src/HOL/Hoare_Parallel/OG_Examples.thy
author wenzelm
Sat Nov 04 15:24:40 2017 +0100 (21 months ago)
changeset 67003 49850a679c2c
parent 64267 b9a1486e79be
child 67443 3abf6a722518
permissions -rw-r--r--
more robust sorted_entries;
     1 section \<open>Examples\<close>
     2 
     3 theory OG_Examples imports OG_Syntax begin
     4 
     5 subsection \<open>Mutual Exclusion\<close>
     6 
     7 subsubsection \<open>Peterson's Algorithm I\<close>
     8 
     9 text \<open>Eike Best. "Semantics of Sequential and Parallel Programs", page 217.\<close>
    10 
    11 record Petersons_mutex_1 =
    12  pr1 :: nat
    13  pr2 :: nat
    14  in1 :: bool
    15  in2 :: bool
    16  hold :: nat
    17 
    18 lemma Petersons_mutex_1:
    19   "\<parallel>- \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1 \<and> \<acute>pr2=0 \<and> \<not>\<acute>in2 \<rbrace>
    20   COBEGIN \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1\<rbrace>
    21   WHILE True INV \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1\<rbrace>
    22   DO
    23   \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1\<rbrace> \<langle> \<acute>in1:=True,,\<acute>pr1:=1 \<rangle>;;
    24   \<lbrace>\<acute>pr1=1 \<and> \<acute>in1\<rbrace>  \<langle> \<acute>hold:=1,,\<acute>pr1:=2 \<rangle>;;
    25   \<lbrace>\<acute>pr1=2 \<and> \<acute>in1 \<and> (\<acute>hold=1 \<or> \<acute>hold=2 \<and> \<acute>pr2=2)\<rbrace>
    26   AWAIT (\<not>\<acute>in2 \<or> \<not>(\<acute>hold=1)) THEN \<acute>pr1:=3 END;;
    27   \<lbrace>\<acute>pr1=3 \<and> \<acute>in1 \<and> (\<acute>hold=1 \<or> \<acute>hold=2 \<and> \<acute>pr2=2)\<rbrace>
    28    \<langle>\<acute>in1:=False,,\<acute>pr1:=0\<rangle>
    29   OD \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1\<rbrace>
    30   \<parallel>
    31   \<lbrace>\<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace>
    32   WHILE True INV \<lbrace>\<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace>
    33   DO
    34   \<lbrace>\<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace> \<langle> \<acute>in2:=True,,\<acute>pr2:=1 \<rangle>;;
    35   \<lbrace>\<acute>pr2=1 \<and> \<acute>in2\<rbrace> \<langle>  \<acute>hold:=2,,\<acute>pr2:=2 \<rangle>;;
    36   \<lbrace>\<acute>pr2=2 \<and> \<acute>in2 \<and> (\<acute>hold=2 \<or> (\<acute>hold=1 \<and> \<acute>pr1=2))\<rbrace>
    37   AWAIT (\<not>\<acute>in1 \<or> \<not>(\<acute>hold=2)) THEN \<acute>pr2:=3  END;;
    38   \<lbrace>\<acute>pr2=3 \<and> \<acute>in2 \<and> (\<acute>hold=2 \<or> (\<acute>hold=1 \<and> \<acute>pr1=2))\<rbrace>
    39     \<langle>\<acute>in2:=False,,\<acute>pr2:=0\<rangle>
    40   OD \<lbrace>\<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace>
    41   COEND
    42   \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1 \<and> \<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace>"
    43 apply oghoare
    44 \<comment>\<open>104 verification conditions.\<close>
    45 apply auto
    46 done
    47 
    48 subsubsection \<open>Peterson's Algorithm II: A Busy Wait Solution\<close>
    49 
    50 text \<open>Apt and Olderog. "Verification of sequential and concurrent Programs", page 282.\<close>
    51 
    52 record Busy_wait_mutex =
    53  flag1 :: bool
    54  flag2 :: bool
    55  turn  :: nat
    56  after1 :: bool
