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| (* Title: BDD | |
| Author: Veronika Ortner and Norbert Schirmer, 2004 | |
| Maintainer: Norbert Schirmer, norbert.schirmer at web de | |
| License: LGPL | |
| *) | |
| (* | |
| ShareReduceRepListProof.thy | |
| Copyright (C) 2004 Veronika Ortner and Norbert Schirmer | |
| Some rights reserved, TU Muenchen | |
| This library is free software; you can redistribute it and/or modify | |
| it under the terms of the GNU Lesser General Public License as | |
| published by the Free Software Foundation; either version 2.1 of the | |
| License, or (at your option) any later version. | |
| This library is distributed in the hope that it will be useful, but | |
| WITHOUT ANY WARRANTY; without even the implied warranty of | |
| MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| Lesser General Public License for more details. | |
| You should have received a copy of the GNU Lesser General Public | |
| License along with this library; if not, write to the Free Software | |
| Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 | |
| USA | |
| *) | |
| section \<open>Proof of Procedure ShareReduceRepList\<close> | |
| theory ShareReduceRepListProof imports ShareRepProof begin | |
| lemma (in ShareReduceRepList_impl) ShareReduceRepList_modifies: | |
| shows "\<forall>\<sigma>. \<Gamma>\<turnstile>{\<sigma>} PROC ShareReduceRepList (\<acute>nodeslist) | |
| {t. t may_only_modify_globals \<sigma> in [rep]}" | |
| apply (hoare_rule HoarePartial.ProcRec1) | |
| apply (vcg spec=modifies) | |
| done | |
| lemma hd_filter_app: "\<lbrakk>filter P xs \<noteq> []; zs=xs@ys\<rbrakk> \<Longrightarrow> | |
| hd (filter P zs) = hd (filter P xs)" | |
| by (induct xs arbitrary: n m) auto | |
| lemma (in ShareReduceRepList_impl) ShareReduceRepList_spec_total: | |
| defines "var_eq \<equiv> (\<lambda>ns var. (\<forall>no1 \<in> set ns. \<forall>no2 \<in> set ns. no1\<rightarrow>var = no2\<rightarrow>var))" | |
| shows | |
| "\<forall>\<sigma> ns. \<Gamma>\<turnstile>\<^sub>t | |
| \<lbrace>\<sigma>. List \<acute>nodeslist \<acute>next ns \<and> | |
| (\<forall>no \<in> set ns. | |
| no \<noteq> Null \<and> ((no\<rightarrow>\<acute>low = Null) = (no\<rightarrow>\<acute>high = Null)) \<and> | |
| no\<rightarrow>\<acute>low \<notin> set ns \<and> no\<rightarrow>\<acute>high \<notin> set ns \<and> | |
| (isLeaf_pt no \<acute>low \<acute>high = (no\<rightarrow>\<acute>var \<le> 1)) \<and> | |
| (no\<rightarrow>\<acute>low \<noteq> Null \<longrightarrow> (no\<rightarrow>\<acute>low)\<rightarrow>\<acute>rep \<noteq> Null) \<and> | |
| ((\<acute>rep \<propto> \<acute>low) no \<notin> set ns)) \<and> | |
| var_eq ns \<acute>var\<rbrace> | |
| PROC ShareReduceRepList (\<acute>nodeslist) | |
| \<lbrace>(\<forall>no. no \<notin> set ns \<longrightarrow> no\<rightarrow>\<^bsup>\<sigma>\<^esup>rep = no\<rightarrow>\<acute>rep) \<and> | |
| (\<forall>no \<in> set ns. no\<rightarrow>\<acute>rep \<noteq> Null \<and> | |
| (if ((\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>low) no = (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>high) no \<and> no\<rightarrow> \<^bsup>\<sigma>\<^esup>low \<noteq> Null) | |
| then (no\<rightarrow>\<acute>rep = (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>low) no ) | |
| else ((no\<rightarrow>\<acute>rep) \<in> set ns \<and> no\<rightarrow>\<acute>rep\<rightarrow>\<acute>rep = no\<rightarrow>\<acute>rep \<and> | |
| (\<forall> no1 \<in> set ns. | |
| ((\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>high) no1 = (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>high) no \<and> | |
| (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>low) no1 = (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>low) no) = (no\<rightarrow>\<acute>rep = no1\<rightarrow>\<acute>rep)))))\<rbrace>" | |
| apply (hoare_rule HoareTotal.ProcNoRec1) | |
| apply (hoare_rule anno= | |
| " \<acute>node :== \<acute>nodeslist;; | |
| WHILE (\<acute>node \<noteq> Null ) | |
| INV \<lbrace>\<exists>prx sfx. List \<acute>node \<acute>next sfx \<and> | |
| List \<acute>nodeslist \<acute>next ns \<and> ns=prx@sfx \<and> | |
| (\<forall>no \<in> set ns. | |
| no \<noteq> Null \<and> ((no\<rightarrow>\<^bsup>\<sigma>\<^esup>low = Null) = (no\<rightarrow>\<^bsup>\<sigma>\<^esup>high = Null)) \<and> | |
| no\<rightarrow>\<^bsup>\<sigma>\<^esup>low \<notin> set ns \<and> no\<rightarrow>\<^bsup>\<sigma>\<^esup>high \<notin> set ns \<and> | |
| (isLeaf_pt no \<^bsup>\<sigma>\<^esup>low \<^bsup>\<sigma>\<^esup>high = (no\<rightarrow>\<^bsup>\<sigma>\<^esup>var \<le> 1)) \<and> | |
| (no\<rightarrow>\<^bsup>\<sigma>\<^esup>low \<noteq> Null \<longrightarrow> (no\<rightarrow>\<^bsup>\<sigma>\<^esup>low)\<rightarrow>\<^bsup>\<sigma>\<^esup>rep \<noteq> Null) \<and> | |
| ((\<^bsup>\<sigma>\<^esup>rep \<propto> \<^bsup>\<sigma>\<^esup>low) no \<notin> set ns)) \<and> | |
| var_eq ns \<acute>var \<and> | |
| (\<forall>no. no \<notin> set prx \<longrightarrow> no\<rightarrow>\<^bsup>\<sigma>\<^esup>rep = no \<rightarrow>\<acute>rep) \<and> | |
| (\<forall> no \<in> set prx. no\<rightarrow>\<acute>rep \<noteq> Null \<and> | |
| (if ((\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>low) no = (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>high) no \<and> no\<rightarrow>\<^bsup>\<sigma>\<^esup>low \<noteq> Null) | |
| then (no\<rightarrow>\<acute>rep = (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>low) no ) | |
| else ((no\<rightarrow>\<acute>rep)=hd (filter (\<lambda>sn. repNodes_eq sn no \<^bsup>\<sigma>\<^esup>low \<^bsup>\<sigma>\<^esup>high \<acute>rep) | |
| prx) \<and> | |
| ((no\<rightarrow>\<acute>rep)\<rightarrow>\<acute>rep) = no\<rightarrow>\<acute>rep \<and> | |
| (\<forall>no1 \<in> set prx. | |
| ((\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>high) no1 = (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>high) no \<and> | |
| (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>low) no1 = (\<acute>rep \<propto> \<^bsup>\<sigma>\<^esup>low) no) = | |
| (no\<rightarrow>\<acute>rep = no1\<rightarrow>\<acute>rep))))) \<and> | |
| \<acute>nodeslist= \<^bsup>\<sigma>\<^esup>nodeslist \<and> \<acute>high=\<^bsup>\<sigma>\<^esup>high \<and> \<acute>low=\<^bsup>\<sigma>\<^esup>low \<and> \<acute>var=\<^bsup>\<sigma>\<^esup>var\<rbrace> | |
| VAR MEASURE (length (list \<acute>node \<acute>next)) | |
| DO | |
| IF (\<not> isLeaf_pt \<acute>node \<acute>low \<acute>high \<and> | |
| \<acute>node \<rightarrow> \<acute>low \<rightarrow> \<acute>rep = \<acute>node \<rightarrow> \<acute>high \<rightarrow> \<acute>rep ) | |
| THEN \<acute>node \<rightarrow> \<acute>rep :== \<acute>node \<rightarrow> \<acute>low \<rightarrow> \<acute>rep | |
| ELSE CALL ShareRep (\<acute>nodeslist , \<acute>node) | |
| FI;; | |
| \<acute>node :==\<acute>node \<rightarrow> \<acute>next | |
| OD" in HoareTotal.annotateI) | |
| apply (vcg spec=spec_total) | |
| apply (rule_tac x="[]" in exI) | |
| apply (rule_tac x="ns" in exI) | |
| using [[simp_depth_limit = 2]] | |
| apply (simp (no_asm_use)) | |
| prefer 2 | |
| using [[simp_depth_limit = 4]] | |
| apply (clarsimp) | |
| prefer 2 | |
| apply (rule conjI) | |
| apply clarify | |
| apply (rule conjI) | |
| apply (clarsimp simp add: List_list) (* termination *) | |
| apply (simp only: List_not_Null simp_thms triv_forall_equality) | |
| apply clarify | |
| apply (simp only: triv_forall_equality) | |
| apply (rename_tac sfx) | |
| apply (rule_tac x="prx@[node]" in exI) | |
| apply (rule_tac x="sfx" in exI) | |
| apply (rule conjI) | |
| apply assumption | |
| apply (rule conjI) | |
| apply (simp (no_asm)) | |
| apply (rule conjI) | |
| apply (assumption) | |
| prefer 2 | |
| apply clarify | |
| apply (simp only: List_not_Null simp_thms triv_forall_equality) | |
| apply clarify | |
| apply (simp only: triv_forall_equality) | |
| apply (rename_tac sfx) | |
| apply (rule_tac x="prx@node#sfx" in exI) (* Precondition for ShareRep *) | |
| apply (rule conjI) | |
| apply assumption | |
| apply (rule conjI) | |
| apply (rule ballI) | |
| apply (frule_tac x=no in bspec, assumption) | |
| apply (drule_tac x=node in bspec) | |
| apply (simp (no_asm_use)) | |
| apply (elim conjE) | |
| apply (rule conjI) | |
| apply assumption | |
| apply (rule conjI) | |
| apply assumption | |
| apply (unfold var_eq_def) | |
| apply (drule_tac x=node in bspec, simp) | |
| apply (drule_tac x=no in bspec,assumption) | |
| apply (simp add: isLeaf_pt_def ) | |
| apply (rule conjI) | |
| apply (simp (no_asm)) | |
| apply (clarify) | |
| apply (rule conjI) | |
| apply (subgoal_tac "List node next (node#sfx)") (* termination *) | |
| apply (simp only: List_list) | |
| apply (simp (no_asm)) | |
| apply (simp (no_asm_simp)) | |
| apply (rule_tac x="prx@[node]" in exI) | |
| apply (rule_tac x="sfx" in exI) | |
| apply (rule conjI) | |
| apply assumption | |
| apply (rule conjI) | |
| apply (simp (no_asm)) | |
| apply (rule conjI) | |
| apply (assumption) | |
| using [[simp_depth_limit = 100]] | |
| proof - (* From invariant to postcondition *) | |
| fix var low high rep nodeslist ns repa "next" no | |
| assume ns: "List nodeslist next ns" | |
| assume no_in_ns: "no \<in> set ns" | |
| assume while_inv: "\<forall>no\<in>set ns. | |
| repa no \<noteq> Null \<and> | |
| (if (repa \<propto> low) no = (repa \<propto> high) no \<and> high no \<noteq> Null | |
| then repa no = (repa \<propto> low) no | |
| else repa no = hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa] \<and> | |
| repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set ns. | |
| ((repa \<propto> high) no1 = (repa \<propto> high) no \<and> | |
| (repa \<propto> low) no1 = (repa \<propto> low) no) = | |
| (repa no = repa no1)))" | |
| assume pre: "\<forall>no\<in>set ns. | |
| no \<noteq> Null \<and> | |
| (low no = Null) = (high no = Null) \<and> | |
| low no \<notin> set ns \<and> | |
| high no \<notin> set ns \<and> | |
| isLeaf_pt no low high = (var no \<le> Suc 0) \<and> | |
| (low no \<noteq> Null \<longrightarrow> rep (low no) \<noteq> Null) \<and> (rep \<propto> low) no \<notin> set ns" | |
| assume same_var: "\<forall>no1\<in>set ns. \<forall>no2\<in>set ns. var no1 = var no2" | |
| assume share_case: "(repa \<propto> low) no = (repa \<propto> high) no \<longrightarrow> high no = Null" | |
| assume unmodif: "\<forall>no. no \<notin> set ns \<longrightarrow> rep no = repa no" | |
| show "hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa] \<in> set ns \<and> | |
| repa (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) = | |
| hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]" | |
| proof - | |
| from no_in_ns pre obtain | |
| no_nNull: " no \<noteq> Null" and | |
| no_balanced: "(low no = Null) = (high no = Null)" and | |
| isLeaf_var: "isLeaf_pt no low high = (var no \<le> Suc 0)" | |
| by blast | |
| have repNodes_eq_same_node: "repNodes_eq no no low high repa" | |
| by (simp add: repNodes_eq_def) | |
| from no_in_ns have ns_nempty: "ns \<noteq> []" | |
| by auto | |
| from no_in_ns repNodes_eq_same_node | |
| have repNodes_not_empty: "[sn\<leftarrow>ns . repNodes_eq sn no low high repa] \<noteq> []" | |
| by (rule filter_not_empty) | |
| then have hd_term_in_ns: "hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa] \<in> set ns" | |
| by (rule hd_filter_in_list) | |
| with while_inv obtain | |
| repa_hd_nNull: "repa (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) \<noteq> Null" | |
