then show "f x = g x" by simp
```
```   157 qed
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```   158
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```   159
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```   160 subsection {* The Disjoint Sum of Sets *}
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```   161
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```   162 definition Plus :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a + 'b) set" (infixr "<+>" 65) where
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```   163   "A <+> B = Inl ` A \<union> Inr ` B"
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```   164
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```   165 hide_const (open) Plus --"Valuable identifier"
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```   166
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```   167 lemma InlI [intro!]: "a \<in> A \<Longrightarrow> Inl a \<in> A <+> B"
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```   168 by (simp add: Plus_def)
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```   169
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```   170 lemma InrI [intro!]: "b \<in> B \<Longrightarrow> Inr b \<in> A <+> B"
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```   171 by (simp add: Plus_def)
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```   172
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```   173 text {* Exhaustion rule for sums, a degenerate form of induction *}
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```   174
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```   175 lemma PlusE [elim!]:
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```   176   "u \<in> A <+> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> u = Inl x \<Longrightarrow> P) \<Longrightarrow> (\<And>y. y \<in> B \<Longrightarrow> u = Inr y \<Longrightarrow> P) \<Longrightarrow> P"
```
```   177 by (auto simp add: Plus_def)
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```   178
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```   179 lemma Plus_eq_empty_conv [simp]: "A <+> B = {} \<longleftrightarrow> A = {} \<and> B = {}"
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```   180 by auto
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```   181
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```   182 lemma UNIV_Plus_UNIV [simp]: "UNIV <+> UNIV = UNIV"
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```   183 proof (rule set_eqI)
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```   184   fix u :: "'a + 'b"
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```   185   show "u \<in> UNIV <+> UNIV \<longleftrightarrow> u \<in> UNIV" by (cases u) auto
```
```   186 qed
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```   187
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```   188 hide_const (open) Suml Sumr Projl Projr
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```   189
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```   190 end
```