Cfg optimizations (unreachable blocks and block coalescing)

BoogieLang/Ast.thy

@@ -0,0 +1,217 @@
+section \<open>Semantics of the AST\<close>
+
+theory Ast
+  imports Main Semantics Lang BackedgeElim
+
+begin
+
+subsection \<open>AST definition\<close>
+
+type_synonym name = string
+type_synonym label = string
+type_synonym guard = expr
+type_synonym invariant = expr
+
+datatype transfer_cmd
+ = Goto label
+ | Return
+
+datatype parsed_structured_cmd
+ = ParsedIf "guard option" "bigblock list" "bigblock list"
+ | ParsedWhile "guard option" "invariant list" "bigblock list"
+ | ParsedBreak nat
+ | WhileWrapper parsed_structured_cmd
+
+and bigblock 
+ = BigBlock "name option" "cmd list" "parsed_structured_cmd option" "transfer_cmd option"
+
+
+text \<open>A Boogie statement represented as an AST is a list of \<^typ>\<open>bigblock\<close>\<close>
+
+type_synonym ast = "bigblock list"
+
+subsection \<open>AST semantics\<close>
+
+text \<open>We define a continuation-based small-step semantics.\<close>
+
+datatype cont 
+ = KStop 
+ | KSeq "bigblock" cont
+ | KEndBlock cont 
+
+type_synonym 'a ast_config = "bigblock * cont * ('a state)"
+
+fun convert_list_to_cont :: "bigblock list \<Rightarrow> cont \<Rightarrow> cont" where
+    "convert_list_to_cont [] cont0 = cont0"
+  | "convert_list_to_cont (x#xs) cont0 = KSeq x (convert_list_to_cont xs cont0)"
+
+text\<open>auxillary function to find the label a Goto statement is referring to\<close>
+fun find_label :: "label \<Rightarrow> bigblock list \<Rightarrow> cont \<Rightarrow> ((bigblock * cont) option)" where
+    "find_label lbl [] cont = None" 
+  | "find_label lbl ((BigBlock bb_name cmds None None) # []) cont = 
+      (if (Some lbl = bb_name) then (Some ((BigBlock bb_name cmds None None), cont)) else (None))" 
+  | "find_label lbl ((BigBlock bb_name cmds None None) # bbs) cont = 
+      (if (Some lbl = bb_name) 
+        then (Some ((BigBlock bb_name cmds None None), (convert_list_to_cont ( bbs) cont))) 
+        else (find_label lbl bbs cont))" 
+  | "find_label lbl ((BigBlock bb_name cmds (Some (ParsedIf guard then_bbs else_bbs)) None) # bbs) cont = 
+      (if (Some lbl = bb_name) 
+        then (Some ((BigBlock bb_name cmds (Some (ParsedIf guard then_bbs else_bbs)) None), (convert_list_to_cont ( bbs) cont))) 
+        else (if (find_label lbl then_bbs cont \<noteq> None) 
+                then (find_label lbl (then_bbs @ bbs) cont) 
+                else (find_label lbl (else_bbs @ bbs) cont)))" 
+  | "find_label lbl ((BigBlock bb_name cmds (Some (ParsedWhile guard invariants body_bbs)) None) # bbs) cont = 
+      (if (Some lbl = bb_name) 
+        then (Some ((BigBlock bb_name cmds (Some (ParsedWhile guard invariants body_bbs)) None), (convert_list_to_cont ( bbs) cont))) 
+        else (if (find_label lbl body_bbs cont \<noteq> None) 
+                then (find_label lbl body_bbs (convert_list_to_cont ((bbs)@[(BigBlock None [] (Some (ParsedWhile guard invariants body_bbs)) None)]) cont)) 
+                else (find_label lbl bbs cont)))"  
+  | "find_label lbl ((BigBlock bb_name cmds (Some (ParsedBreak n)) None) # bbs) cont = 
+      (if (Some lbl = bb_name) 
+        then (Some ((BigBlock bb_name cmds (Some (ParsedBreak n)) None), (convert_list_to_cont ( bbs) cont))) 
+        else (find_label lbl bbs cont))" 
