src/HOL/Bali/AxExample.thy
 author schirmer Mon Jan 28 17:00:19 2002 +0100 (2002-01-28) changeset 12854 00d4a435777f child 12857 a4386cc9b1c3 permissions -rw-r--r--
Isabelle/Bali sources;
```     1 (*  Title:      isabelle/Bali/AxExample.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     David von Oheimb
```
```     4     Copyright   2000 Technische Universitaet Muenchen
```
```     5 *)
```
```     6 header {* Example of a proof based on the Bali axiomatic semantics *}
```
```     7
```
```     8 theory AxExample = AxSem + Example:
```
```     9
```
```    10 constdefs
```
```    11   arr_inv :: "st \<Rightarrow> bool"
```
```    12  "arr_inv \<equiv> \<lambda>s. \<exists>obj a T el. globs s (Stat Base) = Some obj \<and>
```
```    13                               values obj (Inl (arr, Base)) = Some (Addr a) \<and>
```
```    14                               heap s a = Some \<lparr>tag=Arr T 2,values=el\<rparr>"
```
```    15
```
```    16 lemma arr_inv_new_obj:
```
```    17 "\<And>a. \<lbrakk>arr_inv s; new_Addr (heap s)=Some a\<rbrakk> \<Longrightarrow> arr_inv (gupd(Inl a\<mapsto>x) s)"
```
```    18 apply (unfold arr_inv_def)
```
```    19 apply (force dest!: new_AddrD2)
```
```    20 done
```
```    21
```
```    22 lemma arr_inv_set_locals [simp]: "arr_inv (set_locals l s) = arr_inv s"
```
```    23 apply (unfold arr_inv_def)
```
```    24 apply (simp (no_asm))
```
```    25 done
```
```    26
```
```    27 lemma arr_inv_gupd_Stat [simp]:
```
```    28   "Base \<noteq> C \<Longrightarrow> arr_inv (gupd(Stat C\<mapsto>obj) s) = arr_inv s"
```
```    29 apply (unfold arr_inv_def)
```
```    30 apply (simp (no_asm_simp))
```
```    31 done
```
```    32
```
```    33 lemma ax_inv_lupd [simp]: "arr_inv (lupd(x\<mapsto>y) s) = arr_inv s"
```
```    34 apply (unfold arr_inv_def)
```
```    35 apply (simp (no_asm))
```
```    36 done
```
```    37
```
```    38
```
```    39 declare split_if_asm [split del]
```
```    40 declare lvar_def [simp]
```
```    41
```
```    42 ML {*
```
```    43 fun inst1_tac s t = instantiate_tac [(s,t)];
```
```    44 val ax_tac = REPEAT o rtac allI THEN'
```
```    45              resolve_tac(thm "ax_Skip"::thm "ax_StatRef"::thm "ax_MethdN"::
```
```    46                          thm "ax_Alloc"::thm "ax_Alloc_Arr"::
```
```    47                          thm "ax_SXAlloc_Normal"::
```
```    48                          funpow 7 tl (thms "ax_derivs.intros"))
```
```    49 *}
```
```    50
```
```    51
```
```    52 theorem ax_test: "tprg,({}::'a triple set)\<turnstile>
```
```    53   {Normal (\<lambda>Y s Z::'a. heap_free four s \<and> \<not>initd Base s \<and> \<not> initd Ext s)}
```
```    54   .test [Class Base]. {\<lambda>Y s Z. abrupt s = Some (Xcpt (Std IndOutBound))}"
```
```    55 apply (unfold test_def arr_viewed_from_def)
```
```    56 apply (tactic "ax_tac 1" (*;;*))
```
```    57 defer
```
```    58 apply  (tactic "ax_tac 1" (* Try *))
```
```    59 defer
```
```    60 apply    (tactic {* inst1_tac "Q1"
```
```    61                  "\<lambda>Y s Z. arr_inv (snd s) \<and> tprg,s\<turnstile>catch SXcpt NullPointer" *})
```
```    62 prefer 2
```
```    63 apply    simp
```
```    64 apply   (rule_tac P' = "Normal (\<lambda>Y s Z. arr_inv (snd s))" in conseq1)
```
```    65 prefer 2
```
```    66 apply    clarsimp
```
```    67 apply   (rule_tac Q' = "(\<lambda>Y s Z. ?Q Y s Z)\<leftarrow>=False\<down>=\<diamondsuit>" in conseq2)
```
```    68 prefer 2
```
```    69 apply    simp
```
```    70 apply   (tactic "ax_tac 1" (* While *))
```
```    71 prefer 2