| by auto | |
| let ?hd = "hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]" | |
| from hd_term_in_ns pre obtain | |
| hd_nNull: " ?hd \<noteq> Null" and | |
| hd_balanced: | |
| "(low (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) = Null) = | |
| (high (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) = Null)" and | |
| hd_isLeaf_var: | |
| "isLeaf_pt (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) low high = | |
| (var (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) \<le> Suc 0)" | |
| by blast | |
| have "repa (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) = | |
| hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]" | |
| proof (cases "high no = Null") | |
| case True | |
| with no_balanced have "low no = Null" | |
| by simp | |
| with True have no_Leaf: "isLeaf_pt no low high" | |
| by (simp add: isLeaf_pt_def) | |
| with isLeaf_var have varno: "var no <= 1" | |
| by simp | |
| from same_var [rule_format, OF no_in_ns hd_term_in_ns] varno | |
| have "var (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) \<le> 1" | |
| by simp | |
| with hd_isLeaf_var have | |
| "isLeaf_pt (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) low high" | |
| by simp | |
| with while_inv hd_term_in_ns repNodes_not_empty show ?thesis | |
| apply (simp add: isLeaf_pt_def) | |
| apply (erule_tac x= | |
| "hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]" in ballE) | |
| prefer 2 | |
| apply simp | |
| apply (simp (no_asm_use) add: repNodes_eq_def) | |
| apply (rule filter_hd_P_rep_indep) | |
| apply (simp (no_asm_simp)) | |
| apply (simp (no_asm_simp)) | |
| apply assumption | |
| done | |
| next | |
| assume hno_nNull: "high no \<noteq> Null" | |
| with share_case have repchildren_neq: "(repa \<propto> low) no \<noteq> (repa \<propto> high) no" | |
| by simp | |
| from repNodes_not_empty have | |
| "repNodes_eq (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) no low high repa" | |
| by (rule hd_filter_prop) | |
| then | |
| have "(repa \<propto> low) (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) = | |
| (repa \<propto> low) no \<and> | |
| (repa \<propto> high) (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) = | |
| (repa \<propto> high) no" | |
| by (simp add: repNodes_eq_def) | |
| with repchildren_neq have | |
| "(repa \<propto> low) (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]) | |
| \<noteq> (repa \<propto> high) (hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa])" | |
| by simp | |
| with while_inv hd_term_in_ns repNodes_not_empty show ?thesis | |
| apply (simp add: isLeaf_pt_def) | |
| apply (erule_tac x= | |
| "hd [sn\<leftarrow>ns . repNodes_eq sn no low high repa]" in ballE) | |
| prefer 2 | |
| apply simp | |
| apply (simp (no_asm_use) add: repNodes_eq_def) | |
| apply (rule filter_hd_P_rep_indep) | |
| apply simp | |
| apply fastforce | |
| apply fastforce | |
| done | |
| qed | |
| with hd_term_in_ns | |
| show ?thesis | |
| by simp | |
| qed | |
| next | |
| (* invariant to invariant, THEN part -- REDUCING*) | |
| fix var low high rep nodeslist repa "next" node prx sfx | |
| assume ns: "List nodeslist next (prx @ node # sfx)" | |
| assume sfx: "List (next node) next sfx" | |
| assume node_not_Null: "node \<noteq> Null" | |
| assume nodes_balanced_ordered: "\<forall>no\<in>set (prx @ node # sfx). | |
| no \<noteq> Null \<and> | |
| (low no = Null) = (high no = Null) \<and> | |
| low no \<notin> set (prx @ node # sfx) \<and> | |
| high no \<notin> set (prx @ node # sfx) \<and> | |
| isLeaf_pt no low high = (var no \<le> (1::nat)) \<and> | |
| (low no \<noteq> Null \<longrightarrow> rep (low no) \<noteq> Null) \<and> | |
| (rep \<propto> low) no \<notin> set (prx @ node # sfx)" | |
| assume all_nodes_same_var: "\<forall>no1\<in>set (prx @ node # sfx). | |
| \<forall>no2\<in>set (prx @ node # sfx). var no1 = var no2" | |
| assume rep_repa_nc: "\<forall>no. no \<notin> set prx \<longrightarrow> rep no = repa no" | |
| assume while_inv: "\<forall>no\<in>set prx. | |
| repa no \<noteq> Null \<and> | |
| (if (repa \<propto> low) no = (repa \<propto> high) no \<and> low no \<noteq> Null | |
| then repa no = (repa \<propto> low) no | |
| else repa no = hd [sn\<leftarrow>prx . repNodes_eq sn no low high repa] \<and> | |
| repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set prx. | |
| ((repa \<propto> high) no1 = (repa \<propto> high) no \<and> | |
| (repa \<propto> low) no1 = (repa \<propto> low) no) = | |
| (repa no = repa no1)))" | |
| assume not_Leaf: "\<not> isLeaf_pt node low high" | |
| assume repchildren_eq_nln: "repa (low node) = repa (high node)" | |
| show "(\<forall>no. no \<notin> set (prx @ [node]) \<longrightarrow> | |
| rep no = (repa(node := repa (high node))) no) \<and> | |
| (\<forall>no\<in>set (prx @ [node]). | |
| (repa(node := repa (high node))) no \<noteq> Null \<and> | |
| (if (repa(node := repa (high node)) \<propto> low) no = | |
| (repa(node := repa (high node)) \<propto> high) no \<and> | |
| low no \<noteq> Null | |
| then (repa(node := repa (high node))) no = | |
| (repa(node := repa (high node)) \<propto> low) no | |
| else (repa(node := repa (high node))) no = | |
| hd [sn\<leftarrow>prx @ [node] . | |
| repNodes_eq sn no low high | |
| (repa(node := repa (high node)))] \<and> | |
| (repa(node := repa (high node))) | |
| ((repa(node := repa (high node))) no) = | |
| (repa(node := repa (high node))) no \<and> | |
| (\<forall>no1\<in>set (prx @ [node]). | |
| ((repa(node := repa (high node)) \<propto> high) no1 = | |
| (repa(node := repa (high node)) \<propto> high) no \<and> | |
| (repa(node := repa (high node)) \<propto> low) no1 = | |
| (repa(node := repa (high node)) \<propto> low) no) = | |
| ((repa(node := repa (high node))) no = | |
| (repa(node := repa (high node))) no1))))" | |
| (is "?NodesUnmodif \<and> ?NodesModif") | |
| proof - | |
| \<comment> \<open>This proof was originally conducted without the | |
| substitution @{term "repa (low node) = repa (high node)"} preformed. | |
| So don't be confused if we show everythin for @{text "repa (low node)"}.\<close> | |
| from rep_repa_nc | |
| have nodes_unmodif: ?NodesUnmodif | |
| by auto | |
| hence rep_Sucna_nc: | |
| "(\<forall>no. no \<notin> set (prx @ [node]) | |
| \<longrightarrow> rep no = (repa(node := repa (low (node )))) no)" | |
| by auto | |
| have nodes_modif: ?NodesModif (is "\<forall>no\<in>set (prx @ [node]). ?P no \<and> ?Q no") | |
| proof (rule ballI) | |
| fix no | |
| assume no_in_take_Sucna: " no \<in> set (prx @ [node])" | |
| show "?P no \<and> ?Q no" | |
| proof (cases "no = node") | |
| case False | |
| note no_noteq_nln=this | |
| with no_in_take_Sucna | |
| have no_in_take_n: "no \<in> set prx" | |
| by auto | |
| with no_in_take_n while_inv obtain | |
| repa_no_nNull: " repa no \<noteq> Null" and | |
| repa_cases: "(if (repa \<propto> low) no = (repa \<propto> high) no \<and> low no \<noteq> Null | |
| then repa no = (repa \<propto> low) no | |
| else repa no = hd [sn\<leftarrow>prx . repNodes_eq sn no low high repa] | |
| \<and> repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set prx. ((repa \<propto> high) no1 = (repa \<propto> high) no | |
| \<and> (repa \<propto> low) no1 = (repa \<propto> low) no) | |
| = (repa no = repa no1)))" | |
| using [[simp_depth_limit = 2]] | |
| by auto | |
| from no_in_take_n | |
| have no_in_nodeslist: "no \<in> set (prx @ node # sfx)" | |
| by auto | |
| from repa_no_nNull no_noteq_nln have ext_repa_nNull: "?P no" | |
| by auto | |
| from no_in_nodeslist nodes_balanced_ordered obtain | |
| nln_nNull: "node \<noteq> Null" and | |
| nln_balanced_children: "(low node = Null) = (high node = Null)" and | |
| lnln_notin_nodeslist: "low node \<notin> set (prx @ node # sfx)" and | |
| hnln_notin_nodeslist: "high node \<notin> set (prx @ node # sfx)" and | |
| isLeaf_var_nln: "isLeaf_pt node low high = (var node \<le> 1)" and | |
| node_nNull_rap_nNull_nln: "(low node \<noteq> Null | |
| \<longrightarrow> rep (low node) \<noteq> Null)" and | |
| nln_varrep_le_var: "(rep \<propto> low) node \<notin> set (prx @ node # sfx)" | |
| apply (erule_tac x="node" in ballE) | |
| apply auto | |
| done | |
| from no_in_nodeslist nodes_balanced_ordered no_in_take_Sucna | |
| obtain | |
| no_nNull: "no \<noteq> Null" and | |
| balanced_children: "(low no = Null) = (high no = Null)" and | |
| lno_notin_nodeslist: "low no \<notin> set (prx @ node # sfx)" and | |
| hno_notin_nodeslist: "high no \<notin> set (prx @ node # sfx)" and | |
| isLeaf_var_no: "isLeaf_pt no low high = (var no \<le> 1)" and | |
| node_nNull_rep_nNull: "(low no \<noteq> Null \<longrightarrow> rep (low no) \<noteq> Null)" and | |
| varrep_le_var: "(rep \<propto> low) no \<notin> set (prx @ node # sfx)" | |
| apply - | |
| apply (erule_tac x=no in ballE) | |
| apply auto | |
| done | |
| from lno_notin_nodeslist | |
| have ext_rep_null_comp_low: | |
| "(repa (node := repa (low node)) \<propto> low) no = (repa \<propto> low) no" | |
| by (auto simp add: null_comp_def) | |
| from hno_notin_nodeslist | |
| have ext_rep_null_comp_high: | |
| "(repa (node := repa (low node)) \<propto> high) no = (repa \<propto> high) no" | |
| by (auto simp add: null_comp_def) | |
| have share_reduce_if: "?Q no" | |
| proof (cases "(repa (node := repa (low node)) \<propto> low) no = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> low no \<noteq> Null") | |
| case True | |
| then obtain | |
| red_case: "(repa (node := repa (low node)) \<propto> low) no = | |
| (repa(node := repa (low node)) \<propto> high) no" and | |
| lno_nNull: "low no \<noteq> Null" | |
| by simp | |
| from lno_nNull balanced_children have hno_nNull: "high no \<noteq> Null" | |
| by simp | |
| from True ext_rep_null_comp_low ext_rep_null_comp_high | |
| have repchildren_eq_no: "(repa \<propto> low) no = (repa \<propto> high) no" | |
| by simp | |
| with repa_cases lno_nNull have "repa no = (repa \<propto> low) no" | |
| by auto | |
| with ext_rep_null_comp_low no_noteq_nln | |
| have "(repa(node := repa (low node))) no = | |
| (repa (node := repa (low node)) \<propto> low) no" | |
| by simp | |
| with True repchildren_eq_nln show ?thesis | |
| by auto | |
| next | |
| assume share_case_ext: | |
| " \<not> ((repa(node := repa (low node)) \<propto> low) no = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> low no \<noteq> Null)" | |
| from not_Leaf isLeaf_var_nln | |
| have "1 < var node" | |
| by simp | |
| with all_nodes_same_var | |
| have all_nodes_nl_Suc0_l_var: "\<forall>x \<in> set (prx @ node # sfx). 1 < var x" | |
| using [[simp_depth_limit=1]] | |
| by auto | |
| with nodes_balanced_ordered | |
| have all_nodes_nl_noLeaf: | |
| "\<forall>x \<in> set (prx @ node # sfx). \<not> isLeaf_pt x low high" | |
| apply - | |
| apply rule | |
| apply (drule_tac x=x in bspec,assumption) | |
| apply (drule_tac x=x in bspec,assumption) | |
| apply auto | |
| done | |
| from nodes_balanced_ordered | |
| have all_nodes_nl_balanced: | |
| "\<forall>x \<in> set (prx @ node # sfx). (low x = Null) = (high x = Null)" | |
| apply - | |
| apply rule | |
| apply (drule_tac x=x in bspec,assumption) | |
| apply auto | |
| done | |
| from all_nodes_nl_Suc0_l_var no_in_nodeslist | |
| have Suc0_l_var_no: "1 < var no" | |
| by auto | |
| with isLeaf_var_no have no_nLeaf: " \<not> isLeaf_pt no low high" | |
| by simp | |