    57  after2 :: bool
    58 
    59 lemma Busy_wait_mutex:
    60  "\<parallel>-  \<lbrace>True\<rbrace>
    61   \<acute>flag1:=False,, \<acute>flag2:=False,,
    62   COBEGIN \<lbrace>\<not>\<acute>flag1\<rbrace>
    63         WHILE True
    64         INV \<lbrace>\<not>\<acute>flag1\<rbrace>
    65         DO \<lbrace>\<not>\<acute>flag1\<rbrace> \<langle> \<acute>flag1:=True,,\<acute>after1:=False \<rangle>;;
    66            \<lbrace>\<acute>flag1 \<and> \<not>\<acute>after1\<rbrace> \<langle> \<acute>turn:=1,,\<acute>after1:=True \<rangle>;;
    67            \<lbrace>\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace>
    68             WHILE \<not>(\<acute>flag2 \<longrightarrow> \<acute>turn=2)
    69             INV \<lbrace>\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace>
    70             DO \<lbrace>\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace> SKIP OD;;
    71            \<lbrace>\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>flag2 \<and> \<acute>after2 \<longrightarrow> \<acute>turn=2)\<rbrace>
    72             \<acute>flag1:=False
    73         OD
    74        \<lbrace>False\<rbrace>
    75   \<parallel>
    76      \<lbrace>\<not>\<acute>flag2\<rbrace>
    77         WHILE True
    78         INV \<lbrace>\<not>\<acute>flag2\<rbrace>
    79         DO \<lbrace>\<not>\<acute>flag2\<rbrace> \<langle> \<acute>flag2:=True,,\<acute>after2:=False \<rangle>;;
    80            \<lbrace>\<acute>flag2 \<and> \<not>\<acute>after2\<rbrace> \<langle> \<acute>turn:=2,,\<acute>after2:=True \<rangle>;;
    81            \<lbrace>\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace>
    82             WHILE \<not>(\<acute>flag1 \<longrightarrow> \<acute>turn=1)
    83             INV \<lbrace>\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace>
    84             DO \<lbrace>\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace> SKIP OD;;
    85            \<lbrace>\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>flag1 \<and> \<acute>after1 \<longrightarrow> \<acute>turn=1)\<rbrace>
    86             \<acute>flag2:=False
    87         OD
    88        \<lbrace>False\<rbrace>
    89   COEND
    90   \<lbrace>False\<rbrace>"
    91 apply oghoare
    92 \<comment>\<open>122 vc\<close>
    93 apply auto
    94 done
    95 
    96 subsubsection \<open>Peterson's Algorithm III: A Solution using Semaphores\<close>
    97 
    98 record  Semaphores_mutex =
    99  out :: bool
   100  who :: nat
   101 
   102 lemma Semaphores_mutex:
   103  "\<parallel>- \<lbrace>i\<noteq>j\<rbrace>
   104   \<acute>out:=True ,,
   105   COBEGIN \<lbrace>i\<noteq>j\<rbrace>
   106        WHILE True INV \<lbrace>i\<noteq>j\<rbrace>
   107        DO \<lbrace>i\<noteq>j\<rbrace> AWAIT \<acute>out THEN  \<acute>out:=False,, \<acute>who:=i END;;
   108           \<lbrace>\<not>\<acute>out \<and> \<acute>who=i \<and> i\<noteq>j\<rbrace> \<acute>out:=True OD
   109        \<lbrace>False\<rbrace>
   110   \<parallel>
   111        \<lbrace>i\<noteq>j\<rbrace>
   112        WHILE True INV \<lbrace>i\<noteq>j\<rbrace>
   113        DO \<lbrace>i\<noteq>j\<rbrace> AWAIT \<acute>out THEN  \<acute>out:=False,,\<acute>who:=j END;;
   114           \<lbrace>\<not>\<acute>out \<and> \<acute>who=j \<and> i\<noteq>j\<rbrace> \<acute>out:=True OD