```
```    72 apply    (rule ax_impossible [THEN conseq1], clarsimp)
```
```    73 apply   (rule_tac P' = "Normal ?P" in conseq1)
```
```    74 prefer 2
```
```    75 apply    clarsimp
```
```    76 apply   (tactic "ax_tac 1")
```
```    77 apply   (tactic "ax_tac 1" (* AVar *))
```
```    78 prefer 2
```
```    79 apply    (rule ax_subst_Val_allI)
```
```    80 apply    (tactic {* inst1_tac "P'21" "\<lambda>u a. Normal (?PP a\<leftarrow>?x) u" *})
```
```    81 apply    (simp del: avar_def2 peek_and_def2)
```
```    82 apply    (tactic "ax_tac 1")
```
```    83 apply   (tactic "ax_tac 1")
```
```    84       (* just for clarification: *)
```
```    85 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)
```
```    86 prefer 2
```
```    87 apply    (clarsimp simp add: split_beta)
```
```    88 apply   (tactic "ax_tac 1" (* FVar *))
```
```    89 apply    (tactic "ax_tac 2" (* StatRef *))
```
```    90 apply   (rule ax_derivs.Done [THEN conseq1])
```
```    91 apply   (clarsimp simp add: arr_inv_def inited_def in_bounds_def)
```
```    92 defer
```
```    93 apply  (rule ax_SXAlloc_catch_SXcpt)
```
```    94 apply  (rule_tac Q' = "(\<lambda>Y (x, s) Z. x = Some (Xcpt (Std NullPointer)) \<and> arr_inv s) \<and>. heap_free two" in conseq2)
```
```    95 prefer 2
```
```    96 apply   (simp add: arr_inv_new_obj)
```
```    97 apply  (tactic "ax_tac 1")
```
```    98 apply  (rule_tac C = "Ext" in ax_Call_known_DynT)
```
```    99 apply     (unfold DynT_prop_def)
```
```   100 apply     (simp (no_asm))
```
```   101 apply    (intro strip)
```
```   102 apply    (rule_tac P' = "Normal ?P" in conseq1)
```
```   103 apply     (tactic "ax_tac 1" (* Methd *))
```
```   104 apply     (rule ax_thin [OF _ empty_subsetI])
```
```   105 apply     (simp (no_asm) add: body_def2)
```
```   106 apply     (tactic "ax_tac 1" (* Body *))
```
```   107 (* apply       (rule_tac [2] ax_derivs.Abrupt) *)
```
```   108 defer
```
```   109 apply      (simp (no_asm))
```
```   110 apply      (tactic "ax_tac 1")
```
```   111 apply      (tactic "ax_tac 1") (* Ass *)
```
```   112 prefer 2
```
```   113 apply       (rule ax_subst_Var_allI)
```
```   114 apply       (tactic {* inst1_tac "P'27" "\<lambda>a vs l vf. ?PP a vs l vf\<leftarrow>?x \<and>. ?p" *})
```
```   115 apply       (rule allI)
```
```   116 apply       (tactic {* simp_tac (simpset() delloop "split_all_tac" delsimps [thm "peek_and_def2"]) 1 *})
```
```   117 apply       (rule ax_derivs.Abrupt)
```
```   118 apply      (simp (no_asm))
```
```   119 apply      (tactic "ax_tac 1" (* FVar *))
```
```   120 apply       (tactic "ax_tac 2", tactic "ax_tac 2", tactic "ax_tac 2")
```
```   121 apply      (tactic "ax_tac 1")
```
```   122 apply     clarsimp
```
```   123 apply     (tactic {* inst1_tac "R14" "\<lambda>a'. Normal ((\<lambda>Vals:vs (x, s) Z. arr_inv s \<and> inited Ext (globs s) \<and> a' \<noteq> Null \<and> hd vs = Null) \<and>. heap_free two)" *})
```
```   124 prefer 5
```
```   125 apply     (rule ax_derivs.Done [THEN conseq1], force)
```
```   126 apply    force
```
```   127 apply   (rule ax_subst_Val_allI)
```
```   128 apply   (tactic {* inst1_tac "P'33" "\<lambda>u a. Normal (?PP a\<leftarrow>?x) u" *})
```
```   129 apply   (simp (no_asm) del: peek_and_def2)
```
```   130 apply   (tactic "ax_tac 1")
```
```   131 prefer 2
```
```   132 apply   (rule ax_subst_Val_allI)
```
```   133 apply    (tactic {* inst1_tac "P'4" "\<lambda>aa v. Normal (?QQ aa v\<leftarrow>?y)" *})
```
```   134 apply    (simp del: peek_and_def2)
```
```   135 apply    (tactic "ax_tac 1")
