81 lines
3.1 KiB
Plaintext
81 lines
3.1 KiB
Plaintext
(*
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* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
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*
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* SPDX-License-Identifier: BSD-2-Clause
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*)
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theory Sep_Attribs
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imports Separation_Algebra Sep_Tactic_Helpers
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begin
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text\<open>Beyond the tactics above, there is also a set of attributes implemented to make proving
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things in separation logic easier. These rules should be considered internals and are not
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intended for direct use.\<close>
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lemma sep_curry_atomised: "\<lbrakk>(\<And>s. (P \<and>* Q) s \<longrightarrow> R s); P s \<rbrakk> \<Longrightarrow> (Q \<longrightarrow>* R) s"
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by (clarsimp simp: sep_conj_sep_impl)
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lemma sep_remove_pure_imp_sep_imp: "( P \<longrightarrow>* (\<lambda>s. P' \<longrightarrow> Q s)) s \<Longrightarrow> P' \<Longrightarrow> (P \<longrightarrow>* Q) s"
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by (clarsimp)
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lemma sep_backward: "\<lbrakk>\<And>s. P s \<longrightarrow> (Q \<and>* T) s; (P \<and>* (Q \<longrightarrow>* R)) s \<rbrakk> \<Longrightarrow> (T \<and>* R) s"
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by (metis sep_conj_commute sep_conj_impl1 sep_mp_frame)
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lemma sep_remove_conj: "\<lbrakk>(P \<and>* R) s ; Q\<rbrakk> \<Longrightarrow> ((\<lambda>s. P s \<and> Q) \<and>* R) s "
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apply (clarsimp)
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done
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lemma curry: "(P \<longrightarrow> Q \<longrightarrow> R) \<Longrightarrow> (P \<and> Q) \<longrightarrow> R"
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apply (safe)
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done
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ML \<open>
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local
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fun atomize_thm ctxt thm = Conv.fconv_rule (Object_Logic.atomize ctxt) thm
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fun setup_simpset ctxt = put_simpset HOL_basic_ss ctxt addsimps [(sym OF [@{thm sep_conj_assoc}])]
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fun simp ctxt thm = simplify (setup_simpset ctxt) thm
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fun REPEAT_TRYOF_N _ thm2 0 = thm2
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| REPEAT_TRYOF_N thm1 thm2 n = REPEAT_TRYOF_N thm1 (thm1 OF [thm2]) (n-1)
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fun REPEAT_TRYOF'_N thm1 _ 0 = thm1
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| REPEAT_TRYOF'_N thm1 thm2 n = REPEAT_TRYOF'_N (thm1 OF [thm2]) thm2 (n-1)
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fun attribute_thm ctxt thm thm' =
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REPEAT_TRYOF_N @{thm sep_remove_pure_imp_sep_imp} (thm OF [atomize_thm ctxt thm']) (Thm.nprems_of thm' - 1)
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fun attribute_thm' thm ctxt thm' =
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thm OF [REPEAT_TRYOF_N @{thm curry} (thm' |> atomize_thm ctxt o simp ctxt) (Thm.nprems_of thm' - 1)]
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in
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(*
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By attributing a theorem with [sep_curry], we can now take a rule (A \<and>* B) \<Longrightarrow> C and turn it into A \<Longrightarrow> (B \<longrightarrow>* C)
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*)
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fun sep_curry_inner ctxt = attribute_thm ( ctxt) @{thm sep_curry_atomised}
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val sep_curry = Thm.rule_attribute [] (fn ctxt => sep_curry_inner (Context.proof_of ctxt))
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(*
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The attribute sep_back takes a rule of the form A \<Longrightarrow> B and returns a rule (A \<and>* (B \<longrightarrow>* R)) \<Longrightarrow> R.
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The R then matches with any conclusion. If the theorem is of form (A \<and>* B) \<Longrightarrow> C, it is advised to
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use sep_curry on the theorem first, and then sep_back. This aids sep_cancel in simplifying the result.
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*)
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fun backward ctxt thm =
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REPEAT_TRYOF'_N (attribute_thm' @{thm sep_backward} ctxt thm) @{thm sep_remove_conj} (Thm.nprems_of thm - 1)
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fun backward' ctxt thm = backward (Context.proof_of ctxt) thm
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val sep_backward = Thm.rule_attribute [] (backward')
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end
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\<close>
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attribute_setup sep_curry = \<open>Scan.succeed sep_curry\<close>
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attribute_setup sep_backward = \<open>Scan.succeed sep_backward\<close>
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end
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