   115        \<lbrace>False\<rbrace>
   116   COEND
   117   \<lbrace>False\<rbrace>"
   118 apply oghoare
   119 \<comment>\<open>38 vc\<close>
   120 apply auto
   121 done
   122 
   123 subsubsection \<open>Peterson's Algorithm III: Parameterized version:\<close>
   124 
   125 lemma Semaphores_parameterized_mutex:
   126  "0<n \<Longrightarrow> \<parallel>- \<lbrace>True\<rbrace>
   127   \<acute>out:=True ,,
   128  COBEGIN
   129   SCHEME [0\<le> i< n]
   130     \<lbrace>True\<rbrace>
   131      WHILE True INV \<lbrace>True\<rbrace>
   132       DO \<lbrace>True\<rbrace> AWAIT \<acute>out THEN  \<acute>out:=False,, \<acute>who:=i END;;
   133          \<lbrace>\<not>\<acute>out \<and> \<acute>who=i\<rbrace> \<acute>out:=True OD
   134     \<lbrace>False\<rbrace>
   135  COEND
   136   \<lbrace>False\<rbrace>"
   137 apply oghoare
   138 \<comment>\<open>20 vc\<close>
   139 apply auto
   140 done
   141 
   142 subsubsection\<open>The Ticket Algorithm\<close>
   143 
   144 record Ticket_mutex =
   145  num :: nat
   146  nextv :: nat
   147  turn :: "nat list"
   148  index :: nat
   149 
   150 lemma Ticket_mutex:
   151  "\<lbrakk> 0<n; I=\<guillemotleft>n=length \<acute>turn \<and> 0<\<acute>nextv \<and> (\<forall>k l. k<n \<and> l<n \<and> k\<noteq>l
   152     \<longrightarrow> \<acute>turn!k < \<acute>num \<and> (\<acute>turn!k =0 \<or> \<acute>turn!k\<noteq>\<acute>turn!l))\<guillemotright> \<rbrakk>
   153    \<Longrightarrow> \<parallel>- \<lbrace>n=length \<acute>turn\<rbrace>
   154    \<acute>index:= 0,,
   155    WHILE \<acute>index < n INV \<lbrace>n=length \<acute>turn \<and> (\<forall>i<\<acute>index. \<acute>turn!i=0)\<rbrace>
   156     DO \<acute>turn:= \<acute>turn[\<acute>index:=0],, \<acute>index:=\<acute>index +1 OD,,
   157   \<acute>num:=1 ,, \<acute>nextv:=1 ,,
   158  COBEGIN
   159   SCHEME [0\<le> i< n]
   160     \<lbrace>\<acute>I\<rbrace>
   161      WHILE True INV \<lbrace>\<acute>I\<rbrace>
   162       DO \<lbrace>\<acute>I\<rbrace> \<langle> \<acute>turn :=\<acute>turn[i:=\<acute>num],, \<acute>num:=\<acute>num+1 \<rangle>;;
   163          \<lbrace>\<acute>I\<rbrace> WAIT \<acute>turn!i=\<acute>nextv END;;
   164          \<lbrace>\<acute>I \<and> \<acute>turn!i=\<acute>nextv\<rbrace> \<acute>nextv:=\<acute>nextv+1
   165       OD
   166     \<lbrace>False\<rbrace>
   167  COEND
   168   \<lbrace>False\<rbrace>"
   169 apply oghoare
   170 \<comment>\<open>35 vc\<close>
   171 apply simp_all
   172 \<comment>\<open>16 vc\<close>
   173 apply(tactic \<open>ALLGOALS (clarify_tac @{context})\<close>)
   174 \<comment>\<open>11 vc\<close>
   175 apply simp_all
   176 apply(tactic \<open>ALLGOALS (clarify_tac @{context})\<close>)
   177 \<comment>\<open>10 subgoals left\<close>
   178 apply(erule less_SucE)
   179  apply simp
   180 apply simp
   181 \<comment>\<open>9 subgoals left\<close>
   182 apply(case_tac "i=k")
   183  apply force
   184 apply simp
   185 apply(case_tac "i=l")
   186  apply force
   187 apply force
   188 \<comment>\<open>8 subgoals left\<close>
   189 prefer 8
   190 apply force
   191 apply force
   192 \<comment>\<open>6 subgoals left\<close>
   193 prefer 6
   194 apply(erule_tac x=j in allE)
   195 apply fastforce
   196 \<comment>\<open>5 subgoals left\<close>