+  | "find_label lbl ((BigBlock bb_name cmds (Some (WhileWrapper while_loop)) None) # bbs) cont = 
+      find_label lbl ((BigBlock bb_name cmds (Some while_loop) None) # bbs) cont"
+  | "find_label lbl ((BigBlock bb_name cmds None (Some transfer_stmt)) # bbs) cont = 
+      (if (Some lbl = bb_name) 
+        then (Some ((BigBlock bb_name cmds None (Some transfer_stmt)), (convert_list_to_cont ( bbs) cont))) 
+        else (find_label lbl bbs cont))"
+  | "find_label lbl ((BigBlock bb_name cmds (Some s) (Some t)) # bbs) cont = None"
+
+fun get_state :: "'a ast_config \<Rightarrow> 'a state"
+  where
+    "get_state (bb, cont, s1) = s1"
+
+fun is_final :: "'a ast_config \<Rightarrow> bool" 
+  where
+    "is_final ((BigBlock bb_name [] None None), KStop, s1) = True"
+  | "is_final other = False"
+
+text \<open>Small-step semantics\<close>
+
+inductive red_bigblock :: "'a absval_ty_fun \<Rightarrow> 'm proc_context \<Rightarrow> var_context \<Rightarrow> 'a fun_interp \<Rightarrow> rtype_env  \<Rightarrow> ast \<Rightarrow> 'a ast_config \<Rightarrow> 'a ast_config \<Rightarrow> bool" 
+  ("_,_,_,_,_,_ \<turnstile> (\<langle>_\<rangle> \<longrightarrow>/ _)" [51,0,0,0] 81)
+  for A :: "'a absval_ty_fun" and M :: "'m proc_context" and \<Lambda> :: var_context and \<Gamma> :: "'a fun_interp" and \<Omega> :: rtype_env and T :: ast
+  where
+    RedSimpleCmds: 
+    "\<lbrakk>(A,M,\<Lambda>,\<Gamma>,\<Omega> \<turnstile> \<langle>cs, (Normal n_s)\<rangle> [\<rightarrow>] s1) \<and> (cs \<noteq> Nil) \<rbrakk> 
+      \<Longrightarrow> A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>((BigBlock bb_name cs str_cmd tr_cmd), cont0, Normal n_s)\<rangle> \<longrightarrow> 
+                              ((BigBlock bb_name [] str_cmd tr_cmd), cont0, s1)"  
+
+  | RedFailure_or_Magic: 
+    "\<lbrakk> (s1 = Magic) \<or> (s1 = Failure); \<not> (is_final ((BigBlock bb_name [] str_cmd tr_cmd), cont0, s1))  \<rbrakk> 
+      \<Longrightarrow> A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>((BigBlock bb_name [] str_cmd tr_cmd), cont0, s1)\<rangle> \<longrightarrow> 
+                          ((BigBlock bb_name [] None None), KStop, s1)"
+
+  | RedSkip: 
+    "A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>((BigBlock bb_name [] None None), (KSeq b cont0), Normal n_s)\<rangle> \<longrightarrow> 
+                    (b, cont0, Normal n_s)"
+
+  | RedSkipEndBlock: 
+    "A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>((BigBlock bb_name [] None None),  (KEndBlock cont0),  Normal n_s)\<rangle> \<longrightarrow> 
+                    ((BigBlock bb_name [] None None), cont0, Normal n_s)"
+
+  | RedReturn: 
+    "A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>(BigBlock bb_name [] None (Some Return), cont0, Normal n_s)\<rangle> \<longrightarrow> 
+                          ((BigBlock bb_name [] None None), KStop, Normal n_s)"
+
+  | RedParsedIfTrue: 
+    "\<lbrakk>\<And> b. bb_guard = (Some b) \<Longrightarrow>  A,\<Lambda>,\<Gamma>,\<Omega> \<turnstile> \<langle>b, n_s\<rangle> \<Down> LitV (LBool True) \<rbrakk>  
+        \<Longrightarrow> A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>((BigBlock bb_name [] 
+                                (Some (ParsedIf bb_guard (then_hd # then_bbs) elsebigblocks)) None), cont0, Normal n_s)\<rangle> \<longrightarrow> 