| with balanced_children have lno_nNull: "low no \<noteq> Null" | |
| by (simp add: isLeaf_pt_def) | |
| with balanced_children have hno_nNull: "high no \<noteq> Null" | |
| by simp | |
| with share_case_ext ext_rep_null_comp_low ext_rep_null_comp_high lno_nNull | |
| have repchildren_neq_no: "(repa \<propto> low) no \<noteq> (repa \<propto> high) no" | |
| by (simp add: null_comp_def) | |
| with repa_cases | |
| have share_case_inv: | |
| "repa no = hd [sn\<leftarrow>prx . repNodes_eq sn no low high repa] \<and> | |
| repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set prx. ((repa \<propto> high) no1 = (repa \<propto> high) no \<and> | |
| (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1))" | |
| by auto | |
| then have repa_no: "repa no = hd [sn\<leftarrow>prx . repNodes_eq sn no low high repa]" | |
| by simp | |
| from Suc0_l_var_no have "\<forall>x \<in> set (prx @ node # sfx). 1 < var no" | |
| by auto | |
| from no_in_take_n have "[sn\<leftarrow>prx . repNodes_eq sn no low high repa] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then have "repNodes_eq | |
| (hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa]) no low high repa" | |
| by (rule hd_filter_prop) | |
| with repa_no | |
| have rep_children_eq_no_repa_no: | |
| "(repa \<propto> low) (repa no) = (repa \<propto> low) no \<and> | |
| (repa \<propto> high) (repa no) = (repa \<propto> high) no" | |
| by (simp add: repNodes_eq_def) | |
| from lno_notin_nodeslist rep_repa_nc | |
| have rep_repa_nc_low_no: "rep (low no) = repa (low no)" | |
| apply - | |
| apply (erule_tac x="low no" in allE) | |
| apply auto | |
| done | |
| have "\<forall>x \<in> set (prx @ [node]). | |
| repNodes_eq x no low high (repa(node := repa (low node))) = | |
| repNodes_eq x no low high repa" | |
| proof (rule ballI, unfold repNodes_eq_def) | |
| fix x | |
| assume x_in_take_Sucn: " x \<in> set (prx @ [node])" | |
| hence x_in_nodeslist: "x \<in> set (prx @ node # sfx)" | |
| by auto | |
| with all_nodes_nl_noLeaf nodes_balanced_ordered | |
| have children_nNull_x: "low x \<noteq> Null \<and> high x \<noteq> Null" | |
| apply - | |
| apply (drule_tac x=x in bspec,assumption) | |
| apply (drule_tac x=x in bspec,assumption) | |
| apply (auto simp add: isLeaf_pt_def) | |
| done | |
| from x_in_nodeslist nodes_balanced_ordered | |
| have "low x \<notin> set (prx @ node # sfx) \<and> high x \<notin> set (prx @ node # sfx)" | |
| apply - | |
| apply (drule_tac x=x in bspec,assumption) | |
| apply auto | |
| done | |
| with lno_notin_nodeslist hno_notin_nodeslist | |
| children_nNull_x lno_nNull hno_nNull | |
| show "((repa(node := repa (low node)) \<propto> high) x = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> | |
| (repa(node := repa (low node)) \<propto> low) x = | |
| (repa(node := repa (low node)) \<propto> low) no) = | |
| ((repa \<propto> high) x = (repa \<propto> high) no \<and> | |
| (repa \<propto> low) x = (repa \<propto> low) no)" | |
| by (simp add: null_comp_def) | |
| qed | |
| then have filter_extrep_rep: | |
| "[sn\<leftarrow>(prx @ [node]). repNodes_eq sn no low high | |
| (repa(node := repa (low node)))] = | |
| [sn\<leftarrow>(prx @ [node]) . repNodes_eq sn no low high repa]" | |
| by (rule P_eq_list_filter) | |
| from no_in_take_n | |
| have filter_n_notempty: "[sn\<leftarrow>prx. repNodes_eq sn no low high repa] \<noteq> []" | |
| apply (rule filter_not_empty) | |
| apply (simp add: repNodes_eq_def) | |
| done | |
| then have "hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa] = | |
| hd [sn\<leftarrow>prx@[node]. repNodes_eq sn no low high repa]" | |
| by auto | |
| with no_noteq_nln filter_extrep_rep repa_no | |
| have ext_repa_no: "(repa(node:= repa (low node))) no = | |
| hd [sn\<leftarrow>prx@[node] . repNodes_eq sn no low high | |
| (repa(node := repa (low node)))]" | |
| by simp | |
| have "(repa(node := repa (low node))) (repa no) = repa no" | |
| proof (cases "repa no = node") | |
| case True | |
| note rno_nln=this | |
| from rep_repa_nc_low_no rep_children_eq_no_repa_no lno_nNull | |
| node_nNull_rep_nNull | |
| have low_rep_no_nNull: "low (repa no) \<noteq> Null" | |
| apply (simp add: null_comp_def) | |
| apply auto | |
| done | |
| with nodes_balanced_ordered rno_nln | |
| have high_rap_no_nNull: "high (repa no) \<noteq> Null" | |
| apply - | |
| apply (drule_tac x="repa no" in bspec) | |
| apply auto | |
| done | |
| with low_rep_no_nNull rno_nln rep_children_eq_no_repa_no | |
| have "repa (low node) = (repa \<propto> low) no \<and> | |
| repa (high node) = (repa \<propto> high) no" | |
| by (simp add: null_comp_def) | |
| with repchildren_eq_nln have " (repa \<propto> low) no = (repa \<propto> high) no" | |
| by simp | |
| with repchildren_neq_no show ?thesis | |
| by simp | |
| next | |
| assume rno_not_nln: "repa no \<noteq> node" | |
| from share_case_inv have "repa (repa no) = repa no" | |
| by auto | |
| with rno_not_nln show ?thesis | |
| by simp | |
| qed | |
| with no_noteq_nln have ext_repa_ext_repa: | |
| "(repa(node := repa (low node))) | |
| ((repa(node := repa (low node))) no) | |
| = (repa(node := repa (low node))) no" | |
| by simp | |
| have "(\<forall>no1\<in>set (prx@[node]). | |
| ((repa(node := repa (low node)) \<propto> high) no1 = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> | |
| (repa(node := repa (low node)) \<propto> low) no1 = | |
| (repa(node := repa (low node)) \<propto> low) no) = | |
| ((repa(node := repa (low node))) no = | |
| (repa(node := repa (low node))) no1))" | |
| proof (rule ballI) | |
| fix no1 | |
| assume no1_in_take_Sucn: " no1 \<in> set (prx@[node])" | |
| hence no1_in_nodeslist: "no1 \<in> set (prx @ node # sfx)" | |
| by auto | |
| show "((repa(node := repa (low node)) \<propto> high) no1 = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> | |
| (repa(node := repa (low node)) \<propto> low) no1 = | |
| (repa(node := repa (low node)) \<propto> low) no) = | |
| ((repa(node := repa (low node))) no = | |
| (repa(node := repa (low node))) no1)" | |
| proof (cases "no1 = node") | |
| case True | |
| show ?thesis | |
| proof (rule, elim conjE) | |
| assume ext_repa_no_no1: | |
| "(repa(node := repa (low node))) no = | |
| (repa(node := repa (low node))) no1" | |
| with True no_noteq_nln | |
| have repa_no_repa_low_nln: "repa no = repa (low node)" | |
| by simp | |
| from filter_n_notempty | |
| have repa_no_in_take_n: | |
| "hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa] | |
| \<in> set prx " | |
| apply - | |
| apply (rule hd_filter_in_list) | |
| apply auto | |
| done | |
| with repa_no | |
| have repa_no_in_nodeslist: "repa no \<in> set (prx @ node # sfx)" | |
| by auto | |
| from lnln_notin_nodeslist rep_repa_nc | |
| have rep_repa_low_nln: "rep (low node) = repa (low node)" | |
| by auto | |
| from all_nodes_nl_noLeaf nln_balanced_children | |
| have "low node \<noteq> Null" | |
| by (auto simp add: isLeaf_pt_def) | |
| with rep_repa_low_nln lnln_notin_nodeslist lno_nNull | |
| nln_varrep_le_var | |
| have "repa (low node) \<notin> set (prx @ node # sfx)" | |
| by (simp add: null_comp_def) | |
| with repa_no_repa_low_nln repa_no_in_nodeslist | |
| show "(repa(node := repa (low node)) \<propto> high) no1 = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> | |
| (repa(node := repa (low node)) \<propto> low) no1 = | |
| (repa(node := repa (low node)) \<propto> low) no" | |
| by simp | |
| next | |
| assume no_no1_high: | |
| "(repa(node := repa (low node)) \<propto> high) no1 = | |
| (repa(node := repa (low node)) \<propto> high) no" | |
| assume no_no1_low: | |
| "(repa(node := repa (low node)) \<propto> low) no1 = | |
| (repa(node := repa (low node)) \<propto> low) no" | |
| from True repchildren_eq_nln | |
| have repachildren_eq_no1: " repa (low no1) = repa (high no1)" | |
| by simp | |
| from not_Leaf True nln_balanced_children | |
| have children_nNull_no1: "(low no1) \<noteq> Null \<and> high no1 \<noteq> Null" | |
| by (simp add: isLeaf_pt_def) | |
| with repachildren_eq_no1 | |
| have repchildren_eq_no1: "(repa \<propto> low) no1 = (repa \<propto> high) no1" | |
| by (simp add: null_comp_def) | |
| from no_no1_low children_nNull_no1 lno_nNull | |
| lnln_notin_nodeslist lno_notin_nodeslist True | |
| have rep_low_eq_no_no1: "(repa \<propto> low) no1 = (repa \<propto> low) no" | |
| by (simp add: null_comp_def) | |
| from no_no1_high children_nNull_no1 hno_nNull | |
| hnln_notin_nodeslist hno_notin_nodeslist True | |
| have rep_high_eq_no_no1: "(repa \<propto> high) no1 = (repa \<propto> high) no" | |
| by (simp add: null_comp_def) | |
| with rep_low_eq_no_no1 repchildren_eq_no1 | |
| have "(repa \<propto> low) no = (repa \<propto> high) no" | |
| by simp | |
| with repchildren_neq_no | |
| show "(repa(node := repa (low node))) no = | |
| (repa(node := repa (low node))) no1" | |
| by simp | |
| qed | |
| next | |
| assume no1_neq_nln: "no1 \<noteq> node" | |
| from no1_in_nodeslist nodes_balanced_ordered | |
| have children_notin_nl_no1: | |
| "low no1 \<notin> set (prx @ node # sfx) \<and> high no1 \<notin> set (prx @ node # sfx)" | |
| apply - | |
| apply (drule_tac x=no1 in bspec,assumption) | |
| by auto | |
| from no1_neq_nln no1_in_take_Sucn | |
| have no1_in_take_n: "no1 \<in> set prx" | |
| by auto | |
| from no1_in_nodeslist all_nodes_nl_noLeaf all_nodes_nl_balanced | |
| have children_nNull_no1: "(low no1) \<noteq> Null \<and> high no1 \<noteq> Null" | |
| by (auto simp add: isLeaf_pt_def) | |
| show ?thesis | |
| proof (rule, elim conjE) | |
| assume ext_repa_high_no1_no: | |
| "(repa(node := repa (low node)) \<propto> high) no1 | |
| = (repa(node := repa (low node)) \<propto> high) no" | |
| assume ext_repa_low_no1_no: | |
| "(repa(node := repa (low node)) \<propto> low) no1 | |
| = (repa(node := repa (low node)) \<propto> low) no" | |
| from children_nNull_no1 hno_nNull ext_repa_high_no1_no | |
| children_notin_nl_no1 | |
| hno_notin_nodeslist | |
| have repa_high_no1_no: "(repa \<propto> high) no1 = (repa \<propto> high) no" | |
| by (simp add: null_comp_def) | |
| from children_nNull_no1 lno_nNull ext_repa_low_no1_no | |
| children_notin_nl_no1 lno_notin_nodeslist | |
| have repa_low_no1_no: "(repa \<propto> low) no1 = (repa \<propto> low) no" | |
| by (simp add: null_comp_def) | |
| from repchildren_neq_no repa_high_no1_no repa_low_no1_no | |
| have "(repa \<propto> low) no1 \<noteq> (repa \<propto> high) no1" | |
| by simp | |
| from no1_in_take_n share_case_inv repa_high_no1_no repa_low_no1_no | |
| have "repa no = repa no1" | |
| by auto | |
| with no_noteq_nln no1_neq_nln | |
| show " (repa(node := repa (low node))) no = | |
| (repa(node := repa (low node))) no1" | |
| by simp | |
| next | |
| assume "(repa(node := repa (low node))) no = | |
| (repa(node := repa (low node))) no1" | |
| with no_noteq_nln no1_neq_nln | |
| have "repa no = repa no1" | |
| by simp | |
| with share_case_inv no1_in_take_n | |
| have "((repa \<propto> high) no1 = (repa \<propto> high) no \<and> | |
| (repa \<propto> low) no1 = (repa \<propto> low) no)" | |
| by auto | |
| with children_notin_nl_no1 children_nNull_no1 lno_notin_nodeslist | |
| hno_notin_nodeslist lno_nNull hno_nNull | |
| show "(repa(node := repa (low node)) \<propto> high) no1 = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> | |
| (repa(node := repa (low node)) \<propto> low) no1 = | |
| (repa(node := repa (low node)) \<propto> low) no" | |
| by (auto simp add: null_comp_def) | |
| qed | |
