(* Title: HOL/Statespace/DistinctTreeProver.thy Author: Norbert Schirmer, TU Muenchen *) section ‹Distinctness of Names in a Binary Tree \label{sec:DistinctTreeProver}› theory DistinctTreeProver imports Main begin text ‹A state space manages a set of (abstract) names and assumes that the names are distinct. The names are stored as parameters of a locale and distinctness as an assumption. The most common request is to proof distinctness of two given names. We maintain the names in a balanced binary tree and formulate a predicate that all nodes in the tree have distinct names. This setup leads to logarithmic certificates. › subsection ‹The Binary Tree› datatype 'a tree = Node "'a tree" 'a bool "'a tree" | Tip text ‹The boolean flag in the node marks the content of the node as deleted, without having to build a new tree. We prefer the boolean flag to an option type, so that the ML-layer can still use the node content to facilitate binary search in the tree. The ML code keeps the nodes sorted using the term order. We do not have to push ordering to the HOL level.› subsection ‹Distinctness of Nodes› primrec set_of :: "'a tree ⇒ 'a set" where "set_of Tip = {}" | "set_of (Node l x d r) = (if d then {} else {x}) ∪ set_of l ∪ set_of r" primrec all_distinct :: "'a tree ⇒ bool" where "all_distinct Tip = True" | "all_distinct (Node l x d r) = ((d ∨ (x ∉ set_of l ∧ x ∉ set_of r)) ∧ set_of l ∩ set_of r = {} ∧ all_distinct l ∧ all_distinct r)" text ‹Given a binary tree @{term "t"} for which @{const all_distinct} holds, given two different nodes contained in the tree, we want to write a ML function that generates a logarithmic certificate that the content of the nodes is distinct. We use the following lemmas to achieve this.› lemma all_distinct_left: "all_distinct (Node l x b r) ⟹ all_distinct l" by simp lemma all_distinct_right: "all_distinct (Node l x b r) ⟹ all_distinct r" by simp lemma distinct_left: "all_distinct (Node l x False r) ⟹ y ∈ set_of l ⟹ x ≠ y" by auto lemma distinct_right: "all_distinct (Node l x False r) ⟹ y ∈ set_of r ⟹ x ≠ y" by auto lemma distinct_left_right: "all_distinct (Node l z b r) ⟹ x ∈ set_of l ⟹ y ∈ set_of r ⟹ x ≠ y" by auto lemma in_set_root: "x ∈ set_of (Node l x False r)" by simp lemma in_set_left: "y ∈ set_of l ⟹ y ∈ set_of (Node l x False r)" by simp lemma in_set_right: "y ∈ set_of r ⟹ y ∈ set_of (Node l x False r)" by simp lemma swap_neq: "x ≠ y ⟹ y ≠ x" by blast lemma neq_to_eq_False: "x≠y ⟹ (x=y)≡False" by simp subsection ‹Containment of Trees› text ‹When deriving a state space from other ones, we create a new name tree which contains all the names of the parent state spaces and assume the predicate @{const all_distinct}. We then prove that the new locale interprets all parent locales. Hence we have to show that the new distinctness assumption on all names implies the distinctness assumptions of the parent locales. This proof is implemented in ML. We do this efficiently by defining a kind of containment check of trees by ``subtraction''. We subtract the parent tree from the new tree. If this succeeds we know that @{const all_distinct} of the new tree implies @{const all_distinct} of the parent tree. The resulting certificate is of the order @{term "n * log(m)"} where @{term "n"} is the size of the (smaller) parent tree and @{term "m"} the size of the (bigger) new tree.