+                            (then_hd, (convert_list_to_cont ( then_bbs) cont0), Normal n_s)"
+
+  | RedParsedIfFalse: 
+    "\<lbrakk>\<And>b. bb_guard = (Some b) \<Longrightarrow>  A,\<Lambda>,\<Gamma>,\<Omega> \<turnstile> \<langle>b, n_s\<rangle> \<Down> LitV (LBool False) \<rbrakk> 
+        \<Longrightarrow> A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>((BigBlock bb_name []
+                                (Some (ParsedIf bb_guard thenbigblocks (else_hd # else_bbs))) None), cont0, Normal n_s)\<rangle> \<longrightarrow>
+                            (else_hd, (convert_list_to_cont ( else_bbs) cont0), Normal n_s)"
+
+  | RedParsedWhileWrapper: 
+    "A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> 
+      \<langle>((BigBlock bb_name [] 
+              (Some (WhileWrapper str)) None), cont0,  Normal n_s)\<rangle> \<longrightarrow>
+        ((BigBlock bb_name [] 
+                (Some str) None), (KEndBlock cont0),  Normal n_s)"
+
+  | RedParsedWhile_InvFail: 
+    "\<lbrakk>\<And> b. bb_guard = (Some b) \<Longrightarrow> A,\<Lambda>,\<Gamma>,\<Omega> \<turnstile> \<langle>b, n_s\<rangle> \<Down> LitV (LBool True); 
+       bb_invariants = invs1@[I]@invs2;
+       expr_all_sat A \<Lambda> \<Gamma> \<Omega> n_s invs1;
+       A,\<Lambda>,\<Gamma>,\<Omega> \<turnstile> \<langle>I, n_s\<rangle> \<Down> BoolV False \<rbrakk> 
+       \<Longrightarrow> A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> 
+           \<langle>((BigBlock bb_name [] 
+                    (Some (ParsedWhile bb_guard bb_invariants (bb_hd # body_bbs))) None), cont0,  Normal n_s)\<rangle> \<longrightarrow> 
+              ((BigBlock bb_name [] None None), KStop, Failure)"
+
+  | RedParsedWhileTrue: 
+    "\<lbrakk>\<And> b.  bb_guard = (Some b) \<Longrightarrow> A,\<Lambda>,\<Gamma>,\<Omega> \<turnstile> \<langle>b, n_s\<rangle> \<Down> LitV (LBool True); 
+       (expr_all_sat A \<Lambda> \<Gamma> \<Omega> n_s bb_invariants) \<rbrakk> 
+       \<Longrightarrow> A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> 
+            \<langle>((BigBlock bb_name [] 
+                     (Some (ParsedWhile bb_guard bb_invariants (bb_hd # body_bbs))) None), cont0,  Normal n_s)\<rangle> \<longrightarrow> 
+              (bb_hd, convert_list_to_cont ((body_bbs)@[(BigBlock bb_name [] (Some (ParsedWhile bb_guard bb_invariants (bb_hd # body_bbs))) None)]) cont0, Normal n_s)"
+
+
+  | RedParsedWhileFalse: 
+    "\<lbrakk>\<And> b. bb_guard = (Some b) \<Longrightarrow> A,\<Lambda>,\<Gamma>,\<Omega> \<turnstile> \<langle>b, n_s\<rangle> \<Down> LitV (LBool False);
+       (expr_all_sat A \<Lambda> \<Gamma> \<Omega> n_s bb_invariants) \<rbrakk>  
+       \<Longrightarrow> A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>((BigBlock bb_name [] 
+                                (Some (ParsedWhile bb_guard bb_invariants bigblocks)) None), cont0, Normal n_s)\<rangle> \<longrightarrow> 
+                          ((BigBlock bb_name [] None None), cont0, Normal n_s)"
+
+  | RedBreak0: 
+    "A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>((BigBlock bb_name [] (Some (ParsedBreak 0)) None), (KEndBlock cont0), Normal n_s)\<rangle> \<longrightarrow> 
+                          ((BigBlock bb_name [] None None), cont0, Normal n_s)"
+
+  | RedBreakN: 
