src/HOL/Bali/AxExample.thy
author hoelzl
Fri Feb 19 13:40:50 2016 +0100 (2016-02-19)
changeset 62378 85ed00c1fe7c
parent 62042 6c6ccf573479
child 62390 842917225d56
permissions -rw-r--r--
generalize more theorems to support enat and ennreal
     1 (*  Title:      HOL/Bali/AxExample.thy
     2     Author:     David von Oheimb
     3 *)
     4 
     5 subsection \<open>Example of a proof based on the Bali axiomatic semantics\<close>
     6 
     7 theory AxExample
     8 imports AxSem Example
     9 begin
    10 
    11 definition
    12   arr_inv :: "st \<Rightarrow> bool" where
    13  "arr_inv = (\<lambda>s. \<exists>obj a T el. globs s (Stat Base) = Some obj \<and>
    14                               values obj (Inl (arr, Base)) = Some (Addr a) \<and>
    15                               heap s a = Some \<lparr>tag=Arr T 2,values=el\<rparr>)"
    16 
    17 lemma arr_inv_new_obj: 
    18 "\<And>a. \<lbrakk>arr_inv s; new_Addr (heap s)=Some a\<rbrakk> \<Longrightarrow> arr_inv (gupd(Inl a\<mapsto>x) s)"
    19 apply (unfold arr_inv_def)
    20 apply (force dest!: new_AddrD2)
    21 done
    22 
    23 lemma arr_inv_set_locals [simp]: "arr_inv (set_locals l s) = arr_inv s"
    24 apply (unfold arr_inv_def)
    25 apply (simp (no_asm))
    26 done
    27 
    28 lemma arr_inv_gupd_Stat [simp]: 
    29   "Base \<noteq> C \<Longrightarrow> arr_inv (gupd(Stat C\<mapsto>obj) s) = arr_inv s"
    30 apply (unfold arr_inv_def)
    31 apply (simp (no_asm_simp))
    32 done
    33 
    34 lemma ax_inv_lupd [simp]: "arr_inv (lupd(x\<mapsto>y) s) = arr_inv s"
    35 apply (unfold arr_inv_def)
    36 apply (simp (no_asm))
    37 done
    38 
    39 
    40 declare split_if_asm [split del]
    41 declare lvar_def [simp]
    42 
    43 ML \<open>
    44 fun inst1_tac ctxt s t xs st =
    45   (case AList.lookup (op =) (rev (Term.add_var_names (Thm.prop_of st) [])) s of
    46     SOME i => PRIMITIVE (Rule_Insts.read_instantiate ctxt [(((s, i), Position.none), t)] xs) st
    47   | NONE => Seq.empty);
    48 
    49 fun ax_tac ctxt =
    50   REPEAT o resolve_tac ctxt [allI] THEN'
    51   resolve_tac ctxt
    52     @{thms ax_Skip ax_StatRef ax_MethdN ax_Alloc ax_Alloc_Arr ax_SXAlloc_Normal ax_derivs.intros(8-)};
    53 \<close>
    54 
    55 
    56 theorem ax_test: "tprg,({}::'a triple set)\<turnstile> 
    57   {Normal (\<lambda>Y s Z::'a. heap_free four s \<and> \<not>initd Base s \<and> \<not> initd Ext s)} 
    58   .test [Class Base]. 
    59   {\<lambda>Y s Z. abrupt s = Some (Xcpt (Std IndOutBound))}"
    60 apply (unfold test_def arr_viewed_from_def)
    61 apply (tactic "ax_tac @{context} 1" (*;;*))
    62 defer (* We begin with the last assertion, to synthesise the intermediate
    63          assertions, like in the fashion of the weakest
    64          precondition. *)
    65 apply  (tactic "ax_tac @{context} 1" (* Try *))
    66 defer
    67 apply    (tactic \<open>inst1_tac @{context} "Q" 
    68                  "\<lambda>Y s Z. arr_inv (snd s) \<and> tprg,s\<turnstile>catch SXcpt NullPointer" []\<close>)
    69 prefer 2
    70 apply    simp
    71 apply   (rule_tac P' = "Normal (\<lambda>Y s Z. arr_inv (snd s))" in conseq1)
    72 prefer 2
    73 apply    clarsimp
    74 apply   (rule_tac Q' = "(\<lambda>Y s Z. Q Y s Z)\<leftarrow>=False\<down>=\<diamondsuit>" and Q = Q for Q in conseq2)
    75 prefer 2
    76 apply    simp
    77 apply   (tactic "ax_tac @{context} 1" (* While *))
    78 prefer 2
    79 apply    (rule ax_impossible [THEN conseq1], clarsimp)
    80 apply   (rule_tac P' = "Normal P" and P = P for P in conseq1)
    81 prefer 2
    82 apply    clarsimp