| qed | |
| qed | |
| from ext_repa_ext_repa ext_repa_no share_case_ext repchildren_eq_nln this | |
| show ?thesis | |
| using [[simp_depth_limit=4]] | |
| by auto | |
| qed | |
| with ext_repa_nNull show ?thesis | |
| by auto | |
| next | |
| assume no_nln: "no = node" | |
| hence no_in_nodeslist: "no \<in> set (prx @ node # sfx)" | |
| by simp | |
| from no_nln not_Leaf no_in_nodeslist | |
| nodes_balanced_ordered [rule_format, OF this] obtain | |
| low_no_nNull: "low no \<noteq> Null" and | |
| high_no_nNull: "high no \<noteq> Null" and | |
| rep_low_no_nNull: "rep (low no) \<noteq> Null" and | |
| lno_notin_nl: "low no \<notin> set (prx @ node # sfx)" and | |
| hno_notin_nl: "high no \<notin> set (prx @ node # sfx)" and | |
| children_nNull_no: "(low no \<noteq> Null) \<and> (high no \<noteq> Null)" | |
| apply (unfold isLeaf_pt_def) | |
| apply blast | |
| done | |
| then have "low no \<notin> set prx" | |
| by auto | |
| with rep_repa_nc no_nln rep_low_no_nNull | |
| have "(repa(node := repa (low node))) no \<noteq> Null" | |
| by simp | |
| moreover | |
| have "(if (repa(node := repa (low node)) \<propto> low) no = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> low no \<noteq> Null | |
| then (repa(node := repa (low node))) no = | |
| (repa(node := repa (low node)) \<propto> low) no | |
| else (repa(node := repa (low node))) no = | |
| hd [sn\<leftarrow>prx@[node]. repNodes_eq sn no low high | |
| (repa(node := repa (low node)))] \<and> | |
| (repa(node := repa (low node))) | |
| ((repa(node := repa (low node))) no) = | |
| (repa(node := repa (low node))) no \<and> | |
| (\<forall>no1\<in>set (prx@[node]). | |
| ((repa(node := repa (low node)) \<propto> high) no1 = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> | |
| (repa(node := repa (low node)) \<propto> low) no1 = | |
| (repa(node := repa (low node)) \<propto> low) no) = | |
| ((repa(node := repa (low node))) no = | |
| (repa(node := repa (low node))) no1)))" | |
| proof (cases "(repa(node := repa (low node)) \<propto> low) no = | |
| (repa(node := repa (low node)) \<propto> high) no \<and> low no \<noteq> Null") | |
| case True | |
| note red_case=this | |
| with children_nNull_no lno_notin_nl hno_notin_nl | |
| have "(repa \<propto> low) no = (repa \<propto> high) no" | |
| by (auto simp add: null_comp_def) | |
| from children_nNull_no lno_notin_nl | |
| have ext_repa_eq_repa_low: "(repa(node := repa (low node)) \<propto> low) no | |
| = (repa \<propto> low) no " | |
| by (auto simp add: null_comp_def) | |
| from children_nNull_no hno_notin_nl | |
| have ext_repa_eq_repa_high: | |
| "(repa(node := repa (low node)) \<propto> high) no | |
| = (repa \<propto> high) no " | |
| by (auto simp add: null_comp_def) | |
| from no_nln children_nNull_no | |
| have "repa (low node) = (repa \<propto> low) no" | |
| by (simp add: null_comp_def) | |
| with red_case ext_repa_eq_repa_high ext_repa_eq_repa_low no_nln | |
| show ?thesis | |
| using [[simp_depth_limit=2]] | |
| by (auto simp del: null_comp_not_Null) | |
| next | |
| assume share_case: " \<not> ((repa(node := repa (low node)) \<propto> low) no | |
| = (repa(node := repa (low node)) \<propto> high) no \<and> low no \<noteq> Null)" | |
| with low_no_nNull have "(repa(node := repa (low node)) \<propto> low) no | |
| \<noteq> (repa(node := repa (low node)) \<propto> high) no" | |
| by simp | |
| with children_nNull_no lno_notin_nl hno_notin_nl | |
| have "(repa \<propto> low) no \<noteq> (repa \<propto> high) no" | |
| by (auto simp add: null_comp_def) | |
| with children_nNull_no have "repa (low no) \<noteq> repa (high no)" | |
| by (simp add: null_comp_def) | |
| with repchildren_eq_nln no_nln show ?thesis | |
| by simp | |
| qed | |
| ultimately show ?thesis | |
| using repchildren_eq_nln | |
| apply - | |
| apply (simp only:) | |
| apply (simp (no_asm)) | |
| done | |
| qed | |
| qed | |
| from nodes_unmodif nodes_modif | |
| show ?thesis by iprover | |
| qed | |
| next | |
| fix var low high rep nodeslist repa "next" node prx sfx repb | |
| assume ns: "List nodeslist next (prx @ node # sfx)" | |
| assume sfx: "List (next node) next sfx" | |
| assume nodes_balanced_ordered: "\<forall>no\<in>set (prx @ node # sfx). | |
| no \<noteq> Null \<and> | |
| (low no = Null) = (high no = Null) \<and> | |
| low no \<notin> set (prx @ node # sfx) \<and> | |
| high no \<notin> set (prx @ node # sfx) \<and> | |
| isLeaf_pt no low high = (var no \<le> (1::nat)) \<and> | |
| (low no \<noteq> Null \<longrightarrow> rep (low no) \<noteq> Null) \<and> | |
| (rep \<propto> low) no \<notin> set (prx @ node # sfx)" | |
| assume all_nodes_same_var: "\<forall>no1\<in>set (prx @ node # sfx). | |
| \<forall>no2\<in>set (prx @ node # sfx). var no1 = var no2" | |
| assume rep_repa_nc: "\<forall>no. no \<notin> set prx \<longrightarrow> rep no = repa no" | |
| assume while_inv: "\<forall>no\<in>set prx. | |
| repa no \<noteq> Null \<and> | |
| (if (repa \<propto> low) no = (repa \<propto> high) no \<and> low no \<noteq> Null | |
| then repa no = (repa \<propto> low) no | |
| else repa no = hd [sn\<leftarrow>prx . repNodes_eq sn no low high repa] \<and> | |
| repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set prx. | |
| ((repa \<propto> high) no1 = (repa \<propto> high) no \<and> | |
| (repa \<propto> low) no1 = (repa \<propto> low) no) = | |
| (repa no = repa no1)))" | |
| assume share_cond: | |
| "\<not> (\<not> isLeaf_pt node low high \<and> repa (low node) = repa (high node))" | |
| assume repb_node: | |
| "repb node = hd [sn\<leftarrow>prx @ node # sfx . repNodes_eq sn node low high repa]" | |
| assume repa_repb_nc: "\<forall>pt. pt \<noteq> node \<longrightarrow> repa pt = repb pt" | |
| assume var_repb_node: "var (repb node) = var node" | |
| show "(\<forall>no. no \<notin> set (prx @ [node]) \<longrightarrow> rep no = repb no) \<and> | |
| (\<forall>no\<in>set (prx @ [node]). | |
| repb no \<noteq> Null \<and> | |
| (if (repb \<propto> low) no = (repb \<propto> high) no \<and> low no \<noteq> Null | |
| then repb no = (repb \<propto> low) no | |
| else repb no = | |
| hd [sn\<leftarrow>prx @ [node] . repNodes_eq sn no low high repb] \<and> | |
| repb (repb no) = repb no \<and> | |
| (\<forall>no1\<in>set (prx @ [node]). | |
| ((repb \<propto> high) no1 = (repb \<propto> high) no \<and> | |
| (repb \<propto> low) no1 = (repb \<propto> low) no) = | |
| (repb no = repb no1))))" | |
| proof - | |
| have rep_repb_nc: "(\<forall>no. no \<notin> set (prx @ [node]) \<longrightarrow> rep no = repb no)" | |
| proof (intro allI impI) | |
| fix no | |
| assume no_notin_take_Sucn: "no \<notin> set (prx @ [node])" | |
| with rep_repa_nc | |
| have rep_repa_nc_Sucn: "rep no = repa no" | |
| by auto | |
| from no_notin_take_Sucn have "no \<noteq> node" | |
| by auto | |
| with repa_repb_nc have "repa no = repb no" | |
| by auto | |
| with rep_repa_nc_Sucn show "rep no = repb no" | |
| by simp | |
| qed | |
| moreover | |
| have repb_no_share_def: | |
| "(\<forall>no\<in>set (prx @ [node]). | |
| \<not> ((repb \<propto> low) no = (repb \<propto> high) no \<and> low no \<noteq> Null) \<longrightarrow> | |
| repb no = hd [sn\<leftarrow>(prx @ [node]) . repNodes_eq sn no low high repb])" | |
| proof (intro ballI impI) | |
| fix no | |
| assume no_in_take_Sucn: " no \<in> set (prx @ [node])" | |
| assume share_prop: "\<not> ((repb \<propto> low) no = (repb \<propto> high) no \<and> low no \<noteq> Null)" | |
| from share_prop have share_or: | |
| "(repb \<propto> low) no \<noteq> (repb \<propto> high) no \<or> low no = Null" | |
| using [[simp_depth_limit=2]] | |
| by simp | |
| from no_in_take_Sucn have no_in_nl: "no \<in> set (prx @ node # sfx)" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF this] obtain | |
| no_nNull: "no \<noteq> Null" and | |
| balanced_no: "(low no = Null) = (high no = Null)" and | |
| lno_notin_nl: "low no \<notin> set (prx @ node # sfx)" and | |
| hno_notin_nl: "high no \<notin> set (prx @ node # sfx)" and | |
| isLeaf_var_no: "isLeaf_pt no low high = (var no \<le> 1)" | |
| by auto | |
| have nodes_notin_nl_neq_nln: "\<forall>p. p \<notin> set (prx @ node # sfx) \<longrightarrow> p \<noteq> node " | |
| by auto | |
| show " repb no = hd [sn\<leftarrow>(prx @ [node]). repNodes_eq sn no low high repb]" | |
| proof (cases "no = node") | |
| case False | |
| note no_notin_nl=this | |
| with no_in_take_Sucn have no_in_take_n: "no \<in> set prx" | |
| by auto | |
| from False repa_repb_nc have repb_repa_no: "repb no = repa no" | |
| by auto | |
| with while_inv [rule_format, OF no_in_take_n] no_in_take_n obtain | |
| repa_no_nNull: "repa no \<noteq> Null" and | |
| while_share_red_exp: | |
| "(if (repa \<propto> low) no = (repa \<propto> high) no \<and> low no \<noteq> Null | |
| then repa no = (repa \<propto> low) no | |
| else repa no = hd [sn\<leftarrow>prx . repNodes_eq sn no low high repa] \<and> | |
| repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set prx. ((repa \<propto> high) no1 = (repa \<propto> high) no \<and> | |
| (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1)))" | |
| using [[simp_depth_limit = 2]] | |
| by auto | |
| from no_in_take_n | |
| have filter_take_n_notempty: "[sn\<leftarrow>prx. | |
| repNodes_eq sn no low high repa] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then have hd_term_n_Sucn: | |
| "hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa] | |
| = hd [sn\<leftarrow>prx@[node] . repNodes_eq sn no low high repa]" | |
| by auto | |
| thus ?thesis | |
| proof (cases "low no = Null") | |
| case True | |
| note lno_Null=this | |
| with balanced_no have hno_Null: "high no = Null" | |
| by simp | |
| from lno_Null hno_Null have isLeaf_no: "isLeaf_pt no low high" | |
| by (simp add: isLeaf_pt_def) | |
| from True while_share_red_exp | |
| have while_low_Null: | |
| "repa no = hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa] \<and> | |
| repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set prx. ((repa \<propto> high) no1 = (repa \<propto> high) no | |
| \<and> (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1))" | |
| by auto | |
| have all_nodes_in_nl_Leafs: | |
| "\<forall>x \<in> set (prx @ node # sfx). isLeaf_pt x low high" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nodeslist: "x \<in> set (prx @ node # sfx)" | |
| from isLeaf_no isLeaf_var_no have "var no \<le> 1" | |
| by simp | |
| with all_nodes_same_var [rule_format, OF x_in_nodeslist no_in_nl] | |
| have "var x \<le> 1" | |
| by simp | |
| with nodes_balanced_ordered [rule_format, OF x_in_nodeslist] | |
| show "isLeaf_pt x low high" | |
| by (auto simp add: isLeaf_pt_def) | |
| qed | |
| have "\<forall> x \<in> set (prx@[node]). repNodes_eq x no low high repb | |
| = repNodes_eq x no low high repa" | |
| proof (rule ballI) | |
| fix x | |
| assume x_in_take_Sucn: "x \<in> set (prx@[node])" | |
| hence x_in_nodeslist: "x \<in> set (prx @ node # sfx)" | |
| by auto | |
| with all_nodes_in_nl_Leafs have "isLeaf_pt x low high" | |
| by auto | |
| with isLeaf_no repa_repb_nc show "repNodes_eq x no low high repb | |
| = repNodes_eq x no low high repa" | |
| by (simp add: repNodes_eq_def null_comp_def isLeaf_pt_def) | |
| qed | |
| then have " [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repa] | |
| = [sn\<leftarrow>(prx@[node]) . repNodes_eq sn no low high repb]" | |
| apply - | |
| apply (rule P_eq_list_filter) | |
| apply simp | |
| done | |
