author  krauss 
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child 21211  5370cfbf3070 
permissions  rwrr 
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(* Title: HOL/FunDef.thy 
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ID: $Id$ 

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Author: Alexander Krauss, TU Muenchen 

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A package for general recursive function definitions. 

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*) 

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theory FunDef 
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imports Accessible_Part Datatype Recdef 
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uses 
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("Tools/function_package/sum_tools.ML") 
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("Tools/function_package/fundef_common.ML") 
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("Tools/function_package/fundef_lib.ML") 
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("Tools/function_package/inductive_wrap.ML") 
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("Tools/function_package/context_tree.ML") 
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("Tools/function_package/fundef_prep.ML") 
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("Tools/function_package/fundef_proof.ML") 
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("Tools/function_package/termination.ML") 
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("Tools/function_package/mutual.ML") 
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("Tools/function_package/pattern_split.ML") 
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("Tools/function_package/fundef_package.ML") 
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("Tools/function_package/fundef_datatype.ML") 
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("Tools/function_package/auto_term.ML") 
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begin 
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section {* Wellfoundedness and Accessibility: Predicate versions *} 
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constdefs 
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wfP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) => bool" 
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"wfP(r) == (!P. (!x. (!y. r y x > P(y)) > P(x)) > (!x. P(x)))" 
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lemma wfP_induct: 
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"[ wfP r; 
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!!x.[ ALL y. r y x > P(y) ] ==> P(x) 
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] ==> P(a)" 
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by (unfold wfP_def, blast) 
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lemmas wfP_induct_rule = wfP_induct [rule_format, consumes 1, case_names less] 
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definition in_rel_def[simp]: 
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"in_rel R x y == (x, y) \<in> R" 
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lemma wf_in_rel: 
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"wf R \<Longrightarrow> wfP (in_rel R)" 
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unfolding wfP_def wf_def in_rel_def . 
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inductive2 accP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" 
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for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" 
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intros 
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accPI: "(!!y. r y x ==> accP r y) ==> accP r x" 
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theorem accP_induct: 
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assumes major: "accP r a" 
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assumes hyp: "!!x. accP r x ==> \<forall>y. r y x > P y ==> P x" 
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shows "P a" 
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apply (rule major [THEN accP.induct]) 
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apply (rule hyp) 
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apply (rule accPI) 
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apply fast 
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apply fast 
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done 
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theorems accP_induct_rule = accP_induct [rule_format, induct set: accP] 
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theorem accP_downward: "accP r b ==> r a b ==> accP r a" 
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apply (erule accP.cases) 
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apply fast 
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done 
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lemma accP_subset: 
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assumes sub: "\<And>x y. R1 x y \<Longrightarrow> R2 x y" 
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shows "\<And>x. accP R2 x \<Longrightarrow> accP R1 x" 
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proof 
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fix x assume "accP R2 x" 
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then show "accP R1 x" 
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proof (induct x) 
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fix x 
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assume ih: "\<And>y. R2 y x \<Longrightarrow> accP R1 y" 
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with sub show "accP R1 x" 
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by (blast intro:accPI) 
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qed 
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qed 
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lemma accP_subset_induct: 
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assumes subset: "\<And>x. D x \<Longrightarrow> accP R x" 
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and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z" 
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and "D x" 
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and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x" 
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shows "P x" 
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proof  
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from subset and `D x` 
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have "accP R x" . 
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then show "P x" using `D x` 
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proof (induct x) 
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fix x 
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assume "D x" 
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and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y" 
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with dcl and istep show "P x" by blast 
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qed 
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qed 
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section {* Definitions with default value *} 
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definition 
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THE_default :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" 