   197 prefer 5
   198 apply(case_tac [!] "j=k")
   199 \<comment>\<open>10 subgoals left\<close>
   200 apply simp_all
   201 apply(erule_tac x=k in allE)
   202 apply force
   203 \<comment>\<open>9 subgoals left\<close>
   204 apply(case_tac "j=l")
   205  apply simp
   206  apply(erule_tac x=k in allE)
   207  apply(erule_tac x=k in allE)
   208  apply(erule_tac x=l in allE)
   209  apply force
   210 apply(erule_tac x=k in allE)
   211 apply(erule_tac x=k in allE)
   212 apply(erule_tac x=l in allE)
   213 apply force
   214 \<comment>\<open>8 subgoals left\<close>
   215 apply force
   216 apply(case_tac "j=l")
   217  apply simp
   218 apply(erule_tac x=k in allE)
   219 apply(erule_tac x=l in allE)
   220 apply force
   221 apply force
   222 apply force
   223 \<comment>\<open>5 subgoals left\<close>
   224 apply(erule_tac x=k in allE)
   225 apply(erule_tac x=l in allE)
   226 apply(case_tac "j=l")
   227  apply force
   228 apply force
   229 apply force
   230 \<comment>\<open>3 subgoals left\<close>
   231 apply(erule_tac x=k in allE)
   232 apply(erule_tac x=l in allE)
   233 apply(case_tac "j=l")
   234  apply force
   235 apply force
   236 apply force
   237 \<comment>\<open>1 subgoals left\<close>
   238 apply(erule_tac x=k in allE)
   239 apply(erule_tac x=l in allE)
   240 apply(case_tac "j=l")
   241  apply force
   242 apply force
   243 done
   244 
   245 subsection\<open>Parallel Zero Search\<close>
   246 
   247 text \<open>Synchronized Zero Search. Zero-6\<close>
   248 
   249 text \<open>Apt and Olderog. "Verification of sequential and concurrent Programs" page 294:\<close>
   250 
   251 record Zero_search =
   252    turn :: nat
   253    found :: bool
   254    x :: nat
   255    y :: nat
   256 
   257 lemma Zero_search:
   258   "\<lbrakk>I1= \<guillemotleft> a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> (a<\<acute>x \<and> f(\<acute>x)=0) \<or> (\<acute>y\<le>a \<and> f(\<acute>y)=0))
   259       \<and> (\<not>\<acute>found \<and> a<\<acute> x \<longrightarrow> f(\<acute>x)\<noteq>0) \<guillemotright> ;
   260     I2= \<guillemotleft>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> (a<\<acute>x \<and> f(\<acute>x)=0) \<or> (\<acute>y\<le>a \<and> f(\<acute>y)=0))
   261       \<and> (\<not>\<acute>found \<and> \<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0) \<guillemotright> \<rbrakk> \<Longrightarrow>
   262   \<parallel>- \<lbrace>\<exists> u. f(u)=0\<rbrace>
   263   \<acute>turn:=1,, \<acute>found:= False,,
   264   \<acute>x:=a,, \<acute>y:=a+1 ,,
   265   COBEGIN \<lbrace>\<acute>I1\<rbrace>
   266        WHILE \<not>\<acute>found
   267        INV \<lbrace>\<acute>I1\<rbrace>
   268        DO \<lbrace>a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<rbrace>
   269           WAIT \<acute>turn=1 END;;
   270           \<lbrace>a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<rbrace>
   271           \<acute>turn:=2;;
   272           \<lbrace>a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<rbrace>
   273           \<langle> \<acute>x:=\<acute>x+1,,
   274             IF f(\<acute>x)=0 THEN \<acute>found:=True ELSE SKIP FI\<rangle>
   275        OD;;
   276        \<lbrace>\<acute>I1  \<and> \<acute>found\<rbrace>
   277        \<acute>turn:=2
