59 lines
1.8 KiB
Plaintext
59 lines
1.8 KiB
Plaintext
(*
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* Copyright 2014, General Dynamics C4 Systems
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*
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* SPDX-License-Identifier: GPL-2.0-only
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*)
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(* Title: Confinement_S
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* Description: Rephrasing of the confinement proof using the concept of islands.
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*)
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theory Islands_S
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imports Confine_S
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begin
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definition
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island :: "state \<Rightarrow> entity_id \<Rightarrow> entity_id set" where
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"island s x \<equiv> {e\<^sub>i. s \<turnstile> x \<leftrightarrow>* e\<^sub>i}"
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definition
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island_caps :: "state \<Rightarrow> entity_id \<Rightarrow> cap set" where
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"island_caps s x \<equiv> \<Union>(caps_of s ` island s x)"
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lemma island_caps_def2:
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"island_caps s x \<equiv> \<Union> e \<in> island s x. caps_of s e"
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by(simp add: island_caps_def)
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lemma island_caps_def3:
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"island_caps s x = \<Union>(direct_caps_of s ` island s x)"
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apply (clarsimp simp: island_caps_def)
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apply rule
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apply (clarsimp simp: island_def caps_of_def)
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apply (drule store_connected_directly_tgs_connected)
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apply (metis directly_tgs_connected_rtrancl_into_rtrancl)
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apply (fastforce simp: caps_of_def store_connected_def)
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done
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lemma island_caps_dom:
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"island_caps s e\<^sub>x \<le>cap c =
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(\<forall>e\<^sub>i. (e\<^sub>x, e\<^sub>i) \<in> tgs_connected s \<longrightarrow> caps_of s e\<^sub>i \<le>cap c)"
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by (auto simp add: island_caps_def caps_dominated_by_def island_def)
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lemma authority_confinement_islands:
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"\<lbrakk>s' \<in> execute cmds s;
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island_caps s x \<le>cap c\<rbrakk>
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\<Longrightarrow> island_caps s' x \<le>cap c"
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apply (simp add: island_caps_dom)
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apply clarsimp
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apply (frule (1) tgs_connected_preserved)
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apply (subst (asm) tgs_connected_comm_eq)
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apply (erule authority_confinement)
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apply clarsimp
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apply (erule_tac x=e\<^sub>i' in allE)
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apply (erule impE)
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apply (metis (opaque_lifting, no_types) tgs_connected_comm_eq tgs_connected_def rtrancl_trans)
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apply clarsimp
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done
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end
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