| with hd_term_n_Sucn while_low_Null repb_repa_no show ?thesis | |
| by auto | |
| next | |
| assume lno_nNull: " low no \<noteq> Null" | |
| with balanced_no have hno_nNull: "high no \<noteq> Null" | |
| by simp | |
| with lno_nNull have no_nLeaf: "\<not> isLeaf_pt no low high" | |
| by (simp add: isLeaf_pt_def) | |
| with isLeaf_var_no have Sucn_s_varno: "1 < var no" | |
| by auto | |
| with no_in_nl all_nodes_same_var | |
| have all_nodes_nl_var: "\<forall> x \<in> set (prx @ node # sfx). 1 < var x" | |
| apply - | |
| apply (rule ballI) | |
| apply (drule_tac x=no in bspec,assumption) | |
| apply (drule_tac x=x in bspec,assumption) | |
| apply auto | |
| done | |
| with nodes_balanced_ordered | |
| have all_nodes_nl_nLeaf: | |
| "\<forall>x \<in> set (prx @ node # sfx). \<not> isLeaf_pt x low high" | |
| apply - | |
| apply (rule ballI) | |
| apply (drule_tac x=x in bspec,assumption) | |
| apply (drule_tac x=x in bspec,assumption) | |
| apply auto | |
| done | |
| from lno_nNull share_or | |
| have repbchildren_eq_no: "(repb \<propto> low) no \<noteq> (repb \<propto> high) no" | |
| by simp | |
| with lno_nNull hno_nNull lno_notin_nl hno_notin_nl repa_repb_nc | |
| nodes_notin_nl_neq_nln | |
| have repachildren_eq_no: "(repa \<propto> low) no \<noteq> (repa \<propto> high) no" | |
| using [[simp_depth_limit=2]] | |
| by (simp add: null_comp_def) | |
| with while_share_red_exp | |
| have repa_no_def: | |
| "repa no = hd [sn\<leftarrow>prx . repNodes_eq sn no low high repa] " | |
| by auto | |
| with no_notin_nl repa_repb_nc | |
| have "repb no = hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa] " | |
| by simp | |
| with hd_term_n_Sucn | |
| have repb_no_hd_term_repa: "repb no = | |
| hd [sn\<leftarrow>prx@[node] . repNodes_eq sn no low high repa] " | |
| by simp | |
| have "\<forall>x \<in> set (prx@[node]). | |
| repNodes_eq x no low high repa = repNodes_eq x no low high repb" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_take_Sucn: "x \<in> set (prx@[node]) " | |
| hence x_in_nodeslist: "x \<in> set (prx @ node # sfx)" | |
| by auto | |
| with all_nodes_nl_nLeaf have x_nLeaf: "\<not> isLeaf_pt x low high" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF x_in_nodeslist] obtain | |
| balanced_x: "(low x = Null) = (high x = Null)" and | |
| lx_notin_nl: "low x \<notin> set (prx @ node # sfx)" and | |
| hx_notin_nl: "high x \<notin> set (prx @ node # sfx)" | |
| by auto | |
| with nodes_notin_nl_neq_nln lno_notin_nl hno_notin_nl lno_nNull | |
| hno_nNull repa_repb_nc | |
| show " repNodes_eq x no low high repa = repNodes_eq x no low high repb" | |
| by (simp add: repNodes_eq_def null_comp_def) | |
| qed | |
| then have " [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repa] = | |
| [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| apply - | |
| apply (rule P_eq_list_filter) | |
| apply auto | |
| done | |
| with repb_no_hd_term_repa show ?thesis | |
| by simp | |
| qed | |
| next | |
| assume no_nln: "no = node" | |
| with repb_node have repb_no_def: "repb no = | |
| hd [sn\<leftarrow>(prx @ node # sfx). repNodes_eq sn node low high repa]" | |
| by simp | |
| show ?thesis | |
| proof (cases "isLeaf_pt no low high") | |
| case True | |
| note isLeaf_no=this | |
| have "\<forall>x \<in> set (prx @ node # sfx). repNodes_eq x no low high repb | |
| = repNodes_eq x no low high repa" | |
| proof (rule ballI) | |
| fix x | |
| assume x_in_nodeslist: "x \<in> set (prx @ node # sfx)" | |
| have all_nodes_in_nl_Leafs: | |
| "\<forall>x \<in> set (prx @ node # sfx). isLeaf_pt x low high" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nodeslist: " x \<in> set (prx @ node # sfx)" | |
| from isLeaf_no isLeaf_var_no have "var no \<le> 1" | |
| by simp | |
| with all_nodes_same_var [rule_format, OF x_in_nodeslist no_in_nl] | |
| have "var x \<le> 1" | |
| by simp | |
| with nodes_balanced_ordered [rule_format, OF x_in_nodeslist] | |
| show "isLeaf_pt x low high" | |
| by (auto simp add: isLeaf_pt_def) | |
| qed | |
| with x_in_nodeslist have "isLeaf_pt x low high" | |
| by auto | |
| with isLeaf_no repa_repb_nc | |
| show "repNodes_eq x no low high repb = repNodes_eq x no low high repa" | |
| by (simp add: repNodes_eq_def null_comp_def isLeaf_pt_def) | |
| qed | |
| with repb_no_def no_nln have repb_no_whole_nl: "repb no = | |
| hd [sn\<leftarrow> (prx @ node # sfx). repNodes_eq sn node low high repb]" | |
| apply - | |
| apply (subgoal_tac | |
| "[sn\<leftarrow> (prx@node#sfx). repNodes_eq sn node low high repa] | |
| = [sn\<leftarrow>(prx @ node # sfx) . repNodes_eq sn node low high repb]") | |
| apply simp | |
| apply (rule P_eq_list_filter) | |
| apply auto | |
| done | |
| from no_in_take_Sucn no_nln | |
| have "[sn\<leftarrow> (prx@[node]). repNodes_eq sn node low high repb] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then | |
| have "hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn node low high repb] = | |
| hd [sn\<leftarrow>(prx @ node # sfx). repNodes_eq sn node low high repb]" | |
| apply - | |
| apply (rule hd_filter_app [symmetric]) | |
| apply auto | |
| done | |
| with repb_no_whole_nl no_nln show ?thesis | |
| by simp | |
| next | |
| assume no_nLeaf: " \<not> isLeaf_pt no low high" | |
| with share_or balanced_no have "(repb \<propto> low) no \<noteq> (repb \<propto> high) no" | |
| using [[simp_depth_limit=2]] | |
| by (simp add: isLeaf_pt_def) | |
| from no_nLeaf share_cond no_nln have "repa (low no) \<noteq> repa (high no)" | |
| by auto | |
| with no_nLeaf balanced_no have "(repa \<propto> low) no \<noteq> (repa \<propto> high) no " | |
| by (simp add: null_comp_def isLeaf_pt_def) | |
| have "\<forall> x \<in> set (prx@node#sfx). repNodes_eq x no low high repb | |
| = repNodes_eq x no low high repa" | |
| proof (rule ballI) | |
| fix x | |
| assume x_in_nodeslist: " x \<in> set (prx@node#sfx)" | |
| have all_nodes_in_nl_Leafs: | |
| "\<forall>x \<in> set (prx@node#sfx). \<not> isLeaf_pt x low high" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nodeslist: " x \<in> set (prx@node#sfx)" | |
| from no_nLeaf isLeaf_var_no have "1 < var no " | |
| by simp | |
| with all_nodes_same_var [rule_format, OF x_in_nodeslist no_in_nl] | |
| have "1 < var x" | |
| by auto | |
| with nodes_balanced_ordered [rule_format, OF x_in_nodeslist] | |
| show "\<not> isLeaf_pt x low high" | |
| apply (unfold isLeaf_pt_def) | |
| apply fastforce | |
| done | |
| qed | |
| with x_in_nodeslist have x_nLeaf: "\<not> isLeaf_pt x low high" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF x_in_nodeslist] | |
| have "(low x = Null) = (high x = Null) | |
| \<and> low x \<notin> set (prx@node#sfx) \<and> high x \<notin> set (prx@node#sfx)" | |
| by auto | |
| with x_nLeaf balanced_no no_nLeaf repa_repb_nc | |
| nodes_notin_nl_neq_nln lno_notin_nl hno_notin_nl | |
| show "repNodes_eq x no low high repb = repNodes_eq x no low high repa" | |
| using [[simp_depth_limit=2]] | |
| by (simp add: repNodes_eq_def null_comp_def isLeaf_pt_def) | |
| qed | |
| with repb_no_def no_nln | |
| have repb_no_whole_nl: | |
| "repb no = hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn node low high repb]" | |
| apply - | |
| apply (subgoal_tac | |
| "[sn\<leftarrow>(prx@node#sfx). repNodes_eq sn node low high repa] | |
| = [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn node low high repb]") | |
| apply simp | |
| apply (rule P_eq_list_filter) | |
| apply auto | |
| done | |
| from no_in_take_Sucn no_nln | |
| have "[sn\<leftarrow>(prx@[node]) . repNodes_eq sn node low high repb] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then have | |
| "hd [sn\<leftarrow> (prx@[node]) . repNodes_eq sn node low high repb] = | |
| hd [sn\<leftarrow>(prx@node#sfx) . repNodes_eq sn node low high repb]" | |
| apply - | |
| apply (rule hd_filter_app [symmetric]) | |
| apply auto | |
| done | |
| with repb_no_whole_nl no_nln show ?thesis | |
| by simp | |
| qed | |
| qed | |
| qed | |
| have repb_no_red_def: "(\<forall>no\<in>set (prx@[node]).(repb \<propto> low) no = (repb \<propto> high) no | |
| \<and> low no \<noteq> Null \<longrightarrow> repb no = (repb \<propto> low) no)" | |
| proof (intro ballI impI) | |
| fix no | |
| assume no_in_take_Sucn: "no \<in> set (prx@[node])" | |
| assume red_cond_no: " (repb \<propto> low) no = (repb \<propto> high) no \<and> low no \<noteq> Null" | |
| from no_in_take_Sucn have no_in_nl: "no \<in> set (prx@node#sfx)" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF this]obtain | |
| no_nNull: "no \<noteq> Null" and | |
| balanced_no: "(low no = Null) = (high no = Null)" and | |
| lno_notin_nl: "low no \<notin> set (prx@node#sfx)" and | |
| hno_notin_nl: "high no \<notin> set (prx@node#sfx)" and | |
| isLeaf_var_no: "isLeaf_pt no low high = (var no \<le> 1)" | |
| by auto | |
| have nodes_notin_nl_neq_nln: "\<forall> p. p \<notin> set (prx@node#sfx) \<longrightarrow> p \<noteq> node" | |
| by auto | |
| show " repb no = (repb \<propto> low) no" | |
| proof (cases "no = node") | |
| case False | |
| note no_notin_nl=this | |
| with no_in_take_Sucn have no_in_take_n: "no \<in> set prx" | |
| by auto | |
| from False repa_repb_nc have repb_repa_no: "repb no = repa no" | |
| by auto | |
| with while_inv [rule_format, OF no_in_take_n] obtain | |
| repa_no_nNull: "repa no \<noteq> Null" and | |
| while_share_red_exp: | |
| "(if (repa \<propto> low) no = (repa \<propto> high) no \<and> low no \<noteq> Null | |
| then repa no = (repa \<propto> low) no | |
| else repa no = hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa] \<and> | |
| repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set prx. ((repa \<propto> high) no1 = (repa \<propto> high) no \<and> | |
| (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1)))" | |
| using [[simp_depth_limit=2]] | |
| by auto | |
| from red_cond_no nodes_notin_nl_neq_nln lno_notin_nl | |
| hno_notin_nl while_share_red_exp balanced_no repa_repb_nc | |
| have red_repa_no: "repa no = (repa \<propto> low) no" | |
| by (auto simp add: null_comp_def) | |
| from red_cond_no nodes_notin_nl_neq_nln lno_notin_nl repa_repb_nc | |
| have "(repb \<propto> low) no = (repa \<propto> low) no" | |
| by (auto simp add: null_comp_def) | |
| with red_repa_no no_notin_nl balanced_no repa_repb_nc | |
| have "repb no = (repb \<propto> low) no" | |
| by auto | |
| with red_cond_no show ?thesis | |
| by auto | |
| next | |
| assume "no = node" | |
| with share_cond | |
| have share_cond_pre: | |
| "isLeaf_pt no low high \<or> repa (low no) \<noteq> repa (high no)" | |
| by simp | |
| show ?thesis | |
| proof (cases "isLeaf_pt no low high") | |
| case True | |
| with red_cond_no show ?thesis | |
| by (simp add: isLeaf_pt_def) | |
| next | |
| assume no_nLeaf: "\<not> isLeaf_pt no low high" | |
| with share_cond_pre | |
| have "repa (low no) \<noteq> repa (high no)" | |
| by simp | |
| with no_nLeaf lno_notin_nl hno_notin_nl nodes_notin_nl_neq_nln | |
| balanced_no repa_repb_nc | |
| have "repb (low no) \<noteq> repb (high no)" | |
| using [[simp_depth_limit=2]] | |
| by (auto simp add: isLeaf_pt_def) | |
| with no_nLeaf balanced_no have "(repb \<propto> low) no \<noteq> (repb \<propto> high) no" | |
| by (simp add: null_comp_def isLeaf_pt_def) | |
| with red_cond_no show ?thesis | |
| by simp | |
| qed | |
| qed | |
| qed | |
| have while_while: "(\<forall>no\<in>set (prx@[node]). | |
| repb no \<noteq> Null \<and> | |
