author  huffman 
Tue, 09 Dec 2008 15:31:38 0800  
changeset 29025  8c8859c0d734 
parent 28524  644b62cf678f 
child 29183  f1648e009dc1 
permissions  rwrr 
10213  1 
(* Title: HOL/Sum_Type.thy 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1992 University of Cambridge 

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*) 

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header{*The Disjoint Sum of Two Types*} 
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theory Sum_Type 
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imports Typedef Fun 
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begin 
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text{*The representations of the two injections*} 
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constdefs 

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Inl_Rep :: "['a, 'a, 'b, bool] => bool" 
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"Inl_Rep == (%a. %x y p. x=a & p)" 
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Inr_Rep :: "['b, 'a, 'b, bool] => bool" 
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"Inr_Rep == (%b. %x y p. y=b & ~p)" 
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global 
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typedef (Sum) 

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('a, 'b) "+" (infixr "+" 10) 
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= "{f. (? a. f = Inl_Rep(a::'a))  (? b. f = Inr_Rep(b::'b))}" 
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by auto 
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local 

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text{*abstract constants and syntax*} 
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constdefs 
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Inl :: "'a => 'a + 'b" 
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"Inl == (%a. Abs_Sum(Inl_Rep(a)))" 
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Inr :: "'b => 'a + 'b" 
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"Inr == (%b. Abs_Sum(Inr_Rep(b)))" 
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Plus :: "['a set, 'b set] => ('a + 'b) set" (infixr "<+>" 65) 
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"A <+> B == (Inl`A) Un (Inr`B)" 
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{*disjoint sum for sets; the operator + is overloaded with wrong type!*} 
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Part :: "['a set, 'b => 'a] => 'a set" 
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"Part A h == A Int {x. ? z. x = h(z)}" 
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{*for selecting out the components of a mutually recursive definition*} 
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(** Inl_Rep and Inr_Rep: Representations of the constructors **) 
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(*This counts as a nonemptiness result for admitting 'a+'b as a type*) 
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lemma Inl_RepI: "Inl_Rep(a) : Sum" 
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by (auto simp add: Sum_def) 
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lemma Inr_RepI: "Inr_Rep(b) : Sum" 
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by (auto simp add: Sum_def) 
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lemma inj_on_Abs_Sum: "inj_on Abs_Sum Sum" 
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apply (rule inj_on_inverseI) 
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apply (erule Abs_Sum_inverse) 
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done 
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subsection{*Freeness Properties for @{term Inl} and @{term Inr}*} 
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text{*Distinctness*} 
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lemma Inl_Rep_not_Inr_Rep: "Inl_Rep(a) ~= Inr_Rep(b)" 
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by (auto simp add: Inl_Rep_def Inr_Rep_def expand_fun_eq) 
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lemma Inl_not_Inr [iff]: "Inl(a) ~= Inr(b)" 
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apply (simp add: Inl_def Inr_def) 
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apply (rule inj_on_Abs_Sum [THEN inj_on_contraD]) 
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apply (rule Inl_Rep_not_Inr_Rep) 
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apply (rule Inl_RepI) 
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apply (rule Inr_RepI) 
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done 
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lemmas Inr_not_Inl = Inl_not_Inr [THEN not_sym, standard] 
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declare Inr_not_Inl [iff] 
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lemmas Inl_neq_Inr = Inl_not_Inr [THEN notE, standard] 
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lemmas Inr_neq_Inl = sym [THEN Inl_neq_Inr, standard] 
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text{*Injectiveness*} 
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lemma Inl_Rep_inject: "Inl_Rep(a) = Inl_Rep(c) ==> a=c" 
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by (auto simp add: Inl_Rep_def expand_fun_eq) 
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lemma Inr_Rep_inject: "Inr_Rep(b) = Inr_Rep(d) ==> b=d" 
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by (auto simp add: Inr_Rep_def expand_fun_eq) 
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lemma inj_Inl [simp]: "inj_on Inl A" 
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apply (simp add: Inl_def) 
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apply (rule inj_onI) 
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apply (erule inj_on_Abs_Sum [THEN inj_onD, THEN Inl_Rep_inject]) 
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apply (rule Inl_RepI) 
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apply (rule Inl_RepI) 
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done 
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lemmas Inl_inject = inj_Inl [THEN injD, standard] 
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lemma inj_Inr [simp]: "inj_on Inr A" 
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apply (simp add: Inr_def) 
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apply (rule inj_onI) 
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apply (erule inj_on_Abs_Sum [THEN inj_onD, THEN Inr_Rep_inject]) 
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apply (rule Inr_RepI) 
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apply (rule Inr_RepI) 
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done 
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lemmas Inr_inject = inj_Inr [THEN injD, standard] 
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lemma Inl_eq [iff]: "(Inl(x)=Inl(y)) = (x=y)" 
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by (blast dest!: Inl_inject) 
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lemma Inr_eq [iff]: "(Inr(x)=Inr(y)) = (x=y)" 
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by (blast dest!: Inr_inject) 
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subsection {* Projections *} 
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definition 
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"sum_case f g x = 
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(if (\<exists>!y. x = Inl y) 
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then f (THE y. x = Inl y) 
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else g (THE y. x = Inr y))" 
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definition "Projl x = sum_case id undefined x" 
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definition "Projr x = sum_case undefined id x" 

