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theory TM_Assemble
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imports Hoare_tm StateMonad AList
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"~~/src/HOL/Library/FinFun_Syntax"
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"~~/src/HOL/Library/Sublist"
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LetElim
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begin
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section {* The assembler based on Benton's x86 paper *}
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text {*
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The problem with the assembler is that it is too slow to be useful.
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*}
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primrec pass1 :: "tpg \<Rightarrow> (unit, (nat \<times> nat \<times> (nat \<rightharpoonup> nat))) SM"
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where
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"pass1 (TInstr ai) = sm_map (\<lambda> (pos, lno, lmap). (pos + 1, lno, lmap))" |
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"pass1 (TSeq p1 p2) = do {pass1 p1; pass1 p2 }" |
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"pass1 (TLocal fp) = do { lno \<leftarrow> tap (\<lambda> (pos, lno, lmap). lno);
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sm_map (\<lambda> (pos, lno, lmap). (pos, lno+1, lmap));
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pass1 (fp lno) }" |
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"pass1 (TLabel l) = sm_map ((\<lambda> (pos, lno, lmap). (pos, lno, lmap(l \<mapsto> pos))))"
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declare pass1.simps[simp del]
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type_synonym ('a, 'b) alist = "('a \<times> 'b) list"
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primrec pass2 :: "tpg \<Rightarrow> (nat \<rightharpoonup> nat) \<Rightarrow> (unit, (nat \<times> nat \<times> (nat, tm_inst) alist)) SM"
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where
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"pass2 (TInstr ai) lmap = sm_map (\<lambda> (pos, lno, prog). (pos + 1, lno, (pos, ai)#prog))" |
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"pass2 (TSeq p1 p2) lmap = do {pass2 p1 lmap; pass2 p2 lmap}" |
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"pass2 (TLocal fp) lmap = do { lno \<leftarrow> tap (\<lambda> (pos, lno, prog). lno);
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sm_map (\<lambda> (pos, lno, prog). (pos, lno + 1, prog));
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(case (lmap lno) of
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Some l => pass2 (fp l) lmap |
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None => (raise ''Undefined label''))} " |
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"pass2 (TLabel l) lmap = do { pos \<leftarrow> tap (\<lambda> (pos, lno, prog). pos);
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if (l = pos) then return ()
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else (raise ''Label mismatch'') }"
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declare pass2.simps[simp del]
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definition "assembleM i tpg =
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do {(x, (pos, lno, lmap)) \<leftarrow> execute (pass1 tpg) (i, 0, empty);
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execute (pass2 tpg lmap) (i, 0, [])}"
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definition
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"assemble i tpg = Option.map (\<lambda> (x, (j, lno, prog)).(prog, j)) (assembleM i tpg)"
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lemma tprog_set_union:
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assumes "(fst ` set pg3) \<inter> (fst ` set pg2) = {}"
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shows "tprog_set (map_of pg3 ++ map_of pg2) = tprog_set (map_of pg3) \<union> tprog_set (map_of pg2)"
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proof -
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from assms have "dom (map_of pg3) \<inter> dom (map_of pg2) = {}"
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by (metis dom_map_of_conv_image_fst)
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hence map_comm: "map_of pg3 ++ map_of pg2 = map_of pg2 ++ map_of pg3"
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by (metis map_add_comm)
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show ?thesis
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proof
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show "tprog_set (map_of pg3 ++ map_of pg2) \<subseteq> tprog_set (map_of pg3) \<union> tprog_set (map_of pg2)"
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proof
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fix x
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assume " x \<in> tprog_set (map_of pg3 ++ map_of pg2)"
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then obtain i inst where h:
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"x = TC i inst"
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"(map_of pg3 ++ map_of pg2) i = Some inst"
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apply (unfold tprog_set_def)
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by (smt mem_Collect_eq)
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from map_add_SomeD[OF h(2)] h(1)
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show " x \<in> tprog_set (map_of pg3) \<union> tprog_set (map_of pg2)"
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apply (unfold tprog_set_def)
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by (smt mem_Collect_eq sup1CI sup_Un_eq)
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qed
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next
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show "tprog_set (map_of pg3) \<union> tprog_set (map_of pg2) \<subseteq> tprog_set (map_of pg3 ++ map_of pg2)"
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proof
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fix x
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assume " x \<in> tprog_set (map_of pg3) \<union> tprog_set (map_of pg2)"
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then obtain i inst
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where h: "x = TC i inst" "map_of pg3 i = Some inst \<or> map_of pg2 i = Some inst"
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apply (unfold tprog_set_def)
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by (smt Un_iff mem_Collect_eq)
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from h(2)
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show "x \<in> tprog_set (map_of pg3 ++ map_of pg2)"
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proof
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assume "map_of pg2 i = Some inst"
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hence "(map_of pg3 ++ map_of pg2) i = Some inst"
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by (metis map_add_find_right)
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with h(1) show ?thesis
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apply (unfold tprog_set_def)
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by (smt mem_Collect_eq)
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next
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assume "map_of pg3 i = Some inst"
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hence "(map_of pg2 ++ map_of pg3) i = Some inst"
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by (metis map_add_find_right)
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with h(1) show ?thesis
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apply (unfold map_comm)
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apply (unfold tprog_set_def)
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by (smt mem_Collect_eq)
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qed
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qed
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qed
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qed
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lemma assumes "assemble i c = Some (prog, j)"
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shows "(i:[c]:j) (tprog_set (map_of prog))"
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proof -
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from assms obtain x lno
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where "(assembleM i c) = Some (x, (j, lno, prog))"
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apply(unfold assemble_def)
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by (cases "(assembleM i c)", auto)
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then obtain y pos lno' lmap where
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"execute (pass1 c) (i, 0, empty) = Some (y, (pos, lno', lmap))"
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"execute (pass2 c lmap) (i, 0, []) = Some (x, (j, lno, prog))"
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apply (unfold assembleM_def)
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by (cases "execute (pass1 c) (i, 0, Map.empty)", auto simp:Option.bind.simps)
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hence mid: "effect (pass1 c) (i, 0, empty) (pos, lno', lmap) y"
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"effect (pass2 c lmap) (i, 0, []) (j, lno, prog) x"
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by (auto intro:effectI)
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{ fix lnos lmap lmap' prog1 prog2
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assume "effect (pass2 c lmap') (i, lnos, prog1) (j, lno, prog2) x"
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hence "\<exists> prog. (prog2 = prog@prog1 \<and> (i:[c]:j) (tprog_set (map_of prog)) \<and>
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(\<forall> k \<in> fst ` (set prog). i \<le> k \<and> k < j) \<and> i \<le> j)"
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proof(induct c arbitrary:lmap' i lnos prog1 j lno prog2 x)
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case (TInstr instr lmap' i lnos prog1 j lno prog2 x)
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thus ?case
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apply (auto simp: effect_def assemble_def assembleM_def execute.simps sm_map_def sm_def
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tprog_set_def tassemble_to.simps sg_def pass1.simps pass2.simps
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split:if_splits)
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by (cases instr, auto)
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next
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case (TLabel l lmap' i lnos prog1 j lno prog2 x)
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thus ?case
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apply (rule_tac x = "[]" in exI)
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apply (unfold tassemble_to.simps)
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by (auto simp: effect_def assemble_def assembleM_def execute.simps sm_map_def sm_def
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tprog_set_def tassemble_to.simps sg_def pass1.simps pass2.simps tap_def bind_def
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return_def raise_def sep_empty_def set_ins_def
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split:if_splits)
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next
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case (TSeq c1 c2 lmap' i lnos prog1 j lno prog2 x)
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from TSeq(3)
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obtain h' r where
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"effect (pass2 c1 lmap') (i, lnos, prog1) h' r"
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"effect (pass2 c2 lmap') h' (j, lno, prog2) x"
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apply (unfold pass2.simps)
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by (auto elim!:effect_elims)
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then obtain pos1 lno1 pg1
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where h:
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"effect (pass2 c1 lmap') (i, lnos, prog1) (pos1, lno1, pg1) r"
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"effect (pass2 c2 lmap') (pos1, lno1, pg1) (j, lno, prog2) x"
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by (cases h', auto)
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from TSeq(1)[OF h(1)] TSeq(2)[OF h(2)]
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obtain pg2 pg3
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where hh: "pg1 = pg2 @ prog1 \<and> (i :[ c1 ]: pos1) (tprog_set (map_of pg2))"
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"(\<forall>k\<in> fst ` (set pg2). i \<le> k \<and> k < pos1)"
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"i \<le> pos1"
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"prog2 = pg3 @ pg1 \<and> (pos1 :[ c2 ]: j) (tprog_set (map_of pg3))"
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"(\<forall>k\<in>fst ` (set pg3). pos1 \<le> k \<and> k < j)"
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"pos1 \<le> j"
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by auto
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thus ?case
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apply (rule_tac x = "pg3 @ pg2" in exI, auto)
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apply (unfold tassemble_to.simps)
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apply (rule_tac x = pos1 in EXS_intro)
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my_block have
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"(tprog_set (map_of pg2 ++ map_of pg3)) = tprog_set (map_of pg2) \<union> tprog_set (map_of pg3)"
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proof(rule tprog_set_union)
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from hh(2, 5) show "fst ` set pg2 \<inter> fst ` set pg3 = {}"
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by (smt disjoint_iff_not_equal)
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qed
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my_block_end
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apply (unfold this, insert this)
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my_block
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have "tprog_set (map_of pg2) \<inter> tprog_set (map_of pg3) = {}"
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proof -
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{ fix x
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assume h: "x \<in> tprog_set (map_of pg2)" "x \<in> tprog_set (map_of pg3)"
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then obtain i inst where "x = TC i inst"
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"map_of pg2 i = Some inst"
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"map_of pg3 i = Some inst"
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apply (unfold tprog_set_def)
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by (smt mem_Collect_eq tresource.inject(2))
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hence "(i, inst) \<in> set pg2" "(i, inst) \<in> set pg3"
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by (metis map_of_SomeD)+
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with hh(2, 5)
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have "False"
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by (smt rev_image_eqI)
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} thus ?thesis by auto
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qed
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my_block_end
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apply (insert this)
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apply (fold set_ins_def)
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by (rule sep_conjI, assumption+, simp)
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next
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case (TLocal body lmap' i lnos prog1 j lno prog2 x)
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from TLocal(2)
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obtain l where h:
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"lmap' lnos = Some l"
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"effect (pass2 (body l) lmap') (i, Suc lnos, prog1) (j, lno, prog2) ()"
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apply (unfold pass2.simps)
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by (auto elim!:effect_elims split:option.splits simp:sm_map_def)
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from TLocal(1)[OF this(2)]
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obtain pg where hh: "prog2 = pg @ prog1 \<and> (i :[ body l ]: j) (tprog_set (map_of pg))"
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"(\<forall>k\<in> fst ` (set pg). i \<le> k \<and> k < j)"
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"i \<le> j"
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by auto
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thus ?case
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apply (rule_tac x = pg in exI, auto)
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apply (unfold tassemble_to.simps)
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by (rule_tac x = l in EXS_intro, auto)
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qed
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} from this[OF mid(2)] show ?thesis by auto
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qed
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definition "valid_tpg tpg = (\<forall> i. \<exists> j prog. assemble i tpg = Some (j, prog))"
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section {* A new method based on DB indexing *}
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text {*
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In this section, we introduced a new method based on DB-indexing to provide a quick check of
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assemblebility of TM assmbly programs in the format of @{text "tpg"}. The
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lemma @{text "ct_left_until_zero"} at the end shows how the well-formedness of @{text "left_until_zero"}
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is proved in a modular way.
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*}
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datatype cpg =
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CInstr tm_inst
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| CLabel nat
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| CSeq cpg cpg
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| CLocal cpg
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datatype status = Free | Bound
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definition "lift_b t i j = (if (j \<ge> t) then (j + i) else j)"
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fun lift_t :: "nat \<Rightarrow> nat \<Rightarrow> cpg \<Rightarrow> cpg"
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where "lift_t t i (CInstr ((act0, l0), (act1, l1))) =
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(CInstr ((act0, lift_b t i l0), (act1, lift_b t i l1)))" |
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"lift_t t i (CLabel l) = CLabel (lift_b t i l)" |
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"lift_t t i (CSeq c1 c2) = CSeq (lift_t t i c1) (lift_t t i c2)" |
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"lift_t t i (CLocal c) = CLocal (lift_t (t + 1) i c)"
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definition "lift0 (i::nat) cpg = lift_t 0 i cpg"
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definition "perm_b t i j k = (if ((k::nat) = i \<and> i < t \<and> j < t) then j else
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if (k = j \<and> i < t \<and> j < t) then i else k)"
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lemma inj_perm_b: "inj (perm_b t i j)"
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proof(induct rule:injI)
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case (1 x y)
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thus ?case
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by (unfold perm_b_def, auto split:if_splits)
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qed
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fun perm :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> cpg \<Rightarrow> cpg"
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where "perm t i j (CInstr ((act0, l0), (act1, l1))) =
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(CInstr ((act0, perm_b t i j l0), (act1, perm_b t i j l1)))" |
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"perm t i j (CLabel l) = CLabel (perm_b t i j l)" |
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"perm t i j (CSeq c1 c2) = CSeq (perm t i j c1) (perm t i j c2)" |
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"perm t i j (CLocal c) = CLocal (perm (t + 1) (i + 1) (j + 1) c)"
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definition "map_idx f sts = map (\<lambda> k. sts!(f (nat k))) [0 .. int (length sts) - 1]"
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definition "perm_s i j sts = map_idx (perm_b (length sts) i j) sts"
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value "perm_s 2 5 [(0::int), 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]"
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lemma "perm_s 2 20 [(0::int), 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] = x"
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apply (unfold perm_s_def map_idx_def perm_b_def, simp add:upto.simps)
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oops
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lemma upto_len: "length [i .. j] = (if j < i then 0 else (nat (j - i + 1)))"
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proof(induct i j rule:upto.induct)
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case (1 i j)
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show ?case
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proof(cases "j < i")
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case True
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thus ?thesis by simp
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next
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case False
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hence eq_ij: "[i..j] = i # [i + 1..j]" by (simp add:upto.simps)
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|
|
284 |
from 1 False
|
|
|
285 |
show ?thesis
|
|
|
286 |
by (auto simp:eq_ij)
|
|
|
287 |
qed
|
|
|
288 |
qed
|
|
|
289 |
|
|
|
290 |
lemma perm_s_len: "length (perm_s i j sts') = length sts'"
|
|
|
291 |
apply (unfold perm_s_def map_idx_def)
|
|
|
292 |
by (smt Nil_is_map_conv length_0_conv length_greater_0_conv length_map neq_if_length_neq upto_len)
|
|
|
293 |
|
|
|
294 |
fun c2t :: "nat list \<Rightarrow> cpg \<Rightarrow> tpg" where
|
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|
295 |
"c2t env (CInstr ((act0, st0), (act1, st1))) = TInstr ((act0, env!st0), (act1, env!st1))" |
|
|
|
296 |
"c2t env (CLabel l) = TLabel (env!l)" |
|
|
|
297 |
"c2t env (CSeq cpg1 cpg2) = TSeq (c2t env cpg1) (c2t env cpg2)" |
|
|
|
298 |
"c2t env (CLocal cpg) = TLocal (\<lambda> x. c2t (x#env) cpg)"
|
|
|
299 |
|
|
|
300 |
instantiation status :: minus
|
|
|
301 |
begin
|
|
|
302 |
fun minus_status :: "status \<Rightarrow> status \<Rightarrow> status" where
|
|
|
303 |
"minus_status Bound Bound = Free" |
|
|
|
304 |
"minus_status Bound Free = Bound" |
|
|
|
305 |
"minus_status Free x = Free "
|
|
|
306 |
instance ..