   278        \<lbrace>\<acute>I1 \<and> \<acute>found\<rbrace>
   279   \<parallel>
   280       \<lbrace>\<acute>I2\<rbrace>
   281        WHILE \<not>\<acute>found
   282        INV \<lbrace>\<acute>I2\<rbrace>
   283        DO \<lbrace>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<rbrace>
   284           WAIT \<acute>turn=2 END;;
   285           \<lbrace>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<rbrace>
   286           \<acute>turn:=1;;
   287           \<lbrace>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<rbrace>
   288           \<langle> \<acute>y:=(\<acute>y - 1),,
   289             IF f(\<acute>y)=0 THEN \<acute>found:=True ELSE SKIP FI\<rangle>
   290        OD;;
   291        \<lbrace>\<acute>I2 \<and> \<acute>found\<rbrace>
   292        \<acute>turn:=1
   293        \<lbrace>\<acute>I2 \<and> \<acute>found\<rbrace>
   294   COEND
   295   \<lbrace>f(\<acute>x)=0 \<or> f(\<acute>y)=0\<rbrace>"
   296 apply oghoare
   297 \<comment>\<open>98 verification conditions\<close>
   298 apply auto
   299 \<comment>\<open>auto takes about 3 minutes !!\<close>
   300 done
   301 
   302 text \<open>Easier Version: without AWAIT.  Apt and Olderog. page 256:\<close>
   303 
   304 lemma Zero_Search_2:
   305 "\<lbrakk>I1=\<guillemotleft> a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> (a<\<acute>x \<and> f(\<acute>x)=0) \<or> (\<acute>y\<le>a \<and> f(\<acute>y)=0))
   306     \<and> (\<not>\<acute>found \<and> a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<guillemotright>;
   307  I2= \<guillemotleft>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> (a<\<acute>x \<and> f(\<acute>x)=0) \<or> (\<acute>y\<le>a \<and> f(\<acute>y)=0))
   308     \<and> (\<not>\<acute>found \<and> \<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<guillemotright>\<rbrakk> \<Longrightarrow>
   309   \<parallel>- \<lbrace>\<exists>u. f(u)=0\<rbrace>
   310   \<acute>found:= False,,
   311   \<acute>x:=a,, \<acute>y:=a+1,,
   312   COBEGIN \<lbrace>\<acute>I1\<rbrace>
   313        WHILE \<not>\<acute>found
   314        INV \<lbrace>\<acute>I1\<rbrace>
   315        DO \<lbrace>a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<rbrace>
   316           \<langle> \<acute>x:=\<acute>x+1,,IF f(\<acute>x)=0 THEN  \<acute>found:=True ELSE  SKIP FI\<rangle>
   317        OD
   318        \<lbrace>\<acute>I1 \<and> \<acute>found\<rbrace>
   319   \<parallel>
   320       \<lbrace>\<acute>I2\<rbrace>
   321        WHILE \<not>\<acute>found
   322        INV \<lbrace>\<acute>I2\<rbrace>
   323        DO \<lbrace>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<rbrace>
   324           \<langle> \<acute>y:=(\<acute>y - 1),,IF f(\<acute>y)=0 THEN  \<acute>found:=True ELSE  SKIP FI\<rangle>
   325        OD
   326        \<lbrace>\<acute>I2 \<and> \<acute>found\<rbrace>
   327   COEND
   328   \<lbrace>f(\<acute>x)=0 \<or> f(\<acute>y)=0\<rbrace>"
   329 apply oghoare
   330 \<comment>\<open>20 vc\<close>
   331 apply auto
   332 \<comment>\<open>auto takes aprox. 2 minutes.\<close>
   333 done
   334 
   335 subsection \<open>Producer/Consumer\<close>
   336 
   337 subsubsection \<open>Previous lemmas\<close>
   338 
   339 lemma nat_lemma2: "\<lbrakk> b = m*(n::nat) + t; a = s*n + u; t=u; b-a < n \<rbrakk> \<Longrightarrow> m \<le> s"