+    "A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> 
+           \<langle>((BigBlock bb_name [] (Some (ParsedBreak n)) None), (KSeq b cont0), Normal n_s)\<rangle> \<longrightarrow> 
+            ((BigBlock None [] (Some (ParsedBreak n)) None), cont0, Normal n_s)"
+
+  | RedBreakNPlus1: 
+    "A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> 
+           \<langle>((BigBlock bb_name [] (Some (ParsedBreak (n + 1))) None), (KEndBlock cont0), Normal n_s)\<rangle> \<longrightarrow> 
+            ((BigBlock None [] (Some (ParsedBreak n)) None), cont0, Normal n_s)"
+
+  | RedGoto: 
+    "\<lbrakk> (find_label label T KStop) = Some (found_bigblock, found_cont) \<rbrakk> 
+        \<Longrightarrow> A,M,\<Lambda>,\<Gamma>,\<Omega>,T \<turnstile> \<langle>((BigBlock bb_name [] None (Some (Goto label))), cont0, Normal n_s)\<rangle> \<longrightarrow> 
+                            (found_bigblock, found_cont, (Normal n_s))"
+
+abbreviation red_bigblock_k_step :: "'a absval_ty_fun \<Rightarrow> 'm proc_context \<Rightarrow> var_context \<Rightarrow> 'a fun_interp \<Rightarrow> rtype_env \<Rightarrow> ast \<Rightarrow> 'a ast_config \<Rightarrow> nat \<Rightarrow> 'a ast_config \<Rightarrow> bool"
+  ("_,_,_,_,_,_ \<turnstile>_ -n\<longrightarrow>^_/ _" [51,0,0,0,0] 81)
+where "red_bigblock_k_step A M \<Lambda> \<Gamma> \<Omega> T c1 n c2 \<equiv> ((red_bigblock A M \<Lambda> \<Gamma> \<Omega> T)^^n) c1 c2"
+
+subsection \<open>Procedure Correctness\<close>
+
+text\<open>defining correctness of the AST\<close>
+
+fun convert_ast_to_program_point :: "ast \<Rightarrow> bigblock \<times> cont" where
+    "convert_ast_to_program_point [] = ((BigBlock None [] None None), KStop)"
+  | "convert_ast_to_program_point (b#bs) = (b, convert_list_to_cont bs KStop)"
+
+fun init_ast :: "ast \<Rightarrow> 'a nstate \<Rightarrow> 'a ast_config"
+  where
+    "init_ast [] ns1 = ((BigBlock None [] None None), KStop, Normal ns1)"
+  | "init_ast (b#bbs) ns1 = (b, convert_list_to_cont ( bbs) KStop, Normal ns1)"
+
+definition valid_configuration 
+  where "valid_configuration A \<Lambda> \<Gamma> \<Omega> posts bb cont state \<equiv> 
+         state \<noteq> Failure \<and> 
+         (is_final (bb, cont, state) \<longrightarrow> (\<forall>ns'. state = Normal ns' \<longrightarrow> expr_all_sat A \<Lambda> \<Gamma> \<Omega> ns' posts))"
+
+definition proc_body_satisfies_spec :: "'a absval_ty_fun \<Rightarrow> 'm proc_context \<Rightarrow> var_context \<Rightarrow> 'a fun_interp \<Rightarrow> rtype_env \<Rightarrow> expr list \<Rightarrow> expr list \<Rightarrow> ast \<Rightarrow> 'a nstate \<Rightarrow> bool"
+  where "proc_body_satisfies_spec A M \<Lambda> \<Gamma> \<Omega> pres posts ast ns \<equiv>
+         expr_all_sat A \<Lambda> \<Gamma> \<Omega> ns pres \<longrightarrow> 
+          (\<forall> bb cont state. (rtranclp (red_bigblock A M \<Lambda> \<Gamma> \<Omega> ast) (init_ast ast ns) (bb, cont, state)) \<longrightarrow> 
+                    valid_configuration A \<Lambda> \<Gamma> \<Omega> posts bb cont state)"
+
+fun proc_all_pres :: "ast procedure \<Rightarrow> expr list"
+  where "proc_all_pres p = map fst (proc_pres p)"
+
+fun proc_checked_posts :: "ast procedure \<Rightarrow> expr list"
+  where "proc_checked_posts p = map fst (filter (\<lambda>x. \<not> snd(x)) (proc_posts p))"
+
+end
+