|
|
|
307 |
end
|
|
|
308 |
|
|
|
309 |
instantiation status :: plus
|
|
|
310 |
begin
|
|
|
311 |
fun plus_status :: "status \<Rightarrow> status \<Rightarrow> status" where
|
|
|
312 |
"plus_status Free x = x" |
|
|
|
313 |
"plus_status Bound x = Bound"
|
|
|
314 |
instance ..
|
|
|
315 |
end
|
|
|
316 |
|
|
|
317 |
instantiation list :: (plus)plus
|
|
|
318 |
begin
|
|
|
319 |
fun plus_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
|
|
|
320 |
"plus_list [] ys = []" |
|
|
|
321 |
"plus_list (x#xs) [] = []" |
|
|
|
322 |
"plus_list (x#xs) (y#ys) = ((x + y)#plus_list xs ys)"
|
|
|
323 |
instance ..
|
|
|
324 |
end
|
|
|
325 |
|
|
|
326 |
instantiation list :: (minus)minus
|
|
|
327 |
begin
|
|
|
328 |
fun minus_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
|
|
|
329 |
"minus_list [] ys = []" |
|
|
|
330 |
"minus_list (x#xs) [] = []" |
|
|
|
331 |
"minus_list (x#xs) (y#ys) = ((x - y)#minus_list xs ys)"
|
|
|
332 |
instance ..
|
|
|
333 |
end
|
|
|
334 |
|
|
|
335 |
(* consts castr :: "nat list \<Rightarrow> nat \<Rightarrow> cpg \<Rightarrow> nat \<Rightarrow> tassert"
|
|
|
336 |
|
|
|
337 |
definition "castr env i cpg j = (i:[c2t env cpg]:j)" *)
|
|
|
338 |
|
|
|
339 |
(*
|
|
|
340 |
definition
|
|
|
341 |
"c2p sts i cpg j = (\<forall> x. ((length x = length sts \<and>
|
|
|
342 |
(\<forall> k < length sts. sts!k = Bound \<longrightarrow> (\<exists> f. x!k = f i)))
|
|
|
343 |
\<longrightarrow> (\<exists> s. (i:[(c2t x cpg)]:j) s)))"
|
|
|
344 |
*)
|
|
|
345 |
|
|
|
346 |
definition
|
|
|
347 |
"c2p sts i cpg j =
|
|
|
348 |
(\<exists> f. \<forall> x. ((length x = length sts \<and>
|
|
|
349 |
(\<forall> k < length sts. sts!k = Bound \<longrightarrow> (x!k = f i k)))
|
|
|
350 |
\<longrightarrow> (\<exists> s. (i:[(c2t x cpg)]:j) s)))"
|
|
|
351 |
|
|
|
352 |
fun wf_cpg_test :: "status list \<Rightarrow> cpg \<Rightarrow> (bool \<times> status list)" where
|
|
|
353 |
"wf_cpg_test sts (CInstr ((a0, l0), (a1, l1))) = ((l0 < length sts \<and> l1 < length sts), sts)" |
|
|
|
354 |
"wf_cpg_test sts (CLabel l) = ((l < length sts) \<and> sts!l = Free, sts[l:=Bound])" |
|
|
|
355 |
"wf_cpg_test sts (CSeq c1 c2) = (let (b1, sts1) = wf_cpg_test sts c1;
|
|
|
356 |
(b2, sts2) = wf_cpg_test sts1 c2 in
|
|
|
357 |
(b1 \<and> b2, sts2))" |
|
|
|
358 |
"wf_cpg_test sts (CLocal body) = (let (b, sts') = (wf_cpg_test (Free#sts) body) in
|
|
|
359 |
(b, tl sts'))"
|
|
|
360 |
|
|
|
361 |
instantiation status :: order
|
|
|
362 |
begin
|
|
|
363 |
definition less_eq_status_def: "((st1::status) \<le> st2) = (st1 = Free \<or> st2 = Bound)"
|
|
|
364 |
definition less_status_def: "((st1::status) < st2) = (st1 \<le> st2 \<and> st1 \<noteq> st2)"
|
|
|
365 |
instance
|
|
|
366 |
proof
|
|
|
367 |
fix x y
|
|
|
368 |
show "(x < (y::status)) = (x \<le> y \<and> \<not> y \<le> x)"
|
|
|
369 |
by (metis less_eq_status_def less_status_def status.distinct(1))
|
|
|
370 |
next
|
|
|
371 |
fix x show "x \<le> (x::status)"
|
|
|
372 |
by (metis (full_types) less_eq_status_def status.exhaust)
|
|
|
373 |
next
|
|
|
374 |
fix x y z
|
|
|
375 |
assume "x \<le> y" "y \<le> (z::status)" show "x \<le> (z::status)"
|
|
|
376 |
by (metis `x \<le> y` `y \<le> z` less_eq_status_def status.distinct(1))
|
|
|
377 |
next
|
|
|
378 |
fix x y
|
|
|
379 |
assume "x \<le> y" "y \<le> (x::status)" show "x = y"
|
|
|
380 |
by (metis `x \<le> y` `y \<le> x` less_eq_status_def status.distinct(1))
|
|
|
381 |
qed
|
|
|
382 |
end
|
|
|
383 |
|
|
|
384 |
instantiation list :: (order)order
|
|
|
385 |
begin
|
|
|
386 |
definition "((sts::('a::order) list) \<le> sts') =
|
|
|
387 |
((length sts = length sts') \<and> (\<forall> i < length sts. sts!i \<le> sts'!i))"
|
|
|
388 |
definition "((sts::('a::order) list) < sts') = ((sts \<le> sts') \<and> sts \<noteq> sts')"
|
|
|
389 |
|
|
|
390 |
lemma anti_sym: assumes h: "x \<le> (y::'a list)" "y \<le> x"
|
|
|
391 |
shows "x = y"
|
|
|
392 |
proof -
|
|
|
393 |
from h have "length x = length y"
|
|
|
394 |
by (metis less_eq_list_def)
|
|
|
395 |
moreover from h have " (\<forall> i < length x. x!i = y!i)"
|
|
|
396 |
by (metis (full_types) antisym_conv less_eq_list_def)
|
|
|
397 |
ultimately show ?thesis
|
|
|
398 |
by (metis nth_equalityI)
|
|
|
399 |
qed
|
|
|
400 |
|
|
|
401 |
lemma refl: "x \<le> (x::('a::order) list)"
|
|
|
402 |
apply (unfold less_eq_list_def)
|
|
|
403 |
by (metis order_refl)
|
|
|
404 |
|
|
|
405 |
instance
|
|
|
406 |
proof
|
|
|
407 |
fix x y
|
|
|
408 |
show "((x::('a::order) list) < y) = (x \<le> y \<and> \<not> y \<le> x)"
|
|
|
409 |
proof
|
|
|
410 |
assume h: "x \<le> y \<and> \<not> y \<le> x"
|
|
|
411 |
have "x \<noteq> y"
|
|
|
412 |
proof
|
|
|
413 |
assume "x = y" with h have "\<not> (x \<le> x)" by simp
|
|
|
414 |
with refl show False by auto
|
|
|
415 |
qed
|
|
|
416 |
moreover from h have "x \<le> y" by blast
|
|
|
417 |
ultimately show "x < y" by (auto simp:less_list_def)
|
|
|
418 |
next
|
|
|
419 |
assume h: "x < y"
|
|
|
420 |
hence hh: "x \<le> y"
|
|
|
421 |
by (metis TM_Assemble.less_list_def)
|
|
|
422 |
moreover have "\<not> y \<le> x"
|
|
|
423 |
proof
|
|
|
424 |
assume "y \<le> x"
|
|
|
425 |
from anti_sym[OF hh this] have "x = y" .
|
|
|
426 |
with h show False
|
|
|
427 |
by (metis less_list_def)
|
|
|
428 |
qed
|
|
|
429 |
ultimately show "x \<le> y \<and> \<not> y \<le> x" by auto
|
|
|
430 |
qed
|
|
|
431 |
next
|
|
|
432 |
fix x from refl show "(x::'a list) \<le> x" .
|
|
|
433 |
next
|
|
|
434 |
fix x y assume "(x::'a list) \<le> y" "y \<le> x"
|
|
|
435 |
from anti_sym[OF this] show "x = y" .
|
|
|
436 |
next
|
|
|
437 |
fix x y z
|
|
|
438 |
assume h: "(x::'a list) \<le> y" "y \<le> z"
|
|
|
439 |
show "x \<le> z"
|
|
|
440 |
proof -
|
|
|
441 |
from h have "length x = length z" by (metis TM_Assemble.less_eq_list_def)
|
|
|
442 |
moreover from h have "\<forall> i < length x. x!i \<le> z!i"
|
|
|
443 |
by (metis TM_Assemble.less_eq_list_def order_trans)
|
|
|
444 |
ultimately show "x \<le> z"
|
|
|
445 |
by (metis TM_Assemble.less_eq_list_def)
|
|
|
446 |
qed
|
|
|
447 |
qed
|
|
|
448 |
end
|
|
|
449 |
|
|
|
450 |
lemma sts_bound_le: "sts \<le> sts[l := Bound]"
|
|
|
451 |
proof -
|
|
|
452 |
have "length sts = length (sts[l := Bound])"
|
|
|
453 |
by (metis length_list_update)
|
|
|
454 |
moreover have "\<forall> i < length sts. sts!i \<le> (sts[l:=Bound])!i"
|
|
|
455 |
proof -
|
|
|
456 |
{ fix i
|
|
|
457 |
assume "i < length sts"
|
|
|
458 |
have "sts ! i \<le> sts[l := Bound] ! i"
|
|
|
459 |
proof(cases "l < length sts")
|
|
|
460 |
case True
|
|
|
461 |
note le_l = this
|
|
|
462 |
show ?thesis
|
|
|
463 |
proof(cases "l = i")
|
|
|
464 |
case True with le_l
|
|
|
465 |
have "sts[l := Bound] ! i = Bound" by auto
|
|
|
466 |
thus ?thesis by (metis less_eq_status_def)
|
|
|
467 |
next
|
|
|
468 |
case False
|
|
|
469 |
with le_l have "sts[l := Bound] ! i = sts!i" by auto
|
|
|
470 |
thus ?thesis by auto
|
|
|
471 |
qed
|
|
|
472 |
next
|
|
|
473 |
case False
|
|
|
474 |
hence "sts[l := Bound] = sts" by auto
|
|
|
475 |
thus ?thesis by auto
|
|
|
476 |
qed
|
|
|
477 |
} thus ?thesis by auto
|
|
|
478 |
qed
|
|
|
479 |
ultimately show ?thesis by (metis less_eq_list_def)
|
|
|
480 |
qed
|
|
|
481 |
|
|
|
482 |
lemma sts_tl_le:
|
|
|
483 |
assumes "sts \<le> sts'"
|
|
|
484 |
shows "tl sts \<le> tl sts'"
|
|
|
485 |
proof -
|
|
|
486 |
from assms have "length (tl sts) = length (tl sts')"
|
|
|
487 |
by (metis (hide_lams, no_types) length_tl less_eq_list_def)
|
|
|
488 |
moreover from assms have "\<forall> i < length (tl sts). (tl sts)!i \<le> (tl sts')!i"
|
|
|
489 |
by (smt calculation length_tl less_eq_list_def nth_tl)
|
|
|
490 |
ultimately show ?thesis
|
|
|
491 |
by (metis less_eq_list_def)
|
|
|
492 |
qed
|
|
|
493 |
|
|
|
494 |
lemma wf_cpg_test_le:
|
|
|
495 |
assumes "wf_cpg_test sts cpg = (True, sts')"
|
|
|
496 |
shows "sts \<le> sts'" using assms
|
|
|
497 |
proof(induct cpg arbitrary:sts sts')
|
|
|
498 |
case (CInstr instr sts sts')
|
|
|
499 |
obtain a0 l0 a1 l1 where eq_instr: "instr = ((a0, l0), (a1, l1))"
|
|
|
500 |
by (metis prod.exhaust)
|
|
|
501 |
from CInstr[unfolded this]
|
|
|
502 |
show ?case by simp
|
|
|
503 |
next
|
|
|
504 |
case (CLabel l sts sts')
|
|
|
505 |
thus ?case by (auto simp:sts_bound_le)
|
|
|
506 |
next
|
|
|
507 |
case (CLocal body sts sts')
|
|
|
508 |
from this(2)
|
|
|
509 |
obtain sts1 where h: "wf_cpg_test (Free # sts) body = (True, sts1)" "sts' = tl sts1"
|
|
|
510 |
by (auto split:prod.splits)
|
|
|
511 |
from CLocal(1)[OF this(1)] have "Free # sts \<le> sts1" .
|
|
|
512 |
from sts_tl_le[OF this]
|
|
|
513 |
have "sts \<le> tl sts1" by simp
|
|
|
514 |
from this[folded h(2)]
|
|
|
515 |
show ?case .