    83 apply   (tactic "ax_tac @{context} 1")
    84 apply   (tactic "ax_tac @{context} 1" (* AVar *))
    85 prefer 2
    86 apply    (rule ax_subst_Val_allI)
    87 apply    (tactic \<open>inst1_tac @{context} "P'" "\<lambda>a. Normal (PP a\<leftarrow>x)" ["PP", "x"]\<close>)
    88 apply    (simp del: avar_def2 peek_and_def2)
    89 apply    (tactic "ax_tac @{context} 1")
    90 apply   (tactic "ax_tac @{context} 1")
    91       (* just for clarification: *)
    92 apply   (rule_tac Q' = "Normal (\<lambda>Var:(v, f) u ua. fst (snd (avar tprg (Intg 2) v u)) = Some (Xcpt (Std IndOutBound)))" in conseq2)
    93 prefer 2
    94 apply    (clarsimp simp add: split_beta)
    95 apply   (tactic "ax_tac @{context} 1" (* FVar *))
    96 apply    (tactic "ax_tac @{context} 2" (* StatRef *))
    97 apply   (rule ax_derivs.Done [THEN conseq1])
    98 apply   (clarsimp simp add: arr_inv_def inited_def in_bounds_def)
    99 defer
   100 apply  (rule ax_SXAlloc_catch_SXcpt)
   101 apply  (rule_tac Q' = "(\<lambda>Y (x, s) Z. x = Some (Xcpt (Std NullPointer)) \<and> arr_inv s) \<and>. heap_free two" in conseq2)
   102 prefer 2
   103 apply   (simp add: arr_inv_new_obj)
   104 apply  (tactic "ax_tac @{context} 1") 
   105 apply  (rule_tac C = "Ext" in ax_Call_known_DynT)
   106 apply     (unfold DynT_prop_def)
   107 apply     (simp (no_asm))
   108 apply    (intro strip)
   109 apply    (rule_tac P' = "Normal P" and P = P for P in conseq1)
   110 apply     (tactic "ax_tac @{context} 1" (* Methd *))
   111 apply     (rule ax_thin [OF _ empty_subsetI])
   112 apply     (simp (no_asm) add: body_def2)
   113 apply     (tactic "ax_tac @{context} 1" (* Body *))
   114 (* apply       (rule_tac [2] ax_derivs.Abrupt) *)
   115 defer
   116 apply      (simp (no_asm))
   117 apply      (tactic "ax_tac @{context} 1") (* Comp *)
   118             (* The first statement in the  composition 
   119                  ((Ext)z).vee = 1; Return Null 
   120                 will throw an exception (since z is null). So we can handle
   121                 Return Null with the Abrupt rule *)
   122 apply       (rule_tac [2] ax_derivs.Abrupt)
   123              
   124 apply      (rule ax_derivs.Expr) (* Expr *)
   125 apply      (tactic "ax_tac @{context} 1") (* Ass *)
   126 prefer 2
   127 apply       (rule ax_subst_Var_allI)
   128 apply       (tactic \<open>inst1_tac @{context} "P'" "\<lambda>a vs l vf. PP a vs l vf\<leftarrow>x \<and>. p" ["PP", "x", "p"]\<close>)
   129 apply       (rule allI)
   130 apply       (tactic \<open>simp_tac (@{context} delloop "split_all_tac" delsimps [@{thm peek_and_def2}, @{thm heap_def2}, @{thm subst_res_def2}, @{thm normal_def2}]) 1\<close>)
   131 apply       (rule ax_derivs.Abrupt)
   132 apply      (simp (no_asm))
   133 apply      (tactic "ax_tac @{context} 1" (* FVar *))
   134 apply       (tactic "ax_tac @{context} 2", tactic "ax_tac @{context} 2", tactic "ax_tac @{context} 2")
   135 apply      (tactic "ax_tac @{context} 1")
   136 apply     (tactic \<open>inst1_tac @{context} "R" "\<lambda>a'. Normal ((\<lambda>Vals:vs (x, s) Z. arr_inv s \<and> inited Ext (globs s) \<and> a' \<noteq> Null \<and> vs = [Null]) \<and>. heap_free two)" []\<close>)
   137 apply     fastforce
   138 prefer 4
   139 apply    (rule ax_derivs.Done [THEN conseq1],force)
   140 apply   (rule ax_subst_Val_allI)
   141 apply   (tactic \<open>inst1_tac @{context} "P'" "\<lambda>a. Normal (PP a\<leftarrow>x)" ["PP", "x"]\<close>)
   142 apply   (simp (no_asm) del: peek_and_def2 heap_free_def2 normal_def2 o_apply)
   143 apply   (tactic "ax_tac @{context} 1")
   144 prefer 2