| (if (repb \<propto> low) no = (repb \<propto> high) no \<and> low no \<noteq> Null | |
| then repb no = (repb \<propto> low) no | |
| else repb no = hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb] \<and> | |
| repb (repb no) = repb no \<and> | |
| (\<forall>no1\<in>set ((prx@[node])). ((repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no) = (repb no = repb no1))))" | |
| (is "\<forall>no\<in>set (prx@[node]). ?P no \<and> ?Q no") | |
| proof (intro ballI) | |
| fix no | |
| assume no_in_take_Sucn: "no \<in> set (prx@[node])" | |
| hence no_in_nl: "no \<in> set (prx@node#sfx)" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF this] obtain | |
| no_nNull: "no \<noteq> Null" and | |
| balanced_no: "(low no = Null) = (high no = Null)" and | |
| lno_notin_nl: "low no \<notin> set (prx@node#sfx)" and | |
| hno_notin_nl: "high no \<notin> set (prx@node#sfx)" and | |
| isLeaf_var_no: "isLeaf_pt no low high = (var no \<le> 1)" | |
| by auto | |
| from no_in_take_Sucn | |
| have filter_take_Sucn_not_empty: | |
| "[sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then have hd_filter_Sucn_in_Sucn: | |
| "hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb] \<in> | |
| set (prx@[node])" | |
| by (rule hd_filter_in_list) | |
| have nodes_notin_nl_neq_nln: "\<forall>p. p \<notin> set (prx@node#sfx) \<longrightarrow> p \<noteq> node" | |
| by auto | |
| show "?P no \<and> ?Q no" | |
| proof (cases "no = node") | |
| case False | |
| note no_notin_nl=this | |
| with no_in_take_Sucn | |
| have no_in_take_n: "no \<in> set prx" | |
| by auto | |
| from False repa_repb_nc have repb_repa_no: "repb no = repa no" | |
| by auto | |
| with while_inv [rule_format, OF no_in_take_n] obtain | |
| repa_no_nNull: "repa no \<noteq> Null" and | |
| while_share_red_exp: | |
| "(if (repa \<propto> low) no = (repa \<propto> high) no \<and> low no \<noteq> Null | |
| then repa no = (repa \<propto> low) no | |
| else repa no = hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa] \<and> | |
| repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set prx. ((repa \<propto> high) no1 = (repa \<propto> high) no \<and> | |
| (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1)))" | |
| using [[simp_depth_limit=2]] | |
| by auto | |
| from repb_repa_no repa_no_nNull have repb_no_nNull: "?P no" | |
| by simp | |
| have "?Q no" | |
| proof (cases "(repb \<propto> low) no = (repb \<propto> high) no \<and> low no \<noteq> Null") | |
| case True | |
| with no_in_take_Sucn repb_no_red_def show ?thesis | |
| by auto | |
| next | |
| assume share_case_repb: | |
| " \<not> ((repb \<propto> low) no = (repb \<propto> high) no \<and> low no \<noteq> Null)" | |
| with repb_no_share_def no_in_take_Sucn | |
| have repb_no_def: "repb no = hd [sn\<leftarrow> (prx@[node]). | |
| repNodes_eq sn no low high repb]" | |
| by auto | |
| with share_case_repb | |
| have "(repb \<propto> low) no \<noteq> (repb \<propto> high) no \<or> low no = Null" | |
| using [[simp_depth_limit=2]] | |
| by simp | |
| thus ?thesis | |
| proof (cases "low no = Null") | |
| case True | |
| note lno_Null=this | |
| with balanced_no have hno_Null: "high no = Null" | |
| by simp | |
| from lno_Null hno_Null have isLeaf_no: "isLeaf_pt no low high" | |
| by (simp add: isLeaf_pt_def) | |
| from True while_share_red_exp | |
| have while_low_Null: | |
| "repa no = hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa] \<and> | |
| repa (repa no) = repa no \<and> | |
| (\<forall>no1\<in>set prx. ((repa \<propto> high) no1 = (repa \<propto> high) no | |
| \<and> (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1))" | |
| by auto | |
| from no_in_take_n | |
| have "[sn\<leftarrow>prx. repNodes_eq sn no low high repa] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then have hd_term_n_Sucn: "hd [sn\<leftarrow>prx. repNodes_eq sn no low high repa] = | |
| hd [sn\<leftarrow>(prx@[node]) . repNodes_eq sn no low high repa]" | |
| apply - | |
| apply (rule hd_filter_app [symmetric]) | |
| apply auto | |
| done | |
| have all_nodes_in_nl_Leafs: | |
| "\<forall>x \<in> set (prx@node#sfx). isLeaf_pt x low high" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nodeslist: " x \<in> set (prx@node#sfx)" | |
| from isLeaf_no isLeaf_var_no have "var no \<le> 1" | |
| by simp | |
| with all_nodes_same_var [rule_format, OF x_in_nodeslist no_in_nl] | |
| have "var x \<le> 1" | |
| by simp | |
| with nodes_balanced_ordered [rule_format, OF x_in_nodeslist] | |
| show "isLeaf_pt x low high" | |
| by (auto simp add: isLeaf_pt_def) | |
| qed | |
| from no_in_take_Sucn have | |
| filter_Sucn_no_notempty: | |
| "[sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then have hd_term_in_take_Sucn: | |
| "hd [sn\<leftarrow>(prx@[node]) . repNodes_eq sn no low high repb] | |
| \<in> set (prx@[node])" | |
| by (rule hd_filter_in_list) | |
| then have hd_term_in_nl: | |
| "hd [sn\<leftarrow>(prx@[node]) . repNodes_eq sn no low high repb] | |
| \<in> set (prx@node#sfx)" | |
| by auto | |
| with all_nodes_in_nl_Leafs | |
| have hd_term_Leaf: "isLeaf_pt (hd [sn\<leftarrow> (prx@[node]). | |
| repNodes_eq sn no low high repb]) low high " | |
| by auto | |
| from while_low_Null have "repa (repa no) = repa no" | |
| by auto | |
| with no_notin_nl repa_repb_nc | |
| have repa_repb_no_repb: "repa (repb no) = repb no" | |
| by auto | |
| have repb_repb_no: "repb (repb no) = repb no" | |
| proof (cases "repb no = node") | |
| case False | |
| with repa_repb_nc repa_repb_no_repb show ?thesis | |
| by auto | |
| next | |
| assume repb_no_nln: " repb no = node" | |
| with hd_term_Leaf isLeaf_no all_nodes_in_nl_Leafs | |
| have nested_hd_repa_repb: | |
| "hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn | |
| (hd [sn\<leftarrow>(prx@[node]) . repNodes_eq sn no low high repb]) | |
| low high repa] = | |
| hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn | |
| ( hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repb]" | |
| by (simp add: isLeaf_pt_def repNodes_eq_def null_comp_def) | |
| from hd_term_in_take_Sucn | |
| have "[sn\<leftarrow>(prx@[node]). repNodes_eq sn | |
| (hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repb] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then have "hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn | |
| ( hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repb] = | |
| hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn | |
| ( hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repb]" | |
| apply - | |
| apply (rule hd_filter_app [symmetric]) | |
| apply auto | |
| done | |
| then have hd_term_nodeslist_Sucn: | |
| "hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn | |
| ( hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repb] = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn | |
| ( hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repb]" | |
| by simp | |
| from no_in_take_Sucn filter_Sucn_no_notempty | |
| have filter_filter: "hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn | |
| (hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repb] = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| apply - | |
| apply (rule filter_hd_P_rep_indep) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| from repb_no_def repb_no_nln repb_node | |
| have "repb (repb no) = hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn | |
| ( hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repa]" | |
| by simp | |
| with nested_hd_repa_repb | |
| have "repb (repb no) = hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn | |
| (hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repb]" | |
| by simp | |
| with hd_term_nodeslist_Sucn | |
| have "repb (repb no) = hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn | |
| ( hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]) | |
| low high repb]" | |
| by simp | |
| with filter_filter | |
| have "repb (repb no) = hd [sn\<leftarrow>(prx@[node]). | |
| repNodes_eq sn no low high repb]" | |
| by simp | |
| with repb_no_def show ?thesis | |
| by simp | |
| qed | |
| have two_nodes_repb: "(\<forall>no1\<in>set (prx@[node]). | |
| ((repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no) = (repb no = repb no1))" | |
| proof (intro ballI) | |
| fix no1 | |
| assume no1_in_take_Sucn: " no1 \<in> set (prx@[node])" | |
| then have "no1 \<in> set (prx@node#sfx)" by auto | |
| with all_nodes_in_nl_Leafs | |
| have isLeaf_no1: "isLeaf_pt no1 low high" | |
| by auto | |
| with isLeaf_no | |
| have repbchildren_eq_no_no1: "(repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no" | |
| by (simp add: null_comp_def isLeaf_pt_def) | |
| from isLeaf_no1 isLeaf_no | |
| have repachildren_eq_no_no1: "(repa \<propto> high) no1 = (repa \<propto> high) no | |
| \<and> (repa \<propto> low) no1 = (repa \<propto> low) no" | |
| by (simp add: null_comp_def isLeaf_pt_def) | |
| from while_low_Null | |
| have while_low_same_rep: "(\<forall>no1\<in>set prx. | |
| ((repa \<propto> high) no1 = (repa \<propto> high) no | |
| \<and> (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1))" | |
| by auto | |
| show "((repb \<propto> high) no1 = (repb \<propto> high) no \<and> | |
| (repb \<propto> low) no1 = (repb \<propto> low) no) = (repb no = repb no1)" | |
| proof (cases "no1 = node") | |
| case False | |
| with no1_in_take_Sucn have "no1 \<in> set prx" | |
| by auto | |
| with while_low_same_rep repachildren_eq_no_no1 | |
| have "repa no = repa no1" | |
| by auto | |
| with repa_repb_nc no_notin_nl False repbchildren_eq_no_no1 | |
| show ?thesis | |
| by auto | |
| next | |
| assume no1_nln: "no1 = node" | |
| hence no1_in_take_Sucn: "no1 \<in> set (prx@[node])" | |
| by auto | |
| hence no1_in_nl: "no1 \<in> set (prx@node#sfx)" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF this] have | |
| balanced_no1: "(low no1 = Null) = (high no1 = Null)" | |
| by auto | |
| with no1_in_take_Sucn repb_no_share_def isLeaf_no1 | |
| have repb_no1: "repb no1 = hd [sn\<leftarrow>(prx@[node]). | |
| repNodes_eq sn no1 low high repb]" | |
| by (auto simp add: isLeaf_pt_def) | |
| from balanced_no1 isLeaf_no1 isLeaf_no balanced_no | |
| have repbchildren_eq_no1_no: "(repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no" | |
| by (simp add: null_comp_def isLeaf_pt_def) | |
| have "\<forall> x \<in> set (prx@[node]). repNodes_eq x no low high repb | |
| = repNodes_eq x no1 low high repb" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_take_Sucn: " x \<in> set (prx@[node])" | |
| with repbchildren_eq_no1_no show "repNodes_eq x no low high repb | |
| = repNodes_eq x no1 low high repb" | |
| by (simp add: repNodes_eq_def) | |
| qed | |
| then have " [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb] | |
| = [sn\<leftarrow>(prx@[node]). repNodes_eq sn no1 low high repb]" | |
| by (rule P_eq_list_filter) | |
| with repb_no_def repb_no1 have repb_no_no1: "repb no = repb no1" | |
| by simp | |
| with repbchildren_eq_no1_no show ?thesis | |
| by simp | |
| qed | |
| qed | |
| with repb_repb_no repb_no_share_def no_in_take_Sucn share_case_repb | |
| show ?thesis | |
| using [[simp_depth_limit=4]] | |
| by auto | |
| next | |
| assume lno_nNull: "low no \<noteq> Null" | |
| with share_case_repb | |
| have repbchildren_neq_no: "(repb \<propto> low) no \<noteq> (repb \<propto> high) no" | |
| by auto | |
| from balanced_no lno_nNull | |
| have hno_nNull: "high no \<noteq> Null" | |
| by simp | |