› primrec delete :: "'a ⇒ 'a tree ⇒ 'a tree option" where "delete x Tip = None" | "delete x (Node l y d r) = (case delete x l of Some l' ⇒ (case delete x r of Some r' ⇒ Some (Node l' y (d ∨ (x=y)) r') | None ⇒ Some (Node l' y (d ∨ (x=y)) r)) | None ⇒ (case delete x r of Some r' ⇒ Some (Node l y (d ∨ (x=y)) r') | None ⇒ if x=y ∧ ¬d then Some (Node l y True r) else None))" lemma delete_Some_set_of: "delete x t = Some t' ⟹ set_of t' ⊆ set_of t" proof (induct t arbitrary: t') case Tip thus ?case by simp next case (Node l y d r) have del: "delete x (Node l y d r) = Some t'" by fact show ?case proof (cases "delete x l") case (Some l') note x_l_Some = this with Node.hyps have l'_l: "set_of l' ⊆ set_of l" by simp show ?thesis proof (cases "delete x r") case (Some r') with Node.hyps have "set_of r' ⊆ set_of r" by simp with l'_l Some x_l_Some del show ?thesis by (auto split: if_split_asm) next case None with l'_l Some x_l_Some del show ?thesis by (fastforce split: if_split_asm) qed next case None note x_l_None = this show ?thesis proof (cases "delete x r") case (Some r') with Node.hyps have "set_of r' ⊆ set_of r" by simp with Some x_l_None del show ?thesis by (fastforce split: if_split_asm) next case None with x_l_None del show ?thesis by (fastforce split: if_split_asm) qed qed qed lemma delete_Some_all_distinct: "delete x t = Some t' ⟹ all_distinct t ⟹ all_distinct t'" proof (induct t arbitrary: t') case Tip thus ?case by simp next case (Node l y d r) have del: "delete x (Node l y d r) = Some t'" by fact have "all_distinct (Node l y d r)" by fact then obtain dist_l: "all_distinct l" and dist_r: "all_distinct r" and d: "d ∨ (y ∉ set_of l ∧ y ∉ set_of r)" and dist_l_r: "set_of l ∩ set_of r = {}" by auto show ?case proof (cases "delete x l") case (Some l') note x_l_Some = this from Node.hyps (1) [OF Some dist_l] have dist_l': "all_distinct l'" by simp from delete_Some_set_of [OF x_l_Some] have l'_l: "set_of l' ⊆ set_of l". show ?thesis proof (cases "delete x r") case (Some r') from Node.hyps (2) [OF Some dist_r] have dist_r': "all_distinct r'" by simp from delete_Some_set_of [OF Some] have "set_of r' ⊆ set_of r". with dist_l' dist_r' l'_l Some x_l_Some del d dist_l_r show ?thesis by fastforce next case None with l'_l dist_l' x_l_Some del d dist_l_r dist_r show ?thesis by fastforce qed next case None note x_l_None = this show ?thesis proof (cases "delete x r") case (Some r') with Node.hyps (2) [OF Some dist_r] have dist_r': "all_distinct r'" by simp from delete_Some_set_of [OF Some] have "set_of r' ⊆ set_of r". with Some dist_r' x_l_None del dist_l d dist_l_r show ?thesis by fastforce next case None with x_l_None del dist_l dist_r d dist_l_r show ?thesis by (fastforce split: if_split_asm) qed qed qed lemma delete_None_set_of_conv: "delete x t = None = (x ∉ set_of t)" proof (induct t) case Tip thus ?case by simp next case (Node l y d r) thus ?case by (auto split: option.splits) qed lemma delete_Some_x_set_of: "delete x t = Some t' ⟹ x ∈ set_of t ∧ x ∉ set_of t'" proof (induct t arbitrary: t') case Tip thus ?case by simp next case (Node l y d r) have del: "delete x (Node l y d r) = Some t'" by fact show ?case proof (cases "delete x l") case (Some l') note x_l_Some = this from Node.hyps (1) [OF Some] obtain x_l: "x ∈ set_of l" "x ∉ set_of l'" by simp show ?thesis proof (cases "delete x r") case (Some r') from Node.hyps (2) [OF Some] obtain x_r: "x ∈ set_of r" "x ∉ set_of r'" by simp from x_r x_l Some x_l_Some del show ?thesis by (clarsimp split: if_split_asm) next case None then have "x ∉ set_of r" by (simp add: delete_None_set_of_conv) with x_l None x_l_Some del show ?thesis by (clarsimp split: if_split_asm) qed next case None note x_l_None = this then have x_notin_l: "x ∉ set_of l" by (simp add: delete_None_set_of_conv) show ?thesis proof (cases "delete x r") case (Some r') from Node.hyps (2) [OF Some] obtain x_r: "x ∈ set_of r" "x ∉ set_of r'" by simp from x_r x_notin_l Some x_l_None del show ?thesis by (clarsimp split: if_split_asm) next case None then have "x ∉ set_of r" by (simp add: delete_None_set_of_conv) with None x_l_None x_notin_l del show ?thesis by (clarsimp split: if_split_asm) qed qed qed primrec subtract :: "'a tree ⇒ 'a tree ⇒ 'a tree option" where "subtract Tip t = Some t" | "subtract (Node l x b r) t = (case delete x t of Some t' ⇒ (case subtract l t' of Some t'' ⇒ subtract r t'' | None ⇒ None) | None ⇒ None)" lemma subtract_Some_set_of_res: "subtract t⇩_{1}t⇩_{2}= Some t ⟹ set_of t ⊆ set_of t⇩_{2}" proof (induct t⇩_{1}arbitrary: t⇩_{2}t) case Tip thus ?case by simp next case (Node l x b r) have sub: "subtract (Node l x b r) t⇩_{2}= Some t" by fact show ?case proof (cases "delete x t⇩_{2}") case (Some t⇩_{2}') note del_x_Some = this from delete_Some_set_of [OF Some] have t2'_t2: "set_of t⇩_{2}' ⊆ set_of t⇩_{2}" . show ?thesis proof (cases "subtract l t⇩_{2}'") case (Some t⇩_{2}'') note sub_l_Some = this from Node.hyps (1) [OF Some] have t2''_t2': "set_of t⇩_{2}'' ⊆ set_of t⇩_{2}'" . show ?thesis proof (cases "subtract r t⇩_{2}''") case (Some t⇩_{2}''') from Node.hyps (2) [OF Some ] have "set_of t⇩_{2}''' ⊆ set_of t⇩_{2}''" . with Some sub_l_Some del_x_Some sub t2''_t2' t2'_t2 show ?thesis by simp next case None with del_x_Some sub_l_Some sub show ?thesis by simp qed next case None with del_x_Some sub show ?thesis by simp qed next case None with sub show ?thesis by simp qed qed lemma subtract_Some_set_of: "subtract t⇩_{1}t⇩_{2}= Some t ⟹ set_of t⇩_{1}⊆ set_of t⇩_{2}" proof (induct t⇩_{1}arbitrary: t⇩_{2}t) case Tip thus ?case by simp next case (Node l x d r) have sub: "subtract (Node l x d r) t⇩_{2}= Some t" by fact show ?case proof (cases "delete x t⇩_{2}") case (Some t⇩_{2}') note del_x_Some = this from delete_Some_set_of [OF Some] have t2'_t2: "set_of t⇩_{2}' ⊆ set_of t⇩_{2}" . from delete_None_set_of_conv [of x t⇩_{2}] Some have x_t2: "x ∈ set_of t⇩_{2}" by simp show ?thesis proof (cases "subtract l t⇩_{2}'") case (Some t⇩_{2}'') note sub_l_Some = this from Node.hyps (1) [OF Some] have l_t2': "set_of l ⊆ set_of t⇩_{2}'" . from subtract_Some_set_of_res [OF Some] have t2''_t2': "set_of