   340 proof -
   341   assume "b = m*(n::nat) + t" "a = s*n + u" "t=u"
   342   hence "(m - s) * n = b - a" by (simp add: diff_mult_distrib)
   343   also assume "\<dots> < n"
   344   finally have "m - s < 1" by simp
   345   thus ?thesis by arith
   346 qed
   347 
   348 lemma mod_lemma: "\<lbrakk> (c::nat) \<le> a; a < b; b - c < n \<rbrakk> \<Longrightarrow> b mod n \<noteq> a mod n"
   349 apply(subgoal_tac "b=b div n*n + b mod n" )
   350  prefer 2  apply (simp add: div_mult_mod_eq [symmetric])
   351 apply(subgoal_tac "a=a div n*n + a mod n")
   352  prefer 2
   353  apply(simp add: div_mult_mod_eq [symmetric])
   354 apply(subgoal_tac "b - a \<le> b - c")
   355  prefer 2 apply arith
   356 apply(drule le_less_trans)
   357 back
   358  apply assumption
   359 apply(frule less_not_refl2)
   360 apply(drule less_imp_le)
   361 apply (drule_tac m = "a" and k = n in div_le_mono)
   362 apply(safe)
   363 apply(frule_tac b = "b" and a = "a" and n = "n" in nat_lemma2, assumption, assumption)
   364 apply assumption
   365 apply(drule order_antisym, assumption)
   366 apply(rotate_tac -3)
   367 apply(simp)
   368 done
   369 
   370 
   371 subsubsection \<open>Producer/Consumer Algorithm\<close>
   372 
   373 record Producer_consumer =
   374   ins :: nat
   375   outs :: nat
   376   li :: nat
   377   lj :: nat
   378   vx :: nat
   379   vy :: nat
   380   buffer :: "nat list"
   381   b :: "nat list"
   382 
   383 text \<open>The whole proof takes aprox. 4 minutes.\<close>
   384 
   385 lemma Producer_consumer:
   386   "\<lbrakk>INIT= \<guillemotleft>0<length a \<and> 0<length \<acute>buffer \<and> length \<acute>b=length a\<guillemotright> ;
   387     I= \<guillemotleft>(\<forall>k<\<acute>ins. \<acute>outs\<le>k \<longrightarrow> (a ! k) = \<acute>buffer ! (k mod (length \<acute>buffer))) \<and>
   388             \<acute>outs\<le>\<acute>ins \<and> \<acute>ins-\<acute>outs\<le>length \<acute>buffer\<guillemotright> ;
   389     I1= \<guillemotleft>\<acute>I \<and> \<acute>li\<le>length a\<guillemotright> ;
   390     p1= \<guillemotleft>\<acute>I1 \<and> \<acute>li=\<acute>ins\<guillemotright> ;
   391     I2 = \<guillemotleft>\<acute>I \<and> (\<forall>k<\<acute>lj. (a ! k)=(\<acute>b ! k)) \<and> \<acute>lj\<le>length a\<guillemotright> ;
   392     p2 = \<guillemotleft>\<acute>I2 \<and> \<acute>lj=\<acute>outs\<guillemotright> \<rbrakk> \<Longrightarrow>
   393   \<parallel>- \<lbrace>\<acute>INIT\<rbrace>
   394  \<acute>ins:=0,, \<acute>outs:=0,, \<acute>li:=0,, \<acute>lj:=0,,
   395  COBEGIN \<lbrace>\<acute>p1 \<and> \<acute>INIT\<rbrace>
   396    WHILE \<acute>li <length a
   397      INV \<lbrace>\<acute>p1 \<and> \<acute>INIT\<rbrace>
   398    DO \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a\<rbrace>
   399        \<acute>vx:= (a ! \<acute>li);;
   400       \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a \<and> \<acute>vx=(a ! \<acute>li)\<rbrace>
   401         WAIT \<acute>ins-\<acute>outs < length \<acute>buffer END;;
   402       \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a \<and> \<acute>vx=(a ! \<acute>li)
   403          \<and> \<acute>ins-\<acute>outs < length \<acute>buffer\<rbrace>
   404        \<acute>buffer:=(list_update \<acute>buffer (\<acute>ins mod (length \<acute>buffer)) \<acute>vx);;