```
```   136 apply   (tactic "ax_tac 1")
```
```   137 apply  (tactic "ax_tac 1")
```
```   138 apply  (tactic "ax_tac 1")
```
```   139 (* end method call *)
```
```   140 apply (simp (no_asm))
```
```   141     (* just for clarification: *)
```
```   142 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>
```
```   143  invocation_declclass tprg IntVir s (the (locals s (VName e))) (ClassT Base)
```
```   144      \<lparr>name = foo, parTs = [Class Base]\<rparr> = Ext)) \<and>. initd Ext \<and>. heap_free two)"
```
```   145   in conseq2)
```
```   146 prefer 2
```
```   147 apply  clarsimp
```
```   148 apply (tactic "ax_tac 1")
```
```   149 apply (tactic "ax_tac 1")
```
```   150 defer
```
```   151 apply  (rule ax_subst_Var_allI)
```
```   152 apply  (tactic {* inst1_tac "P'14" "\<lambda>u vf. Normal (?PP vf \<and>. ?p) u" *})
```
```   153 apply  (simp (no_asm) del: split_paired_All peek_and_def2)
```
```   154 apply  (tactic "ax_tac 1" (* NewC *))
```
```   155 apply  (tactic "ax_tac 1" (* ax_Alloc *))
```
```   156      (* just for clarification: *)
```
```   157 apply  (rule_tac Q' = "Normal ((\<lambda>Y s Z. arr_inv (store s) \<and> vf=lvar (VName e) (store s)) \<and>. heap_free tree \<and>. initd Ext)" in conseq2)
```
```   158 prefer 2
```
```   159 apply   (simp add: invocation_declclass_def dynmethd_def)
```
```   160 apply   (unfold dynlookup_def)
```
```   161 apply   (simp add: dynmethd_Ext_foo)
```
```   162 apply   (force elim!: arr_inv_new_obj atleast_free_SucD atleast_free_weaken)
```
```   163      (* begin init *)
```
```   164 apply  (rule ax_InitS)
```
```   165 apply     force
```
```   166 apply    (simp (no_asm))
```
```   167 apply   (tactic {* simp_tac (simpset() delloop "split_all_tac") 1 *})
```
```   168 apply   (rule ax_Init_Skip_lemma)
```
```   169 apply  (tactic {* simp_tac (simpset() delloop "split_all_tac") 1 *})
```
```   170 apply  (rule ax_InitS [THEN conseq1] (* init Base *))
```
```   171 apply      force
```
```   172 apply     (simp (no_asm))
```
```   173 apply    (unfold arr_viewed_from_def)
```
```   174 apply    (rule allI)
```
```   175 apply    (rule_tac P' = "Normal ?P" in conseq1)
```
```   176 apply     (tactic {* simp_tac (simpset() delloop "split_all_tac") 1 *})
```
```   177 apply     (tactic "ax_tac 1")
```
```   178 apply     (tactic "ax_tac 1")
```
```   179 apply     (rule_tac [2] ax_subst_Var_allI)
```
```   180 apply      (tactic {* inst1_tac "P'29" "\<lambda>vf l vfa. Normal (?P vf l vfa)" *})
```
```   181 apply     (tactic {* simp_tac (simpset() delloop "split_all_tac" delsimps [split_paired_All, thm "peek_and_def2"]) 2 *})
```
```   182 apply      (tactic "ax_tac 2" (* NewA *))
```
```   183 apply       (tactic "ax_tac 3" (* ax_Alloc_Arr *))
```
```   184 apply       (tactic "ax_tac 3")
```
```   185 apply      (tactic {* inst1_tac "P" "\<lambda>vf l vfa. Normal (?P vf l vfa\<leftarrow>\<diamondsuit>)" *})
```
```   186 apply      (tactic {* simp_tac (simpset() delloop "split_all_tac") 2 *})
```
```   187 apply      (tactic "ax_tac 2")
```
```   188 apply     (tactic "ax_tac 1" (* FVar *))
```
```   189 apply      (tactic "ax_tac 2" (* StatRef *))
```
```   190 apply     (rule ax_derivs.Done [THEN conseq1])
```
```   191 apply     (tactic {* inst1_tac "Q22" "\<lambda>vf. Normal ((\<lambda>Y s Z. vf=lvar (VName e) (snd s)) \<and>. heap_free four \<and>. initd Base \<and>. initd Ext)" *})
```
```   192 apply     (clarsimp split del: split_if)
```
```   193 apply     (frule atleast_free_weaken [THEN atleast_free_weaken])
