author  bulwahn 
Thu, 12 Nov 2009 20:38:57 +0100  
changeset 33649  854173fcd21c 
child 33870  5b0d23d2c08f 
permissions  rwrr 
33649
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(* Author: Stefan Berghofer, Lukas Bulwahn, TU Muenchen *) 
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header {* A tabled implementation of the reflexive transitive closure *} 
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theory Transitive_Closure_Table 
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imports Main 
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begin 
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inductive rtrancl_path :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> bool" 
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for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" 
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where 
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base: "rtrancl_path r x [] x" 
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 step: "r x y \<Longrightarrow> rtrancl_path r y ys z \<Longrightarrow> rtrancl_path r x (y # ys) z" 
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lemma rtranclp_eq_rtrancl_path: "r\<^sup>*\<^sup>* x y = (\<exists>xs. rtrancl_path r x xs y)" 
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proof 
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assume "r\<^sup>*\<^sup>* x y" 
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then show "\<exists>xs. rtrancl_path r x xs y" 
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proof (induct rule: converse_rtranclp_induct) 
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case 1 
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have "rtrancl_path r y [] y" by (rule rtrancl_path.base) 
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then show ?case .. 
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next 
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case (2 x z) 
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from `\<exists>xs. rtrancl_path r z xs y` 
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obtain xs where "rtrancl_path r z xs y" .. 
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with `r x z` have "rtrancl_path r x (z # xs) y" 
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by (rule rtrancl_path.step) 
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then show ?case .. 
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qed 
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next 
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assume "\<exists>xs. rtrancl_path r x xs y" 
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then obtain xs where "rtrancl_path r x xs y" .. 
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then show "r\<^sup>*\<^sup>* x y" 
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proof induct 
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case (base x) 
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show ?case by (rule rtranclp.rtrancl_refl) 
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next 
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case (step x y ys z) 
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from `r x y` `r\<^sup>*\<^sup>* y z` show ?case 
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by (rule converse_rtranclp_into_rtranclp) 
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qed 
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qed 
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lemma rtrancl_path_trans: 
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assumes xy: "rtrancl_path r x xs y" 
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and yz: "rtrancl_path r y ys z" 
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shows "rtrancl_path r x (xs @ ys) z" using xy yz 
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proof (induct arbitrary: z) 
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case (base x) 
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then show ?case by simp 
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next 
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case (step x y xs) 
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then have "rtrancl_path r y (xs @ ys) z" 
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by simp 
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with `r x y` have "rtrancl_path r x (y # (xs @ ys)) z" 
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by (rule rtrancl_path.step) 
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then show ?case by simp 
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qed 
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lemma rtrancl_path_appendE: 
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assumes xz: "rtrancl_path r x (xs @ y # ys) z" 
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obtains "rtrancl_path r x (xs @ [y]) y" and "rtrancl_path r y ys z" using xz 
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proof (induct xs arbitrary: x) 
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case Nil 
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then have "rtrancl_path r x (y # ys) z" by simp 
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then obtain xy: "r x y" and yz: "rtrancl_path r y ys z" 
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by cases auto 
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from xy have "rtrancl_path r x [y] y" 
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by (rule rtrancl_path.step [OF _ rtrancl_path.base]) 
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then have "rtrancl_path r x ([] @ [y]) y" by simp 
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then show ?thesis using yz by (rule Nil) 
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next 
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case (Cons a as) 
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then have "rtrancl_path r x (a # (as @ y # ys)) z" by simp 
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then obtain xa: "r x a" and az: "rtrancl_path r a (as @ y # ys) z" 
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by cases auto 
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show ?thesis 
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proof (rule Cons(1) [OF _ az]) 
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assume "rtrancl_path r y ys z" 
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assume "rtrancl_path r a (as @ [y]) y" 
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with xa have "rtrancl_path r x (a # (as @ [y])) y" 
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by (rule rtrancl_path.step) 
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then have "rtrancl_path r x ((a # as) @ [y]) y" 
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by simp 
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then show ?thesis using `rtrancl_path r y ys z` 
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by (rule Cons(2)) 
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qed 
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qed 
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lemma rtrancl_path_distinct: 
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assumes xy: "rtrancl_path r x xs y" 
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obtains xs' where "rtrancl_path r x xs' y" and "distinct (x # xs')" using xy 
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proof (induct xs rule: measure_induct_rule [of length]) 
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case (less xs) 
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show ?case 
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proof (cases "distinct (x # xs)") 
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case True 
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with `rtrancl_path r x xs y` show ?thesis by (rule less) 
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next 
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case False 
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then have "\<exists>as bs cs a. x # xs = as @ [a] @ bs @ [a] @ cs" 
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by (rule not_distinct_decomp) 
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then obtain as bs cs a where xxs: "x # xs = as @ [a] @ bs @ [a] @ cs" 
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by iprover 
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show ?thesis 
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proof (cases as) 
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case Nil 
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with xxs have x: "x = a" and xs: "xs = bs @ a # cs" 
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by auto 
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from x xs `rtrancl_path r x xs y` have cs: "rtrancl_path r x cs y" 