| with repbchildren_neq_no lno_nNull repa_repb_nc | |
| lno_notin_nl hno_notin_nl nodes_notin_nl_neq_nln | |
| have repachildren_neq_no: "(repa \<propto> low) no \<noteq> (repa \<propto> high) no" | |
| using [[simp_depth_limit=2]] | |
| by (auto simp add: null_comp_def) | |
| with while_share_red_exp | |
| have repa_while_inv: "repa (repa no) = repa no | |
| \<and> (\<forall>no1\<in>set prx. ((repa \<propto> high) no1 = (repa \<propto> high) no | |
| \<and> (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1))" | |
| by auto | |
| from lno_nNull hno_nNull | |
| have no_nLeaf: "\<not> isLeaf_pt no low high" | |
| by (simp add: isLeaf_pt_def) | |
| have all_nodes_in_nl_nLeafs: | |
| "\<forall>x \<in> set (prx@node#sfx). \<not> isLeaf_pt x low high" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nodeslist: " x \<in> set (prx@node#sfx)" | |
| from no_nLeaf isLeaf_var_no have "1 < var no " | |
| by simp | |
| with all_nodes_same_var [rule_format, OF x_in_nodeslist no_in_nl] | |
| have "1 < var x" | |
| by simp | |
| with nodes_balanced_ordered [rule_format, OF x_in_nodeslist] | |
| show " \<not> isLeaf_pt x low high" | |
| using [[simp_depth_limit = 2]] | |
| by (auto simp add: isLeaf_pt_def) | |
| qed | |
| have repb_repb_no: "repb (repb no) = repb no" | |
| proof - | |
| from repa_while_inv no_notin_nl repa_repb_nc | |
| have "repa (repb no) = repb no" | |
| by simp | |
| from hd_filter_Sucn_in_Sucn repb_no_def | |
| have repb_no_in_take_Sucn: "repb no \<in> set (prx@[node])" | |
| by simp | |
| hence repb_no_in_nl: "repb no \<in> set (prx@node#sfx)" | |
| by auto | |
| from all_nodes_in_nl_nLeafs repb_no_in_nl | |
| have repb_no_nLeaf: "\<not> isLeaf_pt (repb no) low high" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF repb_no_in_nl] | |
| have "(low (repb no) = Null) = (high (repb no) = Null) | |
| \<and> low (repb no) \<notin> set (prx@node#sfx) \<and> | |
| high (repb no) \<notin> set (prx@node#sfx)" | |
| by auto | |
| from filter_take_Sucn_not_empty | |
| have " repNodes_eq (hd [sn\<leftarrow>(prx@[node]). | |
| repNodes_eq sn no low high repb]) no low high repb" | |
| by (rule hd_filter_prop) | |
| with repb_no_def have "repNodes_eq (repb no) no low high repb" | |
| by simp | |
| then have "(repb \<propto> low) (repb no) = (repb \<propto> low) no | |
| \<and> (repb \<propto> high) (repb no) = (repb \<propto> high) no" | |
| by (simp add: repNodes_eq_def) | |
| with repbchildren_neq_no have "(repb \<propto> low) (repb no) | |
| \<noteq> (repb \<propto> high) (repb no)" | |
| by simp | |
| with repb_no_in_take_Sucn repb_no_share_def | |
| have repb_repb_no_double_hd: | |
| "repb (repb no) = hd [sn\<leftarrow>(prx@[node]). | |
| repNodes_eq sn (repb no) low high repb]" | |
| by auto | |
| from filter_take_Sucn_not_empty | |
| have " hd [sn\<leftarrow>(prx@[node]). | |
| repNodes_eq sn (repb no) low high repb] = repb no" | |
| apply (simp only: repb_no_def ) | |
| apply (rule filter_hd_P_rep_indep) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| with repb_repb_no_double_hd show ?thesis | |
| by simp | |
| qed | |
| have "(\<forall>no1\<in>set (prx@[node]). | |
| ((repb \<propto> high) no1 = (repb \<propto> high) no \<and> | |
| (repb \<propto> low) no1 = (repb \<propto> low) no) = (repb no = repb no1))" | |
| proof (intro ballI) | |
| fix no1 | |
| assume no1_in_take_Sucn: "no1 \<in> set (prx@[node])" | |
| hence no1_in_nl: "no1 \<in> set (prx@node#sfx)" | |
| by auto | |
| from all_nodes_in_nl_nLeafs no1_in_nl | |
| have no1_nLeaf: "\<not> isLeaf_pt no1 low high" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF no1_in_nl] | |
| have no1_props: "(low no1 = Null) = (high no1 = Null) | |
| \<and> low no1 \<notin> set (prx@node#sfx) \<and> high no1 \<notin> set (prx@node#sfx)" | |
| by auto | |
| show "((repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no) = (repb no = repb no1)" | |
| proof (cases "no1 = node") | |
| case False | |
| note no1_neq_nln=this | |
| with no1_in_take_Sucn | |
| have no1_in_take_n: "no1 \<in> set prx" | |
| by auto | |
| with repa_while_inv have "((repa \<propto> high) no1 = (repa \<propto> high) no | |
| \<and> (repa \<propto> low) no1 = (repa \<propto> low) no) = (repa no = repa no1)" | |
| by fastforce | |
| with no1_props no1_nLeaf no_nLeaf balanced_no lno_notin_nl | |
| hno_notin_nl nodes_notin_nl_neq_nln no_notin_nl | |
| no1_neq_nln repa_repb_nc | |
| show ?thesis | |
| using [[simp_depth_limit=1]] | |
| by (auto simp add: null_comp_def isLeaf_pt_def) | |
| next | |
| assume no1_nln: " no1 = node" | |
| show ?thesis | |
| proof | |
| assume repbchildren_eq_no1_no: | |
| "(repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no" | |
| with repbchildren_neq_no | |
| have "(repb \<propto> high) no1 \<noteq> (repb \<propto> low) no1" | |
| by auto | |
| with repb_no_share_def no1_in_take_Sucn | |
| have repb_no1_def: " repb no1 = hd [sn\<leftarrow>(prx@[node]). | |
| repNodes_eq sn no1 low high repb]" | |
| by auto | |
| have filter_no1_eq_filter_no: "[sn\<leftarrow>(prx@[node]). | |
| repNodes_eq sn no1 low high repb] = | |
| [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| proof - | |
| have "\<forall>x \<in> set (prx@[node]). | |
| repNodes_eq x no1 low high repb = | |
| repNodes_eq x no low high repb" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_take_Sucn: "x \<in> set (prx@[node])" | |
| with repbchildren_eq_no1_no | |
| show "repNodes_eq x no1 low high repb = | |
| repNodes_eq x no low high repb" | |
| by (simp add: repNodes_eq_def) | |
| qed | |
| then show ?thesis | |
| by (rule P_eq_list_filter) | |
| qed | |
| with repb_no1_def repb_no_def show " repb no = repb no1" | |
| by simp | |
| next | |
| assume repb_no_no1_eq: "repb no = repb no1" | |
| from no1_nln repb_node repb_no_def have repb_no1_def: | |
| "repb no1 = | |
| hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn node low high repa]" | |
| by auto | |
| with no1_nln repb_no_def repb_no_no1_eq | |
| have repb_Sucn_repa_nl_hd: " hd [sn\<leftarrow>(prx@[node]). | |
| repNodes_eq sn no low high repb] = | |
| hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no1 low high repa]" | |
| by simp | |
| from filter_take_Sucn_not_empty | |
| have " hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb] | |
| = hd [sn\<leftarrow>(prx@node#sfx) . repNodes_eq sn no low high repb]" | |
| apply - | |
| apply (rule hd_filter_app [symmetric]) | |
| apply auto | |
| done | |
| then have hd_Sucn_hd_whole_list: | |
| "hd [sn\<leftarrow>(prx@[node]) . | |
| repNodes_eq sn no low high repb] = | |
| hd [sn\<leftarrow> (prx@node#sfx). repNodes_eq sn no low high repb]" | |
| by simp | |
| have hd_nl_repb_repa: | |
| "[sn\<leftarrow> (prx@node#sfx). repNodes_eq sn no low high repb] | |
| = [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no low high repa]" | |
| proof - | |
| have "\<forall>x \<in> set (prx@node#sfx). | |
| repNodes_eq x no low high repb = | |
| repNodes_eq x no low high repa" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nl: "x \<in> set (prx@node#sfx)" | |
| from all_nodes_in_nl_nLeafs x_in_nl | |
| have x_nLeaf: "\<not> isLeaf_pt x low high" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF x_in_nl] | |
| have x_props: "(low x = Null) = (high x = Null) \<and> | |
| low x \<notin> set (prx@node#sfx) \<and> high x \<notin> set (prx@node#sfx)" | |
| by auto | |
| with x_nLeaf lno_nNull hno_nNull lno_notin_nl hno_notin_nl | |
| nodes_notin_nl_neq_nln repa_repb_nc | |
| show "repNodes_eq x no low high repb = | |
| repNodes_eq x no low high repa" | |
| using [[simp_depth_limit=1]] | |
| by (simp add: repNodes_eq_def isLeaf_pt_def null_comp_def) | |
| qed | |
| then show ?thesis | |
| by (rule P_eq_list_filter) | |
| qed | |
| with repb_Sucn_repa_nl_hd hd_Sucn_hd_whole_list | |
| have filter_nl_no_no1: | |
| "hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no low high repa] | |
| = hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no1 low high repa]" | |
| by simp | |
| from no_in_nl have filter_no_not_empty: | |
| "[sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no low high repa] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| from no1_in_nl have filter_no1_not_empty: | |
| "[sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no1 low high repa] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| from repb_no_def hd_Sucn_hd_whole_list hd_nl_repb_repa | |
| have "repb no = | |
| hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no low high repa]" | |
| by simp | |
| with hd_filter_prop [OF filter_no_not_empty ] | |
| have repNodes_no_repa: "repNodes_eq (repb no) no low high repa" | |
| by auto | |
| from repb_no1_def no1_nln | |
| have | |
| "repb no1 = hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no1 | |
| low high repa]" | |
| by simp | |
| with hd_filter_prop [OF filter_no1_not_empty ] | |
| have "repNodes_eq (repb no1) no1 low high repa" | |
| by auto | |
| with filter_nl_no_no1 repNodes_no_repa repb_no_no1_eq | |
| have "(repa \<propto> high) no1 = | |
| (repa \<propto> high) no \<and> (repa \<propto> low) no1 = (repa \<propto> low) no" | |
| by (simp add: repNodes_eq_def) | |
| with hno_nNull no1_props no1_nLeaf lno_nNull lno_notin_nl | |
| hno_notin_nl nodes_notin_nl_neq_nln repa_repb_nc | |
| show "(repb \<propto> high) no1 = | |
| (repb \<propto> high) no \<and> (repb \<propto> low) no1 = (repb \<propto> low) no" | |
| using [[simp_depth_limit=1]] | |
| by (auto simp add: isLeaf_pt_def null_comp_def) | |
| qed | |
| qed | |
| qed | |
| with repb_repb_no repb_no_share_def share_case_repb no_in_take_Sucn | |
| show ?thesis | |
| using [[simp_depth_limit=1]] | |
| by auto | |
| qed | |
| qed | |
| with repb_no_nNull show ?thesis | |
| by simp | |
| next | |
| assume no_nln: "no = node" | |
| with repb_node have repb_no_def: | |
| "repb no = hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no low high repa]" | |
| by simp | |
| from no_nln have "no \<in> set (prx@node#sfx)" | |
| by auto | |
| then have filter_nl_repa_not_empty: | |
| "[sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no low high repa] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then have hd_filter_nl_in_nl: | |
| "hd [sn\<leftarrow>(prx@node#sfx). repNodes_eq sn no low high repa] \<in> set (prx@node#sfx)" | |
| by (rule hd_filter_in_list) | |
| with repb_no_def | |
| have repb_no_in_nodeslist: "repb no \<in> set (prx@node#sfx)" | |
| by simp | |
| from nodes_balanced_ordered [rule_format,OF this] | |
| have repb_no_nNull: "repb no \<noteq> Null" | |
| by auto | |
| from share_cond no_nln have share_cond_or: | |
| "isLeaf_pt no low high \<or> repa (low no) \<noteq> repa (high no)" | |
| by auto | |
| have share_reduce_if: " (if (repb \<propto> low) no = (repb \<propto> high) no \<and> low no \<noteq> Null | |
| then repb no = (repb \<propto> low) no | |
| else repb no = hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb] \<and> | |
| repb (repb no) = repb no | |
| \<and> (\<forall>no1\<in>set (prx@[node]). ((repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no) = (repb no = repb no1)))" | |
| proof (cases "isLeaf_pt no low high") | |
| case True | |
| note isLeaf_no=this | |
| then have lno_Null: "low no = Null" by (simp add: isLeaf_pt_def) | |
| from isLeaf_no no_in_take_Sucn repb_no_share_def | |
| have repb_no_repb_def: "repb no | |
| = hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| by (auto simp add: isLeaf_pt_def) | |
