(* Author: Stefan Berghofer, Lukas Bulwahn, TU Muenchen *) section ‹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 show "∃xs. rtrancl_path r x xs y" if "r⇧^{*}⇧^{*}x y" using that 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 show "r⇧^{*}⇧^{*}x y" if "∃xs. rtrancl_path r x xs y" proof - from that obtain xs where "rtrancl_path r x xs y" .. then show ?thesis 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 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')" and "set xs' ⊆ set 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) simp 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" "set cs ⊆ set xs" by (auto elim: rtrancl_path_appendE) from xs have "length cs < length xs" by simp then show ?thesis by (rule less(1))(blast intro: cs less(2) order_trans del: subsetI)+ 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 set: "set ((ds @ [a]) @ cs) ⊆ set xs" by auto from xs have "length ((ds @ [a]) @ cs) < length xs" by simp then show ?thesis by (rule less(1))(blast intro: xy less(2) set[THEN subsetD])+ 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 show "rtrancl_tab r [] x y" if "r⇧^{*}⇧^{*}x y" proof - from that 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 ?thesis by (rule rtrancl_path_imp_rtrancl_tab) qed show "r⇧^{*}⇧^{*}x y" if "rtrancl_tab r [] x y" proof - from that obtain xs where "rtrancl_path r x xs y" by (rule rtrancl_tab_imp_rtrancl_path) then show ?thesis by (auto simp add: rtranclp_eq_rtrancl_path) qed 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 lemma rtrancl_path_Range: "⟦ rtrancl_path R x xs y; z ∈ set xs ⟧ ⟹ Rangep R z" by(induction rule: rtrancl_path.induct) auto lemma rtrancl_path_Range_end: "⟦ rtrancl_path R x xs y; xs ≠ [] ⟧ ⟹ Rangep R y" by(induction rule: rtrancl_path.induct)(auto elim: rtrancl_path.cases) lemma rtrancl_path_nth: "⟦ rtrancl_path R x xs y; i < length xs ⟧ ⟹ R ((x # xs) ! i) (xs ! i)" proof(induction arbitrary: i rule: rtrancl_path.induct) case step thus ?case by(cases i) simp_all qed simp lemma rtrancl_path_last: "⟦ rtrancl_path R x xs y; xs ≠ [] ⟧ ⟹ last xs = y" by(induction rule: rtrancl_path.induct)(auto elim: rtrancl_path.cases) lemma rtrancl_path_mono: "⟦ rtrancl_path R x p y; ⋀x y. R x y ⟹ S x y ⟧ ⟹ rtrancl_path S x p y" by(induction rule: rtrancl_path.induct)(auto intro: rtrancl_path.intros) end