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by (auto elim: rtrancl_path_appendE) 
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from xs have "length cs < length xs" by simp 
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then show ?thesis 
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by (rule less(1)) (iprover intro: cs less(2))+ 
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next 
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case (Cons d ds) 
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with xxs have xs: "xs = ds @ a # (bs @ [a] @ cs)" 
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by auto 
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with `rtrancl_path r x xs y` obtain xa: "rtrancl_path r x (ds @ [a]) a" 
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and ay: "rtrancl_path r a (bs @ a # cs) y" 
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by (auto elim: rtrancl_path_appendE) 
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from ay have "rtrancl_path r a cs y" by (auto elim: rtrancl_path_appendE) 
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with xa have xy: "rtrancl_path r x ((ds @ [a]) @ cs) y" 
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by (rule rtrancl_path_trans) 
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from xs have "length ((ds @ [a]) @ cs) < length xs" by simp 
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then show ?thesis 
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by (rule less(1)) (iprover intro: xy less(2))+ 
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qed 
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qed 
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qed 
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inductive rtrancl_tab :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
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for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" 
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where 
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base: "rtrancl_tab r xs x x" 
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 step: "x \<notin> set xs \<Longrightarrow> r x y \<Longrightarrow> rtrancl_tab r (x # xs) y z \<Longrightarrow> rtrancl_tab r xs x z" 
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lemma rtrancl_path_imp_rtrancl_tab: 
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assumes path: "rtrancl_path r x xs y" 
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and x: "distinct (x # xs)" 
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and ys: "({x} \<union> set xs) \<inter> set ys = {}" 
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143 
shows "rtrancl_tab r ys x y" using path x ys 
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proof (induct arbitrary: ys) 
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case base 
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show ?case by (rule rtrancl_tab.base) 
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next 
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case (step x y zs z) 
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then have "x \<notin> set ys" by auto 
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from step have "distinct (y # zs)" by simp 
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moreover from step have "({y} \<union> set zs) \<inter> set (x # ys) = {}" 
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by auto 
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ultimately have "rtrancl_tab r (x # ys) y z" 
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by (rule step) 
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with `x \<notin> set ys` `r x y` 
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show ?case by (rule rtrancl_tab.step) 
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qed 
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158 

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lemma rtrancl_tab_imp_rtrancl_path: 
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assumes tab: "rtrancl_tab r ys x y" 
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obtains xs where "rtrancl_path r x xs y" using tab 
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proof induct 
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case base 
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from rtrancl_path.base show ?case by (rule base) 
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next 
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case step show ?case by (iprover intro: step rtrancl_path.step) 
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qed 
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168 

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lemma rtranclp_eq_rtrancl_tab_nil: "r\<^sup>*\<^sup>* x y = rtrancl_tab r [] x y" 
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proof 
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assume "r\<^sup>*\<^sup>* x y" 
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then obtain xs where "rtrancl_path r x xs y" 
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by (auto simp add: rtranclp_eq_rtrancl_path) 
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then obtain xs' where xs': "rtrancl_path r x xs' y" 
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and distinct: "distinct (x # xs')" 
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by (rule rtrancl_path_distinct) 
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have "({x} \<union> set xs') \<inter> set [] = {}" by simp 
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with xs' distinct show "rtrancl_tab r [] x y" 
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by (rule rtrancl_path_imp_rtrancl_tab) 
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next 
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assume "rtrancl_tab r [] x y" 
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then obtain xs where "rtrancl_path r x xs y" 
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by (rule rtrancl_tab_imp_rtrancl_path) 
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then show "r\<^sup>*\<^sup>* x y" 
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by (auto simp add: rtranclp_eq_rtrancl_path) 
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qed 
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declare rtranclp_eq_rtrancl_tab_nil [code_unfold] 
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declare rtranclp_eq_rtrancl_tab_nil[THEN iffD2, code_pred_intro] 
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code_pred rtranclp using rtranclp_eq_rtrancl_tab_nil[THEN iffD1] by fastsimp 
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subsection {* A simple example *} 
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datatype ty = A  B  C 
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inductive test :: "ty \<Rightarrow> ty \<Rightarrow> bool" 
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where 
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"test A B" 
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 "test B A" 
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 "test B C" 
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subsubsection {* Invoking with the SML code generator *} 
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code_module Test 
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contains 
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test1 = "test\<^sup>*\<^sup>* A C" 
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test2 = "test\<^sup>*\<^sup>* C A" 
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test3 = "test\<^sup>*\<^sup>* A _" 
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test4 = "test\<^sup>*\<^sup>* _ C" 
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ML "Test.test1" 
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ML "Test.test2" 
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ML "DSeq.list_of Test.test3" 
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ML "DSeq.list_of Test.test4" 
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subsubsection {* Invoking with the predicate compiler and the generic code generator *} 
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code_pred test . 
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values "{x. test\<^sup>*\<^sup>* A C}" 
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values "{x. test\<^sup>*\<^sup>* C A}" 
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values "{x. test\<^sup>*\<^sup>* A x}" 
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values "{x. test\<^sup>*\<^sup>* x C}" 
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hide const test 
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end 
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230 