src/HOL/Sum_Type.thy

 (*  Title:      HOL/Sum_Type.thy
     ID:         $Id$
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1992  University of Cambridge
-
-The disjoint sum of two types.
 *)
 
-Sum_Type = Product_Type +
-
-(* type definition *)
+header{*The Disjoint Sum of Two Types*}
+
+theory Sum_Type
+imports Product_Type
+begin
+
+text{*The representations of the two injections*}
 
 constdefs
-  Inl_Rep       :: ['a, 'a, 'b, bool] => bool
+  Inl_Rep :: "['a, 'a, 'b, bool] => bool"
   "Inl_Rep == (%a. %x y p. x=a & p)"
 
-  Inr_Rep       :: ['b, 'a, 'b, bool] => bool
+  Inr_Rep :: "['b, 'a, 'b, bool] => bool"
   "Inr_Rep == (%b. %x y p. y=b & ~p)"
+
 
 global
 
 typedef (Sum)
   ('a, 'b) "+"          (infixr 10)
     = "{f. (? a. f = Inl_Rep(a::'a)) | (? b. f = Inr_Rep(b::'b))}"
-
-
-(* abstract constants and syntax *)
-
-consts
-  Inl            :: "'a => 'a + 'b"
-  Inr            :: "'b => 'a + 'b"
-
-  (*disjoint sum for sets; the operator + is overloaded with wrong type!*)
-  Plus          :: "['a set, 'b set] => ('a + 'b) set"        (infixr "<+>" 65)
-  Part          :: ['a set, 'b => 'a] => 'a set
+  by auto
 