|
|
|
516 |
next
|
|
|
517 |
case (CSeq cpg1 cpg2 sts sts')
|
|
|
518 |
from this(3)
|
|
|
519 |
show ?case
|
|
|
520 |
by (auto split:prod.splits dest!:CSeq(1, 2))
|
|
|
521 |
qed
|
|
|
522 |
|
|
|
523 |
lemma c2p_assert:
|
|
|
524 |
assumes "(c2p [] i cpg j)"
|
|
|
525 |
shows "\<exists> s. (i :[(c2t [] cpg)]: j) s"
|
|
|
526 |
proof -
|
|
|
527 |
from assms obtain f where
|
|
|
528 |
h [rule_format]:
|
|
|
529 |
"\<forall> x. length x = length [] \<and> (\<forall>k<length []. [] ! k = Bound \<longrightarrow> (x ! k = f i k)) \<longrightarrow>
|
|
|
530 |
(\<exists> s. (i :[ c2t [] cpg ]: j) s)"
|
|
|
531 |
by (unfold c2p_def, auto)
|
|
|
532 |
have "length [] = length [] \<and> (\<forall>k<length []. [] ! k = Bound \<longrightarrow> ([] ! k = f i k))"
|
|
|
533 |
by auto
|
|
|
534 |
from h[OF this] obtain s where "(i :[ c2t [] cpg ]: j) s" by blast
|
|
|
535 |
thus ?thesis by auto
|
|
|
536 |
qed
|
|
|
537 |
|
|
|
538 |
definition "sts_disj sts sts' = ((length sts = length sts') \<and>
|
|
|
539 |
(\<forall> i < length sts. \<not>(sts!i = Bound \<and> sts'!i = Bound)))"
|
|
|
540 |
|
|
|
541 |
lemma length_sts_plus:
|
|
|
542 |
assumes "length (sts1 :: status list) = length sts2"
|
|
|
543 |
shows "length (sts1 + sts2) = length sts1"
|
|
|
544 |
using assms
|
|
|
545 |
proof(induct sts1 arbitrary: sts2)
|
|
|
546 |
case Nil
|
|
|
547 |
thus ?case
|
|
|
548 |
by (metis plus_list.simps(1))
|
|
|
549 |
next
|
|
|
550 |
case (Cons s' sts' sts2)
|
|
|
551 |
thus ?case
|
|
|
552 |
proof(cases "sts2 = []")
|
|
|
553 |
case True
|
|
|
554 |
with Cons show ?thesis by auto
|
|
|
555 |
next
|
|
|
556 |
case False
|
|
|
557 |
then obtain s'' sts'' where eq_sts2: "sts2 = s''#sts''"
|
|
|
558 |
by (metis neq_Nil_conv)
|
|
|
559 |
with Cons
|
|
|
560 |
show ?thesis
|
|
|
561 |
by (metis length_Suc_conv list.inject plus_list.simps(3))
|
|
|
562 |
qed
|
|
|
563 |
qed
|
|
|
564 |
|
|
|
565 |
|
|
|
566 |
lemma nth_sts_plus:
|
|
|
567 |
assumes "i < length ((sts1::status list) + sts2)"
|
|
|
568 |
shows "(sts1 + sts2)!i = sts1!i + sts2!i"
|
|
|
569 |
using assms
|
|
|
570 |
proof(induct sts1 arbitrary:i sts2)
|
|
|
571 |
case (Nil i sts2)
|
|
|
572 |
thus ?case by auto
|
|
|
573 |
next
|
|
|
574 |
case (Cons s' sts' i sts2)
|
|
|
575 |
show ?case
|
|
|
576 |
proof(cases "sts2 = []")
|
|
|
577 |
case True
|
|
|
578 |
with Cons show ?thesis by auto
|
|
|
579 |
next
|
|
|
580 |
case False
|
|
|
581 |
then obtain s'' sts'' where eq_sts2: "sts2 = s''#sts''"
|
|
|
582 |
by (metis neq_Nil_conv)
|
|
|
583 |
with Cons
|
|
|
584 |
show ?thesis
|
|
|
585 |
by (smt list.size(4) nth_Cons' plus_list.simps(3))
|
|
|
586 |
qed
|
|
|
587 |
qed
|
|
|
588 |
|
|
|
589 |
lemma nth_sts_minus:
|
|
|
590 |
assumes "i < length ((sts1::status list) - sts2)"
|
|
|
591 |
shows "(sts1 - sts2)!i = sts1!i - sts2!i"
|
|
|
592 |
using assms
|
|
|
593 |
proof(induct arbitrary:i rule:minus_list.induct)
|
|
|
594 |
case (3 x xs y ys i)
|
|
|
595 |
show ?case
|
|
|
596 |
proof(cases i)
|
|
|
597 |
case 0
|
|
|
598 |
thus ?thesis by simp
|
|
|
599 |
next
|
|
|
600 |
case (Suc k)
|
|
|
601 |
with 3(2) have "k < length (xs - ys)" by auto
|
|
|
602 |
from 3(1)[OF this] and Suc
|
|
|
603 |
show ?thesis
|
|
|
604 |
by auto
|
|
|
605 |
qed
|
|
|
606 |
qed auto
|
|
|
607 |
|
|
|
608 |
fun taddr :: "tresource \<Rightarrow> nat" where
|
|
|
609 |
"taddr (TC i instr) = i"
|
|
|
610 |
|
|
|
611 |
lemma tassemble_to_range:
|
|
|
612 |
assumes "(i :[tpg]: j) s"
|
|
|
613 |
shows "(i \<le> j) \<and> (\<forall> r \<in> s. i \<le> taddr r \<and> taddr r < j)"
|
|
|
614 |
using assms
|
|
|
615 |
proof(induct tpg arbitrary: i j s)
|
|
|
616 |
case (TInstr instr i j s)
|
|
|
617 |
obtain a0 l0 a1 l1 where "instr = ((a0, l0), (a1, l1))"
|
|
|
618 |
by (metis pair_collapse)
|
|
|
619 |
with TInstr
|
|
|
620 |
show ?case
|
|
|
621 |
apply (simp add:tassemble_to.simps sg_def)
|
|
|
622 |
by (smt `instr = ((a0, l0), a1, l1)` cond_eq cond_true_eq1
|
|
|
623 |
empty_iff insert_iff le_refl lessI sep.mult_commute taddr.simps)
|
|
|
624 |
next
|
|
|
625 |
case (TLabel l i j s)
|
|
|
626 |
thus ?case
|
|
|
627 |
apply (simp add:tassemble_to.simps)
|
|
|
628 |
by (smt equals0D pasrt_def set_zero_def)
|
|
|
629 |
next
|
|
|
630 |
case (TSeq c1 c2 i j s)
|
|
|
631 |
from TSeq(3) obtain s1 s2 j' where
|
|
|
632 |
h: "(i :[ c1 ]: j') s1" "(j' :[ c2 ]: j) s2" "s1 ## s2" "s = s1 + s2"
|
|
|
633 |
by (auto simp:tassemble_to.simps elim!:EXS_elim sep_conjE)
|
|
|
634 |
show ?case
|
|
|
635 |
proof -
|
|
|
636 |
{ fix r
|
|
|
637 |
assume "r \<in> s"
|
|
|
638 |
with h (3, 4) have "r \<in> s1 \<or> r \<in> s2"
|
|
|
639 |
by (auto simp:set_ins_def)
|
|
|
640 |
hence "i \<le> j \<and> i \<le> taddr r \<and> taddr r < j"
|
|
|
641 |
proof
|
|
|
642 |
assume " r \<in> s1"
|
|
|
643 |
from TSeq(1)[OF h(1)]
|
|
|
644 |
have "i \<le> j'" "(\<forall>r\<in>s1. i \<le> taddr r \<and> taddr r < j')" by auto
|
|
|
645 |
from this(2)[rule_format, OF `r \<in> s1`]
|
|
|
646 |
have "i \<le> taddr r" "taddr r < j'" by auto
|
|
|
647 |
with TSeq(2)[OF h(2)]
|
|
|
648 |
show ?thesis by auto
|
|
|
649 |
next
|
|
|
650 |
assume "r \<in> s2"
|
|
|
651 |
from TSeq(2)[OF h(2)]
|
|
|
652 |
have "j' \<le> j" "(\<forall>r\<in>s2. j' \<le> taddr r \<and> taddr r < j)" by auto
|
|
|
653 |
from this(2)[rule_format, OF `r \<in> s2`]
|
|
|
654 |
have "j' \<le> taddr r" "taddr r < j" by auto
|
|
|
655 |
with TSeq(1)[OF h(1)]
|
|
|
656 |
show ?thesis by auto
|
|
|
657 |
qed
|
|
|
658 |
} thus ?thesis
|
|
|
659 |
by (smt TSeq.hyps(1) TSeq.hyps(2) h(1) h(2))
|
|
|
660 |
qed
|
|
|
661 |
next
|
|
|
662 |
case (TLocal body i j s)
|
|
|
663 |
from this(2) obtain l s' where "(i :[ body l ]: j) s"
|
|
|
664 |
by (simp add:tassemble_to.simps, auto elim!:EXS_elim)
|
|
|
665 |
from TLocal(1)[OF this]
|
|
|
666 |
show ?case by auto
|
|
|
667 |
qed
|
|
|
668 |
|
|
|
669 |
lemma c2p_seq:
|
|
|
670 |
assumes "c2p sts1 i cpg1 j'"
|
|
|
671 |
"c2p sts2 j' cpg2 j"
|
|
|
672 |
"sts_disj sts1 sts2"
|
|
|
673 |
shows "(c2p (sts1 + sts2) i (CSeq cpg1 cpg2) j)"
|
|
|
674 |
proof -
|
|
|
675 |
from assms(1)[unfolded c2p_def]
|
|
|
676 |
obtain f1 where
|
|
|
677 |
h1[rule_format]:
|
|
|
678 |
"\<forall>x. length x = length sts1 \<and> (\<forall>k<length sts1. sts1 ! k = Bound \<longrightarrow> (x ! k = f1 i k)) \<longrightarrow>
|
|
|
679 |
Ex (i :[ c2t x cpg1 ]: j')" by blast
|
|
|
680 |
from assms(2)[unfolded c2p_def]
|
|
|
681 |
obtain f2 where h2[rule_format]:
|
|
|
682 |
"\<forall>x. length x = length sts2 \<and> (\<forall>k<length sts2. sts2 ! k = Bound \<longrightarrow> (x ! k = f2 j' k)) \<longrightarrow>
|
|
|
683 |
Ex (j' :[ c2t x cpg2 ]: j)" by blast
|
|
|
684 |
from assms(3)[unfolded sts_disj_def]
|
|
|
685 |
have h3: "length sts1 = length sts2"
|
|
|
686 |
and h4[rule_format]:
|
|
|
687 |
"(\<forall>i<length sts1. \<not> (sts1 ! i = Bound \<and> sts2 ! i = Bound))" by auto
|
|
|
688 |
let ?f = "\<lambda> i k. if (sts1!k = Bound) then f1 i k else f2 j' k"
|
|
|
689 |
{ fix x
|
|
|
690 |
assume h5: "length x = length (sts1 + sts2)" and
|
|
|
691 |
h6[rule_format]: "(\<forall>k<length (sts1 + sts2). (sts1 + sts2) ! k = Bound \<longrightarrow> x ! k = ?f i k)"
|
|
|
692 |
obtain s1 where h_s1: "(i :[ c2t x cpg1 ]: j') s1"
|
|
|
693 |
proof(atomize_elim, rule h1)
|
|
|
694 |
from h3 h5 have "length x = length sts1"
|
|
|
695 |
by (metis length_sts_plus)
|
|
|
696 |
moreover {
|
|
|
697 |
fix k assume hh: "k<length sts1" "sts1 ! k = Bound"
|
|
|
698 |
from hh(1) h3 h5 have p1: "k < length (sts1 + sts2)"
|
|
|
699 |
by (metis calculation)
|
|
|
700 |
from h3 hh(2) have p2: "(sts1 + sts2)!k = Bound"
|
|
|
701 |
by (metis nth_sts_plus p1 plus_status.simps(2))
|
|
|
702 |
from h6[OF p1 p2] have "x ! k = (if sts1 ! k = Bound then f1 i k else f2 j' k)" .
|
|
|
703 |
with hh(2)
|
|
|
704 |
have "x ! k = f1 i k" by simp
|
|
|
705 |
} ultimately show "length x = length sts1 \<and>
|
|
|
706 |
(\<forall>k<length sts1. sts1 ! k = Bound \<longrightarrow> (x ! k = f1 i k))"
|
|
|
707 |
by blast
|
|
|
708 |
qed
|
|
|
709 |
obtain s2 where h_s2: "(j' :[ c2t x cpg2 ]: j) s2"
|
|
|
710 |
proof(atomize_elim, rule h2)
|
|
|
711 |
from h3 h5 have "length x = length sts2"
|
|
|
712 |
by (metis length_sts_plus)
|
|
|
713 |
moreover {
|
|
|
714 |
fix k
|
|
|
715 |
assume hh: "k<length sts2" "sts2 ! k = Bound"
|
|
|
716 |
from hh(1) h3 h5 have p1: "k < length (sts1 + sts2)"
|
|
|
717 |
by (metis calculation)
|
|
|
718 |
from hh(1) h3 h5 hh(2) have p2: "(sts1 + sts2)!k = Bound"
|
|
|
719 |
by (metis `length sts1 = length sts2 \<and>
|
|
|
720 |
(\<forall>i<length sts1. \<not> (sts1 ! i = Bound \<and> sts2 ! i = Bound))`
|
|
|
721 |
calculation nth_sts_plus plus_status.simps(1) status.distinct(1) status.exhaust)
|
|
|
722 |
from h6[OF p1 p2] have "x ! k = (if sts1 ! k = Bound then f1 i k else f2 j' k)" .
|
|
|
723 |
moreover from h4[OF hh(1)[folded h3]] hh(2) have "sts1!k \<noteq> Bound" by auto
|
|
|
724 |
ultimately have "x!k = f2 j' k" by simp
|
|
|
725 |
} ultimately show "length x = length sts2 \<and>
|
|
|
726 |
(\<forall>k<length sts2. sts2 ! k = Bound \<longrightarrow> (x ! k = f2 j' k))"
|
|
|
727 |
by blast
|
|
|
728 |
qed
|
|
|
729 |
have h_s12: "s1 ## s2"
|
|
|
730 |
proof -
|
|
|
731 |
{ fix r assume h: "r \<in> s1" "r \<in> s2"
|
|
|
732 |
with h_s1 h_s2
|
|
|
733 |
have "False"by (smt tassemble_to_range)
|
|
|
734 |
} thus ?thesis by (auto simp:set_ins_def)
|
|
|
735 |
qed
|
|
|
736 |
have "(EXS j'. i :[ c2t x cpg1 ]: j' \<and>* j' :[ c2t x cpg2 ]: j) (s1 + s2)"
|
|
|
737 |
proof(rule_tac x = j' in EXS_intro)
|
|
|
738 |
from h_s1 h_s2 h_s12
|
|
|
739 |
show "(i :[ c2t x cpg1 ]: j' \<and>* j' :[ c2t x cpg2 ]: j) (s1 + s2)"
|
|
|
740 |
by (metis sep_conjI)
|
|
|
741 |
qed
|
|
|
742 |
hence "\<exists> s. (i :[ c2t x (CSeq cpg1 cpg2) ]: j) s"
|
|
|
743 |
by (auto simp:tassemble_to.simps)
|
|
|
744 |
}
|
|
|
745 |
hence "\<exists>f. \<forall>x. length x = length (sts1 + sts2) \<and>
|
|
|
746 |
(\<forall>k<length (sts1 + sts2). (sts1 + sts2) ! k = Bound \<longrightarrow> x ! k = f i k) \<longrightarrow>
|
|
|
747 |
Ex (i :[ c2t x (CSeq cpg1 cpg2) ]: j)"
|
|
|
748 |
by (rule_tac x = ?f in exI, auto)
|
|
|
749 |
thus ?thesis
|
|
|
750 |
by(unfold c2p_def, auto)
|
|
|
751 |
qed
|
|
|
752 |
|
|
|
753 |
lemma plus_list_len:
|
|
|
754 |
"length ((sts1::status list) + sts2) = min (length sts1) (length sts2)"
|
|
|
755 |
by(induct rule:plus_list.induct, auto)
|
|
|
756 |
|
|
|
757 |
lemma minus_list_len:
|
|
|
758 |
"length ((sts1::status list) - sts2) = min (length sts1) (length sts2)"
|
|
|
759 |
by(induct rule:minus_list.induct, auto)
|
|
|
760 |
|
|
|
761 |
lemma sts_le_comb:
|
|
|
762 |
assumes "sts1 \<le> sts2"
|
|
|
763 |
and "sts2 \<le> sts3"
|
|
|
764 |
shows "sts_disj (sts2 - sts1) (sts3 - sts2)" and
|
|
|
765 |
"(sts3 - sts1) = (sts2 - sts1) + (sts3 - sts2)"
|
|
|
766 |
proof -
|
|
|
767 |
from assms
|
|
|
768 |
have h1: "length sts1 = length sts2" "\<forall>i<length sts1. sts1 ! i \<le> sts2 ! i"
|
|
|
769 |
and h2: "length sts2 = length sts3" "\<forall>i<length sts1. sts2 ! i \<le> sts3 ! i"
|
|
|
770 |
by (unfold less_eq_list_def, auto)
|
|
|
771 |
have "sts_disj (sts2 - sts1) (sts3 - sts2)"
|
|
|
772 |
proof -
|
|
|
773 |
from h1(1) h2(1)
|
|
|
774 |
have "length (sts2 - sts1) = length (sts3 - sts2)"
|
|
|
775 |
by (metis minus_list_len)
|
|
|
776 |
moreover {
|
|
|
777 |
fix i
|
|
|
778 |
assume lt_i: "i<length (sts2 - sts1)"
|
|
|
779 |
from lt_i h1(1) h2(1) have "i < length sts1"
|
|
|
780 |
by (smt minus_list_len)
|
|
|
781 |
from h1(2)[rule_format, of i, OF this] h2(2)[rule_format, of i, OF this]
|
|
|
782 |
have "sts1 ! i \<le> sts2 ! i" "sts2 ! i \<le> sts3 ! i" .