   145 apply   (rule ax_subst_Val_allI)
   146 apply    (tactic \<open>inst1_tac @{context} "P'" "\<lambda>aa v. Normal (QQ aa v\<leftarrow>y)" ["QQ", "y"]\<close>)
   147 apply    (simp del: peek_and_def2 heap_free_def2 normal_def2)
   148 apply    (tactic "ax_tac @{context} 1")
   149 apply   (tactic "ax_tac @{context} 1")
   150 apply  (tactic "ax_tac @{context} 1")
   151 apply  (tactic "ax_tac @{context} 1")
   152 (* end method call *)
   153 apply (simp (no_asm))
   154     (* just for clarification: *)
   155 apply (rule_tac Q' = "Normal ((\<lambda>Y (x, s) Z. arr_inv s \<and> (\<exists>a. the (locals s (VName e)) = Addr a \<and> obj_class (the (globs s (Inl a))) = Ext \<and> 
   156  invocation_declclass tprg IntVir s (the (locals s (VName e))) (ClassT Base)  
   157      \<lparr>name = foo, parTs = [Class Base]\<rparr> = Ext)) \<and>. initd Ext \<and>. heap_free two)"
   158   in conseq2)
   159 prefer 2
   160 apply  clarsimp
   161 apply (tactic "ax_tac @{context} 1")
   162 apply (tactic "ax_tac @{context} 1")
   163 defer
   164 apply  (rule ax_subst_Var_allI)
   165 apply  (tactic \<open>inst1_tac @{context} "P'" "\<lambda>vf. Normal (PP vf \<and>. p)" ["PP", "p"]\<close>)
   166 apply  (simp (no_asm) del: split_paired_All peek_and_def2 initd_def2 heap_free_def2 normal_def2)
   167 apply  (tactic "ax_tac @{context} 1" (* NewC *))
   168 apply  (tactic "ax_tac @{context} 1" (* ax_Alloc *))
   169      (* just for clarification: *)
   170 apply  (rule_tac Q' = "Normal ((\<lambda>Y s Z. arr_inv (store s) \<and> vf=lvar (VName e) (store s)) \<and>. heap_free three \<and>. initd Ext)" in conseq2)
   171 prefer 2
   172 apply   (simp add: invocation_declclass_def dynmethd_def)
   173 apply   (unfold dynlookup_def)
   174 apply   (simp add: dynmethd_Ext_foo)
   175 apply   (force elim!: arr_inv_new_obj atleast_free_SucD atleast_free_weaken)
   176      (* begin init *)
   177 apply  (rule ax_InitS)
   178 apply     force
   179 apply    (simp (no_asm))
   180 apply   (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 1\<close>)
   181 apply   (rule ax_Init_Skip_lemma)
   182 apply  (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 1\<close>)
   183 apply  (rule ax_InitS [THEN conseq1] (* init Base *))
   184 apply      force
   185 apply     (simp (no_asm))
   186 apply    (unfold arr_viewed_from_def)
   187 apply    (rule allI)
   188 apply    (rule_tac P' = "Normal P" and P = P for P in conseq1)
   189 apply     (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 1\<close>)
   190 apply     (tactic "ax_tac @{context} 1")
   191 apply     (tactic "ax_tac @{context} 1")
   192 apply     (rule_tac [2] ax_subst_Var_allI)
   193 apply      (tactic \<open>inst1_tac @{context} "P'" "\<lambda>vf l vfa. Normal (P vf l vfa)" ["P"]\<close>)
   194 apply     (tactic \<open>simp_tac (@{context} delloop "split_all_tac" delsimps [@{thm split_paired_All}, @{thm peek_and_def2}, @{thm heap_free_def2}, @{thm initd_def2}, @{thm normal_def2}, @{thm supd_lupd}]) 2\<close>)
   195 apply      (tactic "ax_tac @{context} 2" (* NewA *))
   196 apply       (tactic "ax_tac @{context} 3" (* ax_Alloc_Arr *))
   197 apply       (tactic "ax_tac @{context} 3")
   198 apply      (tactic \<open>inst1_tac @{context} "P" "\<lambda>vf l vfa. Normal (P vf l vfa\<leftarrow>\<diamondsuit>)" ["P"]\<close>)
   199 apply      (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 2\<close>)
   200 apply      (tactic "ax_tac @{context} 2")
   201 apply     (tactic "ax_tac @{context} 1" (* FVar *))
   202 apply      (tactic "ax_tac @{context} 2" (* StatRef *))
   203 apply     (rule ax_derivs.Done [THEN conseq1])