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lemma sum_cases[simp]: 
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"sum_case f g (Inl x) = f x" 
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"sum_case f g (Inr y) = g y" 
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unfolding sum_case_def 
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by auto 
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lemma Projl_Inl[simp]: "Projl (Inl x) = x" 
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unfolding Projl_def by simp 
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lemma Projr_Inr[simp]: "Projr (Inr x) = x" 
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unfolding Projr_def by simp 
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144 

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145 

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subsection{*The Disjoint Sum of Sets*} 
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147 

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(** Introduction rules for the injections **) 
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lemma InlI [intro!]: "a : A ==> Inl(a) : A <+> B" 
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by (simp add: Plus_def) 
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152 

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lemma InrI [intro!]: "b : B ==> Inr(b) : A <+> B" 
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by (simp add: Plus_def) 
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155 

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(** Elimination rules **) 
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157 

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lemma PlusE [elim!]: 
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"[ u: A <+> B; 
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!!x. [ x:A; u=Inl(x) ] ==> P; 
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!!y. [ y:B; u=Inr(y) ] ==> P 
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] ==> P" 
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by (auto simp add: Plus_def) 
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164 

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165 

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166 

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text{*Exhaustion rule for sums, a degenerate form of induction*} 
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lemma sumE: 
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"[ !!x::'a. s = Inl(x) ==> P; !!y::'b. s = Inr(y) ==> P 
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] ==> P" 
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apply (rule Abs_Sum_cases [of s]) 
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apply (auto simp add: Sum_def Inl_def Inr_def) 
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done 
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174 

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17026  176 
lemma UNIV_Plus_UNIV [simp]: "UNIV <+> UNIV = UNIV" 
177 
apply (rule set_ext) 

178 
apply(rename_tac s) 

179 
apply(rule_tac s=s in sumE) 

180 
apply auto 

181 
done 

182 

183 

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subsection{*The @{term Part} Primitive*} 
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185 

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lemma Part_eqI [intro]: "[ a : A; a=h(b) ] ==> a : Part A h" 
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by (auto simp add: Part_def) 
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188 

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lemmas PartI = Part_eqI [OF _ refl, standard] 
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190 

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lemma PartE [elim!]: "[ a : Part A h; !!z. [ a : A; a=h(z) ] ==> P ] ==> P" 
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by (auto simp add: Part_def) 
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193 

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194 

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lemma Part_subset: "Part A h <= A" 
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by (auto simp add: Part_def) 
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197 

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lemma Part_mono: "A<=B ==> Part A h <= Part B h" 
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199 
by blast 
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200 

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lemmas basic_monos = basic_monos Part_mono 
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lemma PartD1: "a : Part A h ==> a : A" 
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by (simp add: Part_def) 
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205 

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lemma Part_id: "Part A (%x. x) = A" 
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207 
by blast 
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208 

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lemma Part_Int: "Part (A Int B) h = (Part A h) Int (Part B h)" 
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by blast 
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211 

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lemma Part_Collect: "Part (A Int {x. P x}) h = (Part A h) Int {x. P x}" 
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213 
by blast 
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214 

20588  215 

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216 
ML 
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217 
{* 
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val Inl_RepI = thm "Inl_RepI"; 
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val Inr_RepI = thm "Inr_RepI"; 
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val inj_on_Abs_Sum = thm "inj_on_Abs_Sum"; 
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val Inl_Rep_not_Inr_Rep = thm "Inl_Rep_not_Inr_Rep"; 
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val Inl_not_Inr = thm "Inl_not_Inr"; 
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val Inr_not_Inl = thm "Inr_not_Inl"; 
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val Inl_neq_Inr = thm "Inl_neq_Inr"; 
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val Inr_neq_Inl = thm "Inr_neq_Inl"; 
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val Inl_Rep_inject = thm "Inl_Rep_inject"; 
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val Inr_Rep_inject = thm "Inr_Rep_inject"; 
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val inj_Inl = thm "inj_Inl"; 
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val Inl_inject = thm "Inl_inject"; 
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val inj_Inr = thm "inj_Inr"; 
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val Inr_inject = thm "Inr_inject"; 
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val Inl_eq = thm "Inl_eq"; 
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val Inr_eq = thm "Inr_eq"; 
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val InlI = thm "InlI"; 
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val InrI = thm "InrI"; 
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val PlusE = thm "PlusE"; 
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val sumE = thm "sumE"; 
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val Part_eqI = thm "Part_eqI"; 
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val PartI = thm "PartI"; 
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val PartE = thm "PartE"; 
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val Part_subset = thm "Part_subset"; 
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val Part_mono = thm "Part_mono"; 
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val PartD1 = thm "PartD1"; 
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val Part_id = thm "Part_id"; 
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val Part_Int = thm "Part_Int"; 
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val Part_Collect = thm "Part_Collect"; 
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247 

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val basic_monos = thms "basic_monos"; 
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249 
*} 
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250 

10213  251 
end 