   405       \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a
   406          \<and> (a ! \<acute>li)=(\<acute>buffer ! (\<acute>ins mod (length \<acute>buffer)))
   407          \<and> \<acute>ins-\<acute>outs <length \<acute>buffer\<rbrace>
   408        \<acute>ins:=\<acute>ins+1;;
   409       \<lbrace>\<acute>I1 \<and> \<acute>INIT \<and> (\<acute>li+1)=\<acute>ins \<and> \<acute>li<length a\<rbrace>
   410        \<acute>li:=\<acute>li+1
   411    OD
   412   \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li=length a\<rbrace>
   413   \<parallel>
   414   \<lbrace>\<acute>p2 \<and> \<acute>INIT\<rbrace>
   415    WHILE \<acute>lj < length a
   416      INV \<lbrace>\<acute>p2 \<and> \<acute>INIT\<rbrace>
   417    DO \<lbrace>\<acute>p2 \<and> \<acute>lj<length a \<and> \<acute>INIT\<rbrace>
   418         WAIT \<acute>outs<\<acute>ins END;;
   419       \<lbrace>\<acute>p2 \<and> \<acute>lj<length a \<and> \<acute>outs<\<acute>ins \<and> \<acute>INIT\<rbrace>
   420        \<acute>vy:=(\<acute>buffer ! (\<acute>outs mod (length \<acute>buffer)));;
   421       \<lbrace>\<acute>p2 \<and> \<acute>lj<length a \<and> \<acute>outs<\<acute>ins \<and> \<acute>vy=(a ! \<acute>lj) \<and> \<acute>INIT\<rbrace>
   422        \<acute>outs:=\<acute>outs+1;;
   423       \<lbrace>\<acute>I2 \<and> (\<acute>lj+1)=\<acute>outs \<and> \<acute>lj<length a \<and> \<acute>vy=(a ! \<acute>lj) \<and> \<acute>INIT\<rbrace>
   424        \<acute>b:=(list_update \<acute>b \<acute>lj \<acute>vy);;
   425       \<lbrace>\<acute>I2 \<and> (\<acute>lj+1)=\<acute>outs \<and> \<acute>lj<length a \<and> (a ! \<acute>lj)=(\<acute>b ! \<acute>lj) \<and> \<acute>INIT\<rbrace>
   426        \<acute>lj:=\<acute>lj+1
   427    OD
   428   \<lbrace>\<acute>p2 \<and> \<acute>lj=length a \<and> \<acute>INIT\<rbrace>
   429  COEND
   430  \<lbrace> \<forall>k<length a. (a ! k)=(\<acute>b ! k)\<rbrace>"
   431 apply oghoare
   432 \<comment>\<open>138 vc\<close>
   433 apply(tactic \<open>ALLGOALS (clarify_tac @{context})\<close>)
   434 \<comment>\<open>112 subgoals left\<close>
   435 apply(simp_all (no_asm))
   436 \<comment>\<open>43 subgoals left\<close>
   437 apply(tactic \<open>ALLGOALS (conjI_Tac @{context} (K all_tac))\<close>)
   438 \<comment>\<open>419 subgoals left\<close>
   439 apply(tactic \<open>ALLGOALS (clarify_tac @{context})\<close>)
   440 \<comment>\<open>99 subgoals left\<close>
   441 apply(simp_all only:length_0_conv [THEN sym])
   442 \<comment>\<open>20 subgoals left\<close>
   443 apply (simp_all del:length_0_conv length_greater_0_conv add: nth_list_update mod_lemma)
   444 \<comment>\<open>9 subgoals left\<close>
   445 apply (force simp add:less_Suc_eq)
   446 apply(hypsubst_thin, drule sym)
   447 apply (force simp add:less_Suc_eq)+
   448 done
   449 
   450 subsection \<open>Parameterized Examples\<close>
   451 
   452 subsubsection \<open>Set Elements of an Array to Zero\<close>
   453 
   454 record Example1 =
   455   a :: "nat \<Rightarrow> nat"
   456 
   457 lemma Example1:
   458  "\<parallel>- \<lbrace>True\<rbrace>
   459    COBEGIN SCHEME [0\<le>i<n] \<lbrace>True\<rbrace> \<acute>a:=\<acute>a (i:=0) \<lbrace>\<acute>a i=0\<rbrace> COEND
   460   \<lbrace>\<forall>i < n. \<acute>a i = 0\<rbrace>"
   461 apply oghoare