 local
 
-defs
-  Inl_def       "Inl == (%a. Abs_Sum(Inl_Rep(a)))"
-  Inr_def       "Inr == (%b. Abs_Sum(Inr_Rep(b)))"
-
-  sum_def       "A <+> B == (Inl`A) Un (Inr`B)"
-
-  (*for selecting out the components of a mutually recursive definition*)
-  Part_def      "Part A h == A Int {x. ? z. x = h(z)}"
+
+text{*abstract constants and syntax*}
+
+constdefs
+  Inl :: "'a => 'a + 'b"
+   "Inl == (%a. Abs_Sum(Inl_Rep(a)))"
+
+  Inr :: "'b => 'a + 'b"
+   "Inr == (%b. Abs_Sum(Inr_Rep(b)))"
+
+  Plus :: "['a set, 'b set] => ('a + 'b) set"        (infixr "<+>" 65)
+   "A <+> B == (Inl`A) Un (Inr`B)"
+    --{*disjoint sum for sets; the operator + is overloaded with wrong type!*}
+
+  Part :: "['a set, 'b => 'a] => 'a set"
+   "Part A h == A Int {x. ? z. x = h(z)}"
+    --{*for selecting out the components of a mutually recursive definition*}
+
+
+
+(** Inl_Rep and Inr_Rep: Representations of the constructors **)
+
+(*This counts as a non-emptiness result for admitting 'a+'b as a type*)
+lemma Inl_RepI: "Inl_Rep(a) : Sum"
+by (auto simp add: Sum_def)
+
+lemma Inr_RepI: "Inr_Rep(b) : Sum"
+by (auto simp add: Sum_def)
+
+lemma inj_on_Abs_Sum: "inj_on Abs_Sum Sum"
+apply (rule inj_on_inverseI)
+apply (erule Abs_Sum_inverse)
+done
+
+subsection{*Freeness Properties for @{term Inl} and  @{term Inr}*}
+
+text{*Distinctness*}
+
+lemma Inl_Rep_not_Inr_Rep: "Inl_Rep(a) ~= Inr_Rep(b)"
+by (auto simp add: Inl_Rep_def Inr_Rep_def expand_fun_eq)
+
+lemma Inl_not_Inr [iff]: "Inl(a) ~= Inr(b)"
+apply (simp add: Inl_def Inr_def)
+apply (rule inj_on_Abs_Sum [THEN inj_on_contraD])
+apply (rule Inl_Rep_not_Inr_Rep)
+apply (rule Inl_RepI)
+apply (rule Inr_RepI)
+done
+
+lemmas Inr_not_Inl = Inl_not_Inr [THEN not_sym, standard, iff]
+
+lemmas Inl_neq_Inr = Inl_not_Inr [THEN notE, standard]
+lemmas Inr_neq_Inl = sym [THEN Inl_neq_Inr, standard]
+
+
+text{*Injectiveness*}
+
+lemma Inl_Rep_inject: "Inl_Rep(a) = Inl_Rep(c) ==> a=c"
+by (auto simp add: Inl_Rep_def expand_fun_eq)
+
+lemma Inr_Rep_inject: "Inr_Rep(b) = Inr_Rep(d) ==> b=d"
+by (auto simp add: Inr_Rep_def expand_fun_eq)
+
+lemma inj_Inl: "inj(Inl)"
+apply (simp add: Inl_def)
+apply (rule inj_onI)
+apply (erule inj_on_Abs_Sum [THEN inj_onD, THEN Inl_Rep_inject])
+apply (rule Inl_RepI)
+apply (rule Inl_RepI)
+done
+lemmas Inl_inject = inj_Inl [THEN injD, standard]
+
+lemma inj_Inr: "inj(Inr)"
+apply (simp add: Inr_def)
+apply (rule inj_onI)
+apply (erule inj_on_Abs_Sum [THEN inj_onD, THEN Inr_Rep_inject])
+apply (rule Inr_RepI)
+apply (rule Inr_RepI)
+done
+
+lemmas Inr_inject = inj_Inr [THEN injD, standard]
+
+lemma Inl_eq [iff]: "(Inl(x)=Inl(y)) = (x=y)"
+by (blast dest!: Inl_inject)
+
+lemma Inr_eq [iff]: "(Inr(x)=Inr(y)) = (x=y)"
+by (blast dest!: Inr_inject)
+
+
+subsection{*The Disjoint Sum of Sets*}
+
+(** Introduction rules for the injections **)
+
+lemma InlI [intro!]: "a : A ==> Inl(a) : A <+> B"
+by (simp add: Plus_def)
+
+lemma InrI [intro!]: "b : B ==> Inr(b) : A <+> B"
+by (simp add: Plus_def)
+
+(** Elimination rules **)
+
+lemma PlusE [elim!]: 
+    "[| u: A <+> B;   
+        !!x. [| x:A;  u=Inl(x) |] ==> P;  
+        !!y. [| y:B;  u=Inr(y) |] ==> P  
+     |] ==> P"
+by (auto simp add: Plus_def)
+
+
+
+text{*Exhaustion rule for sums, a degenerate form of induction*}
+lemma sumE: 
+    "[| !!x::'a. s = Inl(x) ==> P;  !!y::'b. s = Inr(y) ==> P  
+     |] ==> P"
+apply (rule Abs_Sum_cases [of s]) 
+apply (auto simp add: Sum_def Inl_def Inr_def)
+done
+
+lemma sum_induct: "[| !!x. P (Inl x); !!x. P (Inr x) |] ==> P x"
+by (rule sumE [of x], auto)
+
+
+subsection{*The @{term Part} Primitive*}
+
+lemma Part_eqI [intro]: "[| a : A;  a=h(b) |] ==> a : Part A h"
+by (auto simp add: Part_def)
+
+lemmas PartI = Part_eqI [OF _ refl, standard]
+
+lemma PartE [elim!]: "[| a : Part A h;  !!z. [| a : A;  a=h(z) |] ==> P |] ==> P"
+by (auto simp add: Part_def)
+
+
+lemma Part_subset: "Part A h <= A"
+by (auto simp add: Part_def)
+
+lemma Part_mono: "A<=B ==> Part A h <= Part B h"
+by blast
+
+lemmas basic_monos = basic_monos Part_mono
+
+lemma PartD1: "a : Part A h ==> a : A"
+by (simp add: Part_def)
+
+lemma Part_id: "Part A (%x. x) = A"
+by blast
+
+lemma Part_Int: "Part (A Int B) h = (Part A h) Int (Part B h)"
+by blast
+
+lemma Part_Collect: "Part (A Int {x. P x}) h = (Part A h) Int {x. P x}"
+by blast
+
+ML
+{*
+val Inl_RepI = thm "Inl_RepI";
+val Inr_RepI = thm "Inr_RepI";
+val inj_on_Abs_Sum = thm "inj_on_Abs_Sum";
+val Inl_Rep_not_Inr_Rep = thm "Inl_Rep_not_Inr_Rep";
+val Inl_not_Inr = thm "Inl_not_Inr";
+val Inr_not_Inl = thm "Inr_not_Inl";
+val Inl_neq_Inr = thm "Inl_neq_Inr";
+val Inr_neq_Inl = thm "Inr_neq_Inl";
+val Inl_Rep_inject = thm "Inl_Rep_inject";
+val Inr_Rep_inject = thm "Inr_Rep_inject";
+val inj_Inl = thm "inj_Inl";
+val Inl_inject = thm "Inl_inject";
+val inj_Inr = thm "inj_Inr";
+val Inr_inject = thm "Inr_inject";
+val Inl_eq = thm "Inl_eq";
+val Inr_eq = thm "Inr_eq";
+val InlI = thm "InlI";
+val InrI = thm "InrI";
+val PlusE = thm "PlusE";
+val sumE = thm "sumE";
+val sum_induct = thm "sum_induct";
+val Part_eqI = thm "Part_eqI";
+val PartI = thm "PartI";
+val PartE = thm "PartE";
+val Part_subset = thm "Part_subset";
+val Part_mono = thm "Part_mono";
+val PartD1 = thm "PartD1";
+val Part_id = thm "Part_id";
+val Part_Int = thm "Part_Int";
+val Part_Collect = thm "Part_Collect";
+
+val basic_monos = thms "basic_monos";
+*}
+
 
 end