|
|
|
783 |
moreover have "(sts2 - sts1) ! i = sts2!i - sts1!i"
|
|
|
784 |
by (metis lt_i nth_sts_minus)
|
|
|
785 |
moreover have "(sts3 - sts2)!i = sts3!i - sts2!i"
|
|
|
786 |
by (metis `length (sts2 - sts1) = length (sts3 - sts2)` lt_i nth_sts_minus)
|
|
|
787 |
ultimately have " \<not> ((sts2 - sts1) ! i = Bound \<and> (sts3 - sts2) ! i = Bound)"
|
|
|
788 |
apply (cases "sts2!i", cases "sts1!i", cases "sts3!i", simp, simp)
|
|
|
789 |
apply (cases "sts3!i", simp, simp)
|
|
|
790 |
apply (cases "sts1!i", cases "sts3!i", simp, simp)
|
|
|
791 |
by (cases "sts3!i", simp, simp)
|
|
|
792 |
} ultimately show ?thesis by (unfold sts_disj_def, auto)
|
|
|
793 |
qed
|
|
|
794 |
moreover have "(sts3 - sts1) = (sts2 - sts1) + (sts3 - sts2)"
|
|
|
795 |
proof(induct rule:nth_equalityI)
|
|
|
796 |
case 1
|
|
|
797 |
show ?case by (metis h1(1) h2(1) length_sts_plus minus_list_len)
|
|
|
798 |
next
|
|
|
799 |
case 2
|
|
|
800 |
{ fix i
|
|
|
801 |
assume lt_i: "i<length (sts3 - sts1)"
|
|
|
802 |
have "(sts3 - sts1) ! i = (sts2 - sts1 + (sts3 - sts2)) ! i" (is "?L = ?R")
|
|
|
803 |
proof -
|
|
|
804 |
have "?R = sts2!i - sts1!i + (sts3!i - sts2!i)"
|
|
|
805 |
by (smt `i < length (sts3 - sts1)` h2(1) minus_list_len nth_sts_minus
|
|
|
806 |
nth_sts_plus plus_list_len)
|
|
|
807 |
moreover have "?L = sts3!i - sts1!i"
|
|
|
808 |
by (metis `i < length (sts3 - sts1)` nth_sts_minus)
|
|
|
809 |
moreover
|
|
|
810 |
have "sts2!i - sts1!i + (sts3!i - sts2!i) = sts3!i - sts1!i"
|
|
|
811 |
proof -
|
|
|
812 |
from lt_i h1(1) h2(1) have "i < length sts1"
|
|
|
813 |
by (smt minus_list_len)
|
|
|
814 |
from h1(2)[rule_format, of i, OF this] h2(2)[rule_format, of i, OF this]
|
|
|
815 |
show ?thesis
|
|
|
816 |
apply (cases "sts2!i", cases "sts1!i", cases "sts3!i", simp, simp)
|
|
|
817 |
apply (cases "sts3!i", simp, simp)
|
|
|
818 |
apply (cases "sts1!i", cases "sts3!i", simp, simp)
|
|
|
819 |
by (cases "sts3!i", simp, simp)
|
|
|
820 |
qed
|
|
|
821 |
ultimately show ?thesis by simp
|
|
|
822 |
qed
|
|
|
823 |
} thus ?case by auto
|
|
|
824 |
qed
|
|
|
825 |
ultimately show "sts_disj (sts2 - sts1) (sts3 - sts2)" and
|
|
|
826 |
"(sts3 - sts1) = (sts2 - sts1) + (sts3 - sts2)" by auto
|
|
|
827 |
qed
|
|
|
828 |
|
|
|
829 |
lemma wf_cpg_test_correct:
|
|
|
830 |
assumes "wf_cpg_test sts cpg = (True, sts')"
|
|
|
831 |
shows "(\<forall> i. \<exists> j. (c2p (sts' - sts) i cpg j))"
|
|
|
832 |
using assms
|
|
|
833 |
proof(induct cpg arbitrary:sts sts')
|
|
|
834 |
case (CInstr instr sts sts')
|
|
|
835 |
obtain a0 l0 a1 l1 where eq_instr: "instr = ((a0, l0), (a1, l1))"
|
|
|
836 |
by (metis prod.exhaust)
|
|
|
837 |
show ?case
|
|
|
838 |
proof(unfold eq_instr c2p_def, clarsimp simp:tassemble_to.simps)
|
|
|
839 |
fix i
|
|
|
840 |
let ?a = "Suc i" and ?f = "\<lambda> i k. i"
|
|
|
841 |
show "\<exists>a f. \<forall>x. length x = length (sts' - sts) \<and>
|
|
|
842 |
(\<forall>k<length (sts' - sts). (sts' - sts) ! k = Bound \<longrightarrow> x ! k = f i k) \<longrightarrow>
|
|
|
843 |
Ex (sg {TC i ((a0, x ! l0), a1, x ! l1)} \<and>* <(a = Suc i)>)"
|
|
|
844 |
proof(rule_tac x = ?a in exI, rule_tac x = ?f in exI, default, clarsimp)
|
|
|
845 |
fix x
|
|
|
846 |
let ?j = "Suc i"
|
|
|
847 |
let ?s = " {TC i ((a0, x ! l0), a1, x ! l1)}"
|
|
|
848 |
have "(sg {TC i ((a0, x ! l0), a1, x ! l1)} \<and>* <(Suc i = Suc i)>) ?s"
|
|
|
849 |
by (simp add:tassemble_to.simps sg_def)
|
|
|
850 |
thus "Ex (sg {TC i ((a0, x ! l0), a1, x ! l1)})" by auto
|
|
|
851 |
qed
|
|
|
852 |
qed
|
|
|
853 |
next
|
|
|
854 |
case (CLabel l sts sts')
|
|
|
855 |
show ?case
|
|
|
856 |
proof
|
|
|
857 |
fix i
|
|
|
858 |
from CLabel
|
|
|
859 |
have h1: "l < length sts"
|
|
|
860 |
and h2: "sts ! l = Free"
|
|
|
861 |
and h3: "sts[l := Bound] = sts'" by auto
|
|
|
862 |
let ?f = "\<lambda> i k. i"
|
|
|
863 |
have "\<exists>a f. \<forall>x. length x = length (sts' - sts) \<and>
|
|
|
864 |
(\<forall>k<length (sts' - sts). (sts' - sts) ! k = Bound \<longrightarrow> x ! k = f (i::nat) k) \<longrightarrow>
|
|
|
865 |
Ex (<(i = a \<and> a = x ! l)>)"
|
|
|
866 |
proof(rule_tac x = i in exI, rule_tac x = ?f in exI, clarsimp)
|
|
|
867 |
fix x
|
|
|
868 |
assume h[rule_format]:
|
|
|
869 |
"\<forall>k<length (sts' - sts). (sts' - sts) ! k = Bound \<longrightarrow> x ! k = i"
|
|
|
870 |
from h1 h3 have p1: "l < length (sts' - sts)"
|
|
|
871 |
by (metis length_list_update min_max.inf.idem minus_list_len)
|
|
|
872 |
from p1 h2 h3 have p2: "(sts' - sts)!l = Bound"
|
|
|
873 |
by (metis h1 minus_status.simps(2) nth_list_update_eq nth_sts_minus)
|
|
|
874 |
from h[OF p1 p2] have "(<(i = x ! l)>) 0"
|
|
|
875 |
by (simp add:set_ins_def)
|
|
|
876 |
thus "\<exists> s. (<(i = x ! l)>) s" by auto
|
|
|
877 |
qed
|
|
|
878 |
thus "\<exists>a. c2p (sts' - sts) i (CLabel l) a"
|
|
|
879 |
by (auto simp:c2p_def tassemble_to.simps)
|
|
|
880 |
qed
|
|
|
881 |
next
|
|
|
882 |
case (CSeq cpg1 cpg2 sts sts')
|
|
|
883 |
show ?case
|
|
|
884 |
proof
|
|
|
885 |
fix i
|
|
|
886 |
from CSeq(3)[unfolded wf_cpg_test.simps]
|
|
|
887 |
show "\<exists> j. c2p (sts' - sts) i (CSeq cpg1 cpg2) j"
|
|
|
888 |
proof(let_elim)
|
|
|
889 |
case (LetE b1 sts1)
|
|
|
890 |
from this(1)
|
|
|
891 |
obtain b2 where h: "(b2, sts') = wf_cpg_test sts1 cpg2" "b1=True" "b2=True"
|
|
|
892 |
by (atomize_elim, unfold Let_def, auto split:prod.splits)
|
|
|
893 |
from wf_cpg_test_le[OF LetE(2)[symmetric, unfolded h(2)]]
|
|
|
894 |
have sts_le1: "sts \<le> sts1" .
|
|
|
895 |
from CSeq(1)[OF LetE(2)[unfolded h(2), symmetric], rule_format, of i]
|
|
|
896 |
obtain j1 where h1: "(c2p (sts1 - sts) i cpg1 j1)" by blast
|
|
|
897 |
from wf_cpg_test_le[OF h(1)[symmetric, unfolded h(3)]]
|
|
|
898 |
have sts_le2: "sts1 \<le> sts'" .
|
|
|
899 |
from sts_le_comb[OF sts_le1 sts_le2]
|
|
|
900 |
have hh: "sts_disj (sts1 - sts) (sts' - sts1)"
|
|
|
901 |
"sts' - sts = (sts1 - sts) + (sts' - sts1)" .
|
|
|
902 |
from CSeq(2)[OF h(1)[symmetric, unfolded h(3)], rule_format, of j1]
|
|
|
903 |
obtain j2 where h2: "(c2p (sts' - sts1) j1 cpg2 j2)" by blast
|
|
|
904 |
have "c2p (sts' - sts) i (CSeq cpg1 cpg2) j2"
|
|
|
905 |
by(unfold hh(2), rule c2p_seq[OF h1 h2 hh(1)])
|
|
|
906 |
thus ?thesis by blast
|
|
|
907 |
qed
|
|
|
908 |
qed
|
|
|
909 |
next
|
|
|
910 |
case (CLocal body sts sts')
|
|
|
911 |
from this(2) obtain b sts1 s where
|
|
|
912 |
h: "wf_cpg_test (Free # sts) body = (True, sts1)"
|
|
|
913 |
"sts' = tl sts1"
|
|
|
914 |
by (unfold wf_cpg_test.simps, auto split:prod.splits)
|
|
|
915 |
from wf_cpg_test_le[OF h(1), unfolded less_eq_list_def] h(2)
|
|
|
916 |
obtain s where eq_sts1: "sts1 = s#sts'"
|
|
|
917 |
by (metis Suc_length_conv list.size(4) tl.simps(2))
|
|
|
918 |
from CLocal(1)[OF h(1)] have p1: "\<forall>i. \<exists>a. c2p (sts1 - (Free # sts)) i body a" .
|
|
|
919 |
show ?case
|
|
|
920 |
proof
|
|
|
921 |
fix i
|
|
|
922 |
from p1[rule_format, of i] obtain j where "(c2p (sts1 - (Free # sts)) i body) j" by blast
|
|
|
923 |
then obtain f where hh [rule_format]:
|
|
|
924 |
"\<forall>x. length x = length (sts1 - (Free # sts)) \<and>
|
|
|
925 |
(\<forall>k<length (sts1 - (Free # sts)). (sts1 - (Free # sts)) ! k = Bound \<longrightarrow> x ! k = f i k) \<longrightarrow>
|
|
|
926 |
(\<exists>s. (i :[ c2t x body ]: j) s)"
|
|
|
927 |
by (auto simp:c2p_def)
|
|
|
928 |
let ?f = "\<lambda> i k. f i (Suc k)"
|
|
|
929 |
have "\<exists>j f. \<forall>x. length x = length (sts' - sts) \<and>
|
|
|
930 |
(\<forall>k<length (sts' - sts). (sts' - sts) ! k = Bound \<longrightarrow> x ! k = f i k) \<longrightarrow>
|
|
|
931 |
(\<exists>s. (i :[ (TL xa. c2t (xa # x) body) ]: j) s)"
|
|
|
932 |
proof(rule_tac x = j in exI, rule_tac x = ?f in exI, default, clarsimp)
|
|
|
933 |
fix x
|
|
|
934 |
assume h1: "length x = length (sts' - sts)"
|
|
|
935 |
and h2: "\<forall>k<length (sts' - sts). (sts' - sts) ! k = Bound \<longrightarrow> x ! k = f i (Suc k)"
|
|
|
936 |
let ?l = "f i 0" let ?x = " ?l#x"
|
|
|
937 |
from wf_cpg_test_le[OF h(1)] have "length (Free#sts) = length sts1"
|
|
|
938 |
by (unfold less_eq_list_def, simp)
|
|
|
939 |
with h1 h(2) have q1: "length (?l # x) = length (sts1 - (Free # sts))"
|
|
|
940 |
by (smt Suc_length_conv length_Suc_conv list.inject list.size(4)
|
|
|
941 |
minus_list.simps(3) minus_list_len tl.simps(2))
|
|
|
942 |
have q2: "(\<forall>k<length (sts1 - (Free # sts)).