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"THE_default d P = (if (\<exists>!x. P x) then (THE x. P x) else d)" 
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lemma THE_defaultI': "\<exists>!x. P x \<Longrightarrow> P (THE_default d P)" 
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by (simp add:theI' THE_default_def) 
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lemma THE_default1_equality: 
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"\<lbrakk>\<exists>!x. P x; P a\<rbrakk> \<Longrightarrow> THE_default d P = a" 
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by (simp add:the1_equality THE_default_def) 
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lemma THE_default_none: 
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"\<not>(\<exists>!x. P x) \<Longrightarrow> THE_default d P = d" 
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by (simp add:THE_default_def) 
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lemma fundef_ex1_existence: 
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assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))" 
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assumes ex1: "\<exists>!y. G x y" 
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shows "G x (f x)" 
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by (simp only:f_def, rule THE_defaultI', rule ex1) 
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lemma fundef_ex1_uniqueness: 
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assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))" 
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assumes ex1: "\<exists>!y. G x y" 
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assumes elm: "G x (h x)" 
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shows "h x = f x" 
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by (simp only:f_def, rule THE_default1_equality[symmetric], rule ex1, rule elm) 
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lemma fundef_ex1_iff: 
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assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))" 
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assumes ex1: "\<exists>!y. G x y" 
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shows "(G x y) = (f x = y)" 
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apply (auto simp:ex1 f_def THE_default1_equality) 
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by (rule THE_defaultI', rule ex1) 
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lemma fundef_default_value: 
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assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))" 
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assumes graph: "\<And>x y. G x y \<Longrightarrow> x \<in> D" 
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assumes "x \<notin> D" 
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shows "f x = d x" 
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proof  
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have "\<not>(\<exists>y. G x y)" 
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proof 
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assume "(\<exists>y. G x y)" 
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with graph and `x\<notin>D` show False by blast 
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qed 
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hence "\<not>(\<exists>!y. G x y)" by blast 
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thus ?thesis 
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unfolding f_def 
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by (rule THE_default_none) 
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qed 
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section {* Projections *} 
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consts 
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lpg::"(('a + 'b) * 'a) set" 
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rpg::"(('a + 'b) * 'b) set" 
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inductive lpg 
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intros 
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"(Inl x, x) : lpg" 
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inductive rpg 
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intros 
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"(Inr y, y) : rpg" 
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definition 
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"lproj x = (THE y. (x,y) : lpg)" 
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"rproj x = (THE y. (x,y) : rpg)" 
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lemma lproj_inl: 
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"lproj (Inl x) = x" 
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by (auto simp:lproj_def intro: the_equality lpg.intros elim: lpg.cases) 
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lemma rproj_inr: 
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"rproj (Inr x) = x" 
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by (auto simp:rproj_def intro: the_equality rpg.intros elim: rpg.cases) 
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lemma P_imp_P: "P \<Longrightarrow> P" . 
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use "Tools/function_package/sum_tools.ML" 
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use "Tools/function_package/fundef_common.ML" 
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use "Tools/function_package/fundef_lib.ML" 
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use "Tools/function_package/inductive_wrap.ML" 
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use "Tools/function_package/context_tree.ML" 
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use "Tools/function_package/fundef_prep.ML" 
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use "Tools/function_package/fundef_proof.ML" 
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use "Tools/function_package/termination.ML" 
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use "Tools/function_package/mutual.ML" 
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use "Tools/function_package/pattern_split.ML" 
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use "Tools/function_package/fundef_package.ML" 
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setup FundefPackage.setup 
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use "Tools/function_package/fundef_datatype.ML" 
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setup FundefDatatype.setup 
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use "Tools/function_package/auto_term.ML" 
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setup FundefAutoTerm.setup 
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216 

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lemmas [fundef_cong] = 
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let_cong if_cong image_cong INT_cong UN_cong bex_cong ball_cong imp_cong 
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19934  222 
lemma split_cong[fundef_cong]: 
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"\<lbrakk> \<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y; p = q \<rbrakk> 

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\<Longrightarrow> split f p = split g q" 

225 
by (auto simp:split_def) 

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227 

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end 