t⇩_{2}'' ⊆ set_of t⇩_{2}'" . show ?thesis proof (cases "subtract r t⇩_{2}''") case (Some t⇩_{2}''') from Node.hyps (2) [OF Some ] have r_t⇩_{2}'': "set_of r ⊆ set_of t⇩_{2}''" . from Some sub_l_Some del_x_Some sub r_t⇩_{2}'' l_t2' t2'_t2 t2''_t2' x_t2 show ?thesis by auto next case None with del_x_Some sub_l_Some sub show ?thesis by simp qed next case None with del_x_Some sub show ?thesis by simp qed next case None with sub show ?thesis by simp qed qed lemma subtract_Some_all_distinct_res: "subtract t⇩_{1}t⇩_{2}= Some t ⟹ all_distinct t⇩_{2}⟹ all_distinct t" proof (induct t⇩_{1}arbitrary: t⇩_{2}t) case Tip thus ?case by simp next case (Node l x d r) have sub: "subtract (Node l x d r) t⇩_{2}= Some t" by fact have dist_t2: "all_distinct t⇩_{2}" by fact show ?case proof (cases "delete x t⇩_{2}") case (Some t⇩_{2}') note del_x_Some = this from delete_Some_all_distinct [OF Some dist_t2] have dist_t2': "all_distinct t⇩_{2}'" . show ?thesis proof (cases "subtract l t⇩_{2}'") case (Some t⇩_{2}'') note sub_l_Some = this from Node.hyps (1) [OF Some dist_t2'] have dist_t2'': "all_distinct t⇩_{2}''" . show ?thesis proof (cases "subtract r t⇩_{2}''") case (Some t⇩_{2}''') from Node.hyps (2) [OF Some dist_t2''] have dist_t2''': "all_distinct t⇩_{2}'''" . from Some sub_l_Some del_x_Some sub dist_t2''' show ?thesis by simp next case None with del_x_Some sub_l_Some sub show ?thesis by simp qed next case None with del_x_Some sub show ?thesis by simp qed next case None with sub show ?thesis by simp qed qed lemma subtract_Some_dist_res: "subtract t⇩_{1}t⇩_{2}= Some t ⟹ set_of t⇩_{1}∩ set_of t = {}" proof (induct t⇩_{1}arbitrary: t⇩_{2}t) case Tip thus ?case by simp next case (Node l x d r) have sub: "subtract (Node l x d r) t⇩_{2}= Some t" by fact show ?case proof (cases "delete x t⇩_{2}") case (Some t⇩_{2}') note del_x_Some = this from delete_Some_x_set_of [OF Some] obtain x_t2: "x ∈ set_of t⇩_{2}" and x_not_t2': "x ∉ set_of t⇩_{2}'" by simp from delete_Some_set_of [OF Some] have t2'_t2: "set_of t⇩_{2}' ⊆ set_of t⇩_{2}" . show ?thesis proof (cases "subtract l t⇩_{2}'") case (Some t⇩_{2}'') note sub_l_Some = this from Node.hyps (1) [OF Some ] have dist_l_t2'': "set_of l ∩ set_of t⇩_{2}'' = {}". from subtract_Some_set_of_res [OF Some] have t2''_t2': "set_of t⇩_{2}'' ⊆ set_of t⇩_{2}'" . show ?thesis proof (cases "subtract r t⇩_{2}''") case (Some t⇩_{2}''') from Node.hyps (2) [OF Some] have dist_r_t2''': "set_of r ∩ set_of t⇩_{2}''' = {}" . from subtract_Some_set_of_res [OF Some] have t2'''_t2'': "set_of t⇩_{2}''' ⊆ set_of t⇩_{2}''". from Some sub_l_Some del_x_Some sub t2'''_t2'' dist_l_t2'' dist_r_t2''' t2''_t2' t2'_t2 x_not_t2' show ?thesis by auto next case None with del_x_Some sub_l_Some sub show ?thesis by simp qed next case None with del_x_Some sub show ?thesis by simp qed next case None with sub show ?thesis by simp qed qed lemma subtract_Some_all_distinct: "subtract t⇩_{1}t⇩_{2}= Some t ⟹ all_distinct t⇩_{2}⟹ all_distinct t⇩_{1}" proof (induct t⇩_{1}arbitrary: t⇩_{2}t) case Tip thus ?case by simp next case (Node l x d r) have sub: "subtract (Node l x d r) t⇩_{2}= Some t" by fact have dist_t2: "all_distinct