|
|
|
943 |
(sts1 - (Free # sts)) ! k = Bound \<longrightarrow> (f i 0 # x) ! k = f i k)"
|
|
|
944 |
proof -
|
|
|
945 |
{ fix k
|
|
|
946 |
assume lt_k: "k<length (sts1 - (Free # sts))"
|
|
|
947 |
and bd_k: "(sts1 - (Free # sts)) ! k = Bound"
|
|
|
948 |
have "(f i 0 # x) ! k = f i k"
|
|
|
949 |
proof(cases "k")
|
|
|
950 |
case (Suc k')
|
|
|
951 |
moreover have "x ! k' = f i (Suc k')"
|
|
|
952 |
proof(rule h2[rule_format])
|
|
|
953 |
from bd_k Suc eq_sts1 show "(sts' - sts) ! k' = Bound" by simp
|
|
|
954 |
next
|
|
|
955 |
from Suc lt_k eq_sts1 show "k' < length (sts' - sts)" by simp
|
|
|
956 |
qed
|
|
|
957 |
ultimately show ?thesis by simp
|
|
|
958 |
qed simp
|
|
|
959 |
} thus ?thesis by auto
|
|
|
960 |
qed
|
|
|
961 |
from conjI[THEN hh[of ?x], OF q1 q2] obtain s
|
|
|
962 |
where h_s: "(i :[ c2t (f i 0 # x) body ]: j) s" by blast
|
|
|
963 |
thus "\<exists>s. (i :[ (TL xa. c2t (xa # x) body) ]: j) s"
|
|
|
964 |
apply (simp add:tassemble_to.simps)
|
|
|
965 |
by (rule_tac x = s in exI, rule_tac x = ?l in EXS_intro, simp)
|
|
|
966 |
qed
|
|
|
967 |
thus "\<exists>j. c2p (sts' - sts) i (CLocal body) j"
|
|
|
968 |
by (auto simp:c2p_def)
|
|
|
969 |
qed
|
|
|
970 |
qed
|
|
|
971 |
|
|
|
972 |
lemma
|
|
|
973 |
assumes "wf_cpg_test [] cpg = (True, sts')"
|
|
|
974 |
and "tpg = c2t [] cpg"
|
|
|
975 |
shows "\<forall> i. \<exists> j s. ((i:[tpg]:j) s)"
|
|
|
976 |
proof
|
|
|
977 |
fix i
|
|
|
978 |
have eq_sts_minus: "(sts' - []) = []"
|
|
|
979 |
by (metis list.exhaust minus_list.simps(1) minus_list.simps(2))
|
|
|
980 |
from wf_cpg_test_correct[OF assms(1), rule_format, of i]
|
|
|
981 |
obtain j where "c2p (sts' - []) i cpg j" by auto
|
|
|
982 |
from c2p_assert [OF this[unfolded eq_sts_minus]]
|
|
|
983 |
obtain s where "(i :[ c2t [] cpg ]: j) s" by blast
|
|
|
984 |
from this[folded assms(2)]
|
|
|
985 |
show " \<exists> j s. ((i:[tpg]:j) s)" by blast
|
|
|
986 |
qed
|
|
|
987 |
|
|
|
988 |
lemma replicate_nth: "(replicate n x @ sts) ! (l + n) = sts!l"
|
|
|
989 |
by (smt length_replicate nth_append)
|
|
|
990 |
|
|
|
991 |
lemma replicate_update:
|
|
|
992 |
"(replicate n x @ sts)[l + n := v] = replicate n x @ sts[l := v]"
|
|
|
993 |
by (smt length_replicate list_update_append)
|
|
|
994 |
|
|
|
995 |
lemma l_n_v_orig:
|
|
|
996 |
assumes "l0 < length env"
|
|
|
997 |
and "t \<le> l0"
|
|
|
998 |
shows "(take t env @ es @ drop t env) ! (l0 + length es) = env ! l0"
|
|
|
999 |
proof -
|
|
|
1000 |
from assms(1, 2) have "t < length env" by auto
|
|
|
1001 |
hence h: "env = take t env @ drop t env"
|
|
|
1002 |
"length (take t env) = t"
|
|
|
1003 |
apply (metis append_take_drop_id)
|
|
|
1004 |
by (smt `t < length env` length_take)
|
|
|
1005 |
with assms(2) have eq_sts_l: "env!l0 = (drop t env)!(l0 - t)"
|
|
|
1006 |
by (metis nth_app)
|
|
|
1007 |
from h(2) have "length (take t env @ es) = t + length es"
|
|
|
1008 |
by (metis length_append length_replicate nat_add_commute)
|
|
|
1009 |
moreover from assms(2) have "t + (length es) \<le> l0 + (length es)" by auto
|
|
|
1010 |
ultimately have "((take t env @ es) @ drop t env)!(l0 + length es) =
|
|
|
1011 |
(drop t env)!(l0+ length es - (t + length es))"
|
|
|
1012 |
by (smt length_replicate length_splice nth_append)
|
|
|
1013 |
with eq_sts_l[symmetric, unfolded assms]
|
|
|
1014 |
show ?thesis by auto
|
|
|
1015 |
qed
|
|
|
1016 |
|
|
|
1017 |
lemma l_n_v:
|
|
|
1018 |
assumes "l < length sts"
|
|
|
1019 |
and "sts!l = v"
|
|
|
1020 |
and "t \<le> l"
|
|
|
1021 |
shows "(take t sts @ replicate n x @ drop t sts) ! (l + n) = v"
|
|
|
1022 |
proof -
|
|
|
1023 |
from l_n_v_orig[OF assms(1) assms(3), of "replicate n x"]
|
|
|
1024 |
and assms(2)
|
|
|
1025 |
show ?thesis by auto
|
|
|
1026 |
qed
|
|
|
1027 |
|
|
|
1028 |
lemma l_n_v_s:
|
|
|
1029 |
assumes "l < length sts"
|
|
|
1030 |
and "t \<le> l"
|
|
|
1031 |
shows "(take t sts @ sts0 @ drop t sts)[l + length sts0 := v] =
|
|
|
1032 |
take t (sts[l:=v])@ sts0 @ drop t (sts[l:=v])"
|
|
|
1033 |
proof -
|
|
|
1034 |
let ?n = "length sts0"
|
|
|
1035 |
from assms(1, 2) have "t < length sts" by auto
|
|
|
1036 |
hence h: "sts = take t sts @ drop t sts"
|
|
|
1037 |
"length (take t sts) = t"
|
|
|
1038 |
apply (metis append_take_drop_id)
|
|
|
1039 |
by (smt `t < length sts` length_take)
|
|
|
1040 |
with assms(2) have eq_sts_l: "sts[l:=v] = take t sts @ drop t sts [(l - t) := v]"
|
|
|
1041 |
by (smt list_update_append)
|
|
|
1042 |
with h(2) have eq_take_drop_t: "take t (sts[l:=v]) = take t sts"
|
|
|
1043 |
"drop t (sts[l:=v]) = (drop t sts)[l - t:=v]"
|
|
|
1044 |
apply (metis append_eq_conv_conj)
|
|
|
1045 |
by (metis append_eq_conv_conj eq_sts_l h(2))
|
|
|
1046 |
from h(2) have "length (take t sts @ sts0) = t + ?n"
|
|
|
1047 |
by (metis length_append length_replicate nat_add_commute)
|
|
|
1048 |
moreover from assms(2) have "t + ?n \<le> l + ?n" by auto
|
|
|
1049 |
ultimately have "((take t sts @ sts0) @ drop t sts)[l + ?n := v] =
|
|
|
1050 |
(take t sts @ sts0) @ (drop t sts)[(l + ?n - (t + ?n)) := v]"
|
|
|
1051 |
by (smt list_update_append replicate_nth)
|
|
|
1052 |
with eq_take_drop_t
|
|
|
1053 |
show ?thesis by auto
|
|
|
1054 |
qed
|
|
|
1055 |
|
|
|
1056 |
lemma l_n_v_s1:
|
|
|
1057 |
assumes "l < length sts"
|
|
|
1058 |
and "\<not> t \<le> l"
|
|
|
1059 |
shows "(take t sts @ sts0 @ drop t sts)[l := v] =
|
|
|
1060 |
take t (sts[l := v]) @ sts0 @ drop t (sts[l := v])"
|
|
|
1061 |
proof(cases "t < length sts")
|
|
|
1062 |
case False
|
|
|
1063 |
hence h: "take t sts = sts" "drop t sts = []"
|
|
|
1064 |
"take t (sts[l:=v]) = sts [l:=v]"
|
|
|
1065 |
"drop t (sts[l:=v]) = []"
|
|
|
1066 |
by auto
|
|
|
1067 |
with assms(1)
|
|
|
1068 |
show ?thesis
|
|
|
1069 |
by (metis list_update_append)
|
|
|
1070 |
next
|
|
|
1071 |
case True
|
|
|
1072 |
with assms(2)
|
|
|
1073 |
have h: "(take t sts)[l:=v] = take t (sts[l:=v])"
|
|
|
1074 |
"(sts[l:=v]) = (take t sts)[l:=v]@drop t sts"
|
|
|
1075 |
"length (take t sts) = t"
|
|
|
1076 |
apply (smt length_list_update length_take nth_equalityI nth_list_update nth_take)
|
|
|
1077 |
apply (smt True append_take_drop_id assms(2) length_take list_update_append1)
|
|
|
1078 |
by (smt True length_take)
|
|
|
1079 |
from h(2,3) have "drop t (sts[l := v]) = drop t sts"
|
|
|
1080 |
by (metis append_eq_conv_conj length_list_update)
|
|
|
1081 |
with h(1)
|
|
|
1082 |
show ?thesis
|
|
|
1083 |
apply simp
|
|
|
1084 |
by (metis assms(2) h(3) list_update_append1 not_leE)
|
|
|
1085 |
qed
|
|
|
1086 |
|
|
|
1087 |
lemma l_n_v_s2:
|
|
|
1088 |
assumes "l0 < length env"
|
|
|
1089 |
and "t \<le> l0"
|
|
|
1090 |
and "\<not> t \<le> l1"
|
|
|
1091 |
shows "(take t env @ es @ drop t env) ! l1 = env ! l1"
|
|
|
1092 |
proof -
|
|
|
1093 |
from assms(1, 2) have "t < length env" by auto
|
|
|
1094 |
hence h: "env = take t env @ drop t env"
|
|
|
1095 |
"length (take t env) = t"
|
|
|
1096 |
apply (metis append_take_drop_id)
|
|
|
1097 |
by (smt `t < length env` length_take)
|
|
|
1098 |
with assms(3) show ?thesis
|
|
|
1099 |
by (smt nth_append)
|
|
|
1100 |
qed
|
|
|
1101 |
|
|
|
1102 |
lemma l_n_v_s3:
|
|
|
1103 |
assumes "l0 < length env"
|
|
|
1104 |
and "\<not> t \<le> l0"
|
|
|
1105 |
shows "(take t env @ es @ drop t env) ! l0 = env ! l0"
|
|
|
1106 |
proof(cases "t < length env")
|
|
|
1107 |
case True
|
|
|
1108 |
hence h: "env = take t env @ drop t env"
|
|
|
1109 |
"length (take t env) = t"
|
|
|
1110 |
apply (metis append_take_drop_id)
|
|
|
1111 |
by (smt `t < length env` length_take)
|
|
|
1112 |
with assms(2) show ?thesis
|
|
|
1113 |
by (smt nth_append)
|
|
|
1114 |
next
|
|
|
1115 |
case False
|
|
|
1116 |
hence "take t env = env" by auto
|
|
|
1117 |
with assms(1) show ?thesis
|
|
|
1118 |
by (metis nth_append)
|
|
|
1119 |
qed
|
|
|
1120 |
|
|
|
1121 |
lemma lift_wf_cpg_test:
|
|
|
1122 |
assumes "wf_cpg_test sts cpg = (True, sts')"
|
|
|
1123 |
shows "wf_cpg_test (take t sts @ sts0 @ drop t sts) (lift_t t (length sts0) cpg) =
|
|
|
1124 |
(True, take t sts' @ sts0 @ drop t sts')"
|
|
|
1125 |
using assms
|
|
|
1126 |
proof(induct cpg arbitrary:t sts0 sts sts')
|
|
|
1127 |
case (CInstr instr t sts0 sts sts')
|
|
|
1128 |
obtain a0 l0 a1 l1 where eq_instr: "instr = ((a0, l0), (a1, l1))"
|
|
|
1129 |
by (metis prod.exhaust)
|
|
|
1130 |
from CInstr
|
|
|
1131 |
show ?case
|
|
|
1132 |
by (auto simp:eq_instr lift_b_def)
|
|
|
1133 |
next
|
|
|
1134 |
case (CLabel l t sts0 sts sts')
|
|
|
1135 |
thus ?case
|
|
|
1136 |
apply (auto simp:lift_b_def
|
|
|
1137 |
replicate_nth replicate_update l_n_v_orig l_n_v_s)
|
|
|
1138 |
apply (metis (mono_tags) diff_diff_cancel length_drop length_rev
|
|
|
1139 |
linear not_less nth_append nth_take rev_take take_all)
|
|
|
1140 |
by (simp add:l_n_v_s1)
|
|
|
1141 |
next
|
|
|
1142 |
case (CSeq c1 c2 sts0 sts sts')
|
|
|
1143 |
thus ?case
|
|
|
1144 |
by (auto simp: lift0_def lift_b_def split:prod.splits)
|
|
|
1145 |
next
|
|
|
1146 |
case (CLocal body t sts0 sts sts')
|
|
|
1147 |
from this(2) obtain sts1 where h: "wf_cpg_test (Free # sts) body = (True, sts1)" "tl sts1 = sts'"
|
|
|
1148 |
by (auto simp:lift0_def lift_b_def split:prod.splits)
|
|
|
1149 |
let ?lift_s = "\<lambda> t sts. take t sts @ sts0 @ drop t sts"
|
|
|
1150 |
have eq_lift_1: "(?lift_s (Suc t) (Free # sts)) = Free#?lift_s t sts"
|
|
|
1151 |
by (simp)
|
|
|
1152 |
from wf_cpg_test_le[OF h(1)] have "length (Free#sts) = length sts1"
|
|
|
1153 |
by (unfold less_eq_list_def, simp)
|
|
|
1154 |
hence eq_sts1: "sts1 = hd sts1 # tl sts1"
|
|
|
1155 |
by (metis append_Nil append_eq_conv_conj hd.simps list.exhaust tl.simps(2))
|
|
|
1156 |
from CLocal(1)[OF h(1), of "Suc t", of "sts0", unfolded eq_lift_1]
|
|
|
1157 |
show ?case
|
|
|
1158 |
apply (simp, subst eq_sts1, simp)
|
|
|
1159 |
apply (simp add:h(2))
|
|
|
1160 |
by (subst eq_sts1, simp add:h(2))
|
|
|
1161 |
qed
|
|
|
1162 |
|
|
|
1163 |
lemma lift_c2t:
|
|
|
1164 |
assumes "wf_cpg_test sts cpg = (True, sts')"
|
|
|
1165 |
and "length env = length sts"
|
|
|
1166 |
shows "c2t (take t env @ es @ drop t env) (lift_t t (length es) cpg) =
|
|
|
1167 |
c2t env cpg"
|
|
|
1168 |
using assms
|
|
|
1169 |
proof(induct cpg arbitrary: t env es sts sts')
|
|
|
1170 |
case (CInstr instr t env es sts sts')
|
|
|
1171 |
obtain a0 l0 a1 l1 where eq_instr: "instr = ((a0, l0), (a1, l1))"
|
|
|
1172 |
by (metis prod.exhaust)
|
|
|
1173 |
from CInstr have h: "l0 < length env" "l1 < length env"
|
|
|
1174 |
by (auto simp:eq_instr)
|
|
|
1175 |
with CInstr(2)
|
|
|
1176 |
show ?case
|
|
|
1177 |
by (auto simp:eq_instr lift_b_def l_n_v_orig l_n_v_s2 l_n_v_s3)
|
|
|
1178 |
next
|
|
|
1179 |
case (CLabel l t env es sts sts')
|
|
|
1180 |
thus ?case
|
|
|
1181 |
by (auto simp:lift_b_def
|
|
|
1182 |
replicate_nth replicate_update l_n_v_orig l_n_v_s3)
|
|
|
1183 |
next
|
|
|
1184 |
case (CSeq c1 c2 t env es sts sts')
|
|
|
1185 |
from CSeq(3) obtain sts1
|
|
|
1186 |
where h: "wf_cpg_test sts c1 = (True, sts1)" "wf_cpg_test sts1 c2 = (True, sts')"
|
|
|
1187 |
by (auto split:prod.splits)
|
|
|
1188 |
from wf_cpg_test_le[OF h(1)] have "length sts = length sts1"
|
|
|
1189 |
by (auto simp:less_eq_list_def)
|
|
|
1190 |
from CSeq(4)[unfolded this] have eq_len_env: "length env = length sts1" .
