74 lines
2.9 KiB
Plaintext
74 lines
2.9 KiB
Plaintext
theory Snapshot
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imports Protocol
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begin
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definition private_frames :: "nat \<Rightarrow> nat \<Rightarrow> pager \<Rightarrow> bool" where
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"private_frames pages slots s \<longleftrightarrow> pages \<le> frontier s \<and>
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(\<forall>k<slots. writable s k \<longrightarrow> pages \<le> ptable s k)"
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lemma initial_private: "private_frames pages slots (initial pages contents)"
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by (simp add: private_frames_def)
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lemma prepare_private:
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"private_frames pages slots s \<Longrightarrow> private_frames pages slots (prepare s k)"
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by (auto simp: private_frames_def prepare_def split: if_splits)
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lemma prepared_frame_private:
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"private_frames pages slots s \<Longrightarrow> k < slots \<Longrightarrow>
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pages \<le> ptable (prepare s k) k"
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by (auto simp: private_frames_def prepare_def)
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lemma store_private:
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"private_frames pages slots s \<Longrightarrow> private_frames pages slots (store s k offset byte)"
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using prepare_private[of pages slots s k]
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by (simp add: private_frames_def)
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lemma prepare_original:
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"private_frames pages slots s \<Longrightarrow> f < pages \<Longrightarrow>
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physical (prepare s k) f = physical s f"
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by (auto simp: private_frames_def prepare_def detach_def)
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theorem store_original:
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assumes owned: "private_frames pages slots s" and valid: "k < slots" and original: "f < pages"
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shows "physical (store s k offset byte) f = physical s f"
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proof -
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have distinct: "f \<noteq> ptable (prepare s k) k"
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using prepared_frame_private[OF owned valid] original by arith
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have preserved: "physical (prepare s k) f = physical s f"
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using prepare_original[OF owned original] .
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show ?thesis
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using distinct preserved by (simp add: store_def Let_def)
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qed
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lemma advance_private:
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"private_frames pages slots s \<Longrightarrow> private_frames pages slots (advance s a)"
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by (cases a; simp add: store_private)
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lemma advance_original:
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"private_frames pages slots s \<Longrightarrow> valid slots bytes a \<Longrightarrow> f < pages \<Longrightarrow>
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physical (advance s a) f = physical s f"
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by (cases a; auto intro: store_original)
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theorem execute_original:
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"private_frames pages slots s \<Longrightarrow> list_all (valid slots bytes) actions \<Longrightarrow>
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f < pages \<Longrightarrow> physical (execute s actions) f = physical s f"
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proof (induction actions arbitrary: s)
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case Nil
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then show ?case by simp
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next
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case (Cons a actions)
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have state: "private_frames pages slots (advance s a)"
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using Cons.prems by (simp add: advance_private)
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have tail: "physical (execute (advance s a) actions) f = physical (advance s a) f"
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using Cons.IH[OF state] Cons.prems by simp
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show ?case
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using tail advance_original Cons.prems by auto
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qed
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corollary original_preserved:
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"list_all (valid slots bytes) actions \<Longrightarrow> f < pages \<Longrightarrow>
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physical (execute (initial pages contents) actions) f = contents f"
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using execute_original[OF initial_private] by simp
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end
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