t⇩_{2}" by fact show ?case proof (cases "delete x t⇩_{2}") case (Some t⇩_{2}') note del_x_Some = this from delete_Some_all_distinct [OF Some dist_t2 ] have dist_t2': "all_distinct t⇩_{2}'" . from delete_Some_set_of [OF Some] have t2'_t2: "set_of t⇩_{2}' ⊆ set_of t⇩_{2}" . from delete_Some_x_set_of [OF Some] obtain x_t2: "x ∈ set_of t⇩_{2}" and x_not_t2': "x ∉ set_of t⇩_{2}'" by simp show ?thesis proof (cases "subtract l t⇩_{2}'") case (Some t⇩_{2}'') note sub_l_Some = this from Node.hyps (1) [OF Some dist_t2' ] have dist_l: "all_distinct l" . from subtract_Some_all_distinct_res [OF Some dist_t2'] have dist_t2'': "all_distinct t⇩_{2}''" . from subtract_Some_set_of [OF Some] have l_t2': "set_of l ⊆ set_of t⇩_{2}'" . from subtract_Some_set_of_res [OF Some] have t2''_t2': "set_of t⇩_{2}'' ⊆ set_of t⇩_{2}'" . from subtract_Some_dist_res [OF Some] have dist_l_t2'': "set_of l ∩ set_of t⇩_{2}'' = {}". show ?thesis proof (cases "subtract r t⇩_{2}''") case (Some t⇩_{2}''') from Node.hyps (2) [OF Some dist_t2''] have dist_r: "all_distinct r" . from subtract_Some_set_of [OF Some] have r_t2'': "set_of r ⊆ set_of t⇩_{2}''" . from subtract_Some_dist_res [OF Some] have dist_r_t2''': "set_of r ∩ set_of t⇩_{2}''' = {}". from dist_l dist_r Some sub_l_Some del_x_Some r_t2'' l_t2' x_t2 x_not_t2' t2''_t2' dist_l_t2'' dist_r_t2''' show ?thesis by auto next case None with del_x_Some sub_l_Some sub show ?thesis by simp qed next case None with del_x_Some sub show ?thesis by simp qed next case None with sub show ?thesis by simp qed qed lemma delete_left: assumes dist: "all_distinct (Node l y d r)" assumes del_l: "delete x l = Some l'" shows "delete x (Node l y d r) = Some (Node l' y d r)" proof - from delete_Some_x_set_of [OF del_l] obtain x: "x ∈ set_of l" by simp with dist have "delete x r = None" by (cases "delete x r") (auto dest:delete_Some_x_set_of) with x show ?thesis using del_l dist by (auto split: option.splits) qed lemma delete_right: assumes dist: "all_distinct (Node l y d r)" assumes del_r: "delete x r = Some r'" shows "delete x (Node l y d r) = Some (Node l y d r')" proof - from delete_Some_x_set_of [OF del_r] obtain x: "x ∈ set_of r" by simp with dist have "delete x l = None" by (cases "delete x l") (auto dest:delete_Some_x_set_of) with x show ?thesis using del_r dist by (auto split: option.splits) qed lemma delete_root: assumes dist: "all_distinct (Node l x False r)" shows "delete x (Node l x False r) = Some (Node l x True r)" proof - from dist have "delete x r = None" by (cases "delete x r") (auto dest:delete_Some_x_set_of) moreover from dist have "delete x l = None" by (cases "delete x l") (auto dest:delete_Some_x_set_of) ultimately show ?thesis using dist by (auto split: option.splits) qed lemma subtract_Node: assumes del: "delete x t = Some t'" assumes sub_l: "subtract l t' = Some t''" assumes sub_r: "subtract r t'' = Some t'''" shows "subtract (Node l x False r) t = Some t'''" using del sub_l sub_r by simp lemma subtract_Tip: "subtract Tip t = Some t" by simp text ‹Now we have all the theorems in place that are needed for the certificate generating ML functions.› ML_file "distinct_tree_prover.ML" end