|
|
|
1191 |
from CSeq(1)[OF h(1) CSeq(4)]
|
|
|
1192 |
CSeq(2)[OF h(2) eq_len_env]
|
|
|
1193 |
show ?case
|
|
|
1194 |
by (auto simp: lift0_def lift_b_def split:prod.splits)
|
|
|
1195 |
next
|
|
|
1196 |
case (CLocal body t env es sts sts')
|
|
|
1197 |
{ fix x
|
|
|
1198 |
from CLocal(2)
|
|
|
1199 |
obtain sts1 where h1: "wf_cpg_test (Free # sts) body = (True, sts1)"
|
|
|
1200 |
by (auto split:prod.splits)
|
|
|
1201 |
from CLocal(3) have "length (x#env) = length (Free # sts)" by simp
|
|
|
1202 |
from CLocal(1)[OF h1 this, of "Suc t"]
|
|
|
1203 |
have "c2t (x # take t env @ es @ drop t env) (lift_t (Suc t) (length es) body) =
|
|
|
1204 |
c2t (x # env) body"
|
|
|
1205 |
by simp
|
|
|
1206 |
} thus ?case by simp
|
|
|
1207 |
qed
|
|
|
1208 |
|
|
|
1209 |
pr 20
|
|
|
1210 |
|
|
|
1211 |
lemma upto_append:
|
|
|
1212 |
assumes "x \<le> y + 1"
|
|
|
1213 |
shows "[x .. y + 1] = [x .. y]@[y + 1]"
|
|
|
1214 |
using assms
|
|
|
1215 |
by (induct x y rule:upto.induct, auto simp:upto.simps)
|
|
|
1216 |
|
|
|
1217 |
lemma nth_upto:
|
|
|
1218 |
assumes "l < length sts"
|
|
|
1219 |
shows "[0..(int (length sts)) - 1]!l = int l"
|
|
|
1220 |
using assms
|
|
|
1221 |
proof(induct sts arbitrary:l)
|
|
|
1222 |
case Nil
|
|
|
1223 |
thus ?case by simp
|
|
|
1224 |
next
|
|
|
1225 |
case (Cons s sts l)
|
|
|
1226 |
from Cons(2)
|
|
|
1227 |
have "0 \<le> (int (length sts) - 1) + 1" by auto
|
|
|
1228 |
from upto_append[OF this]
|
|
|
1229 |
have eq_upto: "[0..int (length sts)] = [0..int (length sts) - 1] @ [int (length sts)]"
|
|
|
1230 |
by simp
|
|
|
1231 |
show ?case
|
|
|
1232 |
proof(cases "l < length sts")
|
|
|
1233 |
case True
|
|
|
1234 |
with Cons(1)[OF True] eq_upto
|
|
|
1235 |
show ?thesis
|
|
|
1236 |
apply simp
|
|
|
1237 |
by (smt nth_append take_eq_Nil upto_len)
|
|
|
1238 |
next
|
|
|
1239 |
case False
|
|
|
1240 |
with Cons(2) have eq_l: "l = length sts" by simp
|
|
|
1241 |
show ?thesis
|
|
|
1242 |
proof(cases sts)
|
|
|
1243 |
case (Cons x xs)
|
|
|
1244 |
have "[0..1 + int (length xs)] = [0 .. int (length xs)]@[1 + int (length xs)]"
|
|
|
1245 |
by (smt upto_append)
|
|
|
1246 |
moreover have "length [0 .. int (length xs)] = Suc (length xs)"
|
|
|
1247 |
by (smt upto_len)
|
|
|
1248 |
moreover note Cons
|
|
|
1249 |
ultimately show ?thesis
|
|
|
1250 |
apply (simp add:eq_l)
|
|
|
1251 |
by (smt nth_Cons' nth_append)
|
|
|
1252 |
qed (simp add:upto_len upto.simps eq_l)
|
|
|
1253 |
qed
|
|
|
1254 |
qed
|
|
|
1255 |
|
|
|
1256 |
lemma map_idx_idx:
|
|
|
1257 |
assumes "l < length sts"
|
|
|
1258 |
shows "(map_idx f sts)!l = sts!(f l)"
|
|
|
1259 |
proof -
|
|
|
1260 |
from assms have lt_l: "l < length [0..int (length sts) - 1]"
|
|
|
1261 |
by (smt upto_len)
|
|
|
1262 |
show ?thesis
|
|
|
1263 |
apply (unfold map_idx_def nth_map[OF lt_l])
|
|
|
1264 |
by (metis assms nat_int nth_upto)
|
|
|
1265 |
qed
|
|
|
1266 |
|
|
|
1267 |
lemma map_idx_len: "length (map_idx f sts) = length sts"
|
|
|
1268 |
apply (unfold map_idx_def)
|
|
|
1269 |
by (smt length_map upto_len)
|
|
|
1270 |
|
|
|
1271 |
lemma map_idx_eq:
|
|
|
1272 |
assumes "\<forall> l < length sts. f l = g l"
|
|
|
1273 |
shows "map_idx f sts = map_idx g sts"
|
|
|
1274 |
proof(induct rule: nth_equalityI)
|
|
|
1275 |
case 1
|
|
|
1276 |
show "length (map_idx f sts) = length (map_idx g sts)"
|
|
|
1277 |
by (metis map_idx_len)
|
|
|
1278 |
next
|
|
|
1279 |
case 2
|
|
|
1280 |
{ fix l
|
|
|
1281 |
assume "l < length (map_idx f sts)"
|
|
|
1282 |
hence "l < length sts"
|
|
|
1283 |
by (metis map_idx_len)
|
|
|
1284 |
from map_idx_idx[OF this] and assms and this
|
|
|
1285 |
have "map_idx f sts ! l = map_idx g sts ! l"
|
|
|
1286 |
by (smt list_eq_iff_nth_eq map_idx_idx map_idx_len)
|
|
|
1287 |
} thus ?case by auto
|
|
|
1288 |
qed
|
|
|
1289 |
|
|
|
1290 |
lemma perm_s_commut: "perm_s i j sts = perm_s j i sts"
|
|
|
1291 |
apply (unfold perm_s_def, rule map_idx_eq, unfold perm_b_def)
|
|
|
1292 |
by smt
|
|
|
1293 |
|
|
|
1294 |
lemma map_idx_id: "map_idx id sts = sts"
|
|
|
1295 |
proof(induct rule:nth_equalityI)
|
|
|
1296 |
case 1
|
|
|
1297 |
from map_idx_len show "length (map_idx id sts) = length sts" .
|
|
|
1298 |
next
|
|
|
1299 |
case 2
|
|
|
1300 |
{ fix l
|
|
|
1301 |
assume "l < length (map_idx id sts)"
|
|
|
1302 |
from map_idx_idx[OF this[unfolded map_idx_len]]
|
|
|
1303 |
have "map_idx id sts ! l = sts ! l" by simp
|
|
|
1304 |
} thus ?case by auto
|
|
|
1305 |
qed
|
|
|
1306 |
|
|
|
1307 |
lemma perm_s_lt_i:
|
|
|
1308 |
assumes "\<not> i < length sts"
|
|
|
1309 |
shows "perm_s i j sts = sts"
|
|
|
1310 |
proof -
|
|
|
1311 |
have "map_idx (perm_b (length sts) i j) sts = map_idx id sts"
|
|
|
1312 |
proof(rule map_idx_eq, default, clarsimp)
|
|
|
1313 |
fix l
|
|
|
1314 |
assume "l < length sts"
|
|
|
1315 |
with assms
|
|
|
1316 |
show "perm_b (length sts) i j l = l"
|
|
|
1317 |
by (unfold perm_b_def, auto)
|
|
|
1318 |
qed
|
|
|
1319 |
with map_idx_id
|
|
|
1320 |
have "map_idx (perm_b (length sts) i j) sts = sts" by simp
|
|
|
1321 |
thus ?thesis by (simp add:perm_s_def)
|
|
|
1322 |
qed
|
|
|
1323 |
|
|
|
1324 |
lemma perm_s_lt:
|
|
|
1325 |
assumes "\<not> i < length sts \<or> \<not> j < length sts"
|
|
|
1326 |
shows "perm_s i j sts = sts"
|
|
|
1327 |
using assms
|
|
|
1328 |
proof
|
|
|
1329 |
assume "\<not> i < length sts"
|
|
|
1330 |
from perm_s_lt_i[OF this] show ?thesis .
|
|
|
1331 |
next
|
|
|
1332 |
assume "\<not> j < length sts"
|
|
|
1333 |
from perm_s_lt_i[OF this, of i, unfolded perm_s_commut]
|
|
|
1334 |
show ?thesis .
|
|
|
1335 |
qed
|
|
|
1336 |
|
|
|
1337 |
lemma perm_s_update_i:
|
|
|
1338 |
assumes "i < length sts"
|
|
|
1339 |
and "j < length sts"
|
|
|
1340 |
shows "sts ! i = perm_s i j sts ! j"
|
|
|
1341 |
proof -
|
|
|
1342 |
from map_idx_idx[OF assms(2)]
|
|
|
1343 |
have "map_idx (perm_b (length sts) i j) sts ! j = sts!(perm_b (length sts) i j j)" .
|
|
|
1344 |
with assms
|
|
|
1345 |
show ?thesis
|
|
|
1346 |
by (auto simp:perm_s_def perm_b_def)
|
|
|
1347 |
qed
|
|
|
1348 |
|
|
|
1349 |
lemma nth_perm_s_neq:
|
|
|
1350 |
assumes "l \<noteq> j"
|
|
|
1351 |
and "l \<noteq> i"
|
|
|
1352 |
and "l < length sts"
|
|
|
1353 |
shows "sts ! l = perm_s i j sts ! l"
|
|
|
1354 |
proof -
|
|
|
1355 |
have "map_idx (perm_b (length sts) i j) sts ! l = sts!(perm_b (length sts) i j l)"
|
|
|
1356 |
by (metis assms(3) map_idx_def map_idx_idx)
|
|
|
1357 |
with assms
|
|
|
1358 |
show ?thesis
|
|
|
1359 |
by (unfold perm_s_def perm_b_def, auto)
|
|
|
1360 |
qed
|
|
|
1361 |
|
|
|
1362 |
lemma map_idx_update:
|
|
|
1363 |
assumes "f j = i"
|
|
|
1364 |
and "inj f"
|
|
|
1365 |
and "i < length sts"
|
|
|
1366 |
and "j < length sts"
|
|
|
1367 |
shows "map_idx f (sts[i:=v]) = map_idx f sts[j := v]"
|
|
|
1368 |
proof(induct rule:nth_equalityI)
|
|
|
1369 |
case 1
|
|
|
1370 |
show "length (map_idx f (sts[i := v])) = length (map_idx f sts[j := v])"
|
|
|
1371 |
by (metis length_list_update map_idx_len)
|
|
|
1372 |
next
|
|
|
1373 |
case 2
|
|
|
1374 |
{ fix l
|
|
|
1375 |
assume lt_l: "l < length (map_idx f (sts[i := v]))"
|
|
|
1376 |
have eq_nth: "sts[i := v] ! f l = map_idx f sts[j := v] ! l"
|
|
|
1377 |
proof(cases "f l = i")
|
|
|
1378 |
case False
|
|
|
1379 |
from lt_l have "l < length sts"
|
|
|
1380 |
by (metis length_list_update map_idx_len)
|
|
|
1381 |
from map_idx_idx[OF this, of f] have " map_idx f sts ! l = sts ! f l" .
|
|
|
1382 |
moreover from False assms have "l \<noteq> j" by auto
|
|
|
1383 |
moreover note False
|
|
|
1384 |
ultimately show ?thesis by simp
|
|
|
1385 |
next
|
|
|
1386 |
case True
|
|
|
1387 |
with assms have eq_l: "l = j"
|
|
|
1388 |
by (metis inj_eq)
|
|
|
1389 |
moreover from lt_l eq_l
|
|
|
1390 |
have "j < length (map_idx f sts[j := v])"
|
|
|
1391 |
by (metis length_list_update map_idx_len)
|
|
|
1392 |
moreover note True assms
|
|
|
1393 |
ultimately show ?thesis by simp
|
|
|
1394 |
qed
|
|
|
1395 |
from lt_l have "l < length (sts[i := v])"
|
|
|
1396 |
by (metis map_idx_len)
|
|
|
1397 |
from map_idx_idx[OF this] eq_nth
|
|
|
1398 |
have "map_idx f (sts[i := v]) ! l = map_idx f sts[j := v] ! l" by simp
|
|
|
1399 |
} thus ?case by auto
|
|
|
1400 |
qed
|
|
|
1401 |
|
|
|
1402 |
lemma perm_s_update:
|
|
|
1403 |
assumes "i < length sts"
|
|
|
1404 |
and "j < length sts"
|
|
|
1405 |
shows "(perm_s i j sts)[i := v] = perm_s i j (sts[j := v])"
|
|
|
1406 |
proof -
|
|
|
1407 |
have "map_idx (perm_b (length (sts[j := v])) i j) (sts[j := v]) =
|
|
|
1408 |
map_idx (perm_b (length (sts[j := v])) i j) sts[i := v]"
|
|
|
1409 |
proof(rule map_idx_update[OF _ _ assms(2, 1)])
|
|
|
1410 |
from inj_perm_b show "inj (perm_b (length (sts[j := v])) i j)" .