   462 apply simp_all
   463 done
   464 
   465 text \<open>Same example with lists as auxiliary variables.\<close>
   466 record Example1_list =
   467   A :: "nat list"
   468 lemma Example1_list:
   469  "\<parallel>- \<lbrace>n < length \<acute>A\<rbrace>
   470    COBEGIN
   471      SCHEME [0\<le>i<n] \<lbrace>n < length \<acute>A\<rbrace> \<acute>A:=\<acute>A[i:=0] \<lbrace>\<acute>A!i=0\<rbrace>
   472    COEND
   473     \<lbrace>\<forall>i < n. \<acute>A!i = 0\<rbrace>"
   474 apply oghoare
   475 apply force+
   476 done
   477 
   478 subsubsection \<open>Increment a Variable in Parallel\<close>
   479 
   480 text \<open>First some lemmas about summation properties.\<close>
   481 (*
   482 lemma Example2_lemma1: "!!b. j<n \<Longrightarrow> (\<Sum>i::nat<n. b i) = (0::nat) \<Longrightarrow> b j = 0 "
   483 apply(induct n)
   484  apply simp_all
   485 apply(force simp add: less_Suc_eq)
   486 done
   487 *)
   488 lemma Example2_lemma2_aux: "!!b. j<n \<Longrightarrow>
   489  (\<Sum>i=0..<n. (b i::nat)) =
   490  (\<Sum>i=0..<j. b i) + b j + (\<Sum>i=0..<n-(Suc j) . b (Suc j + i))"
   491 apply(induct n)
   492  apply simp_all
   493 apply(simp add:less_Suc_eq)
   494  apply(auto)
   495 apply(subgoal_tac "n - j = Suc(n- Suc j)")
   496   apply simp
   497 apply arith
   498 done
   499 
   500 lemma Example2_lemma2_aux2:
   501   "!!b. j\<le> s \<Longrightarrow> (\<Sum>i::nat=0..<j. (b (s:=t)) i) = (\<Sum>i=0..<j. b i)"
   502 apply(induct j)
   503  apply simp_all
   504 done
   505 
   506 lemma Example2_lemma2:
   507  "!!b. \<lbrakk>j<n; b j=0\<rbrakk> \<Longrightarrow> Suc (\<Sum>i::nat=0..<n. b i)=(\<Sum>i=0..<n. (b (j := Suc 0)) i)"
   508 apply(frule_tac b="(b (j:=(Suc 0)))" in Example2_lemma2_aux)
   509 apply(erule_tac  t="sum (b(j := (Suc 0))) {0..<n}" in ssubst)
   510 apply(frule_tac b=b in Example2_lemma2_aux)
   511 apply(erule_tac  t="sum b {0..<n}" in ssubst)
   512 apply(subgoal_tac "Suc (sum b {0..<j} + b j + (\<Sum>i=0..<n - Suc j. b (Suc j + i)))=(sum b {0..<j} + Suc (b j) + (\<Sum>i=0..<n - Suc j. b (Suc j + i)))")
   513 apply(rotate_tac -1)
   514 apply(erule ssubst)
   515 apply(subgoal_tac "j\<le>j")
   516  apply(drule_tac b="b" and t="(Suc 0)" in Example2_lemma2_aux2)
   517 apply(rotate_tac -1)
   518 apply(erule ssubst)
   519 apply simp_all
   520 done
   521 
   522 
   523 record Example2 =
   524  c :: "nat \<Rightarrow> nat"
   525  x :: nat
   526 
   527 lemma Example_2: "0<n \<Longrightarrow>
   528  \<parallel>- \<lbrace>\<acute>x=0 \<and> (\<Sum>i=0..<n. \<acute>c i)=0\<rbrace>
   529  COBEGIN
   530    SCHEME [0\<le>i<n]
   531   \<lbrace>\<acute>x=(\<Sum>i=0..<n. \<acute>c i) \<and> \<acute>c i=0\<rbrace>
   532    \<langle> \<acute>x:=\<acute>x+(Suc 0),, \<acute>c:=\<acute>c (i:=(Suc 0)) \<rangle>
   533   \<lbrace>\<acute>x=(\<Sum>i=0..<n. \<acute>c i) \<and> \<acute>c i=(Suc 0)\<rbrace>
   534  COEND
   535  \<lbrace>\<acute>x=n\<rbrace>"
   536 apply oghoare
   537 apply (simp_all cong del: sum.strong_cong)
   538 apply (tactic \<open>ALLGOALS (clarify_tac @{context})\<close>)
   539 apply (simp_all cong del: sum.strong_cong)
   540    apply(erule (1) Example2_lemma2)
   541   apply(erule (1) Example2_lemma2)
   542  apply(erule (1) Example2_lemma2)
   543 apply(simp)
   544 done
   545 
   546 end