   204 apply     (tactic \<open>inst1_tac @{context} "Q" "\<lambda>vf. Normal ((\<lambda>Y s Z. vf=lvar (VName e) (snd s)) \<and>. heap_free four \<and>. initd Base \<and>. initd Ext)" []\<close>)
   205 apply     (clarsimp split del: split_if)
   206 apply     (frule atleast_free_weaken [THEN atleast_free_weaken])
   207 apply     (drule initedD)
   208 apply     (clarsimp elim!: atleast_free_SucD simp add: arr_inv_def)
   209 apply    force
   210 apply   (tactic \<open>simp_tac (@{context} delloop "split_all_tac") 1\<close>)
   211 apply   (rule ax_triv_Init_Object [THEN peek_and_forget2, THEN conseq1])
   212 apply     (rule wf_tprg)
   213 apply    clarsimp
   214 apply   (tactic \<open>inst1_tac @{context} "P" "\<lambda>vf. Normal ((\<lambda>Y s Z. vf = lvar (VName e) (snd s)) \<and>. heap_free four \<and>. initd Ext)" []\<close>)
   215 apply   clarsimp
   216 apply  (tactic \<open>inst1_tac @{context} "PP" "\<lambda>vf. Normal ((\<lambda>Y s Z. vf = lvar (VName e) (snd s)) \<and>. heap_free four \<and>. Not \<circ> initd Base)" []\<close>)
   217 apply  clarsimp
   218      (* end init *)
   219 apply (rule conseq1)
   220 apply (tactic "ax_tac @{context} 1")
   221 apply clarsimp
   222 done
   223 
   224 (*
   225 while (true) {
   226   if (i) {throw xcpt;}
   227   else i=j
   228 }
   229 *)
   230 lemma Loop_Xcpt_benchmark: 
   231  "Q = (\<lambda>Y (x,s) Z. x \<noteq> None \<longrightarrow> the_Bool (the (locals s i))) \<Longrightarrow>  
   232   G,({}::'a triple set)\<turnstile>{Normal (\<lambda>Y s Z::'a. True)}  
   233   .lab1\<bullet> While(Lit (Bool True)) (If(Acc (LVar i)) (Throw (Acc (LVar xcpt))) Else
   234         (Expr (Ass (LVar i) (Acc (LVar j))))). {Q}"
   235 apply (rule_tac P' = "Q" and Q' = "Q\<leftarrow>=False\<down>=\<diamondsuit>" in conseq12)
   236 apply  safe
   237 apply  (tactic "ax_tac @{context} 1" (* Loop *))
   238 apply   (rule ax_Normal_cases)
   239 prefer 2
   240 apply    (rule ax_derivs.Abrupt [THEN conseq1], clarsimp simp add: Let_def)
   241 apply   (rule conseq1)
   242 apply    (tactic "ax_tac @{context} 1")
   243 apply   clarsimp
   244 prefer 2
   245 apply  clarsimp
   246 apply (tactic "ax_tac @{context} 1" (* If *))
   247 apply  (tactic 
   248   \<open>inst1_tac @{context} "P'" "Normal (\<lambda>s.. (\<lambda>Y s Z. True)\<down>=Val (the (locals s i)))" []\<close>)
   249 apply  (tactic "ax_tac @{context} 1")
   250 apply  (rule conseq1)
   251 apply   (tactic "ax_tac @{context} 1")
   252 apply  clarsimp
   253 apply (rule allI)
   254 apply (rule ax_escape)
   255 apply auto
   256 apply  (rule conseq1)
   257 apply   (tactic "ax_tac @{context} 1" (* Throw *))
   258 apply   (tactic "ax_tac @{context} 1")
   259 apply   (tactic "ax_tac @{context} 1")
   260 apply  clarsimp
   261 apply (rule_tac Q' = "Normal (\<lambda>Y s Z. True)" in conseq2)
   262 prefer 2
   263 apply  clarsimp
   264 apply (rule conseq1)
   265 apply  (tactic "ax_tac @{context} 1")
   266 apply  (tactic "ax_tac @{context} 1")
   267 prefer 2
   268 apply   (rule ax_subst_Var_allI)
   269 apply   (tactic \<open>inst1_tac @{context} "P'" "\<lambda>b Y ba Z vf. \<lambda>Y (x,s) Z. x=None \<and> snd vf = snd (lvar i s)" []\<close>)
   270 apply   (rule allI)
   271 apply   (rule_tac P' = "Normal P" and P = P for P in conseq1)
   272 prefer 2
   273 apply    clarsimp
   274 apply   (tactic "ax_tac @{context} 1")
   275 apply   (rule conseq1)
   276 apply    (tactic "ax_tac @{context} 1")
   277 apply   clarsimp
   278 apply  (tactic "ax_tac @{context} 1")
   279 apply clarsimp
   280 done
   281 
   282 end
   283