|
|
|
1411 |
next
|
|
|
1412 |
from assms show "perm_b (length (sts[j := v])) i j i = j"
|
|
|
1413 |
by (auto simp:perm_b_def)
|
|
|
1414 |
qed
|
|
|
1415 |
hence "map_idx (perm_b (length sts) i j) sts[i := v] =
|
|
|
1416 |
map_idx (perm_b (length (sts[j := v])) i j) (sts[j := v])"
|
|
|
1417 |
by simp
|
|
|
1418 |
thus ?thesis by (simp add:perm_s_def)
|
|
|
1419 |
qed
|
|
|
1420 |
|
|
|
1421 |
lemma perm_s_update_neq:
|
|
|
1422 |
assumes "l \<noteq> i"
|
|
|
1423 |
and "l \<noteq> j"
|
|
|
1424 |
shows "perm_s i j sts[l := v] = perm_s i j (sts[l := v])"
|
|
|
1425 |
proof(cases "i < length sts \<and> j < length sts")
|
|
|
1426 |
case False
|
|
|
1427 |
with perm_s_lt have "perm_s i j sts = sts" by auto
|
|
|
1428 |
moreover have "perm_s i j (sts[l:=v]) = sts[l:=v]"
|
|
|
1429 |
proof -
|
|
|
1430 |
have "length (sts[l:=v]) = length sts" by auto
|
|
|
1431 |
from False[folded this] perm_s_lt
|
|
|
1432 |
show ?thesis by metis
|
|
|
1433 |
qed
|
|
|
1434 |
ultimately show ?thesis by simp
|
|
|
1435 |
next
|
|
|
1436 |
case True
|
|
|
1437 |
note lt_ij = this
|
|
|
1438 |
show ?thesis
|
|
|
1439 |
proof(cases "l < length sts")
|
|
|
1440 |
case False
|
|
|
1441 |
hence "sts[l:=v] = sts" by auto
|
|
|
1442 |
moreover have "perm_s i j sts[l := v] = perm_s i j sts"
|
|
|
1443 |
proof -
|
|
|
1444 |
from False and perm_s_len
|
|
|
1445 |
have "\<not> l < length (perm_s i j sts)" by metis
|
|
|
1446 |
thus ?thesis by auto
|
|
|
1447 |
qed
|
|
|
1448 |
ultimately show ?thesis by simp
|
|
|
1449 |
next
|
|
|
1450 |
case True
|
|
|
1451 |
show ?thesis
|
|
|
1452 |
proof -
|
|
|
1453 |
have "map_idx (perm_b (length (sts[l := v])) i j) (sts[l := v]) =
|
|
|
1454 |
map_idx (perm_b (length (sts[l := v])) i j) sts[l := v]"
|
|
|
1455 |
proof(induct rule:map_idx_update [OF _ inj_perm_b True True])
|
|
|
1456 |
case 1
|
|
|
1457 |
from assms lt_ij
|
|
|
1458 |
show ?case
|
|
|
1459 |
by (unfold perm_b_def, auto)
|
|
|
1460 |
qed
|
|
|
1461 |
thus ?thesis
|
|
|
1462 |
by (unfold perm_s_def, simp)
|
|
|
1463 |
qed
|
|
|
1464 |
qed
|
|
|
1465 |
qed
|
|
|
1466 |
|
|
|
1467 |
lemma perm_sb: "(perm_s i j sts)[perm_b (length sts) i j l := v] = perm_s i j (sts[l := v])"
|
|
|
1468 |
apply(subst perm_b_def, auto simp:perm_s_len perm_s_lt perm_s_update)
|
|
|
1469 |
apply (subst perm_s_commut, subst (2) perm_s_commut, rule_tac perm_s_update, auto)
|
|
|
1470 |
by (rule_tac perm_s_update_neq, auto)
|
|
|
1471 |
|
|
|
1472 |
lemma perm_s_id: "perm_s i i sts = sts" (is "?L = ?R")
|
|
|
1473 |
proof -
|
|
|
1474 |
from map_idx_id have "?R = map_idx id sts" by metis
|
|
|
1475 |
also have "\<dots> = ?L"
|
|
|
1476 |
by (unfold perm_s_def, rule map_idx_eq, unfold perm_b_def, auto)
|
|
|
1477 |
finally show ?thesis by simp
|
|
|
1478 |
qed
|
|
|
1479 |
|
|
|
1480 |
lemma upto_map:
|
|
|
1481 |
assumes "i \<le> j"
|
|
|
1482 |
shows "[i .. j] = i # map (\<lambda> x. x + 1) [i .. (j - 1)]"
|
|
|
1483 |
using assms
|
|
|
1484 |
proof(induct i j rule:upto.induct)
|
|
|
1485 |
case (1 i j)
|
|
|
1486 |
show ?case (is "?L = ?R")
|
|
|
1487 |
proof -
|
|
|
1488 |
from 1(2)
|
|
|
1489 |
have eq_l: "?L = i # [i+1 .. j]" by (simp add:upto.simps)
|
|
|
1490 |
show ?thesis
|
|
|
1491 |
proof(cases "i + 1 \<le> j")
|
|
|
1492 |
case False
|
|
|
1493 |
with eq_l show ?thesis by (auto simp:upto.simps)
|
|
|
1494 |
next
|
|
|
1495 |
case True
|
|
|
1496 |
have "[i + 1..j] = map (\<lambda>x. x + 1) [i..j - 1]"
|
|
|
1497 |
by (smt "1.hyps" Cons_eq_map_conv True upto.simps)
|
|
|
1498 |
with eq_l
|
|
|
1499 |
show ?thesis by simp
|
|
|
1500 |
qed
|
|
|
1501 |
qed
|
|
|
1502 |
qed
|
|
|
1503 |
|
|
|
1504 |
lemma perm_s_cons: "(perm_s (Suc i) (Suc j) (s # sts)) = s#perm_s i j sts"
|
|
|
1505 |
proof -
|
|
|
1506 |
have le_0: "0 \<le> int (length (s # sts)) - 1" by simp
|
|
|
1507 |
have "map (\<lambda>k. (s # sts) ! perm_b (length (s # sts)) (Suc i) (Suc j) (nat k))
|
|
|
1508 |
[0..int (length (s # sts)) - 1] =
|
|
|
1509 |
s # map (\<lambda>k. sts ! perm_b (length sts) i j (nat k)) [0..int (length sts) - 1]"
|
|
|
1510 |
by (unfold upto_map[OF le_0], auto simp:perm_b_def, smt+)
|
|
|
1511 |
thus ?thesis by (unfold perm_s_def map_idx_def, simp)
|
|
|
1512 |
qed
|
|
|
1513 |
|
|
|
1514 |
lemma perm_wf_cpg_test:
|
|
|
1515 |
assumes "wf_cpg_test sts cpg = (True, sts')"
|
|
|
1516 |
shows "wf_cpg_test (perm_s i j sts) (perm (length sts) i j cpg) =
|
|
|
1517 |
(True, perm_s i j sts')"
|
|
|
1518 |
using assms
|
|
|
1519 |
proof(induct cpg arbitrary:t i j sts sts')
|
|
|
1520 |
case (CInstr instr i j sts sts')
|
|
|
1521 |
obtain a0 l0 a1 l1 where eq_instr: "instr = ((a0, l0), (a1, l1))"
|
|
|
1522 |
by (metis prod.exhaust)
|
|
|
1523 |
from CInstr
|
|
|
1524 |
show ?case
|
|
|
1525 |
apply (unfold eq_instr, clarsimp)
|
|
|
1526 |
by (unfold perm_s_len perm_b_def, clarsimp)
|
|
|
1527 |
next
|
|
|
1528 |
case (CLabel l i j sts sts')
|
|
|
1529 |
have "(perm_s i j sts)[perm_b (length sts) i j l := Bound] = perm_s i j (sts[l := Bound])"
|
|
|
1530 |
by (metis perm_sb)
|
|
|
1531 |
with CLabel
|
|
|
1532 |
show ?case
|
|
|
1533 |
apply (auto simp:perm_s_len perm_sb)
|
|
|
1534 |
apply (subst perm_b_def, auto simp:perm_sb)
|
|
|
1535 |
apply (subst perm_b_def, auto simp:perm_s_lt perm_s_update_i)
|
|
|
1536 |
apply (unfold perm_s_id, subst perm_s_commut, simp add: perm_s_update_i[symmetric])
|
|
|
1537 |
apply (simp add:perm_s_update_i[symmetric])
|
|
|
1538 |
by (simp add: nth_perm_s_neq[symmetric])
|
|
|
1539 |
next
|
|
|
1540 |
case (CSeq c1 c2 i j sts sts')
|
|
|
1541 |
thus ?case
|
|
|
1542 |
apply (auto split:prod.splits)
|
|
|
1543 |
apply (metis (hide_lams, no_types) less_eq_list_def prod.inject wf_cpg_test_le)
|
|
|
1544 |
by (metis (hide_lams, no_types) less_eq_list_def prod.inject wf_cpg_test_le)
|
|
|
1545 |
next
|
|
|
1546 |
case (CLocal body i j sts sts')
|
|
|
1547 |
from this(2) obtain sts1 where h: "wf_cpg_test (Free # sts) body = (True, sts1)" "tl sts1 = sts'"
|
|
|
1548 |
by (auto simp:lift0_def lift_b_def split:prod.splits)
|
|
|
1549 |
from wf_cpg_test_le[OF h(1)] have "length (Free#sts) = length sts1"
|
|
|
1550 |
by (unfold less_eq_list_def, simp)
|
|
|
1551 |
hence eq_sts1: "sts1 = hd sts1 # tl sts1"
|
|
|
1552 |
by (metis append_Nil append_eq_conv_conj hd.simps list.exhaust tl.simps(2))
|
|
|
1553 |
from CLocal(1)[OF h(1), of "Suc i" "Suc j"] h(2) eq_sts1
|
|
|
1554 |
show ?case
|
|
|
1555 |
apply (auto split:prod.splits simp:perm_s_cons)
|
|
|
1556 |
by (metis perm_s_cons tl.simps(2))
|
|
|
1557 |
qed
|
|
|
1558 |
|
|
|
1559 |
lemma nth_perm_sb:
|
|
|
1560 |
assumes "l0 < length env"
|
|
|
1561 |
shows "perm_s i j env ! perm_b (length env) i j l0 = env ! l0"
|
|
|
1562 |
by (metis assms nth_perm_s_neq perm_b_def perm_s_commut perm_s_lt perm_s_update_i)
|
|
|
1563 |
|
|
|
1564 |
|
|
|
1565 |
lemma perm_c2t:
|
|
|
1566 |
assumes "wf_cpg_test sts cpg = (True, sts')"
|
|
|
1567 |
and "length env = length sts"
|
|
|
1568 |
shows "c2t (perm_s i j env) (perm (length env) i j cpg) =
|
|
|
1569 |
c2t env cpg"
|
|
|
1570 |
using assms
|
|
|
1571 |
proof(induct cpg arbitrary:i j env sts sts')
|
|
|
1572 |
case (CInstr instr i j env sts sts')
|
|
|
1573 |
obtain a0 l0 a1 l1 where eq_instr: "instr = ((a0, l0), (a1, l1))"
|
|
|
1574 |
by (metis prod.exhaust)
|
|
|
1575 |
from CInstr have h: "l0 < length env" "l1 < length env"
|
|
|
1576 |
by (auto simp:eq_instr)
|
|
|
1577 |
with CInstr(2)
|
|
|
1578 |
show ?case
|
|
|
1579 |
apply (auto simp:eq_instr)
|
|
|
1580 |
by (metis nth_perm_sb)+
|
|
|
1581 |
next
|
|
|
1582 |
case (CLabel l t env es sts sts')
|
|
|
1583 |
thus ?case
|
|
|
1584 |
apply (auto)
|
|
|
1585 |
by (metis nth_perm_sb)
|
|
|
1586 |
next
|
|
|
1587 |
case (CSeq c1 c2 i j env sts sts')
|
|
|
1588 |
from CSeq(3) obtain sts1
|
|
|
1589 |
where h: "wf_cpg_test sts c1 = (True, sts1)" "wf_cpg_test sts1 c2 = (True, sts')"
|
|
|
1590 |
by (auto split:prod.splits)
|
|
|
1591 |
from wf_cpg_test_le[OF h(1)] have "length sts = length sts1"
|
|
|
1592 |
by (auto simp:less_eq_list_def)
|
|
|
1593 |
from CSeq(4)[unfolded this] have eq_len_env: "length env = length sts1" .
|
|
|
1594 |
from CSeq(1)[OF h(1) CSeq(4)]
|
|
|
1595 |
CSeq(2)[OF h(2) eq_len_env]
|
|
|
1596 |
show ?case by auto
|
|
|
1597 |
next
|
|
|
1598 |
case (CLocal body i j env sts sts')
|
|
|
1599 |
{ fix x
|
|
|
1600 |
from CLocal(2, 3)
|
|
|
1601 |
obtain sts1 where "wf_cpg_test (Free # sts) body = (True, sts1)"
|
|
|
1602 |
"length (x#env) = length (Free # sts)"
|
|
|
1603 |
by (auto split:prod.splits)
|
|
|
1604 |
from CLocal(1)[OF this]
|
|
|
1605 |
have "(c2t (x # perm_s i j env) (perm (Suc (length env)) (Suc i) (Suc j) body)) =
|
|
|
1606 |
(c2t (x # env) body)"
|
|
|
1607 |
by (metis Suc_length_conv perm_s_cons)
|
|
|
1608 |
} thus ?case by simp
|
|
|
1609 |
qed
|
|
|
1610 |
|
|
|
1611 |
lemma wf_cpg_test_disj_aux1:
|
|
|
1612 |
assumes "sts_disj sts1 (sts[l := Bound] - sts)"
|
|
|
1613 |
"l < length sts"
|
|
|
1614 |
"sts ! l = Free"
|
|
|
1615 |
shows "(sts1 + sts) ! l = Free"
|
|
|
1616 |
proof -
|
|
|
1617 |
from assms(1)[unfolded sts_disj_def]
|
|
|
1618 |
have h: "length sts1 = length (sts[l := Bound] - sts)"
|
|
|
1619 |
"(\<forall>i<length sts1. \<not> (sts1 ! i = Bound \<and> (sts[l := Bound] - sts) ! i = Bound))"
|
|
|
1620 |
by auto
|
|
|
1621 |
from h(1) assms(2)
|
|
|
1622 |
have lt_l: "l < length sts1"
|
|
|
1623 |
"l < length (sts[l := Bound] - sts)"
|
|
|
1624 |
"l < length (sts1 + sts)"
|
|
|
1625 |
apply (smt length_list_update minus_list_len)
|
|
|
1626 |
apply (smt assms(2) length_list_update minus_list_len)
|
|
|
1627 |
by (smt assms(2) h(1) length_list_update length_sts_plus minus_list_len)
|
|
|
1628 |
from h(2)[rule_format, of l, OF this(1)]
|
|
|
1629 |
have " \<not> (sts1 ! l = Bound \<and> (sts[l := Bound] - sts) ! l = Bound)" .
