(* Author: Stefan Berghofer, Lukas Bulwahn, TU Muenchen *) header {* A table-based implementation of the reflexive transitive closure *} theory Transitive_Closure_Table imports Main begin inductive rtrancl_path :: "('a => 'a => bool) => 'a => 'a list => 'a => bool" for r :: "'a => 'a => bool" where base: "rtrancl_path r x [] x" | step: "r x y ==> rtrancl_path r y ys z ==> rtrancl_path r x (y # ys) z" lemma rtranclp_eq_rtrancl_path: "r⇧^{*}⇧^{*}x y = (∃xs. rtrancl_path r x xs y)" proof assume "r⇧^{*}⇧^{*}x y" then show "∃xs. rtrancl_path r x xs y" proof (induct rule: converse_rtranclp_induct) case base have "rtrancl_path r y [] y" by (rule rtrancl_path.base) then show ?case .. next case (step x z) from `∃xs. rtrancl_path r z xs y` obtain xs where "rtrancl_path r z xs y" .. with `r x z` have "rtrancl_path r x (z # xs) y" by (rule rtrancl_path.step) then show ?case .. qed next assume "∃xs. rtrancl_path r x xs y" then obtain xs where "rtrancl_path r x xs y" .. then show "r⇧^{*}⇧^{*}x y" proof induct case (base x) show ?case by (rule rtranclp.rtrancl_refl) next case (step x y ys z) from `r x y` `r⇧^{*}⇧^{*}y z` show ?case by (rule converse_rtranclp_into_rtranclp) qed qed lemma rtrancl_path_trans: assumes xy: "rtrancl_path r x xs y" and yz: "rtrancl_path r y ys z" shows "rtrancl_path r x (xs @ ys) z" using xy yz proof (induct arbitrary: z) case (base x) then show ?case by simp next case (step x y xs) then have "rtrancl_path r y (xs @ ys) z" by simp with `r x y` have "rtrancl_path r x (y # (xs @ ys)) z" by (rule rtrancl_path.step) then show ?case by simp qed lemma rtrancl_path_appendE: assumes xz: "rtrancl_path r x (xs @ y # ys) z" obtains "rtrancl_path r x (xs @ [y]) y" and "rtrancl_path r y ys z" using xz proof (induct xs arbitrary: x) case Nil then have "rtrancl_path r x (y # ys) z" by simp then obtain xy: "r x y" and yz: "rtrancl_path r y ys z" by cases auto from xy have "rtrancl_path r x [y] y" by (rule rtrancl_path.step [OF _ rtrancl_path.base]) then have "rtrancl_path r x ([] @ [y]) y" by simp then show ?thesis using yz by (rule Nil) next case (Cons a as) then have "rtrancl_path r x (a # (as @ y # ys)) z" by simp then obtain xa: "r x a" and az: "rtrancl_path r a (as @ y # ys) z" by cases auto show ?thesis proof (rule Cons(1) [OF _ az]) assume "rtrancl_path r y ys z" assume "rtrancl_path r a (as @ [y]) y" with xa have "rtrancl_path r x (a # (as @ [y])) y" by (rule rtrancl_path.step) then have "rtrancl_path r x ((a # as) @ [y]) y" by simp then show ?thesis using `rtrancl_path r y ys z` by (rule Cons(2)) qed qed lemma rtrancl_path_distinct: assumes xy: "rtrancl_path r x xs y" obtains xs' where "rtrancl_path r x xs' y" and "distinct (x # xs')" using xy proof (induct xs rule: measure_induct_rule [of length]) case (less xs) show ?case proof (cases "distinct (x # xs)") case True with `rtrancl_path r x xs y` show ?thesis by (rule less) next case False then have "∃as bs cs a. x # xs = as @ [a] @ bs @ [a] @ cs" by (rule not_distinct_decomp) then obtain as bs cs a where xxs: "x # xs = as @ [a] @ bs @ [a] @ cs" by iprover show ?thesis proof (cases as) case Nil with xxs have x: "x = a" and xs: "xs = bs @ a # cs" by auto from x xs `rtrancl_path r x xs y` have cs: "rtrancl_path r x cs y" by (auto elim: rtrancl_path_appendE) from xs have "length cs < length xs" by simp then show ?thesis by (rule less(1)) (iprover intro: cs less(2))+ next case (Cons d ds) with xxs have xs: "xs = ds @ a # (bs @ [a] @ cs)" by auto with `rtrancl_path r x xs y` obtain xa: "rtrancl_path r x (ds @ [a]) a" and ay: "rtrancl_path r a (bs @ a # cs) y" by (auto elim: rtrancl_path_appendE) from ay have "rtrancl_path r a cs y" by (auto elim: rtrancl_path_appendE) with xa have xy: "rtrancl_path r x ((ds @ [a]) @ cs) y" by (rule rtrancl_path_trans) from xs have "length ((ds @ [a]) @ cs) < length xs" by simp then show ?thesis by (rule less(1)) (iprover intro: xy less(2))+ qed qed qed inductive rtrancl_tab :: "('a => 'a => bool) => 'a list => 'a => 'a => bool" for r :: "'a => 'a => bool" where base: "rtrancl_tab r xs x x" | step: "x ∉ set xs ==> r x y ==> rtrancl_tab r (x # xs) y z ==> rtrancl_tab r xs x z" lemma rtrancl_path_imp_rtrancl_tab: assumes path: "rtrancl_path r x xs y" and x: "distinct (x # xs)" and ys: "({x} ∪ set xs) ∩ set ys = {}" shows "rtrancl_tab r ys x y" using path x ys proof (induct arbitrary: ys) case base show ?case by (rule rtrancl_tab.base) next case (step x y zs z) then have "x ∉ set ys" by auto from step have "distinct (y # zs)" by simp moreover from step have "({y} ∪ set zs) ∩ set (x # ys) = {}" by auto ultimately have "rtrancl_tab r (x # ys) y z" by (rule step) with `x ∉ set ys` `r x y` show ?case by (rule rtrancl_tab.step) qed lemma rtrancl_tab_imp_rtrancl_path: assumes tab: "rtrancl_tab r ys x y" obtains xs where "rtrancl_path r x xs y" using tab proof induct case base from rtrancl_path.base show ?case by (rule base) next case step show ?case by (iprover intro: step rtrancl_path.step) qed lemma rtranclp_eq_rtrancl_tab_nil: "r⇧^{*}⇧^{*}x y = rtrancl_tab r [] x y" proof assume "r⇧^{*}⇧^{*}x y" then obtain xs where "rtrancl_path r x xs y" by (auto simp add: rtranclp_eq_rtrancl_path) then obtain xs' where xs': "rtrancl_path r x xs' y" and distinct: "distinct (x # xs')" by (rule rtrancl_path_distinct) have "({x} ∪ set xs') ∩ set [] = {}" by simp with xs' distinct show "rtrancl_tab r [] x y" by (rule rtrancl_path_imp_rtrancl_tab) next assume "rtrancl_tab r [] x y" then obtain xs where "rtrancl_path r x xs y" by (rule rtrancl_tab_imp_rtrancl_path) then show "r⇧^{*}⇧^{*}x y" by (auto simp add: rtranclp_eq_rtrancl_path) qed declare rtranclp_rtrancl_eq[code del] declare rtranclp_eq_rtrancl_tab_nil[THEN iffD2, code_pred_intro] code_pred rtranclp using rtranclp_eq_rtrancl_tab_nil [THEN iffD1] by fastforce end