| from isLeaf_no nodes_balanced_ordered [rule_format, OF no_in_nl] | |
| have var_no: "var no \<le> 1" | |
| by auto | |
| have all_nodes_nl_var_l_1: "\<forall>x \<in> set (prx@node#sfx). var x \<le> 1" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nl: " x \<in> set (prx@node#sfx)" | |
| from all_nodes_same_var [rule_format, OF x_in_nl no_in_nl] var_no | |
| show " var x \<le> 1" | |
| by auto | |
| qed | |
| have all_nodes_nl_Leafs: "\<forall>x \<in> set (prx@node#sfx). isLeaf_pt x low high" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nl: " x \<in> set (prx@node#sfx)" | |
| with all_nodes_nl_var_l_1 have "var x \<le> 1" | |
| by auto | |
| with nodes_balanced_ordered [rule_format, OF x_in_nl ] | |
| show "isLeaf_pt x low high" | |
| by auto | |
| qed | |
| have repb_repb_no: "repb (repb no) = repb no" | |
| proof - | |
| from repb_no_share_def no_in_take_Sucn lno_Null | |
| have repb_no_def: " repb no = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| by auto | |
| with hd_filter_Sucn_in_Sucn | |
| have repb_no_in_take_Sucn: "repb no \<in> set (prx@[node])" | |
| by simp | |
| hence repb_no_in_nl: "repb no \<in> set (prx@[node])" | |
| by auto | |
| with all_nodes_nl_Leafs | |
| have repb_no_Leaf: "isLeaf_pt (repb no) low high" | |
| by auto | |
| with repb_no_in_take_Sucn repb_no_share_def | |
| have repb_repb_no_def: "repb (repb no) = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn (repb no) low high repb] " | |
| by (auto simp add: isLeaf_pt_def) | |
| from filter_take_Sucn_not_empty | |
| show ?thesis | |
| apply (simp only: repb_repb_no_def ) | |
| apply (simp only: repb_no_def) | |
| apply (rule filter_hd_P_rep_indep) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| qed | |
| have two_nodes_repb: "(\<forall>no1\<in>set (prx@[node]). | |
| ((repb \<propto> high) no1 = (repb \<propto> high) no \<and> | |
| (repb \<propto> low) no1 = (repb \<propto> low) no) = (repb no = repb no1))" | |
| proof (intro ballI) | |
| fix no1 | |
| assume no1_in_take_Sucn: "no1 \<in> set (prx@[node])" | |
| from no1_in_take_Sucn | |
| have "no1 \<in> set (prx@node#sfx)" | |
| by auto | |
| with all_nodes_nl_Leafs | |
| have isLeaf_no1: "isLeaf_pt no1 low high" | |
| by auto | |
| with repb_no_share_def no1_in_take_Sucn | |
| have repb_no1_def: "repb no1 = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no1 low high repb]" | |
| by (auto simp add: isLeaf_pt_def) | |
| show "((repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no) = (repb no = repb no1)" | |
| proof | |
| assume repbchildren_eq_no1_no: "(repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no" | |
| have "[sn\<leftarrow>(prx@[node]). repNodes_eq sn no1 low high repb] | |
| = [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| proof - | |
| have "\<forall>x \<in> set (prx@[node]). | |
| repNodes_eq x no1 low high repb = repNodes_eq x no low high repb" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_take_Sucn: " x \<in> set (prx@[node])" | |
| with repbchildren_eq_no1_no | |
| show " repNodes_eq x no1 low high repb = repNodes_eq x no low high repb" | |
| by (simp add: repNodes_eq_def) | |
| qed | |
| then show ?thesis | |
| by (rule P_eq_list_filter) | |
| qed | |
| with repb_no1_def repb_no_repb_def | |
| show "repb no = repb no1" | |
| by simp | |
| next | |
| assume repb_no_no1: "repb no = repb no1" | |
| with isLeaf_no isLeaf_no1 | |
| show "(repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no" | |
| by (simp add: null_comp_def isLeaf_pt_def) | |
| qed | |
| qed | |
| with repb_repb_no lno_Null no_in_take_Sucn repb_no_share_def show ?thesis | |
| by auto | |
| next | |
| assume no_nLeaf: "\<not> isLeaf_pt no low high" | |
| with balanced_no obtain | |
| lno_nNull: "low no \<noteq> Null" and | |
| hno_nNull: "high no \<noteq> Null" | |
| by (simp add: isLeaf_pt_def) | |
| from no_nLeaf nodes_balanced_ordered [rule_format, OF no_in_nl] | |
| have var_no: "1 < var no" | |
| by auto | |
| have all_nodes_nl_var_l_1: "\<forall>x \<in> set (prx@node#sfx). 1 < var x" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nl: " x \<in> set (prx@node#sfx)" | |
| with all_nodes_same_var [rule_format, OF x_in_nl no_in_nl] var_no | |
| show "1 < var x" | |
| by simp | |
| qed | |
| have all_nodes_nl_nLeafs: "\<forall> x \<in> set (prx@node#sfx). \<not> isLeaf_pt x low high" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_nl: " x \<in> set (prx@node#sfx)" | |
| with all_nodes_nl_var_l_1 have "1 < var x" | |
| by auto | |
| with nodes_balanced_ordered [rule_format, OF x_in_nl] show " \<not> isLeaf_pt x low high" | |
| by auto | |
| qed | |
| from no_nLeaf share_cond_or | |
| have repachildren_neq_no: "repa (low no) \<noteq> repa (high no)" | |
| by auto | |
| with lno_nNull hno_nNull | |
| have "(repa \<propto> low) no \<noteq> (repa \<propto> high) no" | |
| by (simp add: null_comp_def) | |
| with repa_repb_nc lno_notin_nl hno_notin_nl | |
| nodes_notin_nl_neq_nln lno_nNull hno_nNull | |
| have repbchildren_neq_no: "(repb \<propto> low) no \<noteq> (repb \<propto> high) no" | |
| using [[simp_depth_limit=1]] | |
| by (auto simp add: null_comp_def) | |
| have repb_repb_no: "repb (repb no) = repb no" | |
| proof - | |
| from repb_no_share_def no_in_take_Sucn repbchildren_neq_no | |
| have repb_no_def: "repb no = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| by auto | |
| from filter_take_Sucn_not_empty | |
| have "repNodes_eq (repb no) no low high repb" | |
| apply (simp only: repb_no_def) | |
| apply (rule hd_filter_prop) | |
| apply simp | |
| done | |
| with repbchildren_neq_no | |
| have repbchildren_neq_repb_no: "(repb \<propto> low) (repb no) \<noteq> (repb \<propto> high) (repb no)" | |
| by (simp add: repNodes_eq_def) | |
| from filter_take_Sucn_not_empty | |
| have "repb no \<in> set (prx@[node])" | |
| apply (simp only: repb_no_def ) | |
| apply (rule hd_filter_in_list) | |
| apply simp | |
| done | |
| with repbchildren_neq_repb_no repb_no_share_def | |
| have repb_repb_no_def: " repb (repb no) = | |
| hd [sn\<leftarrow>(prx@[node]) . repNodes_eq sn (repb no) low high repb] " | |
| by auto | |
| from filter_take_Sucn_not_empty show ?thesis | |
| apply (simp only: repb_repb_no_def ) | |
| apply (simp only: repb_no_def) | |
| apply (rule filter_hd_P_rep_indep) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| qed | |
| have two_nodes_repb: "(\<forall>no1\<in>set (prx@[node]). | |
| ((repb \<propto> high) no1 = (repb \<propto> high) no \<and> | |
| (repb \<propto> low) no1 = (repb \<propto> low) no) = (repb no = repb no1))" | |
| (is "(\<forall>no1\<in>set (prx@[node]). ?P no1)") | |
| proof (intro ballI) | |
| fix no1 | |
| assume no1_in_take_Sucn: " no1 \<in> set (prx@[node])" | |
| hence no1_in_nodeslist: "no1 \<in> set (prx@node#sfx)" | |
| by auto | |
| with all_nodes_nl_nLeafs | |
| have no1_nLeaf: "\<not> isLeaf_pt no1 low high" | |
| by auto | |
| show "?P no1" | |
| proof | |
| assume repbchildren_eq_no1_no: "(repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no" | |
| with repbchildren_neq_no have "(repb \<propto> high) no1 \<noteq> (repb \<propto> low) no1" | |
| by auto | |
| with no1_in_take_Sucn repb_no_share_def have repb_no1_def: "repb no1 = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no1 low high repb]" | |
| by auto | |
| from repb_no_share_def no_in_take_Sucn repbchildren_neq_no | |
| have repb_no_def: "repb no = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| by auto | |
| have "[sn\<leftarrow>(prx@[node]). repNodes_eq sn no1 low high repb] = | |
| [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| proof - | |
| have "\<forall> x \<in> set (prx@[node]). | |
| repNodes_eq x no1 low high repb = repNodes_eq x no low high repb" | |
| proof (intro ballI) | |
| fix x | |
| assume x_in_take_Sucn: " x \<in> set (prx@[node])" | |
| with repbchildren_eq_no1_no | |
| show " repNodes_eq x no1 low high repb = repNodes_eq x no low high repb" | |
| by (simp add: repNodes_eq_def) | |
| qed | |
| then show ?thesis | |
| by (rule P_eq_list_filter) | |
| qed | |
| with repb_no_def repb_no1_def show " repb no = repb no1" | |
| by simp | |
| next | |
| assume repb_no_no1: "repb no = repb no1" | |
| from repb_no_share_def no_in_take_Sucn repbchildren_neq_no | |
| have repb_no_def: "repb no = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no low high repb]" | |
| by auto | |
| from filter_take_Sucn_not_empty | |
| have "repb no \<in> set (prx@[node])" | |
| apply (simp only: repb_no_def) | |
| apply (rule hd_filter_in_list) | |
| apply simp | |
| done | |
| then have repb_no_in_nl: "repb no \<in> set (prx@node#sfx)" | |
| by auto | |
| from filter_take_Sucn_not_empty | |
| have repNodes_repb_no: "repNodes_eq (repb no) no low high repb" | |
| apply (simp only: repb_no_def) | |
| apply (rule hd_filter_prop) | |
| apply simp | |
| done | |
| show "(repb \<propto> high) no1 = (repb \<propto> high) no | |
| \<and> (repb \<propto> low) no1 = (repb \<propto> low) no" | |
| proof (cases "(repb \<propto> low) no1 = (repb \<propto> high) no1") | |
| case True | |
| note red_cond=this | |
| from no1_in_nodeslist all_nodes_nl_nLeafs | |
| have no1_nLeaf: "\<not> isLeaf_pt no1 low high" | |
| by auto | |
| from nodes_balanced_ordered [rule_format, OF no1_in_nodeslist] | |
| have no1_props: "(low no1 \<notin> set (prx@node#sfx)) | |
| \<and> (high no1 \<notin> set (prx@node#sfx)) \<and>(low no1 = Null) = (high no1 = Null) | |
| \<and> ((rep \<propto> low) no1 \<notin> set (prx@node#sfx))" | |
| by auto | |
| with red_cond no1_nLeaf no1_in_take_Sucn repb_no_red_def | |
| have repb_no1_def: "repb no1 = (repb \<propto> low) no1" | |
| by (auto simp add: isLeaf_pt_def) | |
| with no1_nLeaf no1_props have "repb no1 = repb (low no1)" | |
| by (simp add: null_comp_def isLeaf_pt_def) | |
| from no1_props no1_nLeaf have "rep (low no1) \<notin> set (prx@node#sfx)" | |
| by (auto simp add: isLeaf_pt_def null_comp_def) | |
| with rep_repb_nc no1_props | |
| have "repb (low no1) \<notin> set (prx@node#sfx)" | |
| by auto | |
| with repb_no1_def repb_no_no1 no1_props no1_nLeaf | |
| have "repb no \<notin> set (prx@node#sfx)" | |
| by (simp add: isLeaf_pt_def null_comp_def) | |
| with repb_no_in_nl show ?thesis | |
| by simp | |
| next | |
| assume "(repb \<propto> low) no1 \<noteq> (repb \<propto> high) no1" | |
| with repb_no_share_def no1_in_take_Sucn | |
| have repb_no1_def: " repb no1 = | |
| hd [sn\<leftarrow>(prx@[node]). repNodes_eq sn no1 low high repb]" | |
| by auto | |
| from no1_in_take_Sucn | |
| have "[sn\<leftarrow>(prx@[node]). repNodes_eq sn no1 low high repb] \<noteq> []" | |
| apply - | |
| apply (rule filter_not_empty) | |
| apply (auto simp add: repNodes_eq_def) | |
| done | |
| then | |
| have repNodes_repb_no1: "repNodes_eq (repb no1) no1 low high repb" | |
| apply (simp only: repb_no1_def ) | |
| apply (rule hd_filter_prop) | |
| apply simp | |
| done | |
| with repNodes_repb_no repb_no_no1 | |
| have "repNodes_eq no1 no low high repb" | |
| by (simp add: repNodes_eq_def) | |
| then show ?thesis | |
| by (simp add: repNodes_eq_def) | |
| qed | |
| qed | |
| qed | |
| with repb_repb_no repb_no_share_def no_in_take_Sucn repbchildren_neq_no | |
| show ?thesis | |
| using [[simp_depth_limit=2]] | |
| by fastforce | |
| qed | |
| with repb_no_nNull show ?thesis | |
| by simp | |
| qed | |
| qed | |
| with rep_repb_nc show ?thesis | |
| by (intro conjI) | |
| qed | |
| qed | |
| end | |