|
|
|
1630 |
with assms(3) nth_sts_minus[OF lt_l(2)] nth_sts_plus[OF lt_l(3)] assms(2)
|
|
|
1631 |
show ?thesis
|
|
|
1632 |
by (cases "sts1!l", auto)
|
|
|
1633 |
qed
|
|
|
1634 |
|
|
|
1635 |
lemma wf_cpg_test_disj_aux2:
|
|
|
1636 |
assumes "sts_disj sts1 (sts[l := Bound] - sts)"
|
|
|
1637 |
" l < length sts"
|
|
|
1638 |
shows "(sts1 + sts)[l := Bound] = sts1 + sts[l := Bound]"
|
|
|
1639 |
proof -
|
|
|
1640 |
from assms have lt_l: "l < length (sts1 + sts[l:=Bound])"
|
|
|
1641 |
"l < length (sts1 + sts)"
|
|
|
1642 |
apply (smt length_list_update length_sts_plus minus_list_len sts_disj_def)
|
|
|
1643 |
by (smt assms(1) assms(2) length_list_update length_sts_plus minus_list_len sts_disj_def)
|
|
|
1644 |
show ?thesis
|
|
|
1645 |
proof(induct rule:nth_equalityI)
|
|
|
1646 |
case 1
|
|
|
1647 |
show ?case
|
|
|
1648 |
by (smt assms(1) length_list_update length_sts_plus minus_list_len sts_disj_def)
|
|
|
1649 |
next
|
|
|
1650 |
case 2
|
|
|
1651 |
{ fix i
|
|
|
1652 |
assume lt_i: "i < length ((sts1 + sts)[l := Bound])"
|
|
|
1653 |
have " (sts1 + sts)[l := Bound] ! i = (sts1 + sts[l := Bound]) ! i"
|
|
|
1654 |
proof(cases "i = l")
|
|
|
1655 |
case True
|
|
|
1656 |
with nth_sts_plus[OF lt_l(1)] assms(2) nth_sts_plus[OF lt_l(2)] lt_l
|
|
|
1657 |
show ?thesis
|
|
|
1658 |
by (cases "sts1 ! l", auto)
|
|
|
1659 |
next
|
|
|
1660 |
case False
|
|
|
1661 |
from lt_i have "i < length (sts1 + sts)" "i < length (sts1 + sts[l := Bound])"
|
|
|
1662 |
apply auto
|
|
|
1663 |
by (metis length_list_update plus_list_len)
|
|
|
1664 |
from nth_sts_plus[OF this(1)] nth_sts_plus[OF this(2)] lt_i lt_l False
|
|
|
1665 |
show ?thesis
|
|
|
1666 |
by simp
|
|
|
1667 |
qed
|
|
|
1668 |
} thus ?case by auto
|
|
|
1669 |
qed
|
|
|
1670 |
qed
|
|
|
1671 |
|
|
|
1672 |
lemma sts_list_plus_commut:
|
|
|
1673 |
shows "sts1 + sts2 = sts2 + (sts1:: status list)"
|
|
|
1674 |
proof(induct rule:nth_equalityI)
|
|
|
1675 |
case 1
|
|
|
1676 |
show ?case
|
|
|
1677 |
by (metis min_max.inf.commute plus_list_len)
|
|
|
1678 |
next
|
|
|
1679 |
case 2
|
|
|
1680 |
{ fix i
|
|
|
1681 |
assume lt_i1: "i<length (sts1 + sts2)"
|
|
|
1682 |
hence lt_i2: "i < length (sts2 + sts1)"
|
|
|
1683 |
by (smt plus_list_len)
|
|
|
1684 |
from nth_sts_plus[OF this] nth_sts_plus[OF lt_i1]
|
|
|
1685 |
have "(sts1 + sts2) ! i = (sts2 + sts1) ! i"
|
|
|
1686 |
apply simp
|
|
|
1687 |
apply (cases "sts1!i", cases "sts2!i", auto)
|
|
|
1688 |
by (cases "sts2!i", auto)
|
|
|
1689 |
} thus ?case by auto
|
|
|
1690 |
qed
|
|
|
1691 |
|
|
|
1692 |
lemma sts_disj_cons:
|
|
|
1693 |
assumes "sts_disj sts1 sts2"
|
|
|
1694 |
shows "sts_disj (Free # sts1) (s # sts2)"
|
|
|
1695 |
using assms
|
|
|
1696 |
proof -
|
|
|
1697 |
from assms
|
|
|
1698 |
have h: "length sts1 = length sts2"
|
|
|
1699 |
"(\<forall>i<length sts1. \<not> (sts1 ! i = Bound \<and> sts2 ! i = Bound))"
|
|
|
1700 |
by (unfold sts_disj_def, auto)
|
|
|
1701 |
from h(1) have "length (Free # sts1) = length (s # sts2)" by simp
|
|
|
1702 |
moreover {
|
|
|
1703 |
fix i
|
|
|
1704 |
assume lt_i: "i<length (Free # sts1)"
|
|
|
1705 |
have "\<not> ((Free # sts1) ! i = Bound \<and> (s # sts2) ! i = Bound)"
|
|
|
1706 |
proof(cases "i")
|
|
|
1707 |
case 0
|
|
|
1708 |
thus ?thesis by simp
|
|
|
1709 |
next
|
|
|
1710 |
case (Suc k)
|
|
|
1711 |
from h(2)[rule_format, of k] lt_i[unfolded Suc] Suc
|
|
|
1712 |
show ?thesis by auto
|
|
|
1713 |
qed
|
|
|
1714 |
}
|
|
|
1715 |
ultimately show ?thesis by (auto simp:sts_disj_def)
|
|
|
1716 |
qed
|
|
|
1717 |
|
|
|
1718 |
lemma sts_disj_uncomb:
|
|
|
1719 |
assumes "sts_disj sts (sts1 + sts2)"
|
|
|
1720 |
and "sts_disj sts1 sts2"
|
|
|
1721 |
shows "sts_disj sts sts1" "sts_disj sts sts2"
|
|
|
1722 |
using assms
|
|
|
1723 |
apply (smt assms(1) assms(2) length_sts_plus nth_sts_plus plus_status.simps(2) sts_disj_def)
|
|
|
1724 |
by (smt assms(1) assms(2) length_sts_plus nth_sts_plus
|
|
|
1725 |
plus_status.simps(2) sts_disj_def sts_list_plus_commut)
|
|
|
1726 |
|
|
|
1727 |
lemma wf_cpg_test_disj:
|
|
|
1728 |
assumes "wf_cpg_test sts cpg = (True, sts')"
|
|
|
1729 |
and "sts_disj sts1 (sts' - sts)"
|
|
|
1730 |
shows "wf_cpg_test (sts1 + sts) cpg = (True, sts1 + sts')"
|
|
|
1731 |
using assms
|
|
|
1732 |
proof(induct cpg arbitrary:sts sts1 sts')
|
|
|
1733 |
case (CInstr instr sts sts1 sts')
|
|
|
1734 |
obtain a0 l0 a1 l1 where eq_instr: "instr = ((a0, l0), (a1, l1))"
|
|
|
1735 |
by (metis pair_collapse)
|
|
|
1736 |
with CInstr(1) have h: "l0 < length sts" "l1 < length sts" "sts = sts'" by auto
|
|
|
1737 |
with CInstr eq_instr
|
|
|
1738 |
show ?case
|
|
|
1739 |
apply (auto)
|
|
|
1740 |
by (smt length_sts_plus minus_list_len sts_disj_def)+
|
|
|
1741 |
next
|
|
|
1742 |
case (CLabel l sts sts1 sts')
|
|
|
1743 |
thus ?case
|
|
|
1744 |
apply auto
|
|
|
1745 |
apply (smt length_list_update length_sts_plus minus_list_len sts_disj_def)
|
|
|
1746 |
by (auto simp: wf_cpg_test_disj_aux1 wf_cpg_test_disj_aux2)
|
|
|
1747 |
next
|
|
|
1748 |
case (CSeq c1 c2 sts sts1 sts')
|
|
|
1749 |
from CSeq(3) obtain sts''
|
|
|
1750 |
where h: "wf_cpg_test sts c1 = (True, sts'')" "wf_cpg_test sts'' c2 = (True, sts')"
|
|
|
1751 |
by (auto split:prod.splits)
|
|
|
1752 |
from wf_cpg_test_le[OF h(1)] have "length sts = length sts''"
|
|
|
1753 |
by (auto simp:less_eq_list_def)
|
|
|
1754 |
from sts_le_comb[OF wf_cpg_test_le[OF h(1)] wf_cpg_test_le[OF h(2)]]
|
|
|
1755 |
have " sts' - sts = (sts'' - sts) + (sts' - sts'')" "sts_disj (sts'' - sts) (sts' - sts'')"
|
|
|
1756 |
by auto
|
|
|
1757 |
from sts_disj_uncomb[OF CSeq(4)[unfolded this(1)] this(2)]
|
|
|
1758 |
have "sts_disj sts1 (sts'' - sts)" "sts_disj sts1 (sts' - sts'')" .
|
|
|
1759 |
from CSeq(1)[OF h(1) this(1)] CSeq(2)[OF h(2) this(2)]
|
|
|
1760 |
have "wf_cpg_test (sts1 + sts) c1 = (True, sts1 + sts'')"
|
|
|
1761 |
"wf_cpg_test (sts1 + sts'') c2 = (True, sts1 + sts')" .
|
|
|
1762 |
thus ?case
|
|
|
1763 |
by simp
|
|
|
1764 |
next
|
|
|
1765 |
case (CLocal body sts sts1 sts')
|
|
|
1766 |
from this(2)
|
|
|
1767 |
obtain sts'' where h: "wf_cpg_test (Free # sts) body = (True, sts'')" "sts' = tl sts''"
|
|
|
1768 |
by (auto split:prod.splits)
|
|
|
1769 |
from wf_cpg_test_le[OF h(1), unfolded less_eq_list_def] h(2)
|
|
|
1770 |
obtain s where eq_sts'': "sts'' = s#sts'"
|
|
|
1771 |
by (metis Suc_length_conv list.size(4) tl.simps(2))
|
|
|
1772 |
let ?sts = "Free#sts1"
|
|
|
1773 |
from CLocal(3) have "sts_disj ?sts (sts'' - (Free # sts))"
|
|
|
1774 |
apply (unfold eq_sts'', simp)
|
|
|
1775 |
by (metis sts_disj_cons)
|
|
|
1776 |
from CLocal(1)[OF h(1) this] eq_sts''
|
|
|
1777 |
show ?case
|
|
|
1778 |
by (auto split:prod.splits)
|
|
|
1779 |
qed
|
|
|
1780 |
|
|
|
1781 |
section {* Application of the theory above *}
|
|
|
1782 |
|
|
|
1783 |
definition "move_left_skel = CLocal (CSeq (CInstr ((L, 0), (L, 0))) (CLabel 0))"
|
|
|
1784 |
|
|
|
1785 |
lemma wt_move_left: "wf_cpg_test [] move_left_skel = (True, [])"
|
|
|
1786 |
by (unfold move_left_skel_def, simp)
|
|
|
1787 |
|
|
|
1788 |
lemma ct_move_left: "c2t [] move_left_skel = move_left"
|
|
|
1789 |
by (unfold move_left_skel_def move_left_def, simp)
|
|
|
1790 |
|
|
|
1791 |
lemma wf_move_left: "\<forall> i. \<exists> s j. (i:[move_left]:j ) s"
|
|
|
1792 |
proof -
|
|
|
1793 |
from wf_cpg_test_correct[OF wt_move_left] ct_move_left
|
|
|
1794 |
show ?thesis
|
|
|
1795 |
by (unfold c2p_def, simp, metis)
|
|
|
1796 |
qed
|
|
|
1797 |
|
|
|
1798 |
definition "jmp_skel = CInstr ((W0, 0), (W1, 0))"
|
|
|
1799 |
|
|
|
1800 |
lemma wt_jmp: "wf_cpg_test [Free] jmp_skel = (True, [Free])"
|
|
|
1801 |
by (unfold jmp_skel_def, simp)
|
|
|
1802 |
|
|
|
1803 |
lemma ct_jmp: "c2t [l] jmp_skel = (jmp l)"
|
|
|
1804 |
by (unfold jmp_skel_def jmp_def, simp)
|
|
|
1805 |
|
|
|
1806 |
lemma wf_jmp: "\<forall> i. \<exists> s j. (i:[jmp l]:j ) s"
|
|
|
1807 |
proof -
|
|
|
1808 |
from wf_cpg_test_correct[OF wt_jmp] ct_jmp
|
|
|
1809 |
show ?thesis
|
|
|
1810 |
apply (unfold c2p_def, simp)
|
|
|
1811 |
by (metis One_nat_def Suc_eq_plus1 list.size(3) list.size(4))
|
|
|
1812 |
qed
|
|
|
1813 |
|
|
|
1814 |
definition "label_skel = CLabel 0"
|
|
|
1815 |
|
|
|
1816 |
lemma wt_label: "wf_cpg_test [Free] label_skel = (True, [Bound])"
|
|
|
1817 |
by (simp add:label_skel_def)
|
|
|
1818 |
|
|
|
1819 |
lemma ct_label: "c2t [l] label_skel = (TLabel l)"
|
|
|
1820 |
by (simp add:label_skel_def)
|
|
|
1821 |
|
|
|
1822 |
thm if_zero_def
|
|
|
1823 |
|
|
|
1824 |
definition "if_zero_skel = CLocal (CSeq (CInstr ((W0, 1), (W1, 0))) (
|
|
|
1825 |
CLabel 0
|
|
|
1826 |
)
|
|
|
1827 |
)"
|
|
|
1828 |
|
|
|
1829 |
lemma wt_if_zero: "wf_cpg_test [Free] if_zero_skel = (True, [Free])"
|
|
|
1830 |
by (simp add:if_zero_skel_def)
|
|
|
1831 |
|
|
|
1832 |
definition "left_until_zero_skel = CLocal (CLocal (
|
|
|
1833 |
CSeq (CLabel 1) (
|
|
|
1834 |
CSeq if_zero_skel (
|
|
|
1835 |
CSeq move_left_skel (
|
|
|
1836 |
CSeq (lift_t 0 1 jmp_skel) (
|
|
|
1837 |
label_skel
|
|
|
1838 |
))))
|
|
|
1839 |
))"
|
|
|
1840 |
|
|
|
1841 |
lemma w1: "wf_cpg_test [Free, Bound] if_zero_skel = (True, [Free, Bound])"
|
|
|
1842 |
by (simp add:if_zero_skel_def)
|
|
|
1843 |
|
|
|
1844 |
lemma w2: "wf_cpg_test [Free, Bound] move_left_skel = (True, [Free, Bound])"
|
|
|
1845 |
by (simp add:move_left_skel_def)
|
|
|
1846 |
|
|
|
1847 |
lemma w3: "wf_cpg_test [Free, Bound] (lift_t 0 (Suc 0) jmp_skel) =
|
|
|
1848 |
(True, [Free, Bound])"
|
|
|
1849 |
by (simp add:jmp_skel_def lift_b_def)
|
|
|
1850 |
|
|
|
1851 |
lemma w4: "wf_cpg_test [Free, Bound] label_skel = (True, [Bound, Bound])"
|
|
|
1852 |
by (unfold label_skel_def, simp)
|
|
|
1853 |
|
|
|
1854 |
lemma wt_left_until_zero:
|
|
|
1855 |
"wf_cpg_test [] left_until_zero_skel = (True, [])"
|
|
|
1856 |
by (unfold left_until_zero_skel_def, simp add:w1 w2 w3 w4)
|
|
|
1857 |
|
|
|
1858 |
lemma c1: "c2t [xa, x] if_zero_skel = if_zero xa"
|
|
|
1859 |
by (simp add:if_zero_skel_def if_zero_def)
|
|
|
1860 |
|
|
|
1861 |
lemma c2: "c2t [xa, x] move_left_skel = move_left"
|
|
|
1862 |
by (simp add:move_left_skel_def move_left_def)
|
|
|
1863 |
|
|
|
1864 |
lemma c3: "c2t [xa, x] (lift_t 0 (Suc 0) jmp_skel) =
|
|
|
1865 |
jmp x"
|
|
|
1866 |
by (simp add:jmp_skel_def jmp_def lift_b_def)
|
|
|
1867 |
|
|
|
1868 |
lemma c4: "c2t [xa, x] label_skel = TLabel xa"
|
|
|
1869 |
by (simp add:label_skel_def)
|
|
|
1870 |
|
|
|
1871 |
lemma ct_left_until_zero:
|
|
|
1872 |
"c2t [] left_until_zero_skel = left_until_zero"
|
|
|
1873 |
apply (unfold left_until_zero_def left_until_zero_skel_def)
|
|
|
1874 |
by (simp add:c1 c2 c3 c4)
|
|
|
1875 |
|
|
|
1876 |
lemma wf_left_until_zero:
|
|
|
1877 |
"\<forall> i. \<exists> s j. (i:[left_until_zero]:j) s"
|
|
|
1878 |
proof -
|
|
|
1879 |
from wf_cpg_test_correct[OF wt_left_until_zero] ct_left_until_zero
|
|
|
1880 |
show ?thesis
|
|
|
1881 |
apply (unfold c2p_def, simp)
|
|
|
1882 |
by metis
|
|
|
1883 |
qed
|
|
|
1884 |
|
|
|
1885 |
end |