```
```   194 apply     (drule initedD)
```
```   195 apply     (clarsimp elim!: atleast_free_SucD simp add: arr_inv_def)
```
```   196 apply    force
```
```   197 apply   (tactic {* simp_tac (simpset() delloop "split_all_tac") 1 *})
```
```   198 apply   (rule ax_triv_Init_Object [THEN peek_and_forget2, THEN conseq1])
```
```   199 apply     (rule wf_tprg)
```
```   200 apply    clarsimp
```
```   201 apply   (tactic {* inst1_tac "P22" "\<lambda>vf. Normal ((\<lambda>Y s Z. vf = lvar (VName e) (snd s)) \<and>. heap_free four \<and>. initd Ext)" *})
```
```   202 apply   clarsimp
```
```   203 apply  (tactic {* inst1_tac "PP" "\<lambda>vf. Normal ((\<lambda>Y s Z. vf = lvar (VName e) (snd s)) \<and>. heap_free four \<and>. Not \<circ> initd Base)" *})
```
```   204 apply  clarsimp
```
```   205      (* end init *)
```
```   206 apply (rule conseq1)
```
```   207 apply (tactic "ax_tac 1")
```
```   208 apply clarsimp
```
```   209 done
```
```   210
```
```   211 (*
```
```   212 while (true) {
```
```   213   if (i) {throw xcpt;}
```
```   214   else i=j
```
```   215 }
```
```   216 *)
```
```   217 lemma Loop_Xcpt_benchmark:
```
```   218  "Q = (\<lambda>Y (x,s) Z. x \<noteq> None \<longrightarrow> the_Bool (the (locals s i))) \<Longrightarrow>
```
```   219   G,({}::'a triple set)\<turnstile>{Normal (\<lambda>Y s Z::'a. True)}
```
```   220   .lab1\<bullet> While(Lit (Bool True)) (If(Acc (LVar i)) (Throw (Acc (LVar xcpt))) Else
```
```   221         (Expr (Ass (LVar i) (Acc (LVar j))))). {Q}"
```
```   222 apply (rule_tac P' = "Q" and Q' = "Q\<leftarrow>=False\<down>=\<diamondsuit>" in conseq12)
```
```   223 apply  safe
```
```   224 apply  (tactic "ax_tac 1" (* Loop *))
```
```   225 apply   (rule ax_Normal_cases)
```
```   226 prefer 2
```
```   227 apply    (rule ax_derivs.Abrupt [THEN conseq1], clarsimp simp add: Let_def)
```
```   228 apply   (rule conseq1)
```
```   229 apply    (tactic "ax_tac 1")
```
```   230 apply   clarsimp
```
```   231 prefer 2
```
```   232 apply  clarsimp
```
```   233 apply (tactic "ax_tac 1" (* If *))
```
```   234 apply  (tactic
```
```   235   {* inst1_tac "P'21" "Normal (\<lambda>s.. (\<lambda>Y s Z. True)\<down>=Val (the (locals s i)))" *})
```
```   236 apply  (tactic "ax_tac 1")
```
```   237 apply  (rule conseq1)
```
```   238 apply   (tactic "ax_tac 1")
```
```   239 apply  clarsimp
```
```   240 apply (rule allI)
```
```   241 apply (rule ax_escape)
```
```   242 apply auto
```
```   243 apply  (rule conseq1)
```
```   244 apply   (tactic "ax_tac 1" (* Throw *))
```
```   245 apply   (tactic "ax_tac 1")
```
```   246 apply   (tactic "ax_tac 1")
```
```   247 apply  clarsimp
```
```   248 apply (rule_tac Q' = "Normal (\<lambda>Y s Z. True)" in conseq2)
```
```   249 prefer 2
```
```   250 apply  clarsimp
```
```   251 apply (rule conseq1)
```
```   252 apply  (tactic "ax_tac 1")
```
```   253 apply  (tactic "ax_tac 1")
```
```   254 prefer 2
```
```   255 apply   (rule ax_subst_Var_allI)
```
```   256 apply   (tactic {* inst1_tac "P'29" "\<lambda>b Y ba Z vf. \<lambda>Y (x,s) Z. x=None \<and> snd vf = snd (lvar i s)" *})
```
```   257 apply   (rule allI)
```
```   258 apply   (rule_tac P' = "Normal ?P" in conseq1)
```
```   259 prefer 2
```
```   260 apply    clarsimp
```
```   261 apply   (tactic "ax_tac 1")
```
```   262 apply   (rule conseq1)
```
```   263 apply    (tactic "ax_tac 1")
```
```   264 apply   clarsimp
```
```   265 apply  (tactic "ax_tac 1")
```
```   266 apply clarsimp
```
```   267 done
```
```   268
```
```   269 end
```
```   270
```